Exercicios resolvidos James Stewart vol. 2 7ª ed - ingles
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STUDENT SOLUTIONS MANUAL for STEWART' S<br />
DAN CLEGG • BARBARA FRANK
.Student Solutions Manual<br />
for .<br />
MULTIVARIABLE . CALCULUS<br />
SEVENTH EDITION<br />
DAN CLEGG<br />
Palomar College<br />
BARBARA FRANK·<br />
Cape Fear Community College<br />
~- . · BROOKS/COLE<br />
taa. CENGAG ELearning·<br />
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D<br />
PREFACE<br />
This Student Solutions Manual contains detail<strong>ed</strong> solutions to select<strong>ed</strong> exercises in the text<br />
Multivariable Calculus, Seventh Edition (Chapters 10-17 of Calculus, Seventh. Edition, and<br />
Calculus: Early Transcendentals, Sevent,h Edition) by <strong>James</strong> <strong>Stewart</strong>. Specifically, it includes solutions<br />
to the odd-number<strong>ed</strong> exercises in each chapter section, review section, True-False Quiz, and<br />
Problems Plus section. Also includ<strong>ed</strong> are all solutions to the Concept Check questions.<br />
Because of differences between the regular version and the Early Transcendentals version of the<br />
text, some references are given in a dual format. In these cases, readers of the Early Transcendentals<br />
text should use the references denot<strong>ed</strong> by "ET."<br />
Each solution is present<strong>ed</strong> in the context of the corresponding section of the text. In general,<br />
solutions to the initial exercises in<strong>vol</strong>ving a new concept illustrate that concept in more detail; this<br />
knowl<strong>ed</strong>ge is then utiliz<strong>ed</strong> in subsequent solutions. Thus, while the interm<strong>ed</strong>iate steps of a solution<br />
are given, you may ne<strong>ed</strong> to refer back to earlier exercises in the section or prior sections for additional<br />
explanation of the concepts in<strong>vol</strong>v<strong>ed</strong>. Note that, in many cases, different routes to an answer may<br />
exist which are equally valid; also, answers can be express<strong>ed</strong> in different but equivalent forms. Thus,<br />
the goaJ of this manual is not to give the definitive solution to each exercise, but rather to assist you<br />
as a student in understanc#ng the concepts of the text and learning how to apply them to the. challenge<br />
of solving a problem.<br />
We would like to thank <strong>James</strong> <strong>Stewart</strong> for entrusting us with the writing of this manual and offering<br />
suggestions and Kathi Townes of TECH-arts for typesetting and producing this manual as well as<br />
creati ng the illustrations. We also thank Richard Stratton, Liz Covello, and Elizabeth Neustaetter of<br />
Brooks/Cole, Cengage Learning, for their trust, assistance, and patience.<br />
DAN CLEGG<br />
Palomar College<br />
BARBARA FRANK<br />
Cape Fear Community College<br />
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D<br />
ABBREVIATIONS AND SYMBOLS<br />
CD<br />
cu<br />
D<br />
PDT<br />
HA<br />
I<br />
1/D<br />
IP<br />
R<br />
VA<br />
CAS<br />
H<br />
concave downward<br />
concave upward<br />
the domain off<br />
First Derivative Test<br />
horizontal asymptote(s)<br />
interval of convergence<br />
Increasing/Decreasing Test<br />
inflection point(s)<br />
radius of convergence<br />
vertical asymptote(s)<br />
· indicates the use of a computer algebra system.<br />
indicates the use of ljHospital's Rule.<br />
' indicates the use ofFormu'Ja j in the Table of Integrals in the back endpapers.<br />
indicates the use of the substitution { u = sin x, if.u = cos x dx}.<br />
indicates the use of the substitution { u = cos x, du = - sin x dx}.<br />
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D · CONTENTS<br />
0 10 PARAMETRIC EQUATIONS AND POLAR COORDINATES 1<br />
10.1 Curves Defin<strong>ed</strong> by Parametric Equations<br />
10.2 Calculus with Parametric Curves 7<br />
10.3 Polar Coordinates 13<br />
10.4 Areas and Lengths in Polar Coordinates 20<br />
10.5 Conic Sections 26<br />
10.6 Conic Sections in Polar Coordinates 32<br />
Review 35<br />
Problems Plus 43<br />
0 11 INFINITE SEQUENCES AND SERIES 45<br />
11.1 Sequences 45<br />
11.2 Series 51<br />
11.3 The Integral Test and Estimates of Sums 59<br />
11 .4 The Comparison Tests 62<br />
11.5 Alternating Series 65<br />
11 .6 Absolute Convergence and the Ratio and Root Tests 68<br />
11 .7 Strategy for Testing Series 72<br />
11.8 Power Series 74<br />
11.9 Representations of Functions as Power Series 78<br />
11.10 Taylor and Maclaurin Series 83<br />
11.1 1 Applications of Taylor Polynomials 90<br />
Review 97<br />
Problems Plus 105<br />
0 12 VECTQRS AND THE GEOMETRY OF SPACE 111<br />
12.1 Three-Dimensional Coordinate Systems 111<br />
12.2 Vectors 114<br />
12.3 The Dot Product 119<br />
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viii o CONTENTS<br />
12.4 The Cross Product 123<br />
12.5 Equations of Lines and Planes 128<br />
. 12.6 C,ylinders and Quadric Surfaces 135<br />
Review 140<br />
Problems Plus 147<br />
0 13 VECTOR FUNCTIONS 151<br />
13.1 Vector Functions and Space Curves 151<br />
13.2 Derivatives and Integrals of Vector Functions 157<br />
13.3 Arc Length and Curvature 161<br />
13.4 Motion in Space: Velocity and Acceleration 168<br />
Review 173<br />
Problems Plus 179<br />
0 14 PARTIAL DERIVATIVES 183<br />
14.1 Functions of Several Variables 183<br />
14.2 Limits and Continuity 192<br />
14.3 Partial Derivatives 195<br />
14.4 Tangent Planes and Linear. Approximations 203<br />
14.5 The Chain Rule 207<br />
14.6 Directional Derivatives and the Gradient Vector 213<br />
14.7 Maximum and Minimum Values · 220<br />
14.8 Lagrange Multipliers 229<br />
Review 234<br />
Problems Plus 245<br />
0 15 MULTIPLE INTEGRALS 247<br />
15.1 Double Integrals over Rectangles 247<br />
15.2 Iterat<strong>ed</strong> Integrals 249<br />
15.3 Double Integrals over General Regions 251<br />
15.4 Double Integrals in Polar Coordinates 258<br />
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CONTENTS D ix<br />
15.5<br />
15.6<br />
15.7<br />
15.8<br />
15.9<br />
15.10<br />
Applications of Double Integrals 261 '<br />
Surface Area 267<br />
Triple Integrals 269<br />
Triple Integrals in Cylindrical Coordinates<br />
Triple Integrals in Spherical Coordinates<br />
Change ofVariables in Multiple Integrals<br />
Review 289<br />
280<br />
276<br />
285<br />
Problems Plus 297<br />
0 16 . VECTOR CALCULUS 303<br />
16.1 Vector Fields 303<br />
16.2 Line Integrals 305<br />
16.3 The Fundamental Theorem for Line Integrals 310<br />
16.4 Green's Theorem 313<br />
16.5 Curl and Divergence -316<br />
16.6 Parametric Surfaces and Their Areas 321<br />
16.7 Surface Integrals 328<br />
16.8 Stokes' Theorem 333<br />
16.9 The Divergence Theorem 335 .<br />
Review 337<br />
Problems Plus 343<br />
0 17 SECOND-ORDER DIFFERENTIAL EQUATIONS 345<br />
17.1 Second-Order Linear Equations 345 ·<br />
17.2 Nonhomogeneous Linear Equations 347<br />
17.3 Applications Of Second-Order Differential Equations 350<br />
17.4 Series Solutions 352<br />
Review 354<br />
. 0 A.PPENDIX 359<br />
H Complex Numbers 359<br />
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10 D PARAMETRIC EQUATIONS AND POLAR COORDINATES<br />
10.1 Curves Defin<strong>ed</strong> by Parametric Equations<br />
1. X= t 2 + t, y = t 2 - t, -2 S t S 2<br />
t - 2 - 1 0 l 2<br />
X 2 0 0 2 6<br />
y 6 2 0 0 2<br />
y<br />
t = - 2<br />
(2, 6)<br />
X<br />
3. x = cos 2 t, y = 1 - sin t, 0 S t S 7r /2<br />
t 0 7r/6 7r/3 7r/ 2<br />
X 1 3/ 4 1/ 4 0<br />
y 1 1/ 2 1 - 1 ~ 0.13 0<br />
y<br />
r=JI.<br />
6<br />
t=O<br />
(1, 1)<br />
X<br />
5. X = 3 - 4t, y = 2 - 3t<br />
(a)<br />
t - 1 0 1 2<br />
X 7 3 -1 - 5<br />
y 5 2 -1 - 4<br />
y<br />
(7, S)<br />
t= -1<br />
(3, 2)<br />
t = O<br />
X<br />
(b) X= 3 - 4t =} 4t = -X + 3 =} t = -~ X +~ ' SO<br />
y = 2 - 3t = 2 - 3(- ~x + ~ ) = 2 +~ X - £ =} y =~X - ~<br />
(-5, - 4)<br />
1=2<br />
1. x = 1 - e, v = t - 2, - 2 s t s 2<br />
(- 3, 0)<br />
t=2<br />
y<br />
(a)<br />
t - 2 - 1 0 1 2<br />
X -3 0 1 0 -3<br />
y - 4 -3 - 2 - 1 0<br />
X<br />
(1, - 2)<br />
t= O<br />
(b) y = t - 2 =} t = y + 2, SO X= 1 - t 2 = 1 - (y + 2? =}<br />
x = - (y + 2) 2 + 1, or x =:= - y 2 - 4y -e 3, with - 4 S y S 0<br />
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2 D CHAPTER 10 PARAMETRIC EQUATIONS AND POLAR COORDINATES<br />
9. X = Vt, y = 1 - t<br />
(a)<br />
t 0 1 2 3 4<br />
X 0 1 1.414 1.732 2<br />
y 1 0 - 1 - 2 .:....3<br />
(b) x = Vt =} t = x 2 =} y = 1- t = 1 - x 2 . Since t ~ 0, x ~ 0.<br />
So the curve is the right half of the parabola y = 1 - x 2 •<br />
y<br />
(0, I) ,f = O<br />
X<br />
(2, -3) t=4<br />
11. (a) x = sin ~0, y =cos ~(:1, -1r ~ (:1 ~ 1r.<br />
x 2 + y 2 = sin 2 ~8 + cos 2 ~8 = 1. For -1r ~ 0 ~ 0, we have<br />
- 1 ~ x ~ 0 and 0 ~ y ~ 1. For 0 y ~ 0. The graph is a semicircle.<br />
(b)<br />
y<br />
-1 0 I X<br />
13. (a) x = sint, y = csct, 0 < t < ~- y = csct = ~ = !.<br />
. . smt x<br />
For 0 < t < ~.we haveO < x < 1 andy > 1. Thus, the curve is the<br />
portion of the hyperbola y = 1/x withy> 1.<br />
(b)<br />
'\ ,,,<br />
0 X<br />
15. (a) x = e 2 t =}. 2t = ln x =} t. = ~ lnx.<br />
y = t+1 = ~lnx+1.<br />
(b)<br />
y<br />
17. (a) x =sinh t, y = cosh t '=} y 2 - x 2 = cosh 2 t - sinh 2 t = 1. Since<br />
y = cosh t ~ 1, we have the upper branch of the hyperbola y 2 - x 2 = 1.<br />
(b)<br />
y<br />
0 X<br />
. 19. x = 3 + 2 cost, y = 1 + 2 sin t, 1r / 2 ~ t ~ 31r / 2. By Example 4 with r = 2, h = 3, and k = 1, the motion of the particle<br />
takes place on a circle center<strong>ed</strong> at (3, 1) with a radius of2. As t goes from -¥- to 3 ;, the particle starts at the point (3, 3) and<br />
moves counterclockwise along the circle (x - 3) 2 + (y - 1) 2 = 4 to (3, - 1) [one-half of a circle].<br />
21. x = 5sint, y=2cost =} s in t=~ , cost=~· sin 2 t+cos 2 t =1 =} (~r + (~r = l.The motionofthe<br />
particle takes place on an ellipse center<strong>ed</strong> at (0, 0). As t goes from -<br />
1r to 511', the particle starts at the point (0, -2) and moves<br />
clockwise around the ellipse 3 times.<br />
23. We must have 1 ~ x ~ 4 and 2 ~ y ~ 3. So the graph of the curve must be contain<strong>ed</strong> in the rectangle (1, 4] by [2, 3].<br />
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SECTION 10.1 CURVES DEFINED BY PARAMETRIC EQUATIONS D 3<br />
25. When t = - 1,-(x,y) = (0, - 1). As t increases to 0, x decreases to - 1 andy<br />
increases to 0. As t increases from 0 to 1, x increases to 0 and y increases. to 1.<br />
As { increases beyo~d 1, both x and y increase. Fort < - 1, x is positive and<br />
decreasing and y is negative and increasing. We could achieve greater accuracy<br />
by estimating x- and y-values for select<strong>ed</strong> values oft from the given graphs and<br />
(-1,0) X<br />
t=O<br />
t=-1<br />
plotting the correspo~ding points.<br />
27. When t = 0 we see that x = 0 and y = 0, so the curve starts at the origin. As t<br />
y<br />
increases from 0 to~. the graphs show that y increases from 0 to 1 while x<br />
increases from 0 to 1, decreases to 0 and to - 1, then increas.es back to 0, so we<br />
arrive at the point (0, 1). Sinlilarly, as t increases from ~ to 1, y decreases from 1<br />
X<br />
to 0 while x repeats its pattern, and we arrive back at the origin. We could achieve greater accuracy by estimating x- and<br />
y-values for select<strong>ed</strong> values oft from the given graphs and plotting the corresponding points.<br />
29. Use y = t and x = t - 2sin 7ft with at-interval of [-7r, 7r].<br />
31. (a) x = X1 + (x2 - x1)t, y = YI + (y2- Y1)t, 0 5 t 5 1. Clearly the curve passes through P1(X1, Yl) when t = 0 and<br />
through P2(x2, y2) when t = 1. For 0
4 0 CHAPTER 10 PARAMETRIC EQUATIONS AND POLAR COORDINATES<br />
(c) To start at (0, 3) using the original equations, we must have X1 = 0; that is, 2 cos t = 0. Hence, t = ~. So we use<br />
x = 2cost, y = 1 + 2sint, ~ ~ t ~ 3 ;.<br />
Alternatively, if we want t to start at 0, we could change the equations of the curve. For example, we could use<br />
x = - 2 sin t, y = 1 + 2 cost, 0 ~ t ~ 1r.<br />
35. Big circle: It's center<strong>ed</strong> at (2, 2) 'with a radius of 2, so by Example 4, parametric equations are<br />
x = 2 + 2cost, y = 2 + 2 sin t,<br />
Small circles: They are center<strong>ed</strong> at (1, 3) and (3, 3) with a radius ofO.l. By Example 4, parametric equations are<br />
(left)<br />
x ;, 1 + O.l cost,<br />
y.= 3 + 0.1 sin t,<br />
0 ~ t ~ 27T<br />
and<br />
(right)<br />
X = 3 + 0.1 COSt,<br />
y = 3 + 0.1 sint,·<br />
0 ~ t ~ 27T<br />
Semicircle: It's the lower half of a circle center<strong>ed</strong> at (2, 2) with radius 1. By Example 4, parametric equations are<br />
x = 2 + 1 cost, . y = 2 + 1 sin t,<br />
. . .<br />
To get all four graphs on the same screen with a typical graphing calculator, we ne<strong>ed</strong> to change the last t-interval to[O, 21r] in<br />
order to match the others ." We can do this by changing t to 0.5t. This change gives us the upper half. There are several ways to<br />
get the lower half-one is to change the"+" to a "-" in the y-assigr1ment, giving us<br />
x = 2 + 1 cos(0.5t), y = 2 - 1 sin (0.5t ), · 0 ~ t ~ 27T<br />
37. (a)x = t 3 :::? t = x 1 1 3 ,so y=t 2 = x 2 1 3 • (b) x = t 6 :::?·. t = x 1 1 6 , so y= t 4 = x 4 1 6 = x 2 1 3 •<br />
We get the entire curve y = x 2 1 3 travers<strong>ed</strong> in a left to<br />
right direction.<br />
y<br />
x= t ~ y = 1 1<br />
Since x = t 6 2': 0, we only get the right half of the<br />
' .<br />
curve y = x 2 1 3 •<br />
y<br />
0 X<br />
(c) x =e-st = (e- t)3<br />
[so e- t = xl/3],<br />
y = e-2t = (e-t)2 = .(xl/3)2 = x2/3.<br />
If t < 0, then x and y are both larger than 1. !f t > 0, then x and y<br />
are between 0 and 1. Since x > 0 and y > 0, the curve never quite<br />
reaches the origin.<br />
39. The case ¥ < (} < 7T is illustrat<strong>ed</strong>. 0 has coordinates ( rO, r) as in Example 7,<br />
and Q has coordinates (rB, r + r cos(1r- B)) = (r B, 7"(1- cos B))<br />
[sincecos(1r - a) = cos7rc;osa + sin7T sin a· = -cos a ], so.P has<br />
coordinates (rB- rsin(1r - B),r(1-cos·B)) = (r(B :- sin B),r(l - cos B))<br />
[sincesin(1r - a) = sin1rcosa - cos1r.sin a = sin a ]. Again we have the<br />
parametric equations x = r( (} - sin 8), y = r(l - cos B).<br />
y<br />
X<br />
© 2012 CC=nguge Learning. All Rights Res
SECTION 10.1 CURVES DEFINED BY PARAMETRIC EQUATIONS D 5<br />
41. It is apparent that x = JOQJ andy= IQPI = JSTJ. From the diagram,<br />
y<br />
X = I OQI = a cos B and y = I STI = b sine. Thus, the parametric equations are<br />
x = a cos B and y = b sin B. To eliminate B we rearrange: sin B = y / b =><br />
sin 2 B = (y/bl and cosO= x/a =><br />
cos 2 B = (x/a) 2 • Adding the two<br />
X<br />
equations: sin 2 B + cos 2 B = 1 = x 2 /a 2 + y 2 /b 2 .,Thus, we have an ellipse.<br />
43. C = (2a cot B, 2a), so the x-coordinate of Pis x = 2a cot B. Let B = (0, 2a).<br />
Then L.OAB is a rightangle and L.OBA = B, so JOAI '= 2asinB and<br />
A = ( (2a sin B) cos B, (2a sin B) sin B). Thus, they-coordinate of P<br />
is y = 2asin 2 B.<br />
45. (a) 4 There are 2 points of intersection:<br />
( -3, 0) and approximately ( - 2.1, 1.4).<br />
~<br />
2a<br />
-4<br />
(b) A collision point occurs when x1 =.xz and Yl = Y2 for the same t. So solve the equations:<br />
3sint= - 3+cost (1)<br />
2 cost = 1 + sin t (2)<br />
From (2), sin t = 2 cost- 1. Substituting into (1), we get 3(2 cost - 1) = - 3 +cost => 5 cost = 0 (*) =><br />
cost = 0 => t = t or 3 ;. We check that t = 3 2..,. satisfies (1) imd (2) butt = t does not. So the only collision point<br />
occurs when t = 3 2 ", and this gives the point (- 3, 0). [We could check our work by graphing Xt and x 2 together as<br />
functions oft and, on another plot, y 1 and y2 as functions oft. If we do so, we see that the only value oft for which both<br />
pairs of graphs intersect is t ~ 3 ;.]<br />
(c) The circle is center<strong>ed</strong> at (3, 1) instead of (- 3, 1). Th~re are still 2 intersection points: (3, 0) and (2.1, 1.4), but there are<br />
no collision points, since(*) in part (b) becomes 5 cost = 6 => cost = ~ > 1.<br />
47. x = t 2 , y = t 3 - ct. We use a graphing device to produce the graphs for various values of c with -7r ::; t ::; 7r. Note that all<br />
the members of the family are symmetric about the x -axis. For c < 0, the graph does not cross itself, but for c = 0 it has a<br />
cusp at (0, 0) and for c > 0 the graph crosses itself at x = c, so the loop grows larger as c increases.<br />
- I<br />
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6 0 CHAPTER 10 PARAMETRIC EQUATIONS AND POLAR COORDINATES<br />
49. x = t + a cost, y = t + a sin t, a > 0. From the first figure, we see that<br />
curves roughly follow the li ne y = x, and they start having loops when a<br />
is between 1.4 and 1.6. The loops increase in size as a increases.<br />
X<br />
While not requir<strong>ed</strong>, the following is· a solution to determine the exact values for which the curve has a loop,<br />
that is, we seek the values of a for which there exist parameter values t and u such that t < u and<br />
(t+acost,t+asint) = (u + acosu,u+asinu).<br />
y<br />
In the diagram at the left, T denotes the point (t, t), U the point (u, u),<br />
and P the point (t +a cost, t +a sin t) = (u + acosu, u + asinu).<br />
Since PT = PU = a, the triangle PTU is isosceles. Therefore its base<br />
angles, a = LPTU and {3 = LPUT are equal. Since a .= t - i and<br />
(3 = 27r -<br />
3 .;' - u = 5 .;' - u, the relation a = /3. implies that<br />
'!I<br />
4<br />
X<br />
u + t = s; (1).<br />
Since TU = distance{{t, t), (u, u)) = J2(u- t) 2 = y'2 (u- t), we see that<br />
p<br />
cos a= ~TU = (u - t)/ v'2, sou - t = v'2acosa, that is,<br />
PT a<br />
u - t = v'2acos(t- i) (2). Nowcos(t- ~) = sin[f - (t- .;f)] = sinC 4<br />
,.. - t),<br />
so we can rewrite (2) as u - t = v'2 asine 4<br />
,. - t) (2'). Subtracting (2') from (1) and T u - t u-t U<br />
J2 J2<br />
dividing by 2, we obtain t = 3 4 ,.. - ~a sine:- t), or 3 4 ,..- t = 72 sine 4 ,.. - t) (3). 1-Ji(u -r)--!<br />
Since a> 0 and t < u, it follows from (2') that sin(3 4<br />
" - t) > 0. Thus from (3) we see that t < 3 4<br />
,... [We have<br />
'<br />
implicitly assum<strong>ed</strong> that 0 < t < 1r by the way we drew our diagram, but we lost no generality by doing so since replacing t<br />
by t + 27r merely increases x andy by 21r. The curve's basic shape repeats every time we change t by 21r.) Solving for a in<br />
v'2(3'~~'-t) · v'2z<br />
(3), we get a = . ( 3 ! ) . Write z = ~,.. - t . Then a = -.- , where z > 0. Now sin z < z for z > 0, so a > v'2.<br />
sm 4<br />
- t sm z<br />
[As z --+ o+, that is, as t --+ ,..)- , a --+ y'2] . e4<br />
51. Note that all the Lissajous figures are symmetric about the x -axis. The parameters a and b simply stretch the graph in the<br />
x- and y-directions respectively. For a = b = n = 1 the graph is simply a circle with radius 1. For n = 2 the graph crosses<br />
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SECTION 10.2 CALCULUS WITH PARAMETRIC-CURVES 0 7<br />
itself at the origin and there are loops above and b~low the x-axis. In general, the figures have n - 1 points of intersection,<br />
all of which are on the y-axis, and a total of n clos<strong>ed</strong> loops.<br />
-1.1<br />
1.1<br />
t1 =3<br />
n =2<br />
n=l<br />
- 2.1<br />
\- ---+- (a,b) =( ~.2)'<br />
(a, b) = (2, I)<br />
-3.1<br />
(a, b) = (2, 3)<br />
(a, b) = (3, 2)<br />
- 1.1<br />
a=b = l<br />
- 2.1 -3.1<br />
n=2 n =3<br />
10.2 Calculus with Parametric Curves<br />
1. x = t sin t, y = t 2 + t ::}<br />
dy . dx . dy dy I dt 2t + 1<br />
-d = 2t + 1, -d = t cost+ sm t , and -d = d l d = . .<br />
t t x x t t cos t + sm t<br />
z . 3 dy = - 3t2 dx = 4- 2t d dy = dyldt = -3tz -Wh - 1<br />
3. x = 1 + 4t - t , y = 2 - t ; t = 1. dt ' dt , an dx dx I dt 4 - 2t . en t '<br />
( x, y) = ( 4, 1) and dy I dx = - ~, so an equa.tion of the tangent to the curve at the point corresponding to t = 1 is<br />
y- 1 = -~(x- 4), oq/ = - ~x + 7.<br />
5. x = t cos t, y = t sin t; t = 1r.<br />
dy . dx . ' dy dy I dt t cos t + sin t<br />
-d = tcost+smt, -d = t(-smt)+cost,and-d = d ld = . .<br />
t t x x t -tsm t+cost<br />
When t = 1r, (x, y) = ( -1r, 0) and dy I dx. = - 1r I( -1) = 1r, so an equation of the tangent to U1e curve at the point<br />
corresponding to t = 1r is y - 0 = 1r[x -<br />
( -1r) ) , or y = 1rx + 1r 2 •<br />
7. (a) x = 1 + ln t, y = t 2 + 2; (1, 3).<br />
dy dx 1 dy dyldt 2t 2<br />
dt = 2 t, dt = t' and dx = dxldt = 1l t = 2 t · At( 1 • 3 ),<br />
x=l+lnt = 1 ::} lnt = O =? t =1 an~ : = 2,so an equationofthetangentis y-3=2 (x - l),<br />
ory = 2x + 1.<br />
(b)x = 1 + ln t =? lnt=x -1 =? t=e"'-l,soy=t2.+2=(e"'- 1 ) 2 + 2 = e 2 " - 2 +2,andy'=e 2 x-z . 2.<br />
At (1, 3), y' = e 2 (l)- z · 2 = 2, so an equation of the tangent is y - 3 = 2(x - 1), or y = 2x + 1.<br />
9. x = 6sint, y = t2+t; (0,0).<br />
dy dy 1 dt 2t + 1 .<br />
- = d ld = --. The potnt (0, 0) corresponds tot = 0, so the<br />
dx x t 6cost<br />
slope of the tangent at that point is· lJ. An equation of the tangent is therefore<br />
y - 0 = i(x- 0), or y = ix.<br />
© 2012 Ccnt;!lgc Learning. All Rights Rcscr\'Cd. May not be scnnncd, copi<strong>ed</strong>, orduplicotcd. or post<strong>ed</strong> to n publicly ncccssiblc wcbsifc, in whole or in part. .
8 D CHAPTER 10 PARAMETRIC EQUATIONS AND POLAR COORDINATES<br />
2 2 dy _ dy I dt _ 2t + 1 _ 1<br />
.!_<br />
11. x = t + 1 • Y = t + t '* dx - dxfdt - 2t - + 2t<br />
!!:._ (dy)<br />
dt dx<br />
dx/ dt<br />
The curve is CU when ~~ > 0, that is, when t < 0.<br />
dy dyfdt - te- t + e- t = e-t(1- t) = e- 2t( _ t)<br />
13. x = et, y = te-t ::::} dx = dxjdt = et et 1 '*<br />
d 2 y = dt (h;<br />
d(~) .<br />
= e- 2 t(- 1) + (1 - t)(- 2e- 2 t) = e - 2 t(- 1 - 2 + 2t) = e-3t(2t- 3). The curve is CU when<br />
dx 2 dx/dt et et<br />
d 2 y 0 th . h t 3<br />
dx 2<br />
> , at ts, w en > 2 .<br />
15. x = 2sint, y = 3cost, 0 < t < 21r.<br />
. i_OO(dy<br />
- - 3 2<br />
dy _ dy/ dt _ -3sint _ -~ d 2 y _ dt dx _ -2sec t _ - ~ 3<br />
dx- dxjdt - 2cost - 2 tant, so dx2 - dxjdt - 2cos t - 4 sec t.<br />
The curve is CU when sec 3 t < 0 ::::} sect < 0 ::::} cost < 0 ::::} i < t < 3 ; .<br />
. d d<br />
17. X = t 3 - 3t, y = t 2 - 3. dy = 2t, SO dy = 0 ¢> t = 0 ¢><br />
t ' t<br />
(x, y) = (0, - 3). dxd = 3t 2 - 3 = 3(t + 1)(t - 1), so d: dx = 0 <br />
t . t<br />
t = - 1 or 1 <br />
(x, y) = (2, -2) or ( -2, -2). The curve has a horizontal<br />
tangent at (0, - 3) and vertical tangents at (2, - 2) and ( -2, -2).<br />
19. x = cos 9, y = cos 39. The whole curve is trac<strong>ed</strong> out for 0 ~ 9 ~ 1r.<br />
~~ = - 3'sin 39, so ~~ = 0 sin 39 = 0 38 = 0, 1r, 21r, or 37r <br />
'e =0, ~ .<br />
2<br />
3 .,.,or1r (x,y) = (1,1), (~ , - 1 ), (-~ , 1) , or(-1,- 1 ).<br />
dx . dx<br />
d 9<br />
= - sLD 8, so dO = 0 sin 8 = 0 8 = 0 or 1r <br />
dy d.x<br />
(x,y_) = (1, 1) or (-1,-1). Both dO and dO equal 0 when 8 = 0 and 1r.<br />
(- 1,-1)<br />
(J= 1T<br />
2<br />
(-t.1)<br />
9=21T/3<br />
-2<br />
(1,1)<br />
9 = 0<br />
.., fi d th I I 9 0 fi d li dy lim - 3 sin 38 11 lim _g cos 38 9 hi h . th I h 0<br />
10 n e s ope w ten = , we n m dx = . n = ,.. = , w c 1s e same s ope w en = 1r.<br />
o~o o ~o - sm" O-•O -cos"<br />
Thus, the curve has horizontal tangents at ( ~ , -1) and (- t, 1), and there are no vertical tangents.<br />
© 2012 C..ngage Learning. All Righi!< Rcsen'Cd. Moy not be scn nnod, copiod, or duplicotod. or post<strong>ed</strong> to a publicly accessible website, in who le or in port.
SECTION 10.2 CALCULUS WITH PARAMETRIC CURVES 0 9<br />
21. From the graph, it appears that the rightmost point on the curve :J: = t - t 6 , y = et<br />
is about (0.6, 2). To find the exact coordinates, we find the value oft for which the<br />
graph has a vertical tangent, that is, 0 = dxl dt = 1 - 6t 5 {::} t = 11 W.<br />
Hence, the rightmost point is<br />
23. We graph the curve x = t 4 - 2t 3 - 2e, y = t 3 - tin ~he viewing rectangle [-2, 1.1] by [- 0.5, 0.5]. This rectangle<br />
corresponds approximately tot E [ -1, 0.8] .<br />
o.s<br />
- 0.5<br />
We estimate that the curve has horizontal tangents at about (- 1, - 0.4) and (- 0:11, 0.39) and vertica·L tangents at<br />
dy dyl dt· 3t2- 1 .<br />
about (0, 0) and (-0.19, 0.37). We calculate -d = dxl d = 4 3 6 2 . The honzontal tangents occur when<br />
X t t - t - 4 t ·<br />
dy I dt = 3t2 - 1 = 0 1=} t = ± ~ , so both horizontal tangents are shown in our graph. The vertical tange~ts occur when<br />
dxldt = 2t(2t2 - 3t-2) = 0 {::} 2t(2t + 1)(t - . 2) = 0 {::} t = 0, - ~or 2. It seems that we have miss<strong>ed</strong> one vertical<br />
tangent, and inde<strong>ed</strong> if we plot the curve on the t-interval [-1.2, 2.2] we see that there is another vertical tangent at (-8, 6).<br />
25. x = cos t, y = sin tcost. dxl dt = - sint, dyl dt = - sin 2 t + cos 2 t = cos 2t.<br />
(x,y)= (O, O) {::} cost = .O. {::} tisanoddmu lt ipleoflf.Whent=~ ,<br />
dxl dt = - 1 and dyl dt = - 1, so dyl dx = 1. When t = 3 ;<br />
, dxl dt = 1 and<br />
dy I dt = - 1. So dy I dx = - 1. Thus, y = x andy = - x are both tangent to the<br />
curve at (0, 0).<br />
27. x = rB - d sinB, y = r - dcosB.<br />
29. X<br />
dx dy , . dy d sinB<br />
(a) dB = r- dcosll, dB = dsme, so dx = r _ dcosB '<br />
(b) IfO < d < r, then jdcosllj:::; d 0. This shows that dxldll never vanishes,<br />
·so the trochoid can have no vertical tangent if d < r .<br />
= 2t3' y = 1 + 4t - t2 ---'- dy - dy I dt = 4 - 2t N 1 . dy = 1 ~ 4 - 2t = 1. ......,._<br />
_,.. ·dx - dxl dt 6t 2 • ow so ve dx .,_,. 6t2 .....,.<br />
6t 2 + 2t - 4 = 0 {::} 2(3t - 2)(t + 1) = 0 {::} t = ~ or t := - 1. 1ft = ~ . the point is(~~ '<br />
the point is ( -2, -4).<br />
2 :), and ift = -1,<br />
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10 0 CHAPTER 10 PARAMETRIC EQUATIONS AND POLAR COORDINATES<br />
31. By symmetry of the ellipse about the x- andy-axes,<br />
A= 4J 0<br />
a ydx = 4 j~<br />
12<br />
bsinB (-a sin B) dB= 4ab I 0<br />
" 12 sin 2 B dB= 4ab Io" 12 ~(1- cos 28) d8<br />
, /2 I<br />
= 2ab[8- ~ sin2B]~ = 2ab(~) = 1rab<br />
33. The curve x = 1 + et, y = t- t 2 = t(1 - t) intersects the x-axis when y = 0,<br />
y<br />
that is, when t = 0 and t = 1. The corresponding values of x are 2 and 1 + e.<br />
The shad<strong>ed</strong> area is given by<br />
1::1+e (YT- YB) dx = 1::1<br />
[y(t)- OJ x'(t) dt = I;(t - t2)et dt<br />
35. x = r8- dsin8, y = r- dcosB.<br />
= J; tet dt- I 0<br />
1 t 2 et dt = f 0<br />
1<br />
tet dt- [fet] ~ + 2 j; tet dt [Formula 97 or parts]<br />
1 [ ' 1<br />
= 3 Io tet dt- (e - 0) = 3 (t- 1)et] 0<br />
- e [Formula 96 or parts]<br />
= 3[0 - (-1)] - e = 3 ~ e<br />
A= I;~r y dx = J;rr (r- dcosB)(r - dcos 8) dB= I;rr (r 2 - · 2drcos B + d~ cos 2 B) dB<br />
= [r 2 B- 2dr sine+ ~d 2 (8 + ~ .sin 28) ]~7r = 21rr 2 + 1rd 2<br />
37. x=t+e~t, y=t-e-t, O~t~ · 2. dxjdt= 1- e- tanddyjdt=1+e-t,so<br />
(dx/dt) 2 + (dyjdt? = (1- e-t) 2 + (1 + e- t? = 1-2e-t + e- 2 t + 1 + 2e-t + e- 2 t = 2 + 2e- 2 t .<br />
Thus, L = I: './(dx/dt) 2 + (dyjdt) 2 dt =I: v2 + 2e- 2 t dt ~ 3.1416.<br />
39 . . x = t- 2sint, y = 1 - 2 cos t, 0 ? t ~ 47r. &cjdt = 1 -2 cost and dyjdt = 2sint, so<br />
(dx/dt) 2 + (dyjdt) 2 = (1-2 cos t) 2 + (2 sin t) 2 = 1- 4cos t + 4 cos 2 t + 4sin 2 t:::;; 5- 4cost.<br />
Thus, L = I: './(dx/dt) 2 + (dyjdt) 2 dt = Io 4 " v5- 4cost dt ~ 26.7298.<br />
41. x = 1 + 3e, y = 4 + 2t~, o ~ t ~ 1. ~x/dt ~ 6t and dyjdt = 6t 2 , so (dxjdt? + (dy /dt) 2 = 36t2 + 36t 4<br />
0<br />
-I<br />
Thus,<br />
1 . 1 ~<br />
L = Ia './36t 2 + 36t 4 dt = Ia 6t .Jl+i2·dt = 61 JU Gdu) [u = 1 + t 2 , du = 2tdt]<br />
= 3[~u 3 1 2 J: = 2(2 3 ; 2 - 1) = 2(2J2 - 1)<br />
43. x ~ t sin t, y =. t cost,. 0 ~ t ~ 1. ~~ = t cost + sin t and ~~ = - t sin t + cos t, so<br />
( ~~ y + ( ~; y<br />
= e cos 2 t + 2t sin t cost + sin 2 t + t 2 sin 2 t ~ 2t sin t cost + cos 2 t<br />
= t 2 (cos 2 t + sin 2 t) + sin 2 t + cos 2 t = t 2 + l.<br />
1<br />
Thus, L .·= f 0 vt<br />
2<br />
+ 1 dt ~ [~tvt~ + 1 + ~ ln(t + vt 2 + 1 ) ] ~ = 4J2 + ~ ln(1 + J2).<br />
® 2012 Ccngagc Learning. All Righ1s Rescrvcc.l. May no1 be scann<strong>ed</strong>, copi<strong>ed</strong>, or duplicat<strong>ed</strong>, or post<strong>ed</strong> to a publicly accessible website, in whole or in part.
SECTION 10.2 CALCULUS WITH PARAMETRIC CURVES 0 11<br />
45.<br />
( ~~ ) 2 + ( !fJd = [e'(cos t - sin t)f + [et(sin t +cos t)f<br />
= (e') 2 (cos 2 t - 2 cos t sin t + sin 2 t) '<br />
+ (et) 2 (sin 2 t+2sint cost+cos 2 t<br />
= e 2 t(2 cos 2 t + 2 sin 2 t) = 2e 2 t<br />
Thus, L = J~" .J2e2t dt = f 0<br />
" -12 et dt =·-12 [ e' ]~ = -/2 (e,.- 1).<br />
47.<br />
1. 4 The figure shows the curve x = sin t + sin 1. 5t, y = cost for 0 .:::; t 5 47r.<br />
. dx/dt = cost+ 1.5 cos 1.5t and dy/ dt = -sin t, so<br />
-2.1 f->o.o--------,.£-f->'
12 0 CHAPTER 10 PARAMETRIC EQUATIONS AND POLAR COORDINATES<br />
(b) We use the CAS to find the derivatives dx/dt and dy/dt, and then use Theorem 6 to find the arc lengtll. Recent versions<br />
of Maple express the integratJ;" J(dxjdt) 2 + (dyjdt) 2 dt as 88E(2 V2 i), where E (x) is the elliptic i~tegral<br />
{1 v/1- x2t2<br />
Jo v'f=t2 dt and i is the imaginary number yCI.<br />
I<br />
Some earlier versions of Maple (a5 well as Mathematica) cannot do the integral exactly, so we use the command<br />
evalf (Int (sqrt (diff (x, t) - 2+diff (y, t)-2), t=O .. 4*Pi)); to estimate the length, and find that the arc<br />
length is approximately 294.03. Derive's Para_ar c_length function in the utility file Int_apps simplifies the<br />
integral to 11 }~ 4 ,. J -4 cost cos(l~ t ) - 4 sin t s in(l~t) + 5 dt.<br />
57. x=tsint, y=tcost, O~t~Tr/2. dxjdt=tcost +sintanddyjdt= -tsint +cost,so<br />
(dx/dt? + (dyjdt) 2 = t 2 cos 2 t + 2tsin t cost+ sin 2 t + e sin 2 t- 2tsin t cost+ cos 2 t<br />
= t 2 (cos 2 t + sin 2 t) +.sin 2 t + cos 2 t = t 2 + 1<br />
59. x = 1 + tet, y = (t 2 + l)et, 0 ~ t ~ 1.<br />
(~~ ) 2 + (!fltf = (te 1 + et) 2 + [(e + I )et + et(2tW = [et(t + 1}f + [et(e + 2t + 1W<br />
= e 2 t(t + 1? + e 2 t(t + 1) 4 = e 21 (t + 1) 2 [1 + (t + 1) 2 ), so<br />
3<br />
(u 4)<br />
(<br />
= 2Tr J 4 T Ju (<br />
u = Dt<br />
fg 2 + 4, t 2 = (u- 4)/9, J<br />
du) [ du = 18t dt, so t dt = · f.J tlu<br />
- 2!:. [~u5 /2 - §.u3/2] 13 - 2!:. • .L [3u5/2 - 20u3/2] 13<br />
- 81 5 3 4 - 81 15 4<br />
= 1;~5 [(3. 13 2 v'13- 20. 13 VI3) - (3 ·.32 - 20. 8)] = 1;~5 (247 VI3 + 64)<br />
63. x = acos 3 8, y =a sin 3 8, 0 ~ B ~ I· (~~) 2 + (;¥tt) 2 = ( - 3acos 2 8 sinG?+ (3asin 2 8 cos 0) 2 = 9a 2 sin 2 (I cos 2 0.<br />
S = .{ 0<br />
" 12 2Tr · a sin 3 () · 3asin 0 cos() d() = 61ra 2 J 0 rr/ 2 sin~ 0 cos 0 dO = ~1ra 2 [sin 5 0] ~ 12 = ~1ra 2<br />
65. X= 3t2, y = 2t 3 , 0 ~ t ~ 5 . ::} C~l + (~/} = (6t? + (6t 2 ) 2 = 36t 2 (1 + t2) ::}<br />
s = .r; 21rx J(dx/dt)2 + (dyJdt)2 dt = J; 27r(3e)6t vt + t 2 d.t = l81r J; t2J1 + t 2 2tdt<br />
[ u = +t2,] =<br />
1<br />
18Trj·2G(u3/2- u1/2) du = 18Tr ['du5/2- ~u3/2] 26<br />
du = 2t dt 1 5 3 1<br />
= 187r [ ( ~ . 676 J26 - ~ . 26 J26) - ( ~ - ~)] = ¥ 7r ( 949 J26 + 1)<br />
® 201 1 2 Ccngngc Learning. All ltig.hts Rcscn·cd. M.3.y not be scarmcd. copi<strong>ed</strong>, ordupl i c~Jlcd+ or ~st cd to o publicly accessible website. in who le or in part.
SECTION 10.3 POLAR COORDINATES 0 13<br />
67. Iff' is continuous and j' (t) i= 0 for a$ t $ b,. then either!' (t) > 0 for all tin (a, b) or j' (t) < 0 for all t in [a, bj. Thus, f<br />
. . '<br />
is monotonic (in fact, strictly increasing or stri 1 ( fl± - xfl ) xy - xif ' . . ·<br />
dt di = dt ;; = xz ~ dt = 1 + (iJ/x) 2 x2 = xz +iF. Usmg the Cham Rule, and tne<br />
"a·cttllat s- rot (d:r;)2+(!!11..)2dt d .• /(d:r;)2+(!!11..)2 ('2+ •2)1/2 b h<br />
dt dt ~ 'Jt - y dt dt = x y , we ave t at<br />
! i J<br />
0<br />
d¢> _ d¢>/dt _ (xii- xfl ) 1 _ xii - xiJ . _- I d¢> 1-1 xy- xy 1- l±ii - xiJI<br />
ds - ds/dt - :i;2 + y2 (±2 + y2)1/2 - (~2 + 1;2)3/2 · So"'- - ds - (±2 + y2)3/2 - (±2 + y2)3/2 ·<br />
d !( )<br />
. 1 .. 0 d . dy .. d2y<br />
(b) x = x_an y = x ~ x = 'x = an y = dx' y = dx2'<br />
- II· (d~y/dx 2 ) - 0 . (dyjcb:) i - ld 2 y/ dx 2 1<br />
So"' - (1 + (dyjdx) 2 )31 2 - (1 + (dyjdx)2] 31 2 •<br />
71. X = 8 - sin8 ~ :i; = 1----' cosB -~ X = sin8, and y = 1 - cos8 ~ iJ =sine ~ y = cos8. Therefore,<br />
icosB - cos 2 e- sin 2 el icosB - (cos 2 8 + sin 2 8)1 JcosB - II .<br />
"' = = . 2 = . The top of the arch 1s<br />
[(1 - cos B) 2 + sin 2 B)S/ 2 (1 - 2 cos e + cos 2 e + sm 8)31 2 (2 - 2 cos 8)3/2<br />
characteriz<strong>ed</strong> by a horizontal tangent, and from Example 2(b) in Section 10.2, the tangent is horizontal when B = (2n -<br />
l )1r,<br />
d b · B · ti . fi Jcos 1r -11 1-1- 11 1<br />
sotaken=1 an su stttute =1rmto 1e ex~ress ton or ~~;: ~t= ( 2 _ 2 cos1r) 3 / 2 = 12<br />
_ 2<br />
(- 1<br />
))3/ 2<br />
=4·<br />
73. The coordinates of T are .(r cosB, r sin B). Since TP was unwound from<br />
arc T A, T P has length rB. Also LPTQ = LPT R - LQT R = t 1r - 8,<br />
soP has coordinates x = r cos B + r8 cos(~1r ~B) = r(cos B + B sin 8),<br />
y = rsin B - rBsin (~1r - 8) = r(sin8- 8cos9).<br />
X<br />
10.3 Polar Coordinates<br />
1. (a) (2, i) By adding 21r to i, we obtain the point ( 2, 7 ;). The direction<br />
opposite i is 4 3<br />
.,., so (- 2, ~ ) is a point that satisfies the r < 0<br />
· requirement.<br />
© 2012 Cengage ~ing. All Rights Reserv<strong>ed</strong>. May not be scann<strong>ed</strong>, copi<strong>ed</strong>, or duplicat<strong>ed</strong>, or post<strong>ed</strong> too publicly accessible wcbsitc, in whole or in part.
14 0 CHAPTER 10 PARAMETRIC EQUATIONS AND POLAR COORDINATES<br />
(b) (1, - 3 ;)<br />
r > 0: (1, - 3 ; + 21r) = (1, 5 ;)<br />
0<br />
r < 0: (- 1,- 3 r+ 7r) = (-1,~)<br />
_ 311'<br />
4<br />
(c) ( - 1, 2J)<br />
r > 0: (- (- 1), 1J +1r) = (1, 3 ;)<br />
0<br />
,.<br />
2<br />
r < 0: (-1, ~ + 21r) = (- 1, 5 2 ,.)<br />
(-t,f)<br />
3. (a)<br />
X= 1COS7r = 1(- 1) = -1 and<br />
(l, 'TT)<br />
0<br />
'TT<br />
y = 1 sin 1r = 1(0) = 0 give us<br />
the Cartesian coordin~tes ( -1, 0).<br />
(b)<br />
x = 2cos(- 2 3<br />
"' ) = 2(-~) = - 1 and<br />
y = 2sin( - 2 ;) = 2( - ~) ~ ~VS<br />
give us ( -1, -V3).<br />
(c)<br />
x = -2cos 3 ;<br />
= -2( -4) = J2 and<br />
y = - 2sin 3 4 " = -~( 4) = - J2<br />
gives us ( J2, -J2).<br />
5. (a) x = 2 andy = - 2 => r = J2 2 + (-2)2 = 2 J2 and 8 = tan- 1 ( :n := -~ . Since (2, -2) is in the fourth<br />
quadrant, the polar coordinates are (i) (2 J2, 7 ;) and (ii) (-2 J2, 3 ;).<br />
(b) x = - 1 andy = V3 =><br />
r = J< -1) 2 + ( v'3) 2 = 2 and 8 = tan~ 1 ( :1) = 2 :;. Since_( -1, VS) is in the second<br />
quadrant, the polar coordinates are (i) (2, ~)and (ii) ( - 2, 6 ;).<br />
'·<br />
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SECTION 10.3 POLAR COORDINATES 0 15<br />
7. r ~ 1. The curve r = 1 represents a circle with center<br />
0 and radius 1. So r ~ 1 represents the region on or<br />
outside the circle. Note that 8 can take on any value.<br />
9. r ~ 0, 7r/4 S, B S, 37r/ 4 . .<br />
8 = IC represents a line through 0.<br />
0= 37T<br />
4<br />
0='!!..<br />
4<br />
11 2 < < 3 s.,.. < B < r1r<br />
· r •a--a<br />
0 - 11< - 3<br />
r=2<br />
o- s1f<br />
- 3<br />
13. ·Converting the p~lar coordinates (2, 1r / 3) and ( 4, 21r / 3) to Cartesian coordinates gives us (2 cos f, 2 sin f) = (1, J3) and<br />
( 4 cos 2 ; , 4 sin 2 ;) = (- 2, 2 J3). Now use the distance !ormula.<br />
15. r 2 = 5
16 D CHAPTER 10 PARAMETRIC EQUATIONS AND POLAR COORDINATES<br />
27. (a) The description leads jmm<strong>ed</strong>iately to the polar equation 9 = ~.and the Cartesian equation y,; tan (~) x = 7:i xis<br />
slightly more difficult to derive.<br />
(b) The. easier description here is the Cartesian equation x = 3.<br />
29. r = - 2 sin9<br />
r<br />
- 2<br />
(2, 3'11'/2)<br />
31. r = 2(1 +cos B)<br />
(4,0)<br />
2'11' 0<br />
33. r = 9, 9;::: 0 r<br />
(J<br />
35. r = 4sin 39 r<br />
4<br />
./o=f<br />
,..:/ 2 (4.f)<br />
(}<br />
-4<br />
37. r = 2cos49 r<br />
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SECTION 10.3 POLAR COORDINATES 0 17<br />
39. r = 1--2sin8<br />
0=!!.<br />
__________ ..• ----- 6<br />
41. r 2 = 9sin28<br />
3<br />
2..- 0<br />
-3<br />
43. r = 2 + sin 38<br />
r<br />
3<br />
2<br />
(3, ..-j6)<br />
2..<br />
---r--~------~----+- t-~<br />
3..- 7..- 11..- 0<br />
6 6 6<br />
45. r = 1 + 2cos28<br />
IJ<br />
47. For(}=:= 0, 7T, and 27T, r has its minimum ;value of about 0.5. For 0 =~and<br />
We see that the graph has a similar shape for 0 ~ (} ~ 7T and 7T ~ 8 ~ 27T.<br />
3<br />
", r attains its maximum va lue of2.<br />
2<br />
. r 2<br />
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18 0 CHAPTER 10 PARAMETRIC EQUATIONS AND POLAR COORDINATES<br />
49. x = r cos 8 = ( 4 + 2 sec 8) cos 8 = 4 cos 8 + 2. Now, r -+ oo =><br />
(4 + 2sec8)-+ 00 => 8-+ (~) - or 8 -+ e2"' )+ [since we ne<strong>ed</strong> only<br />
consider 0 $ 8 < 271' ], so lim x =<br />
. r-oo 8 -+-rr/2-<br />
lim ( 4 cos 8 + 2) = 2. Also,<br />
T -+ -oo => ( 4 + 2 sec B) -+ -oo => 8 -+ ( ~) + or 8 -+ ( 3 ;)-, so<br />
lim x = lim ( 4 cos 8 + 2) = 2. Therefore, lim x = 2 => x = 2 is a vertical asymptote.<br />
r--oo 8--+rr/2+ T"--+±oo<br />
(6,0)<br />
51 . To show that x = 1 is an asymptote we must prove lim x = 1.<br />
r - ±oo<br />
x = (r) cosO= (sin 8 tan 8) cos 8 = sin 2 ().Now; r -+ oo => sinO tan8-+ oo =><br />
8 -+ (f)- ,so lim x= lim sin 2 8 = l.Aiso,r-+-oo => sinBtanB-->-oo =><br />
r --+oo 0-+'Tr/2- .<br />
x=l<br />
e-+ (~) + ,so lim X = lim sin 2 {;I,;, 1. Therefore, lim X= 1 . => X= 1 is<br />
. r ...... - oo 0--+1fj 2+ ,·-±oo<br />
a vertical asymptote. Also notice that X = sin 2 e ~ 0 for all e, and X = sin 2 {;I $ 1 for all e. And Xi= 1, since the curve is not<br />
defin<strong>ed</strong> at odd multiples of ~. Therefore, the curve lies entirely within the vertical strip 0 $ x < 1.<br />
53. (a) We see that the curve r = 1 + csin 8 crosses itself at the origi n, where r = 0 (in fact the inner loop corresponds to<br />
negative r-values,) so we solve the equation of the lima~on for r = 0 c sin 8 = - 1 sin 8 = - 1/ c. Now if<br />
lei < 1, then this equation has no solution and hence there is no inner loop. But if c < - 1, then on the interval (0, 27T)<br />
the equation has the two solutions 8 = sin- 1 ( -1/c) and 8 = 1r- sin- 1 ( - 1/c), and if c > 1, the solutions are<br />
(} = 1r + sin - 1 (1/c) and ()= 21r- sin- 1 (1(c). Tn each case, r < 0 for (} between the two solutions, indicating a loop.<br />
(b) For 0 < c < 1, the dimple (if it exists) is characteriz<strong>ed</strong> by the fact that y has a local maximum at 8 = 3 ;<br />
. So we<br />
determ.ine for what c-valu~s<br />
d 2 ~ is negative at(} = 3 2~, since by the Second Derivative Test tllis indicates a maximum:<br />
~ . .<br />
y = rsin8 =sinO + c sin 2 (} =><br />
dy<br />
dB = cos8 + 2csinfJ cos8 = cosfJ + csin 28 =>.<br />
d 2 y<br />
- ., = - sin8 + 2ccos 28.<br />
dB~<br />
At 8 = 3 2 ,.., this is equal to - (- 1) + 2c( -1) = 1 - 2c, which is negative only for c > %. A similar argument shows that<br />
for - 1 < c < 0, y only has a local minimum at e = ~ (indicating a dimple) for c < -~ . .<br />
55. r = 2 sin (} => x = r cos(} = 2 sin fJ cos 8 = sin 28, y = 1· sin 8 = 2 sin 2 B =><br />
dy = dyjd(J = 2 · 2 sin fJ cosO= sin 28 = tan 2<br />
B<br />
dx dxjdB cos2fJ·2 cos28<br />
When 8 =<br />
7T dy ( 7r) 7T rr; ·<br />
6 , d.'t =tan 2 · 6 =tan = v 3. [Another method: Use Equation 3.]<br />
3<br />
57. r= l / 8 => x=rcos8 = (cosfJ) j B, y=rsin8=(sinfJ)/ 8 =><br />
.dy - 0 + 7r(- 1) - 'Tr<br />
When 8 = 1r, dx = -(-1) -1r(O) = l<br />
dy dyjd8 sin.O( - 1/ 8 2 ) + (1/ 8) cos 8 8 2 - sin (}+(} cosO<br />
dx = dxj dO = cos8(- 1/ fJ 2 ) - (1/ 8) sinB · (} 2 = - cosO - Osin B<br />
= -1r.<br />
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SECTION 10 .~ POLAR COORDINATES 0 19<br />
59. r = cos 2() => x = r cos() = cos 2() cos(}, y = r sin() = cos 2(} !lin(} =><br />
dx = dx/ d() = cos 2() (- sin(})+ cos() ( -2 sin 2())<br />
dy dy/ df) cos 2(} cos(}+ sin(} (-2 sin 2())<br />
1f dy<br />
When() =<br />
4 , dx -,0,!-( J2---:2 /:---27) +_('-:-v-2 -:::2/,.--!2)~(-_2;,_) - _- J2_2 - 1<br />
O(-J2/ 2) + (J2/ 2}(-2) - -J2 - .<br />
61. r = 3cos0 => x = rcosO = 3cos0 cosO,_ y = rsinO = 3cos0 sinO => ,<br />
~ = -3 sin 2 9 +3cos 2 ,() = 3cos20 = 0 => 2() = ~ or 3 ; 0 = %or 3 ;.<br />
So the tangent is horizontal at ( ~, %) and ( - -32, 3 ;) [same as ( -32, -:a:)] .<br />
~~ = - 6 sin() cos~= -3 sin 20 = 0 => 29 = 0 or 1r 0 = 0 or~· So the tangent is vertical at (3, 0) and (o, ~).<br />
63. r = 1 + cos 0 · => x = r cos 9 = cos() (1 +cos 9), y = r sin 0 = sin 0 (1 + cos 9) =><br />
·.~ = (1 +cosO) cosO- sin 2 0 = 2cos 2 0 + cos9 - 1 = (2cos9 - 1)(cos(J + 1) = 0 => cos()=~ or - 1 =><br />
() = 'i· 1r, or 5 ; => horizontal tangent at(~ , 'i), (0, tr), and(~ ,<br />
~~ = - ( 1 + cos 8) sin (} - cos ()sin() = - sin (} ( 1 + 2 cos 9) = 0 => sin 8 = 0 or cos 9 = - ~ =><br />
5 ; ) .<br />
() = 0, 1r, 2 ;, or 4 ; => vertical tangent ~t (2, 0), (!, 2 ;), and (!, -t;).<br />
Note that the ~gent is horizontal, not vertical when 8 = 7r, since e : j :: = 0.<br />
65. r = a sin() + b cos 9 => r 2 = ar sin() + br cos 8 '=> x 2 + y 2 = ay + bx =><br />
x 2 - bx + (~b? + y 2 - ay + ( ~a) 2 = (~b) 2 + (ta? => (x - ~b) 2 + (y - ~ a? = Ha 2 + b 2 ), and this is a circle<br />
with center (!b, ~a) and radius. ~.Ja 2 + b 2 •<br />
67. r = 1 + 2 sin(8 / 2): The parameter intei-val is [0, 47r]. 69. r = e•inB - 2cos(40).<br />
The parameter interval is [0, 21l'j.<br />
71. r = 1 + cos 999 8. The parameter interval is [0, 2tr).<br />
1.1<br />
/<br />
- 1.1<br />
© 2012
20 D CHAPTER 10 PARAMETRIC EQUATIONS AND POLAR COORDINATES<br />
73. It appears that the graph of r = 1 + sin(~ - "if) is the same shape as<br />
the graph of r = 1 + sine, but ro,tat<strong>ed</strong> counterclockwise about the<br />
origin by "if. Similarly, the graph of r = 1 + sin( e ....,. ~) is rotat<strong>ed</strong> by·<br />
i . In general, the graph of r = f ( e - a) is the same shape as that of<br />
r = f (e), but rotat<strong>ed</strong> counterclockwise through a about the origin.<br />
r = 1 + sin ll<br />
r = l +s~ (o - ~_)<br />
r = 1 + sin(ll - ~)<br />
That is, for any point (ro, eo) on the curve r = j(e), the point<br />
-0.9<br />
(ro, Bo +a) is on the curve r = j(e- a), since ro = f(Oa) = f((eo +a)- a).<br />
75. Consider curves with polar equation r = 1 + c cos e, where c is a real number. If c = 0, we get a circle of radius 1 center<strong>ed</strong> at<br />
the pole. For 0 ~ c s 0.5, the curve gets slightly larger, moves right, and flattens out a bit on the left side. For 0.5 < c < 1,<br />
the left side has a dimple shape. For c = 1, ,the dimple becomes a cusp. For c > 1, there is an internal loop. For c 2: 0, the<br />
rightmost point on the cu;ve is (1 + c, 0). For c < 0, the curves are reflections through the vertical axis of the curves<br />
with c > 0.<br />
c = 0.25 c = 0.75 c = 1<br />
tan¢ - tane<br />
77. tan'lj! = tan(¢- e)= r/> e<br />
1 + tan tan<br />
dy<br />
dy/dB<br />
- - tane dxjde - tane<br />
dx<br />
dy<br />
dyjde<br />
1 + dx ta11e 1 + dxj de tane<br />
_ ~ - ~ tan e ~ (~ ~in e + r cos e) - tan e ( ~ cos B - r sine)<br />
- : + :~ tane - (:;.cos e - r sine) + tan B ( :; sine + r cos e)<br />
c = 2<br />
sin 2 e<br />
rcosB+r ·-<br />
case<br />
dr dr sin 2 e<br />
de case+ de. case<br />
_ rcos 2 B +rsin 2 e r<br />
- dr 2 iJ dr . 2 e = dr I dO<br />
de cos . + dB sm<br />
10.4 Areas and Lengths in Polar Coordinates<br />
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SECTION 10.4 A~~S AND LENGTHS IN POLAR COORDINATES 0 21<br />
3. r 2 = 9sin28, r ~ 0, ·a~ 8::; 7rj2.<br />
1<br />
~/2 1~ /2 / 2<br />
A - lr<br />
2 2 d8<br />
- 1(9sin2B 2 )d8 -l!. -2 [- lcos2B)~ 2 0 ---l!.(- 4<br />
1 - 1)=l!.<br />
2<br />
0 . 0<br />
7. 1' = 4 + 3sin8, -%::; () ::; %·<br />
1<br />
~/2<br />
= ~ (16 + 9sin 2 8) dO<br />
-rr/2<br />
= ~·2} 0<br />
rr/ 2<br />
r /2<br />
[by Theorem 4.5.6(b) [ET 5.5.7(b)]]<br />
[16 + 9 ·~( 1 -cos28))d~ [by Theorem 4.5.6(a) [ET 5.5.7(a)]] .<br />
= 1 ("; - ~cos28) d8 = [~ 1 8 - tsin2BJ; 12 = ( 4 ~" - O)- (0 _ 0) = 4~"<br />
'<br />
9. The area is bound<strong>ed</strong> by r = 2 sin 8 for 8 = 0 to 8 = 7r.<br />
(2, 'TT/ 2)<br />
A= 1"' ~r 2 d8 := ~ 1 -rr (2 sinfJ) 2 dB= ~ 1"' 4sin 2 Bd8<br />
=21" ~( 1 - cos2fJ)dB ~ [e - ~sin2eJ: =1r<br />
Also, note that th is is a circle with radius 1, so its area is 1r(1) 2 = 1r.<br />
" ~r 2 2<br />
d8 = l<br />
11. A = 12<br />
rr H3 + 2cos8) 2 d8=! fo 2 rr(9+12.cos8+4cos 2 8)d8<br />
= ~ 12.,.. [9 + 12cos8 + 4 · t(1.+ cos 28)] dB _(_l ,_'TT--:) H-=-- - --'-:..._:..1----+<br />
{2-rr<br />
= ~ lo (11 + 12 cos 8 + 2 cos 28) d8 = Hne + 12 sin O+ sin 28) ~71'<br />
= H227r) = 111r<br />
13. A = r 71' ~r 2 d8= r rr t{2 + sin4fJ) 2 dfJ = ~ r"(4+4si~4fJ+sin 2 4fJ)d8<br />
Jo lo · Jo ·<br />
3<br />
= ~ fo 2 " I4 + 4sin48 + ~(1- cos88)] dB<br />
2"<br />
= ~ (~ + 4 sin48 - ~ cos88) dB=~ [~B- cos4fJ-<br />
1<br />
fs- sin8B] ~"'<br />
0 .<br />
= ~[(97r- 1) - (- 1)) = ~7('<br />
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22 D CHAPTER 10 PARAMETRIC EQUATIONS AND POLAR COORDINATES<br />
1 ( 3 9 1 . 9] 21r 1 ( ) 3<br />
= 2 2 + 2o sm 10 o = 2 3'1T = 27T<br />
-1.4<br />
17. The curve passes through the pole when r = 0 ==?- 4 cos 30 = 0 ==?- cos 39 = 0 ==?- 39 = ~ + 1rn =?-<br />
9 = % +·in. The part of the shad<strong>ed</strong> loop above the polar axis is trac<strong>ed</strong> out for<br />
9 = 0 to 9 = 7T /6,' so we'll use -7T / 6 and 7T /6 as our limits of integration. r = 4 cos 30<br />
19. r = 0 ==?- sin49 = 0 ==?- 48 = 1rn ==?- 9 = ~n. r=sin 40<br />
r '4 . r'4 r'4<br />
A = Jo Hsin48) 2 dfJ = t Jo sin 2 48d8 = t Jo t(1-cos89)d8<br />
0 . . 0 0<br />
1 (<br />
9<br />
1 ·<br />
89 ]"'I 4 1 (,.) . 1<br />
= 4 - B SlD 0 = 4 4 = Tii7T<br />
21. This is a lima~on, with inner loop trac<strong>ed</strong><br />
r = 1 + 2 sin 8 (rect.)<br />
3 out between 8 = 7 ; and 1 ~" [found by<br />
solving r = 0].<br />
17T ------·<br />
0=6 .<br />
··--•• ,<br />
0= 6<br />
lll'l'<br />
31r/2 . [37f/2 [ 37r/2<br />
[<br />
71r /6 77r /6 71r /6 . .<br />
A = 2 !(1 + 2sin8? d8 = (1 + 4sin8 + 4sin 2 8) d8 = [1 +4sin8 + 4· t{1- cos29)] d9<br />
= [9 - 4cos0 + 20- ~n28J~:~~ = ( 9 ;)- C; + 2 v'3 - 4) = 7T - ¥<br />
23. 2 cos B = 1 => cos 8 = ~ => 8 = :g. or 5 ; .<br />
A = 2 f 0<br />
,. 13 ~[(2 cos8) 2 -<br />
1 2 ) dB= J 0<br />
" 13 (4cos 2 8 - l) d8<br />
= J; ;a { 4 [ ~ (1 + cos 28)] - 1} d8 = J; /S ( 1 + 2 cos 28) d8<br />
= [8 + sin 28)~ 13 = i + 4 r = 2cos0<br />
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SECTION 10.4 AREAS AND LENGTHS IN POLAR COORDINATES 0 23<br />
25. To find the area inside the l ~miniscate r 2 = 8 cos 29 and outside the circle r = 2,<br />
we first note that the two curves intersect when r 2 = 8 cos 29 and ~ = 2,<br />
. I<br />
that is, when cos2B = ~-For -1r < B ::; 1r, cos2B = ~ {::} 2B = ±7r/ 3<br />
r=2<br />
or ± 511' /3 {::}<br />
B = ±1r / 6 or ±57r / 6. The figure shows that the desir<strong>ed</strong> area is<br />
4 times the area between the curves from 0 to 1r /6. Thus,<br />
A = 4 J 0<br />
" /. 6 [~(8 cos2B)- ~(2) 2 ] dB = 8 J" 16 0<br />
(2 cos 2B - 1) dB<br />
[ ] rr~ .<br />
. =8 sin2B - B .=8(VS/2-7r/6) =4\1'3-411'/3<br />
0<br />
27. 3cosB=1+cosB {::} cosB=~ => B=tor-i.<br />
A= 2 f 0<br />
" 13 ~ [(3 cos 8?- (1 +cos 0) 2 ) dB<br />
= J 0<br />
"' 13 (8 cos 2 8-'- 2 cos B- 1) dB = f 0<br />
" 13 [4(1 +cos 2B) - 2 cos B - 1) dB<br />
8 =1!.<br />
I 3<br />
,l'<br />
= J 0<br />
.,. 13 (3 + 4cos28- 2cosB) dB= [3B + 2sin2B- 2sinf!] ~ 1 3<br />
= 1T +v'3-.J3=1T<br />
29. v'3cosB = sinB => J3 = sinBB => tanB = J3 => B = i·<br />
. cos<br />
A -<br />
f "' 1 3 1 (sin B) 2 dB + 1'-"' 12 1 (J3 cos B) 2 d()<br />
-Jo 2 hr/ 3 . 2<br />
- f"'/ 3 1. 1(1 - cos28) dB+ f "/ 2 1 · 3 · 1(1 + cos2B) dB<br />
- Jo 2 2 rr/3 2 2<br />
- 1 [B - 1 · 28]" 13 +!! [B +.! · 28]1f 12<br />
- 4 2 sm o 4 2 sm 7f/3<br />
= t [ ( i - 4) - o] + ~ [ (~ + o)- ( f + 4)]<br />
_ .l!.. _ _il + .!!: _ 1fl _ 5rr _ fl<br />
- 12 16 . 8 16 - 24 4<br />
r = J3cos8<br />
31. sin 2B = cos 2B => sin 2 B = 1 => tan 28 = 1 => 28 = ~ =><br />
cos2B<br />
B=. f =><br />
A= 8 · 2 f0"/8'~ sin 2 2B dB= 8 f 0<br />
"' 18 H1-cos 48) dB<br />
= 4[9- t sin48]~ 18 = 4(f - i ·1) = ~ - 1<br />
33. sin2B = cos2B => tan2B = 1 => 2B = ~ => (} = f<br />
A= 4 f 0 "' 18 ~sin 2B dB [since r 2 = sin 2B)<br />
= f 0 "' 18 2 sin2B dB= [-;- cos 28] ; 18<br />
= - ~ v'2- (- 1) = 1- ~ Y2.<br />
.... ···B=i<br />
© 2012 Ccngage learning. All Riglus Reserv<strong>ed</strong>. May not be scann<strong>ed</strong>, copi<strong>ed</strong>, or duplicat<strong>ed</strong>, or post<strong>ed</strong> to a publicly accessibk website, in whole or in part. ·
24 • 0 CHAPTER 10 PARAMETRIC EQUATIONS AND POLAR COORDINATES<br />
35. The darker shad<strong>ed</strong> region (from 0 = 0 to 0 = 27T / 3) represents ~ of the desir<strong>ed</strong> area plus ~ of the area of the inner loop.<br />
From this area, we'll subtract~ of the area of the inner loop (the lighter shad<strong>ed</strong> region from 0 = 27r /3 to 0 = 7r), and then<br />
double that difference to obtain the desir<strong>ed</strong> ·area.<br />
A = 2 [J;.,. 13 H~ +~os0) 2<br />
dO- g.,. 13 H~ +cos0) 2 dB]<br />
= £ 0<br />
2<br />
.,.<br />
13 (~+cosO+ cos<br />
2<br />
0) dB- f 2<br />
:<br />
. .<br />
= J 0<br />
211'/S [~+cos 0 + ~(1 + cos28)) dO<br />
13 (~ + cos 0"+ cos2 0) dB<br />
- I2:/3 a + cosO+ ~( 1 + cos28)) d(J<br />
2 13<br />
= [~ + sin 8 + ~ + sin 20] " _ [~ + sin 8 + ~ + sin 20 1.,.<br />
4 2 4 0 4 2 4 211'/ 3<br />
- (1!: + .il + 1!: - .:.::2) - (!!: + !!:) + ( 1!: + .il + 1!: - .:.::2)<br />
-6 2 3 8 4 2 6 2 3 8<br />
= ~ + ~ J3 = H 7T+3v'3)<br />
37. The pole is a point of intersection.<br />
1 + sin 8 = 3 sin 0 => 1 = 2 sin (J => sin 0 = ~ =><br />
(J =~ or<br />
5 ;.<br />
The other two ~oints of intersec~ion ~e ( ~, i) and ( f, 5 6 ").<br />
r=l + sinO<br />
39. 2 sin 20 = 1 =? sin 20 = ~ =? 2o = 1!: &.,. 13.,. or 11"<br />
6' 6 ' 6 ' 6 °<br />
By symmetry, the eight points of intersection are given by<br />
(1 8) h (} _ 11' 5,. 13rr d 1771"<br />
, , w ere -<br />
d<br />
12 , 12 , 12 , an 12 , an<br />
r = 2sin20<br />
( - 1, 8), where 0 = i;, 1 t;, 1 : 2<br />
"; and 2 [;.<br />
[There are many ways to describe these points.]<br />
41. The pole is a point of intersection. sin (J = sin 20 = 2 sin(} cos 0 <br />
sinO (1 - 2 cos 9) = 0 sinO= 0 or coslJ = t =><br />
8 = 0, 7T, t. or - t => the other intersectio~ points" are ( ~. t )<br />
and ( ~, 2 ; ) [by symmen:'].<br />
r =sin20<br />
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SECTION 10.4 AREAS AND LENGTHS IN POLAR COORDINATES 0 25<br />
43.<br />
. 3.4 3 y = 2x<br />
r= 2(}<br />
-3<br />
r = 1 +sinO<br />
-1.4 1.4<br />
- 0.3 -3<br />
From th~ first graph, we see that the pole .is one point of intersection. By zooming in or usin~ the cursor, we find the 0-values<br />
of the intersection points to be a:~ 0.88786 ~ 0.89 and :r - a ~ 2.25. (The first of these values may be more easily<br />
estimat<strong>ed</strong> by plotting y = 1 +sin x andy = 2x in rectangular coordinates; see the second graph.) By symmetry, the total<br />
area contain<strong>ed</strong> is twice the area contain<strong>ed</strong> in the first quadrant; that is,<br />
A= 21o. ~(20) 2 dO+ 2[ 1r/Z ~( 1 +sin 0.) 2 d8 = 1"' 40 2 .d8 + l1r/~ [1 + 2sinU + H1- cos 2B)] d8<br />
= (~0 3 ] ~ + [8- 2cosB + (~0 - t sin2B)J: 12 = ~a 3 + [(~+~)-(a- 2cosa + ~a: - t sin2o:)) ~ 3.4645<br />
45. L = 1b Jr2+(d7·jd0) 2 dB= fo7r }(2cos0) 2 +(-2sinB.)2d8<br />
= 17r V 4(cos 2 B + sin 2 B) dB = 1"' ..f4d8 = [2B]; = 21r<br />
As a check, note that the curve is a circle of radius 1, so its circumference is 27r(1) = 21r.<br />
= 12<br />
"' Je 2 (0 2 + 4)d0 = 12<br />
1f BV0 2 +4d0<br />
Now let u = 0 2 + 4, so that du = 20d8 [Ode= t du] and<br />
49. The curve r = cos 4 (0 / 4) is completely trac<strong>ed</strong> with 0 :::; () :::; 47r.<br />
r 2 + (drfd8) 2 = [cos 4 (B/4)] 2 + [4cos 3 (B/ 4) · (-sin(B/4)) · t ] 2<br />
= cos 8 .(B/4) + cos 6 (8/4) sin 2 (B/ 4)<br />
= cos 6 (8/4)[cos 2 (B/4) +sin 2 (0/4)] = cos 6 (B/4)<br />
L = f 0<br />
41 r Jcos 6 (0/4) dO = J 0<br />
4<br />
"' jcos 3 .(0/4) j rj,B .<br />
= 2 f~" cos 8 (B/4)dB [sincecos 3 (B/4) ~ 0 for O :S B $ 27r] = 8J; 12 cos 3 udu [u = tB]<br />
= 8 J; 12 (1 - sin 2 u) cosudu = 8 f 0 1 (1- x 2 ) dx<br />
= 8[x- ~x 3 ) ~ = 8(1- ~) = lf<br />
x sinu, ·]<br />
[ dx = cosudu<br />
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26 0 CHAPTER 111 PARAMETRIC EQUATIONS AND POLAR COORDINATES<br />
51. One loop of the curve r = cos 28 is trac<strong>ed</strong> with - 1r I 4 ~ 8 ~ 1r I 4.<br />
r 2 + (~~) 2 14<br />
= cos 2 28 + (- 2 sin 28) 2 = cos 2 28 + 4sin 2 28 = 1 + 3 sin 2 28 ::::} L /"' Vl + 3 sin 2 28 d8:::::: 2.4221.<br />
-?r/4 -<br />
53. The curver = sin(6sin9) is completely trac<strong>ed</strong> with 0 ~ 8 ~ 1r. r = sin{6sin9) ::::} ~ = cos{6sin9)_-6cos8, so<br />
r 2 : (~;y =s~n 2 (6sin8)+36cos 2 8cos 2 (6sin0) ::::} L L" Jsin 2 (6sin8)+36cos 2 8pos2(6sinO)d8:::::8.0091.<br />
55. (a) From (10.2.6),<br />
s =I: 21ryJ(dxld0)2 + (dyld9) 2 dO<br />
[from the derivation of Equation 10.4.5]<br />
(b) The curve r 2 = cos 28 goes through the pole when cos 28 = 0 ::::}<br />
20 = ~ ::::} 8 = ~. We'll rotate the curve from () = 0 to 8 = ~ and double<br />
this value to obtain the total surface area generat<strong>ed</strong>.<br />
· dr ( dr ) 2 sin 2 28 sin 2 28<br />
r 2 = cos28 ::::} 2r dO= -2sin28 ::::} dO - - <br />
- ~ - cos20'<br />
(o. f)<br />
? = cos28<br />
(1.0)<br />
10.5 Conic Sections<br />
1. x 2 = 6y and x 2 = 4py ::::} 4p = 6 ::::} p = ~<br />
The vertex is (0, 0), the focus is (0, ~),an d the directrix<br />
isy = -~.<br />
3. 2x = - y 2 ::::} y 2 = -2x. 4p = -2 ::::} p = - ~.<br />
'\<br />
The vertex is (0, O),the focus is ( - ~, 0), and the<br />
directrix is x = ~.<br />
y<br />
x=4<br />
6 X X<br />
y= - ~<br />
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SECTION 10.5 CONIC SECTIONS 0 27<br />
5. (x + 2) 2 = 8 (y- 3). 4p = 8, sop= 2. The vertex is<br />
( - 2, 3), the focus is ( -2, 5), and the directrix is y = 1.<br />
y<br />
7. y 2 + 2y + 12x + 25 = 0 =l><br />
y2 + 2y + 1 = - 12x- 24 =;..<br />
(y + 1) 2 = - 12(x + 2). 4p = - 12, sop= -3.<br />
The vertex is (- 2, -1), the focus is (-5, - 1), and the<br />
directrix is x = 1.<br />
-------------------------- __ ,y_~_!_<br />
X<br />
0 :<br />
~ X<br />
I<br />
jx=l<br />
9. The equation has the form y 2 = 4px, where p < 0. Since the parabola passes through (- 1, 1), we have 1 2 = 4p( - 1), so<br />
4p = - 1 and an equation is y 2 = - x or x = - y 2 • 4p = - 1, sop = -i and the focus is ( -;i, 0) while the directrix<br />
. 1<br />
ISX = 4·<br />
ellipse is center<strong>ed</strong> at {0, 0), with vertices at (0, ±2). The foci are (0, ±v'2).<br />
y<br />
2<br />
-..fi<br />
x2<br />
Y2<br />
13. x 2 + 9y 2 = 9 9 + l = 1 =l> a = v'9 = 3, 15. 9x 2 - 18x + 4y 2 = 27 <br />
b = Vi.= 1, c = ~ = v'9=I = J8 = 2v'2.<br />
The ellipse is center<strong>ed</strong> at {0, 0), with vertices (± 3, 0).<br />
The foci are (±2v'2, 0).<br />
y<br />
I<br />
-2<br />
9(x 2 - 2x + 1) + 4y 2 = 27 + 9 <br />
9(x- 1) 2 + 4y 2 = 36 <br />
a = 3, b = 2, c = J5 =:- center (1, 0),<br />
vertices (1 , ± 3}, foci (1, ±J5)<br />
(x- 1)2 y2<br />
4 + g- = 1 =}<br />
- 3<br />
0<br />
3 X<br />
y (1, 3)<br />
- I<br />
3 X<br />
2 2<br />
17. The center is (0, 0), a= 3, and b = 2, so an equation is ~ + Y 9<br />
= 1. c = ~ = J5, so the foci are (0, =;I=J5).<br />
(1,- 3)<br />
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28 0 CHAPTER 10 PARAMETRIC EQUATIONS AND POLARCOORDINATES<br />
yz<br />
xz<br />
19. 25<br />
- 9 = 1 ::::? a = 5, b = 3, c = J25 + 9 = J34 ::::?<br />
center (0, 0), vertices (0, ±5), foci (0, ±J34), asymptotes y = ±~x .<br />
Note: It is helpful to draw a 2a-by-2b rectangle whose center is the center of<br />
the hyperbola. The asymptotes are the extend<strong>ed</strong> diagonals of the rectangle.<br />
x2 y2<br />
21 . x 2 - y 2 = 100 <br />
100 - 100<br />
= 1 ::::? a = b = 10,<br />
c = JlOO + 100 = 10J2 => center (0, 0), vertices (±10, 0),<br />
foci (±10 J2, 0), asymptotes y = .±f§x = ±x<br />
y<br />
X<br />
,.._--!-----",,<br />
,~_; .......<br />
23. 4x 2 - y 2 - 24x - 4y + 28 = 0 <br />
4(x 2 - 6x + 9) - (y 2 + 4y + 4) = -28 + 36- 4 <br />
4(x '_ 3)2 - (y + 2)2 = 4 (x - 3)2 - (y + 2)2 = 1 =><br />
1 4<br />
a = v'f = 1, b = v'4 = 2, c = Vf+4 = J5 ::::?<br />
center (3, - 2), vertices (4, - 2) and (2, -2), foci (3 ± J5, -2),<br />
asymptotes y + 2 = ±2(x- 3).<br />
25. x 2 = y + 1 x 2 = 1(y + 1 ). This is an equation of a parabola with 4p = 1, so p = i. The vertex is (0, -1) and the<br />
focus is (0, - ~).<br />
27. x 2 = 4y- 2y 2 x 2 + 2y 2 - 4y = 0 x 2 + 2(y 2 - 2y + 1) = 2 x 2 + 2(y - 1) 2 = 2 <br />
x; + (y ~ 1 2<br />
)<br />
= l..This is an equation of an ellipse with vertices a~ (±J2, 1) . The foci ~eat (±.,J2=1, 1) = (±1, 1).<br />
29. y 2 + iy = 4x 2 + 3 y 2 + 2y + 1 = 4x 2 + 4 (y + 1? - 4x 2 = 4 (y: 1 2<br />
)<br />
- x 2 = 1. This is an equation<br />
of a hyperbola with vertices (0, - 1 ± 2) = {0, 1) and (0, -3). The foci are at (0, - 1 ± J4+T) = (0, - 1 ± J5).<br />
31. The parabola wi~ vertex {0, 0) and focus (1 , 0) ?pens to the right and hasp= 1, so its equation is y 2 = 4px, or y 2 = 4x. '<br />
33. The distance from. U1e focus ( -4, 0) to the directrix x = 2,is 2 - ( - 4) = 6, so the distance from the focus to the vertex is<br />
~(6) = 3 and the vertex is ( - 1, 0). Since the focus is to the left of the vertex, p = - 3. An equation is y 2 .= 4p(x + 1) =><br />
y 2 = -12(x + 1).<br />
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SECTION 10.5 CONIC SECTIONS 0 29<br />
35. A parabola with vertical axis and vertex (2, 3) has equation y- 3 = a(x- 2?. Since it passes through (I; 5), we have<br />
5-3=a(I-2) 2<br />
'* a =2,soanequationisy -3 =2(x-2?.<br />
37. The ellipse with foci (±2, 0) and vertices (± 5, 0) has center (0, 0) and a horizontal major axis, with a = 5 and c = 2,<br />
x2 1 2<br />
so b 2 = a 2 - c 2 = 25- 4 = 21. An equation is 25<br />
+ ; 1<br />
= 1.<br />
39. Since the vertices are (0, 0) and (0, 8), the ellipse has center (0, 4) with a vertical axis and a = 4. The foci at (0, 2) and (0, 6)<br />
are 2 units from the center, soc= 2 and b = .,Ja 2 - c 2 = .,j4 2 - 2 2 = v'f2. An equation is (x - 2<br />
0) 2 + (y- 4 ) 2 = 1 '*<br />
b a 2<br />
_41. An equation of an ellipse with center (-1,4) and vertex (- 1,0) is (x ! 2<br />
1 )<br />
2<br />
+ (y ~2 4 ) 2 = 1. The focus (-1,6) is 2 units<br />
from the center, soc= 2. Thus, b 2 + 2 2 = 4 2 '* b 2 = 12 and the equation is (x + 1 ? + (y - 4 ) 2 = 1<br />
. , 12 16 .<br />
2 2<br />
43. An equation of a hyperbola with vertices (±3, 0) is ....:. ; ~2 2<br />
= 1. Foci (±5, 0) '* c = 5 and 3 2 + b 2 =5 '*<br />
2<br />
. 2 2<br />
b 2 = 25 - 9 = 16, so the equation is ~ - r6 = 1.<br />
45. The center of a hyperbola with vertices ( -3, - 4) and ( -3, 6) is ( -3, 1), so a = 5 and a.n equation is<br />
(x + 3) 2 . · ·<br />
b 2<br />
= l.Focl(- 3, - 7)and(- 3,9)-* c =8,so5 2 +b 2 =8 2 '* b 2 =64-25=39andthe<br />
( - 1} 2 {x + 3) 2<br />
equation is y - = I<br />
25 39 .<br />
. . 2 • 2<br />
47. The center of a hyperbola with vertices (±3, 0) is (0, 0), so a = 3 and an equation is ~ 2<br />
- t 2<br />
= 1.<br />
b . 2 2<br />
Asymptotes y = ±2x '* - = 2 '* b = 2(3) = 6 and the equation is ~ - JL = 1.<br />
a 9 36<br />
49. In Figure 8, we see that the point on the ellipse closest to a focus is the closer vertex (which is a distance<br />
a - c from it) while the farthest point is the other vertex (at a distance of a+ c). So for this lunar orbit,<br />
(a- c)+ (a+ c) = 2a = (1728 + 110) + (1728 + 314), or a = 1940; and (a+ c)- .(a - c) = 2c = 314 - 110,<br />
or c = 102. Thus, b 2 = a 2 -<br />
x2 y2<br />
c 2 = 3,753,196, and the equation is , , 3 763 600<br />
+ , , =I.<br />
3 753 196<br />
51. (a) Set up the coordinate system so that A is ( -200, 0) and B is (200, 0) .<br />
IPAI-\PBI = (1200)(980) = 1,176,000 ft = 2 1~ mi = 2a -* a= 1 ~ 5 , and c ~ 200 so<br />
b 2 - 2 - 2 - 3,339,375<br />
- c a - 121<br />
121x 2<br />
1,500,625<br />
121y 2 - 1<br />
3,339,375 - .<br />
© 20 12 Ccngoge Leoming. All Rights Rescn'Cil. May not be scann<strong>ed</strong>, copi<strong>ed</strong>. or duplicat<strong>ed</strong>. or post<strong>ed</strong> to a publi
30 0 CHAPTER 10 PARAMETRIC EQUATIONS AND POLAR COORDINATES<br />
(b) Due north of B => x = 200 =><br />
(121}(200) 2<br />
1,500,625<br />
121y 2<br />
y = 133,575 ~ 248<br />
mi<br />
3,339,375 = 1 => 539<br />
53. The function whose graph is the upper branch of this hyperbola is concave upward. The function is<br />
55. (a) If k > 16, then k - 16 > 0, and xk + -k y<br />
2 2<br />
= 1 is an ellipse since it is the sum of two squares on the left side.<br />
- - - 16<br />
2 2<br />
(b) IfO < k < 16, then k - 16 < 0, and xk + k ~ 16<br />
= 1 is a hypel"bola since it is the difference of two squares on the<br />
left side.<br />
(c) If k < 0, then k - 16 < 0, and there is no cw11e since the left side is the sum of two negative tenns, which cannot equal I .<br />
(d) In case (a), a 2 = k, b 2 = k- 16, and c 2 = a 2 - b 2 = 16, so the foci are at (±4, 0}. Jn case (b), k - 16 < 0, so a 2 = k,<br />
b 2 = 16 - k, and c 2 = a 2 + b 2 = 16, and so again the foci are at (±4, 0}.<br />
57. x 2 = 4ptJ => 2x = 4py' => y' = ; , so the tangent line at (x 0 , y 0 ) is<br />
. p<br />
2<br />
y- :; = ~; (x - xo). This line passes through the point (a, - p) on the<br />
y<br />
d . . x6 xo ( ) 4 2 2 2 2 2<br />
1rectnx, so - p -<br />
4 p = 2 p a - xo => - p - x 0 = axo - x 0 <br />
x5 - 2axo - 4p 2 = 0 x5 - 2axo + a 2 = a 2 + 4p 2 <br />
(a,-p)<br />
y= - p<br />
(xo- a) 2 = a 2 + 4p 2 x 0 =a± Ja 2 + 4p 2 • The slopes of the tangent lines at x = a± Ja 2 + 4p 2<br />
a± ..ja 2 + 4p 2<br />
are p , so the product of the two slopes ts<br />
2<br />
showing that the tangent lines are perpendicular.<br />
a+ Ja2 + 4p2 a - Ja2 + 4p2 a2 _ . (a2 + 4p2) -4p2.<br />
-----'----'--- . = = -- = -1<br />
2p 2p 4p2 4p2 '<br />
2 2<br />
59. 9x 2 + 4y 2 = 36 ~ + Y 9<br />
= 1. We use the parametrization x = 2 cost, y = 3 sin t, 0 ~ 't ~ 21r. ·The circumference<br />
is given by<br />
L =I:" J(dx/dt) 2 + (dyjdt)2 dt = ]~ 2 " J( - 2 sin t) 2 + (3 cost)2 dt =I:" )4sin 2 t + 9cos 2 t dt<br />
= I:" v 4 + 5 cos 2 t dt<br />
Now use Simpson's Rule with n = 8, 11t = 2 1T- 0 = ~.an d f(t) = v'4 + 5cos2 t to get<br />
8 . 4<br />
L ~ ss = ~ [J(o) + 4f(*) + 2f(~) + 4Jen + 2J(1r) + 4/( 5 ;) + 21en + 4tCn + J(21T)J ~ 15.9.<br />
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x2 y2 y2 a:2 _ a2 b<br />
61. ;_2 - - = 1 =? - = --- =? y = ±- Jx2- a2.<br />
a b 2 b 2 a 2 a<br />
SECTION 10.5 CONIC SECTIONS 0 31<br />
A= 2 r !!_ J x 2 ·_ a 2 dx ~ 2 b [~ J x2 - a2 - 42 In Jx + J x2 - a2J ] c<br />
la a a 2 2 a·<br />
b<br />
= -[cvc 2 - a 2 -a 2 ln jc + vc 2 - a 2 j + a 2 ln!al)<br />
a<br />
Since a 2 + b 2 = c 2 , c 2 - a 2 = b 2 , and J c2 - a 2 = b.<br />
=!!. [cb- a 2 1n(c + b)+ a 2 !na) = !!_ [ch + a 2 (!na - !n(b +c)))<br />
a<br />
a<br />
= b 2 c/a + abln[a/(b +c)), where c 2 = a 2 + b 2 .<br />
x2<br />
Y2<br />
63. 9x 2 + 4y 2 = 36 '4 + g = 1 =? a= 3, b = 2. By symmetry, x = 0. By Example 2 in Section 7.3, the area o_fthe<br />
top half of the ellipse is H 1rab) = 37T. Solve 9x 2 + 4y 2 = 36 for y to get an equation for the top half of the ell ipse:<br />
9x 2 +4y 2 =36 4y 2 =36-9x 2 ~ y 2 = ~(4-x 2 ) =? y = ~v'4-x 2 .Now<br />
1 1b 1 1 12<br />
1 ( 3 ) 2 3 12<br />
y = - - [f(x)) 2 dx=- - - V 4- x 2 dx= - (4 -x 2 )dx<br />
A a 2 37T - 2 2 2 87T - 2<br />
2 3 [ 1 3] 2 = - · 2 ( 4 - x ) dx = - 4x - - x = -3 ( - 16) = -4<br />
3 ' 12<br />
87T 0 47T 3 0 47T 3 7T<br />
y<br />
so the centroid is (0, 4/ rr).<br />
'ffi . . . I' . I •x 2 y 2 1<br />
65. Dt erenttatmg tmp tctt y, a 2<br />
+ b 2<br />
= =? [y =f. 0]. Thus, the slope ofthe tangent<br />
line at Pis - b:x 1 . The slope of F1P is ___)f];_ and of F2P is ____lt!__ By the formula from Problems Plus, we have<br />
a Y1 X1 + c X1 - c<br />
and<br />
Yl<br />
b 2 Xt<br />
-Xt_+_c + -a2_y_1 _ a 2 yt + b 2 x 1(x1 +c) _ a 2 b 2 + b 2 cx1<br />
tan a = _.:........;.---:-:.-----'-'-=--<br />
1 _ b 2 x1Y1 - a 2 y1 (x1 + c) - b 2 X tY1 - C 2 XtY1 + a 2 CIJ1<br />
a 2 yt (Xt +c)<br />
b 2 (cxt+a 2 ) · b 2<br />
= CIJt(CXt + a 2 ) = cy1<br />
- a 2 y'f. - b 2 x1 (x1 -c) - a 2 b 2 + b 2 cx1 b 2 (ext - a 2 ) b 2<br />
= a 2 y1 (x1- c) - b 2 X 1Y1 = c 2 x1y1 - a 2 C1Jt = CIJ1 (cx1 - a 2 ) = CIJ1<br />
Thus, a = {3.<br />
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32 0 CHAPTER 10 PARAMETRIC EQUATIONS AND POLAR COORDINATES<br />
10.6 Conic Sections in Polar Coordinates<br />
1. The. directrix x = 4 is to the right of the focus at the origin, so we use the form with "+ e cos(}" in the denominator.<br />
<strong>ed</strong> l. 4 4<br />
2<br />
(See Theorem 6 and Figure 2.) An equation is r = (} =<br />
1<br />
(} = (}.<br />
' 1+ ecos 1+ 2 cos 2 +cos<br />
3. The directrix y = 2 is above the focus at the origin, so Wf! use the form w ith "+ e sin(}" in the denominator. An equation is<br />
_ <strong>ed</strong> _ 1.5(2) _ 6<br />
r - 1 + e sin () - 1 + 1.5 sin(} - 2 + 3 sin(}·<br />
5. The vertex ( 4, 3Jr /2) is 4 units below the focus at the origin, so the directrix is 8 units below the focus ( d = 8), and we<br />
use the form with"- e sin(}" in the denominator. e = 1 for a parabola, so an equation is<br />
<strong>ed</strong> 1(8) 8<br />
r = 1 - e sin () = 1 - 1 sin (} = 1 - sin (} ·<br />
7. The directrix r = 4 sec (} (equivalent to r cos(} = 4 or x· = 4) is to the right of the focus at the origin, so we w ill use the form<br />
with"+ e cos()" in the denominator. The distance from the focus to the directrix is d = 4, so an equation is<br />
<strong>ed</strong> ~{4) 2 4<br />
r = 1 + ecosB = 1 + ~cos(}· 2 = 2 +cos((<br />
4<br />
9. r =<br />
. l / 5 4 5<br />
= / where e = .1 and <strong>ed</strong> = .1<br />
5 - 4sinB 1/5. 1 - ~sinB' 5 &<br />
=> d = 1.<br />
(a) Eccentricity = e = ~<br />
(b) Since e = ~ < 1, the conic is an ellipse.<br />
(c) Since"- e sin (}" appears in the denominator, the directrix is below the focus<br />
y<br />
(4, 7r/2)<br />
at the origin, d = IFll = 1, so an equat'ion of the directrix is y = - 1.<br />
(d) The vertices are (4, ¥) and(~ ,<br />
3 ;).<br />
X<br />
-------1 ---------·<br />
(<br />
4 3'1T) y=-1<br />
F2<br />
2 1/3 2/ 3 ' 2<br />
11. r =<br />
3 + 3 sinB · . 1 /~ = l + 1 sinB' wheree=1and <strong>ed</strong>= 3 => d= ~.<br />
(a) Eccentricity= e = 1<br />
(b) Since e = .1, Lhe conic is a parabola.<br />
(c) Since"+ e sin(}" appears in the denom inator, the directrix is above the focus<br />
at the origin. d = IFll = j , so an equation oflhe directrix is y = i·<br />
(d) T~e vertex is at ( ~, ¥), midway between .the focus and directrix.<br />
y<br />
y = 2/ 3<br />
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SECTION 10.6 CONIC SECTIONS IN POLAR COORDINATES 0 33<br />
13. r = : 6 9 · 1 6 3 2<br />
1 = (<br />
+ cos 1<br />
6 1+ 3 cos0<br />
(a) Eccentricity= e = t<br />
(b) Since e = t < 1, the conic is an ellipse.<br />
, where e = ~ and <strong>ed</strong> = ~ :::} d = ~ -<br />
(c) Since "+ e cos e" appears in the denominator, the directrix is to the right of<br />
the focus at the origin. d = !Fll = ~.so an equation of the directrix is<br />
x = ~-<br />
(d) The vertices are ( ~, 0) and ( t, 1r), so the center is midway between them~<br />
that is, ( fn, 1r) .<br />
(f,1T)<br />
I<br />
9 :<br />
x=21<br />
:<br />
!<br />
3 1/ 4 3/ 4 3<br />
15. r = 4 - 8 cos 9 . 1 I 4 = 1 - 2 cos 9' where e = 2 and <strong>ed</strong> = 4<br />
d - 1 - s ·<br />
(a) Eccentricity = e = 2<br />
(b) Since e = 2 > 1, the conic is a hyperbola.<br />
(c) Since "- e cos 0" appears in the denominator, the directrix is to the left of<br />
the focus at the origin. d'= IFll = ~.so an equation of the directrix is<br />
. X - - s·<br />
3<br />
(d) The vertices are ( - ~. 0) and (i, 1r), so the center is midway between them,<br />
that is, (~ , 1r).<br />
17. (a) r = ; .<br />
1 - sm 9 , where e = 2 and <strong>ed</strong> = 1 :::} d = 1 . The eccentricity<br />
2<br />
e = 2 > 1, so the conic is a hyperbola. Since" -e sin B" appears in the<br />
denominator, the directrix is below the focus at the origin. d = IFll = ~.<br />
so an equation of the directrix is y = - ~ . The vertices are (- 1, ~) and<br />
( 1 3 '~~" ) h . "d b th h . ( 2 3 " )<br />
3 , 2 , sot e center IS m1 way etween l'!m, t at 1s, 3 , 2 .<br />
(b) By the discussion that prec<strong>ed</strong>es Example 4, the equation<br />
. 1<br />
IS T = ( 3 ) •<br />
1 - 2sin e- ;<br />
-3<br />
-2<br />
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34 0 CHAPTER 10 PARAMETRIC EQUATIONS AND POLAR COORDINATES<br />
19. Fore < 1 the curve is an ellipse. It is nearly circular when e is close to 0. As e<br />
increases, the graph is stretch<strong>ed</strong> out to the right, and grows larger (that is, its<br />
right-hand focus moves to the right while its left-hand focus remains at the<br />
origin.) At e = 1, the curve becomes a parabola with focus at the origin.<br />
21.IPFI= e iPLI => r =e[d-rcos(7r-9)J = e(d+rcos8) =*<br />
<strong>ed</strong><br />
r(1 - ecos~) = <strong>ed</strong> => r=---....,.<br />
1- ecosO<br />
X<br />
x=-d<br />
23. IPFI = eiP ll => r = e[d- rsin(9 -1r)] = e(d+rsinO) =><br />
r(1 - esin 9) = <strong>ed</strong> =><br />
<strong>ed</strong><br />
r = ---""7<br />
.. 1-esin9<br />
y<br />
X<br />
l • y= - d<br />
25. We are given e = 0.093 and a = 2.28 x 10 8 . By (7), we have<br />
a(1 - e 2 ) 2.28 x 10 8 (1- (0.093) 2 ] . 2.26 x 10 8<br />
r = = ~ -::-----::-=::------::<br />
1 + ecos 8 1 + 0.093 cosO 1 + 0.093cos8<br />
27. Here 2a = length of major axis = 36.18 AU => a = 18.09 AU and e = 0.97. By (7), the equation of the orbit is<br />
18.09(1 - (0.97) 2 ]<br />
r = 1.07 B (&) th · d" fr th h ·<br />
~<br />
. y , e maxtmum 1stance om e comet to t e sun 1s<br />
1 + 0.97 cos 9 1 + 0.97 cos 8<br />
18.09(1 + 0.97) ~ 35.64 AU or about 3.314 billion miles.<br />
29. The minimum distance is at perihelion, where 4.6 x 10 7 = r = a(1 - e) = a(1 - 0.206) = a(0.794} =><br />
a = 4.6 x 10 7 / 0.794. So the maximum distance, which is at aphelion, is<br />
r = a(1 + e) = ( 4.6 x 10 7 / 0.794) (1.206} ~ 7.0 x 10 7 km.<br />
31. From Exercise 29, we have e = 0.206 and a(l- e)= 4.6 x 10 7 km. Thus, a= 4.6 x 10 7 / 0.794. From (7), we can write the<br />
. . 1 - e 2 •<br />
equation of Mercury's orbit as r =a . So smce<br />
1 + ecos 8<br />
dr a(1- e 2 )esin8<br />
d9 (1 + ecos8) 2<br />
-= =><br />
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CHAPTER 10 REVIEW 0 35<br />
the length of the orbit is<br />
L = 12<br />
,. Jr 2 + (dr/d0)2 d(J = a(l- e 2 ) 12,. .Jl(t e 2 + 2 ~)~sO dO:::::: 3.6 x 10 8 km<br />
0 0 +ecos<br />
This seems reasonable, since ,Mercury's orbit is nearly circular, and the circumference of a circle of radius a<br />
is 21ra :::::: 3.6 x 10 8 km.·<br />
10 Review<br />
CONCEPT CHECK<br />
1. (a) A parametric curve is a set of points of the form (x, y) = (J(t), g(t)), where f and g are continuous functions of a<br />
variable t.<br />
(b) Sketching a parametric curve, like sketching the.graph of a function, is difficult to do in general. We can plot points on the<br />
curve by finding f(t) and g(t) for various values oft, either by hand or with a calculator or computer. Sometimes, when<br />
f and g are given by formulas, we can eliminate t from the equations x = j(t) andy = g(t) to get a Cartesian equation<br />
relating x andy. It may be easier to graph that equation than to work with the original formulas for x andy in terms oft.<br />
2. (a) You can find : as a function oft by ca lcu lati ng~~ = ~~J~~ [if dxjdt f: 0].<br />
(b) Calculate the area as I: y dx = I: g(t) j'(t)dt [or I; g(t) !' (t)dt ifthe leftmost point is (!({3), g({J)) ~ther<br />
than (f(a),g(a))].<br />
3. (a) L = I: J(dxjdt)2 + (dyjdt)2 dt =I: vlf'(t))2 + [.g'(t))2 dt<br />
(b) s =:: I: 27ryJ(dxfdt) 2 + (dyfdt) 2 dt =I: 21rg(t) v lf'(t)F + [g'(t)J2 dt .<br />
4. (a) See Figure 5 in Section 10.3.<br />
(b) x = rcosO, y = r sinO<br />
(c) To find a polar representation (r, 0) with r ;?. 0 and 0 s 0 < 27r, first calculate r = ·Jx 2 + y2. TI1en 0 is specifi<strong>ed</strong> by<br />
cosO= xjr and sinO= yjr.<br />
d dy !:.._ (y) !:.._ ( r ::;in 8)<br />
5. (a) Calculate Jx = ~ = ~ = ..::::llJ:;:-- -<br />
dO dO (x) d() (rcosB)<br />
(b) calculate A= I: !r 2 dO= I: tlfCOW d8<br />
(~) sin(} + r cos(}<br />
dr) . , wherer = f(O).<br />
( dO cos () - r sm (}<br />
(c) L =I: J(dx/d8.)2 + (dy/d0) 2 dO= I: Jr 2 + (dr/d0) 2 dO= .r: J[f(0))2 + [!'(0)]2 dO<br />
6. (a) A parabola is a set of points in a plane whose distances from a fix<strong>ed</strong> point F (t11e focus) and a fix<strong>ed</strong> line l (the 'directrix)<br />
are equal.<br />
(b) x 2 = 4py; y 2 = 4px<br />
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36 D CHAPTER 10 PARAMETRIC EQUATIONS AND POLAR COORDINATES<br />
7. (a) An ellipse is a set of points in a plane the sum of whose distances from two fix<strong>ed</strong> points (the foci) is a constant.<br />
2 2<br />
(b) X + _Y_ = 1.<br />
a2 a2 - c2<br />
8. (a) A hyp~rbola is a set of points in a plane the difference of whose distances from two fix<strong>ed</strong> points (the foci) is a constant.<br />
This difference should be interpret<strong>ed</strong> as the larger distance minus the smaller distance.<br />
2 2<br />
(b) ::__ - _ Y_ = 1<br />
a2 c2- a2<br />
(c) y = ±<br />
Jc2 _ o,2<br />
a<br />
x<br />
9. (a) Jfa conic section has focus F and corresponding directrix l, then the eccentricity e is the fix<strong>ed</strong> ratio /PFI / /Pl/ for points<br />
P of the conic section.<br />
(b) e < 1 for an ellipse; e > 1 for a hyperbola; e = 1 for a parabola.<br />
<strong>ed</strong> . <strong>ed</strong> · <strong>ed</strong> <strong>ed</strong><br />
(c) x = d: r = . x = - d:.r = . y = d: 1' = . . y = - d: r = -:----:--::<br />
1 + e cos (} 1 - e cos (} 1 + e sm 8 1 - e sin 8 ·<br />
TRUE-FALSE QUIZ<br />
1. False.<br />
3. False.<br />
Consider the curve defin<strong>ed</strong> by x = f(t) = (t - 1) 3 and y= g(t) = (t- 1) 2 : Then g'(t) = 2(t - 1), so g'(1) = 0,<br />
but its graph bas a vertical tangent when t = 1. Note: The statement is true iff' (1) =I= 0 when l (1) = 0.<br />
For example, iff ( t) = cost and g( t) = sin t for 0 ::; t :S 4rr, then the curve is a circle of radius I, hence its length<br />
is 2rr, but f 0<br />
4<br />
"<br />
J [!'(t)] 2 + [g'(t))2 dt = .{ 0<br />
4<br />
" J(- sin t) 2 +(cos t) 2 dt = f 0<br />
4<br />
" 1 dt = 4rr, since as t increases<br />
from 0 to 4rr, the circle is travers<strong>ed</strong> twice.<br />
. 5. True.<br />
The curve r = 1 - sin 28 is unchang<strong>ed</strong> if we rotate it through 180° about 0 because<br />
1 - sin 2(8 + rr) = 1 - sin(28 + 2rr) = 1 - sin 28. So it's unchang<strong>ed</strong> if we replacer by -r. (See the discussion<br />
after Example 8 in Section 10.3.) In other words, it's the same curve as r = -(1 -sin 20) = sin 28 - 1.<br />
7. False. The first pair of equations gives the portion of the parabola y = x 2 with x ;::: 0, whereas the second pair of equations<br />
traces out the whole parabola y = x 2 •<br />
9. True. By rotating and translating ~he parabola, we can assume it has an equation of the form y = cx 2 , where c > 0.<br />
Tile tangent at the point (a, ca 2 ) is the line y - ca 2 = 2ca(x - a); i.e., y = 2cax - ca 2 • This tangent meets<br />
the parabola at the points (x, cx 2 ) where cx 2 = 2cax - ca 2 • This equation is equivalent to x 2 = 2a~ - a 2<br />
[since c > 0]. But x 2 = 2ax - a 2 x 2 - 2ax + a 2 = 0 (x- a) 2 = 0 x =a <br />
( x, cx 2 ) = (a, ca 2 ). This shows that eacb tangent meets the parabola at exactly one point.<br />
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CHAPTER 10 REVIEW 0 37<br />
EXERCISES<br />
1. X = t 2 + 4t, y = 2 - t, - 4 ~ t ~ 1. ~ = 2 - y, SO<br />
y<br />
X = {2 - y) 2 + 4{2- y) = 4- 4y + y 2 + B- 4y = y 2 - By+ 12 {::}<br />
x + 4 = y 2 -<br />
By+ 16 = (y - 4) 2 . This is part of a parabola with vertex<br />
( -4, 4), opening to the right.<br />
X<br />
3. y = sec() = ~() = .! . Since 0 ~ () ~ 1r /2, 0 < x ~ 1 and y ;::: 1.<br />
COS X<br />
This is part of the hyperbola y = 1/x.<br />
y<br />
\ ,1,1),0-0<br />
0<br />
5. Three different sets of parametric equations for the curve y = -IX are<br />
(i) X = t, y = Vt<br />
(ii) X = t 4 , y = t 2<br />
(iii) x = tan 2 t, y = tan t, 0 ~ t < 1r /2<br />
There are many other sets of equations that also give this curve.<br />
7. (a) The Cartesian coordinates are x = 4 cos 2 ; = 4 (- ~) = -2 and<br />
y = 4sin ;,. = 4( 4) = 2 J3, that is, the point ( - 2,2 J3).<br />
(b) Give~ x = - 3 and y= 3, we haver = J( -3) 2 + 3 2 = JIB = 3 v'2. Also, tan() = ~ '* tan() = ~ , and since<br />
3<br />
( -3, 3) is in the second quadrant,(}= ' 3 4 ". Thus, one set of polar coordinates for ( -3, 3) is (3 V2, 3 ,;'"), and two others are<br />
(3 '2 1 111") and (- 3 '2 7 ,.)<br />
y ~, 4 v. .L.i, 4 .<br />
9. r = 1 - cosO. This cardioid is<br />
symmetric about the polar axis.<br />
r<br />
2<br />
(2, 7T)<br />
7T 27T 0<br />
® 2012 Censagcl..eaming. All Rights Rcscn"
38 0 CHAPTER 10 PARAMETRIC EQUATIONS AND POLAR COORDINATES<br />
11. r = cos 30. This is a<br />
r<br />
three-leav<strong>ed</strong> rose. The curve is<br />
trac<strong>ed</strong> twice.<br />
13. r = 1 +cos 2e. The curve is<br />
sym·metric about the pole and<br />
both the horizontal and vertical<br />
(2, 1r)<br />
(2,0)<br />
axes.<br />
1T<br />
2<br />
1T<br />
31T<br />
2<br />
-1<br />
3<br />
15. r = =} e = 2 > 1, so the conic is a hyperbola. ·de = 3 =><br />
t + 2sin£1<br />
d = ~ and the form "+2 sin(}" imply that the directrix is above the focus at<br />
the origin and bas equation y = ~-The vertices are (1, ~) and ( - 3, 3 ;).<br />
2 .<br />
17. x+y=2 ¢:> rcosll+rsinll = 2 ¢:> r(cos0+sin0)=2 ¢:> r= - - - --<br />
cos(J + s in£1<br />
19. r = (sin £1)/0. As B-. ±oo, r __, 0.<br />
As (} __, 0, r --.. 1. In the first figure,<br />
there are an infinite number of<br />
x -intercepts at x = 1rn, n a nonzero<br />
integer. These correspond to pole<br />
points in the second figure.<br />
-0.25<br />
- 0.75<br />
2 dy dx 1 dy dy / dt 2t 2<br />
21. x = lnt, y = 1 + t ; t = 1. dt = 2t and dt = t' so dx = dxjdt = 1<br />
/t = 2t ·<br />
When t = 1, (x, y) = (0, 2) and dyjdx = 2.<br />
23. r = e- 0 => y = rsin£1 = e- 0 sin(} and x = T"cos O = e- 0 cosO =><br />
dy dyjd£1 *sin(} +rcosO _ -e- 0 sin(}+ e- 0 cos(} - e 0 sine- cosO<br />
dx = dxj d(} = ~;cos(} - rsin(} - -e- 0 cosB - e- 0 sinO · - e 0 = cosB+sine·<br />
dy 0 - (- 1) 1<br />
When (} = 1r, - = = - = - 1.<br />
dx -1 + 0 -1<br />
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CHAPTER 10 REVIEW 0 39<br />
25. x = t + sin t, y = t - cost ~<br />
d (dy)<br />
dt dx<br />
dxjdt<br />
dy dyjdt 1 + sint<br />
dx = d.xjdt = 1 +cos t<br />
{1+cost) cost- {1 +sint)(- sint)<br />
___ _ ___:(,_1,...-+.:..__c_os_t-'-)- 2 _____ = cost+ cos 2 t +sin t + sin 2 t =<br />
l +cost<br />
(l+cost)3<br />
1 + cost + sin t<br />
(1 + cost)3<br />
3 2 .<br />
27. We graph the curve x = t - 3t, y = t + t + 1 for -2.2 :=::; t :=::; 1.2.<br />
By zooming in or using a cursor, we find that the lowest point is about<br />
(1.4, 0. 75). To find the exact values, we find the t-value at which<br />
dyjdt=2t+1 = 0 ¢=> t=~~ , (x,y),;,(lj,~).<br />
29. x·= 2acost- acos2t ~ dx = -2asin t + 2asin2t = 2asint(2cost - 1) = 0 ¢:><br />
dt<br />
sint = 0 or cost = ~ ~ t = O,·t, 1r, orr;.<br />
y = 2asin t - a sin 2t ~ dy = 2a cost - 2a cos 2t = 2a(l +cost- 2 cos 2 t) = 2a(1 -cos t)(1 + 2 cost) = 0 =}<br />
dt<br />
271" 4"<br />
t = 0 '3,or3.<br />
Thus the graph has vertical tangents where t = t. 1r and ~"> 3 " , and horizontal tangents where t = 2 ; and 4 ;. To determine<br />
what the slope is where t = 0, we use !'Hospital's Ru;e to evaluate lim ddy/jddt = 0, so there is a. horizontal tangent there.<br />
t->0 X t<br />
t X y y<br />
0 a 0<br />
3 " ~a :.::]:a<br />
2<br />
2rr<br />
- ~a ~a<br />
3 2<br />
7r - 3a 0<br />
-~a<br />
4 7r<br />
3 -~a 2<br />
51r<br />
~a<br />
_ :.::]:a<br />
3 2<br />
(-3a,O)<br />
(a,O)<br />
)C<br />
31. The curve r 2 = 9 cos 50 has 10 "petals." For instance, for - -fu :=::; 8 :=::; {' 0<br />
, there are two petals, one with r > 0 and one<br />
with r < 0.<br />
33. The curves intersect when 4 cos 8 = 2 ~ cos 8 = ~ ~ 8 = ± i<br />
for -1r :=::; 8 :=::; 1r. The points of intersection are (2, i) and (2,- ~).<br />
r = 4cos 0<br />
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40 D CHAPTER 10 PARAMETRIC EQUATIONS AND POLAR COORDINATES<br />
35. The curves intersect where 2 sin () = sin() + cos () =><br />
sin 0 = cos 0<br />
=> 0 = "i, and also at the origin (at which 0 = 3 4 rr<br />
on the second curve).<br />
A= I 0<br />
" 14 ~(2 sin0) 2 dO+ I:;f 4 ~(sinO+ cos 0) 2 dO<br />
= Io" 14 (1- cos 20) dO+~ I:;t (1 + sin2fJ) dO<br />
= [0 ~ ~ sin20]~ 14 + [~0-:- t cos2BJ!~~ 4 = ~(1r -1)<br />
'<br />
8=f<br />
r = 2 sin 8 r = sin 0 + cos 8<br />
L =I~ .J(dxjdt)2 + (dyjdt) 2 dt = 1~ .J(6t)2 + {6t 2 )2 dt =I: V36t_ 2 + 36t 4 dt = I~ y'36t2 v'1 + t2 dt<br />
=I~ 6\t! v1 + t 2 dt = 6 I~ t v'f+t2 dt = 6 I 1<br />
5 u<br />
1 ;<br />
2<br />
(~du)<br />
= 6. ~ · ~ [u 3 1 2 J: = 2(5 3 1 2 - 1) = 2(5VS -1)<br />
39. L =I:" Jr 2 + (drjdB)2 dB = I:" }(1/0)2 + (-1/0 2 )2 dB= L 2 "' 7<br />
dB<br />
~ [-#+! +In (B + J B2 + 1 )] 2,. = #+I - v' 471"2 + 1 + In ( 271" + ~ )<br />
() 7r: 271" 71" + v' 71"2 + 1<br />
11"<br />
4<br />
s = I 1 27ry.J(dxfdt)2 -t. (dyfdt)2 dt =<br />
4 I1 21r{~t 3 + ~c 2 ) J (2/0) 2 + (t2- t-3)2 dt<br />
= 21r J 4 (.!t 3 + .!c 2 ) . l(t2 + t - 3) 2 dt = 21r f 4 (.!t 5 + !! + l t- 5 ) dt - 21r [...!..t 6 + l!t - 1c 4 ) 4 -<br />
471·295 1r<br />
1 3 2 v 1 3 6 2 - 18 6 8 1 - 1024<br />
43. For all c except - 1, the curve ·is asymptotic to the line x = 1. For<br />
c < -1, the curve bulges to the right near y = 0. As c increases, the<br />
bulge becomes smaller, until at c = - 1 the curve is the straight line x = 1.<br />
As c continues to increase, the curve bulges to the left, until at c = 0 there<br />
is a cusp at the origin. For c > 0, there is a loop to the left of the origin,<br />
whose size and roundness increase as c increases. Note that the x-intercept<br />
of the curve is always -c.<br />
© 2012 Ccngag~ Ltaming. All Rights R.cscrvcd. Muy not be scann<strong>ed</strong>, copi<strong>ed</strong>, or duplicat<strong>ed</strong>, or po!!1etl lo n publicly accessible website, in whole or in part.
CHAPTER 10 REVIEW 0 41<br />
2 2 .<br />
45. ~ + 'lL = 1 is an ellipse with center (0, 0).<br />
9 8<br />
a = 3, b = 2 J2, c = 1 =?<br />
foci (±1, 0), vertices (±3, 0).<br />
y J-<br />
2 2<br />
47. 6y 2 + X - 36y + 55 = 0 ~<br />
6(y 2 - 6y + 9) = - (x + 1) ~<br />
(y - 3) 2 = --[;(x + 1), aparabola with vertex (-1, 3),<br />
opening to the left, p = - f.t =? focus (- ;~, 3) and<br />
directrix x = -~.<br />
y<br />
- 3 3 X<br />
(- 1,3)<br />
0 X<br />
49. The ellipse with foci (±4, 0) and vertices (±5, 0) has center (0, 0) and a horizontal major axis, with a = 5 and c = 4,<br />
2 2<br />
so b 2 = a 2 - c 2 ·= 5 2 - 4 2 = 9. An equation is ~ 5<br />
+ Y 9<br />
= 1.<br />
2 ?<br />
51 . The center of a hyperbola with foci {0, ±4) is (0, 0), so c= 4 and an equation is y 2<br />
- x: = 1.<br />
. a b<br />
a 3 ·<br />
The asymptote y = 3x has slope 3, sob = l - ::::} a= 3b and a 2 + b 2 = c 2 =? {3b? + b 2 = 4 2 =?<br />
10b 2 = 16 ::::}<br />
b 2 = * and so a 2 = 16 - ~ = L 5<br />
2 • Thus, an equation is -<br />
" " 72 5 8 5<br />
y2 x2 5y2 5x2<br />
1 - - - 1 = 1, or - 72 - - 8<br />
= 1.<br />
53. x 2 = -(y - 100) has its vertex at (0,100), so one of the vertices ofthe ellipse is (0, 100). Another form of the equation of a<br />
parabola is x 2 = 4p(y - 100) so 4p(y - 100) = - (y- 100) =? 4p = -1 ::::} p = -~.The refo re the shar<strong>ed</strong> focus is<br />
found at .(0, 3 ~ 9 ) so 2c = 3 ~ 9 -0 ::::} c = 3 ~ 9 and the center of the ellipse is (0, 3 ~ 9 ). So a= 100- 3 ~ 9 = 4 ~ 1 and<br />
2 2 . 2 ( 39!))2<br />
b 2 __ a 2 _ c 2 __ 401 - 399 x y - 8<br />
82<br />
----,=--- = 25. So the equation of th e ell ipse is /1 + a 2<br />
= 1 =?<br />
x 2 (8y - 399? = 1<br />
or 25 + 160,801 ·<br />
55. Directrix x = 4 =? d = 4, so e = ~<br />
<strong>ed</strong> 4<br />
=? r - - ..,..--- -.,<br />
- 1 +ecosfJ- 3+cosfJ'<br />
57. (a) Jf (a, b) lies on the curve, then there is some parameter value t1 such that~ = a and<br />
1 + tl 1 + t l<br />
3 tr ~ =b. lft 1 = 0,<br />
the point is (0, 0), which lies on the line y = x. lft1 =I 0, then the point corresponding tot= ..!. is given by<br />
tl<br />
3(1/tl ) 3ti 3(1/td 3tl<br />
x = 1<br />
+ ( 1<br />
/t<br />
1<br />
) 3 = t~ + 1<br />
= b, Y = 1<br />
+ ( 1<br />
/h) 3 = tt + 1<br />
= a. So (b, a) also lies on the curve. [Another way to see<br />
2<br />
3<br />
this is to do part (e) first; the result is imm<strong>ed</strong>iate.] The curve intersects the line y = x when ~3<br />
= t<br />
1 + t 1+t<br />
t = t 2 ::::} t = 0 or. 1, so the points are (0, 0) and ( ~, ~).<br />
3<br />
::::}<br />
42. D . CHAPTER 10 PARAMETRIC EQUATIONS AND POLAR COORDINATES<br />
(b) dy = ( 1 +t 3 )(Gt) - 3 t2( 3 t 2 ) = Gt - 3 t 4 =Dwhen6t - 3t 4 =3t(.2-t3)=D ·=> t=Dort= 3 '2,so thereare<br />
dt (1+t3)2 (1+t3)2 y.r.<br />
horizontal tangents at (0, D) and ( V2', ?-'4). Using the symmetry from part (a), we see that there are vertical tangents at<br />
(0, D) and ( ?-'4, .v2).<br />
(c) Notice that as t --> -1 +,we have x--> -oo andy --> oo. As t--> -1-, we have x --. oo andy --> - oo. Also<br />
. 3t+3t 2 +{1+t 3 ) {t+1) 3 {t+1) 2<br />
y - {-x - 1) = y + x + 1 =<br />
=<br />
3<br />
3<br />
=<br />
1+ t 1 +t t 2 -t+1<br />
slant asymptote.<br />
-> 0 as t -> -1. Soy = -x - 1 is a<br />
( .d) dx = (I-+ t 3 )(3)- 3t(3t 2 ) = 3 - 6t 3 dy 6t- 3t 4 dy dyjdt t(2 - t 3 )<br />
dt (1 + t3)2 (1 + t3)2 and from part (b) we have dt = (1 + t3)2. So dx = dxjdt = 1- 2t3 .<br />
y<br />
So the curve is concave upward there and has a minimum point at (D, D)<br />
and a maximum point at ( .v2, W). Using this together with the<br />
X<br />
information from parts (a), (b), and (c), we sketch the curve.<br />
(e) x? + y 3 = 1 + t 3 + 1 + t3<br />
(<br />
3t ) 3 ( 3t2 ) 3 27t 3 + 27t 6<br />
.(1 + t3)3<br />
. 2 3<br />
3xy = 3C!\3 )C~t3 ) = ( 1 ~~3 ) 2 ,sox 3 +y 3 =3xy.<br />
27t 3 (1 +t 3 )<br />
(1 + t3)3<br />
'y =-x - l<br />
(f) We start with the equation from part (e) and substitute x = r cos 0, y = r sin 0. Then x 3 + y 3 = 3xy =><br />
r 3 cos 3 0 + 1' 3 sin 3 e ~ 3r 2 cos e sin e. For r =1- 0, this gives r = 3 ~OS e s~: . Dividing numerator and denominator<br />
cos O+ sm 0<br />
3(_1 ) sinO<br />
. cosO · cosO<br />
by cos 3 0, we obtam r = .<br />
sm 3<br />
e<br />
1 +-<br />
cos3 0<br />
(g) The loop corresponds to 0 E (D, ~).so its area is<br />
3sec0 tan B<br />
1 + tan 3 e .<br />
- {"/ 2 r 2 -.! ("/ 2 (3sec0 tan0) 2 - ~ r / 2 sec 2 e tan 2 e - ~ {""' u 2 du<br />
A-} 0<br />
2d0-2} 0<br />
1+tan3 0 d0-2} 0<br />
(l+ tan3 0)2d 0 -2} 0<br />
{1+u3 ) 2<br />
[let u = tan OJ<br />
= lim l!.[-~(l+u3)-l]b =;!<br />
b--+oo 2 3 a 2<br />
(h) By symmetry, the area between the foUum and the line y = -x - 1 is equal to the enclos<strong>ed</strong> area in the third quadrant,<br />
plus twice the enclos<strong>ed</strong> area in the fourth quadrant. The area in the third quadrant is~ . and since y = -x - 1 =><br />
r sine = - r cos 0 - 1 => r = - . e 1 e' the area in the fourth quadrant is<br />
!;JU +cos<br />
1 /_-"'/ 4 [( 1 ) 2 (3sec0 tan0) 2 ] CAS 1. . 1 (1) 3<br />
- - . () () -<br />
2 - rr/2 sm + cos +tan<br />
1<br />
3 () d() = - 2 . Therefore, the total area IS 2 + 2 2 = 2 .<br />
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0 PROBLEMS PLUS<br />
l<br />
lt sin u dx cost dy sin t<br />
t cos u<br />
1. x = --du, y = . --du, so by FTCl, we have dt = -t- and dt = - t - . Vertical tangent lines occur when<br />
1 tL 1 tL<br />
dx = 0 {:} cost = 0. The parameter value corresponding to (x, y) = {0, 0) is t = 1, so tl1e nearest vertical tangent<br />
dt .<br />
occurs when t = ~.<br />
Therefore, tl1e arc length between these points is<br />
7r/2<br />
L=<br />
1<br />
(- dx) 2 + (dy) - 2 dt= 1 1r/Z<br />
1 dt dt 1<br />
2<br />
t . 2 t j '1f/Z<br />
dt<br />
cos ~dt = - ~ [ lnt]7f 12 = ln :!!.<br />
t2 + t2 1 t 1 2<br />
3. ln. terms ofx andy, we have x =?'cosO= (1 + csin B)cosB =cos O+ csinBcosB =cosO + ~cs in2B and<br />
y = rsinB = (1 + csinO)sinB =sinO+ csin 2 B. Now - 1 $sinO $ 1 =><br />
-1 $sinO -t csin 2 0 $ 1 + c $2, so<br />
-1 $ y $ 2. Furthermore, y = 2 when c = 1 and B = ~ , while y = -1 for c = 0 and 0 = 3 ; . Therefore, we ne<strong>ed</strong> a viewing.<br />
rectangle with - 1 $ y $ 2.<br />
To find the x-values, iook at the equation x =cos B + ~ csin 20 and use the fact that sin 20 ~ 0 for 0 $ B $ ~ and<br />
sin 28 $ 0 for - ~ $ B $ 0. [Because r =::, 1 + csin 0 is symmetric about the y-axis, we only ne<strong>ed</strong> to consider<br />
- ~ S 8. $ ~ .) So for - ~ $ B .$ 0, x has a maximum value when c = 0 and then x = cos 0 has a maximum value<br />
of 1 at B = 0. Thus, the maximum value of x must occur on [ 0, ~ ] with c = 1. Then x = cos 0 + ~ sin 20 =><br />
~~ = -sinB +cos2B=- sinB + 1-2sin 2 9 =? j~ = -(2sin9-1)(sin9+1 ) = 0when sin9 = - 1or~<br />
[but sin B # - 1 for 0 $ 9 $ H If sin 9 = ~, then B = * and<br />
x = cos {f + ~sin 3 = ~V3. Thus, the maximum value of xis ~V3, and,<br />
by symmetry, the minimum value is - ~ V3. Therefore, the smallest<br />
viewing rectangle that contains every member of the family of polar curves<br />
r = 1 + csinB, where 0 $ c $ 1, is [-~V3, ~V3] x [-1, 2].<br />
2.1 c=t<br />
5. Without Joss of generality, assume the hyperbola has equation :: - ~: = 1. Use implicit differentiation to get<br />
2 : -<br />
2 Yb2 y' = 0, soy' = b:x. The tangent line at the point (c, d) on the hyperbola has equation y- d = b:dc (x - c).<br />
a a y . a<br />
' 2<br />
The tangent line intersects the asymptote y = !!. x when .!!.x - d = bzdc (x- c)<br />
a a a<br />
=> abdx- a 2 d 2 = b 2 cx - b 2 c 2 =><br />
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44 D CHAPTER 10 PROBLEMS PLUS<br />
a 2 d 2 -b 2 c 2 ad+ bc d h . bad+be ad'-t-be<br />
=> X = b(ad _be) = - -b- an t e y-va 1 Ue IS a --b- = --a- ·<br />
S 1m1 . . 1 ar 1 y, t<br />
h<br />
e tangent 1 .<br />
me mtersects<br />
. y = -ax b<br />
at<br />
(be<br />
--b-,<br />
- ad<br />
--aad<br />
- be )<br />
.<br />
Th<br />
e m1<br />
.d<br />
pomt<br />
.<br />
o<br />
ftl<br />
1ese mtersect1on<br />
. .<br />
pomts<br />
.<br />
1s<br />
.<br />
(<br />
1 (ad+bc be- ad) 1 (ad+ be ad-be)) (12bc 12ad) .<br />
2 --b- + --b- • 2 --a- + --a- = 2 b , 2--;;:- = ( c, d), the pomt of tangency.<br />
Note: Jf y = ·o, then at (±a, 0), the tangent line is x = ± a, and the points of intetsection are clearly equidistant from the point<br />
of tangency.<br />
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11 . D INFINITE SEQUENCES AND SERIES<br />
11.1 Sequences<br />
1. (a) A sequence is an order<strong>ed</strong> list of numbers. It can also be defin<strong>ed</strong> as a function whose domain is the set of positive integers,<br />
(b) The terms a,. approach 8 as n becomes large. In fact, we can make an as close to 8 as we like by taking n sufficiently<br />
large.<br />
(c) The terms an become large as n becomes large. In fact, we can make an as large as we like by taking n sufficiently large.<br />
2n . { 2 4 6 8 10 } { 4 3 8 5 }<br />
3. an = n2 + 1, so the sequence IS 1 + 1' 4 + 1 ' 9 + 1' 16 + 1' 25 + 1' . . . = 1, 5' 5' 17' 13 ' . . . .<br />
(-1)n- l . . { 1 -1 1 -1 1 } { 1 1 1 1 1 } ·<br />
5. an = 5n , so the sequence IS 51' 52' 53 ' 54' 55 '... = 5' - 25' 125'-625' 3125'... .<br />
1 . {1 1 1 1 1 } {11 1 1 1 }<br />
7. ~ = (n + 1)!, so the sequence IS 2!'"3!' 4f• 5!' 6!'... = 2' 6' 24 , 120, 720'... .<br />
9. a 1 = 1, a,,+ 1 = 5an- 3. Each term is defin<strong>ed</strong> in terms of the prec<strong>ed</strong>ing term. a2 = 5a1 - 3 = 5(1)- 3 = 2.<br />
a 3 = 5a2 - 3 = 5(2) - 3 = 7. a4 = 5a3 - 3 = 5(7) - 3 = 32. as = 5a4 - 3 = 5(32) - 3 = 157.<br />
The sequence is {1, 2, 7, 32, 157, ... }.<br />
a3<br />
Q.4 = --=<br />
2/5 2<br />
=<br />
1 + a3 1 + 2/5 7•<br />
a 5 = 1<br />
: 4 a<br />
4<br />
=:' 1<br />
~;/7 = ~- ,The sequence is {2, ~. ~. ~' ~ •... }.<br />
The denominator of the nth term is the nth positive odd integer, so an= - - 1 -.<br />
2n-1<br />
15. { - 3, 2, -1, ~,- ;~, ... }. The first term is -3 and each term is - ~ times the prec<strong>ed</strong>ing one, so a,. = -3 ( -~ f - 1 .<br />
17. { ~, -~,%,- 1 :, 2 5~, ... }. The numerator of the nth term is n 2 and its denominator is n + 1. Including the alternating signs,<br />
n2<br />
wegetan = (-1)n+l __ _<br />
n+1<br />
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46 D CHAPTER 11 INFINITE SEQUENCES AND SERIES<br />
19. 3n<br />
n an= 1+ 6n<br />
1 0.4286<br />
2 0.4615<br />
3 0.4737<br />
' 4 0.4800<br />
5 0.4839<br />
6 0.4865<br />
7 0.4884<br />
8 0.4898<br />
9 0.4909<br />
10 0.4918<br />
0.5<br />
0.4<br />
It appears that lim an = 0.5.<br />
n-+oo<br />
0<br />
. . . . . . . . . .<br />
5 10 n<br />
lim ~ = lim ( 3 n)jn = lim<br />
3<br />
= ~ = .!<br />
n-+oo 1 + 6n n-+oo (1 + 6n)jn n-+oo 1/n + 6 6 2<br />
21 .<br />
n<br />
an= 1 + (-U~<br />
1 0.5000<br />
2 1.2500<br />
3 0.8750<br />
4 1.0625<br />
5 0.9688<br />
6 1.0156 I<br />
7 0.9922<br />
8 1.0039<br />
9 0.9980<br />
10 1.0010<br />
I<br />
It appears that lim a,. = 1.<br />
n-co<br />
. . . . .<br />
0 5 10 n<br />
lim (1+(- ~f) =lim 1+ lim (- ~f=1+0 = 1 s ince<br />
n -oo n-too n-oo ""<br />
23. an = 1 - (0.2)", so lim a,. = 1 - 0 = 1 by (9) . Converges<br />
n - oo<br />
3 + 5n 2 (3 + 5n 2 )jn 2 5 3/n 2 5 + 0<br />
25. a,, = n + n2 = (n + n2)/n2 = 1 + l /n, so a,, -+ i + 0 = 5 as n -+ oo. Converges<br />
27.. Because the natural exponential function is continuous at 0, Theorem 7 enables us to write<br />
lim an= lim e 1 fn = e 1 imn-oo(l /nl = e 0 = 1. Converges<br />
,.,_ex:> n-oo<br />
2mr h lim' b 1' (2mr)/n lim 271' 271' 71' s· . . .,. b<br />
1 + n n-+oo n-+oo + n n n->oo 1 n +<br />
29. I fb n = ---, t en n = 1m ( )/ = /<br />
8 1 8<br />
2<br />
Theorem 7, lim tan ( n 71' 1 8 ) = tan ( lim 2 n1r ) =tan~ = 1. Converges<br />
n --->oo + n n-+oo 1 + 8 n 4<br />
8 = - 8 = - 4 . mce tan IS contmuous at 4 , y<br />
n2 n2 j,;r;J .,jii<br />
31. an = = = , so an -+ oo as n--+ oo since lim .,jii = oo and<br />
vn 3 + 4n ../n3 + 4n/v'n3 Jl + 4jn2 n--->OO<br />
lim j l + 4/ n 2 = 1.<br />
1\-+00<br />
Diverges<br />
1<br />
12 = - 1 (0) = 0, so lim an = 0 by (6). Converges<br />
n--+oo n-+oo 2vn n-+oo n 2 n-oo<br />
33. lim Jan! = lim I (-%"I = - 2<br />
1 lim<br />
1<br />
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SECTION 11.1 SEQUENCES 0 47<br />
35. a,. = cos( n /2). This sequence diverges since the tenns don't approach any particular real number as n---> oo.<br />
I<br />
The terms take on val4es between - 1 and I.<br />
_ (2n - 1)! = (2n - 1}! · = 1 · ___. 0 as n---> oo. C<br />
37· an - (2n + 1)! (2n + 1}(2n}(2n- 1)! (2n + 1}(2n) onverges<br />
1 + e - 2n<br />
--'------ ---> 0 as n---> oo because 1 + e- 2 n ---> 1 and e" - e-n -+ oo.<br />
en- e n<br />
Converges<br />
2 -n n 2 • • x 2 H 2x 11 2<br />
41. an = n e = -. Smce lim - = lim - = lim - = 0, it follows from Theorem 3 that lim an= 0. Converges<br />
en X--+00 e X X-tCO eX X--tOO e:t n-oo<br />
2<br />
43. 0 < cos n < _..!:.._ · [since 0 :s; cos<br />
2 1<br />
n :s; 1], so since lim 2<br />
- 2n - 2" n->oe> " n ·<br />
= 0, { 'c 0 82<br />
n } converges to 0 by the Squeeze Theorem.<br />
2<br />
. ( / ) sin(1/<br />
/ n<br />
n)<br />
.<br />
s·<br />
mce<br />
li<br />
m<br />
sin(1/x)<br />
/ = 1 . tm sin t [ w h ere t = 1 x = 1, tt •O ows · om Theorem 3<br />
45. an = n sm 1 n =<br />
1<br />
that {an } converges to I .<br />
x ..... oo 1 X t-o+ t<br />
/ ] . ., II fr<br />
(<br />
2)"' . ( 2)"<br />
2<br />
= lim = 2 ::::}<br />
x ->oo 1 + 2/x<br />
lim 1 + - = lim e 1 n Y = e 2 , so by Theorem 3, lim 1 + - = e 2 • Converges<br />
!t-oo X :r-oo n -+oo n<br />
( 2n2 1) (2 1/n 2 )<br />
49. an = In(2n 2 + 1)- ln(n 2 + 1) =In n 2 + = In . + / n 1 1 1 2<br />
-+ In 2 as n ---> oo. Converges<br />
51. a.,, = arctan(ln n). Let f(x) = arctan(ln x). Then lim f(x) = ~s ince In x -+ oo as x---> oo and arctan is continuous.<br />
x - oo<br />
Thus, lim a,= lim f(n) = ~-<br />
n-+oo<br />
n -+oo<br />
Converges<br />
53. {0, 1, 0, 0, 1, 0, 0, 0, 1, ... } diverges since the sequence takes on only two values, 0 and I, and never stays arbitrarily close to<br />
either one (or any other value) for n sufficiently large.<br />
n ! 1 2 3 (n- 1) n 1 n n<br />
55. an= 2<br />
n = 2 · 2 · 2 · · · · · - - 2<br />
- · 2 2': 2 · 2 (for n > 1] = 4--> oo as n--+ oo, so {an} diverges.<br />
57. 2~------------~ From the graph, it appears.that the sequence converges to 1.<br />
{ ( -2/e)n} converges to 0 by (7), and hence {1 + ( - 2/e)"}<br />
.. . . .. . .<br />
converges to 1 + 0 = 1.<br />
0<br />
'---~--~--~---~ 21<br />
® 2012 Ccng;,gc Lc011T1ing. AU RighiS R.escn·l.-d. May not be scann<strong>ed</strong>, copi<strong>ed</strong>. or duplicat<strong>ed</strong>. or post<strong>ed</strong> to a publicly accessible website, in whole or in part.
48 D CHAPTER 11 INFINITE SEQUENCES AND SERIES<br />
'<br />
59.<br />
From the graph, it appears that the sequence converges to t.<br />
As n--+ oo,<br />
3/ n<br />
.. ...<br />
2 +2<br />
=><br />
8 + 1/ n<br />
so lim an =~ ·<br />
n -+oo<br />
0 '----~--~--"--'----./ 21<br />
61. 2.~-------------.. From the. graph, it appears 'that the sequence {an} ,;_ { ~ 2 ~o~: } is<br />
..<br />
..<br />
divergent, since it osci llates between 1 and -1 (approximately). To<br />
n2<br />
prove this, suppose that {a,.} converges to L. If bn = --, 2<br />
then<br />
1+n<br />
{bn} converges to 1, and lim abn =!::. = L. But an = cosn, so<br />
n-+oo n 1 bn<br />
63.<br />
- 2<br />
0~-~-~~-----~-J IO<br />
lim abn does not exist. This contradiction shows that {an} diverges.<br />
-oon<br />
•<br />
From the graph, it appears that the sequence approaches 0.<br />
1 · 3 · 5 · · · · · (2n- 1) 1 3 5 2n- 1<br />
O < a,.= n = - · - · - · · .. ·--<br />
(2n) 2n 2n 2n 2n<br />
1 1<br />
: at = 1060, a2 = 1123.60, a3 = 1191.02, a4 = 1262.48, and as = 1338.23.<br />
(b) lim a,. = 1000 lim (1.06)", so the sequence diverges by (9) with r = 1.06 > 1.<br />
n - oo n-+oo<br />
67. (a) We are given that the init!al population is 5000, so Po = 5000. The number of catfish increases by 8% per month and is<br />
decreas<strong>ed</strong> by 300 per month, so Pt = Po + 8%Po - 300 = 1.08Po - 300, H = l.08P1 - 300, and so on. Thus,<br />
P,. = 1.08Pn-t - 300.<br />
(b) Using the recursive formula with Po = 5000, we get P 1 = 5100, P 2 = 520.8, P 3 = 5325 (rounding any portion of a<br />
catfish), P,. = 5451, P 5 = 5587, and P6 = 5734, which is lhe number of catfish in the pond after six monlhs.<br />
69. If lrl ~ 1, then {r"} diverges by (9), so {nrn} diverges also, since lnr''l = n lr"l ~ lr''l· If lrl < 1 then<br />
= lim ___!__I ."'<br />
:::: 0, so lim nr" = 0, and hence { nr"} converges.<br />
:C-tOO :c-oo r-:Z: X-+00 - T T-% %-+00 - n T n-oo .<br />
lim xr"' = lim -=..._ :lb ~ ( In\<br />
whenever lrl < 1.<br />
® 2012 Ccngogc Lcoming. All Rights Rc:sen'Ctl. May not be S
SECTION 11.1 SEQUENCES D 49<br />
71. Since {a,.} is a decreasing sequence, an > an+l for all n ~ 1. Because all of its terms lie between 5 and 8, {an} is a<br />
bound<strong>ed</strong> sequence. By the Monotonic Sequence Theorem,'{ an} is convergent; that is, {an} has a limit L. L must be less than<br />
8 since {an} is decreasing, so 5 $ L < 8.<br />
1 ' d . . 1 1 1 "<br />
73. a,. = 2<br />
n + 3<br />
IS ecreasmg smce an+1 = 2<br />
(n + 1<br />
) + 3<br />
= 2<br />
n + 5<br />
< 2<br />
n + 3<br />
= a1, tOr each n ~ 1. The sequence is<br />
bound<strong>ed</strong> since 0
50 D CHAPTER 11 IN_ FINITE SEQUENCES AND SERIES<br />
85. (a)<br />
50<br />
From the graph, it appears that the sequence { ~~ }<br />
5<br />
converges to 0, that is, lim n 1<br />
= 0.<br />
n-c::o n.<br />
'-~~----~----~---L----~<br />
0 10<br />
(b)<br />
O.Or3 _____________ "'<br />
y= 0.1<br />
7.5 "'--'~--~-----'-----~-------' 12.5<br />
0<br />
y = 0.001<br />
9.5 ' 0.1. From the second graph, it seems that for c = 0.001, the smallest pos~ible value for N<br />
is 11 si n~e n 5 fn! < 0.001 whenever n ~ 12.<br />
87. Theorem 6: If lim lanl = 0 then lim -lanl = 0, and since -ian I ~an ~ Ia,. I, we have that lim an= 0 by the<br />
n -+oo n-+oo n-oo<br />
Squeeze Theorem.<br />
89. To Prove: If lim an = 0 and {bn} is bound<strong>ed</strong>, then lim (a.,b,.) = 0.<br />
tl. -tOO<br />
n -+oo<br />
Proof: Since {bn} is bound<strong>ed</strong>, there is a posi!ive number M such that lb,.l ~ M and hence, lanl lbnl ~ lanl M for<br />
all n ~ 1. Let f: > 0 be given. Since lim an = 0, there is an integer N such that ian - 01 < Me if n > N . Then.<br />
n~oo<br />
la,.bn - Ol = lanbnl = lanl lbnl ~ lanl M = Ia,.- Ol M < ~ · M = f: for all n > N. Since f: was arbitrary,<br />
lim (anbn) = 0.<br />
n -too<br />
91. (a) First we show that a > a1 > b1 > b.<br />
a1 - b1 = ai b -.Jab = ~ ( ~- 2Vllb +b) = ~ ( y'a - -/bf > 0 [since a > b] => a1 > b1 . Also<br />
a- a1 = a-Ha+ b)= Ha - ~) > 0 and b- b1 = b- .Jab= Vii( Vb - 01) < 0, so a >. a1 > b1 >b. In the same<br />
way we can show that a1 > a2 > b2 > b1 and so .the given assertion is true for n = 1. Suppose it is true for n = k, that is,<br />
ak > ak+l > bk+ l > bk. Then<br />
® 20 J2 Cengoge Learning. All Rights Rescr ... cd. Mny not be scann<strong>ed</strong>, copi<strong>ed</strong>, or dupl icat<strong>ed</strong>. or post<strong>ed</strong> to t1 publicly accessible wcbsilc, in whole or in putt.
SECTION 11.2 SERIES 0 51<br />
ak+l- a1.:+2 = ak+l- Hak+l + bk+l) = ~(ak+l<br />
- bk+l) > 0, and<br />
bk+l - bk+2 = bk+1 - Jak+lbk+l = ~ ( ~- Jak+l ) < 0 =* a 1.:+1 > ak+2 > bk+2 > bk+l•<br />
so the assertion is true for n = k + 1. Thus, it is true for a ll n by mathematical induction.<br />
(b) From part (a) we have a > a,. > an+ 1 > bn+ 1 > b.. > b, which shows that both sequences, {an} and { bn}, are<br />
monotonic and bound<strong>ed</strong>. So they are both convergent by the Monotonic Sequence Theorem.<br />
(c) Let lim an =a an d I tm . b ,. =<br />
{3<br />
.<br />
Th<br />
en<br />
li<br />
m an+l = lim -a,.-+b.,<br />
-- 2<br />
n--+oo n--+oo n - oo n--+oo<br />
=><br />
a: +f3<br />
a:=--<br />
2<br />
=><br />
2a: = Q + {3 => Q = {3.<br />
93. (a) Suppose {Pn} converges top. T hen Pn+l = ~<br />
. a+p ...<br />
=><br />
=><br />
bp<br />
p=-<br />
a + p<br />
=><br />
p 2 + ap = bp "* p(p + a - b) = 0 => p = 0 or p = b - a.<br />
(b) Pn+l = ~ = (~);n < (!!.)Pn since 1 + Pn > 1.<br />
a + pn 1 + ....2:: a a<br />
a<br />
(c) By part (b), P1 < (~)Po. P2 < ( ~ )P1 _< ( ~ y P Pn·<br />
For n = 0, we have Pl - po = - -- - po =<br />
bpo Po(b - a - Po)<br />
> 0 since Po < b- a. So Pl > po.<br />
a + Po a+po ·<br />
Now we suppose the assertion is true for n = k, that is, Pk < b- a and Pk+l > Pk · Then<br />
- b _ a_~ __ a(b - a)+ bp~.;- ap~.; - bpk __ a(b- a - Pk)<br />
b- a - Pk+l __.!..____...!~ > 0 because p~; < b - a. So<br />
a+pk a+pk a + pk<br />
bpk+l Pk+l(b- a - Pk+l) .<br />
Pk+l < b - a. And Pk+2 - Pk+l = - Pk+l = > 0 smce Pk+l < b - a. Therefore,<br />
a + Pk+l<br />
a+ Pk+l<br />
Pk+2 > ·Pk+l· Thus, the assertion is true for n = k + 1. It is therefore true for all n by mathematical induction.<br />
A similar proof by induction shows that ifpo > b- a, then Pn > b- a and {Pn} is decreasing.<br />
In either case the sequence {Pn } is bound<strong>ed</strong> and monotonic, so it is convergent by the Monotonic Sequence Theorem.<br />
rt then follows from part (a) that lim Pn = b- a.<br />
n-oo<br />
11.2 Series<br />
1. (a) A sequence is an order<strong>ed</strong> list of numbers whereas a series is the sum of a list of numbers.<br />
(b) A series is convergent if the sequence of partial sums is a convergent sequence. A series is divergent if it is not convergent.<br />
(i) 2012 Ccngagc L
52 0 CHAPTER 11 INFINITE SEQUENCES AND SERIES<br />
00<br />
3. I: an = lim s, = lim [2- 3(0.8)n) = lim 2 - 3 lim (0.8)" = 2 - 3(0) = 2<br />
n=l n-oo n-.oo n--too n-+oo<br />
00<br />
1 1 1 1<br />
5. For I: 3 • a,,= 3· 81 = a1 = 13 = 1, 82 = s1 + a2 = 1 + 23<br />
= 1.125, sa = 82 + aa ~ 1.1620,<br />
n=l n n<br />
s~ = 83 +a~ ~ 1.1777, ss = 84 +as~ 1.1857, so = ss + ao ~ 1.1903, s 7 = 8G + a 7 ~ 1.1932, and<br />
ss = sr + as ~ 1.1952. It appears that the series is convergent.<br />
'<br />
oo n n 1 2<br />
7. For I: ~ ·a,, = ----r.:;· St = a1 i;= - -- = 0.5 82 = St + a2 = 0.5 + ~ ~ 1.3284,<br />
n=l 1 + yn 1 + yn 1 + J1 ' 1 + v2<br />
s3 = s2 +as~ 2.4265, S4 = sa+ a4 :::::: 3.7598, 8::, = 84 + a 5 :::::: 5.3049, 8 6 = s 5 + a 6 :::::: 7.0443,<br />
sr = so+ ar:::::: 8.9644, ss = 87 +as~ 11.0540. It appears that the series is divergent.<br />
9.<br />
n<br />
8n<br />
1 - 2.40000<br />
2 - 1.92000<br />
3 - 2.01600<br />
4 - 1.99680<br />
5 - 2.00064<br />
Ot----+--;---+---+----..----+--i ll<br />
6 - 1.99987 - 3<br />
7 -2.00003<br />
From the graph and the table, it seems that the series converges to - 2. In fact, it is a geometric<br />
8 -1.99999<br />
· ·th<br />
9 - 2.00000<br />
2 d I · · . ~ 12 -2.4 -2.4<br />
sen es w1 a= - .4 an r =-&·so tis sum IS 6 ( -<br />
5 )n = ( 1) = -- = - 2.<br />
. n=l 1 - - & 1.2<br />
10 -2.00000<br />
Note that the dot corresponding to n = 1 is part of both {an} and { Sn}.<br />
TI-86 Note: To graph {an} and { sn}, set your calculator to Param mode and Draw Dot mode. (Draw Dot is under<br />
GRAPH, MORE, FORMT (F3).) Now under E (t) = make the assignments: xt1=t, yt1=12/ ( -5) -t, xt2=t,<br />
yt2=surn seq ( ytl, t, 1, t , 1) . (sum and seq are under LIST, OPS (F5), MORE.) Under WIND use<br />
1 , 10, 1, 0, 1 0, 1, - 3, 1, 1 toobtain a graph s im ilar to the one above. Then use TRACE (F4) to see the values.<br />
11.<br />
n<br />
Sn.<br />
10/------------~<br />
1<br />
2<br />
3<br />
4<br />
5<br />
6<br />
7<br />
8<br />
9<br />
10<br />
0.44721<br />
1.15432<br />
1.98637<br />
2.88080<br />
3.80927<br />
4.75796<br />
5.71948<br />
6.68962<br />
7.66581<br />
8.64639<br />
0....___....__ _<br />
__..__..__<br />
__..___..__, 11<br />
The series f: ~ diverges, since its terms do not approach 0.<br />
n=l n 2 + 4<br />
® 2012 Cengagc Lcaming. All Rights Rcscr.-cd. Mny not be scann<strong>ed</strong>, cor>icd. or duplicat<strong>ed</strong>, or post<strong>ed</strong> to a publicly accessible website, in whole or in p.nrt.
SECTION 11.2 SERIES 0 53<br />
13.<br />
n s.,<br />
1 0.29289<br />
2 0.42265<br />
3 0.50000<br />
4 0.55279<br />
5 0.59175<br />
6 0.62204 o'-~~~~--~~~~~L-~~ u<br />
7 0.64645<br />
From the graph and the table, it seems that the series converges.<br />
8 0.66667<br />
9 0.68377<br />
10 0.69849<br />
so I: 00 (1- - --- 1) = lim ( 1 - - -- 1) = 1<br />
n=l Vn v'n + 1 k-oo y'kTI ·<br />
15. (a) lim an = lim ~ = ~.so the seqitence {an} is convergent by (11.1.1).<br />
n-oo n-oo 3n + 1 3<br />
(b) Since lim an = ~ =I 0, the series f: an is divergent by the Test for Divergence.<br />
n -+oo<br />
n=l<br />
17. 3 - 4 + 1 ; -<br />
64 9<br />
+ ··· is a geometric series with ratio r = -l Since lrl = ~ > 1, the series diverges.<br />
19. 10 - 2 + 0.4 - 0.08 + · · · is a geometric series with ratio - 1<br />
20 = -i· Since lr\ = k < 1, the series converges to<br />
21. f: 6(0.9)n-l is a geometric series with first term a = 6 and ratio r = 0.9. Since lrl = 0.9 < 1, the series converges to<br />
n=l<br />
a 6 6<br />
-- = --=- = 60.<br />
1- 7· 1-0.9 0.1<br />
23. f: ( ~ 3 ~n - l = ~ f: · (-~ ) n- 1 . The latter series is geometric w.ith a= 1 and ratio r =-~.Si nce lrl = ~ < 1, it<br />
n = l 4 4 n=l 4<br />
converges to ( 1 / ) = ~. Thus, tbe given series converges to (-!) ( ~) = t.<br />
1 - - 3 4<br />
oo ?Tn 1 oo (?T)n· . . . . ?T . . .<br />
25. I: n+l = - I: - IS a geometnc senes w1th rat1o r = - 3<br />
. Smce \rl > 1, the senes dtverges.<br />
n=O 3 3 n=O 3<br />
1 1 1 1 1 ~ 1 1 ~ 1 Th' . I . I f h d' h . .<br />
27. - + - + - + - + - + · · · = L- -- = - 3<br />
L- - . IS ts a constant mu t1p e o t e tvergent armoruc senes, so<br />
3 6 9 12 15 n = l 3n n=l n<br />
it diverges.<br />
~ n - 1 . b h "' " o· . lim li n - 1 1 ....£ 0<br />
29. L., ----- dtverges y t e .est .or 1vergence smce . an = m - 3 1<br />
= - 3<br />
r .<br />
n=l 3n - 1 n-oo n-oo n -<br />
© 2012 Ccnll"lle l earning. All RighiS Reserv<strong>ed</strong>. May not be scann<strong>ed</strong>. copi<strong>ed</strong>. or duplicotcd. or poste-d loa publicly a
54 0 CHAPTER 11 INFINITE SEQUENCES AND SERIES<br />
31. Converges.<br />
00<br />
00<br />
00<br />
1 + 2"<br />
( 1 2")<br />
[ (<br />
L:--=L: - + - =L: - 1 ) " +- ( 2 ) " ]<br />
n=1 3" n=1 3n 3" . n=1 3 3<br />
-~ ~ - ! 2-~<br />
- 1 - 1/ 3 + 1 - 2/3 - 2 + - 2<br />
[sum of two convergent geometric series]<br />
00<br />
33. L: o/2 = 2 + v'2 + ?12 + ~ + · · · diverges by the Test for Divergence since<br />
n=l<br />
lim a,. = lim o/2 = lim 2 1 / n = 2° = 1 =/= 0.<br />
n.-oo n-oo n-oo<br />
2 ) .<br />
35. E In ( 2<br />
n 2<br />
+ 1 diverges by the Test for Divergence since<br />
n=l n + 1<br />
37. E (~)" is a geometric series with ratio r = -I ~ 1.047. It diverges because lrl ~ 1.<br />
k=O<br />
00<br />
39. L: arctan n diverges by the Test for Divergence since lim an = lim arctan n = % =/= 0.<br />
n=l n.-+oo n-+oo<br />
1<br />
41 ~ ~ ( 1 1<br />
1 1<br />
)" · · · · h fi<br />
d · s· I I<br />
n = l e n=1 e e e e<br />
th ·<br />
. 6 ~ = 6 - 1s a geometric senes w1t rst term a = - an ratio r = - . mce r = - < 1, e senes converges<br />
1/e 1/e e 1 oo 1 ·<br />
) = 1. Thus, by Theorem 8(ii),<br />
1 - 1 e 1 - 1 e e e - 1 n=1 n n + 1<br />
to --<br />
1<br />
- = --<br />
1<br />
- · - = --.By Example 7, L: (<br />
oo (1 1 · )<br />
00<br />
1 "" · 1 · 1 1 e - 1 e<br />
n~1 en + n(n + 1) = n~l en + n~l n(n + 1) = e- 1 + 1 = e - 1 + e- 1 = e - 1·<br />
43. Using partial fractions, the partial sums of the series E +are<br />
' n =2 n -1<br />
n<br />
s,. = L: .<br />
2 n (<br />
. = L: -.<br />
1<br />
---.<br />
1<br />
-<br />
)<br />
i = 2 (t - 1)(t + 1) i=2 t- 1 t + 1<br />
= (1-!) + (! -!) + (! -!) + ... + ( '-1 - _1) + (-1 - .!.)<br />
3 2 4 3 5 n-3 n- 1 n-2 n<br />
Th . . I . . d 1 1 1 1<br />
n - n<br />
1s sum 1s a te escopmg senes an sn = + - 2<br />
~ -- 1<br />
- - ·<br />
Thus, E --i:-- 1<br />
= l~ Sn = lim (1 +! - - 1 -- .!.) = ~-<br />
n = 2 n - n-+oo n-+00 2 n- 1 n 2<br />
oo 3 n 3 n (1 1 )<br />
45. For the series :E , s,. = :E -. -.-- = L: -:- - -. -<br />
n=1 n(n+3) i =1 t(1 + 3) i=1 \ t+3<br />
[using partial fractions). The latter sum is<br />
(1 - l) + (l ....: l) + (1 - l) + (l - l) + ... + (-1 - l) + (-1 - _1 ) + (-1 - _1 ) + (.!. - --4)<br />
4 2 5 3 6 4 7 n-3 n n -2 n + 1 n-1 n+2 n n+3<br />
= 1 + 2 1 + 3- 1 1 1 1 n+1 - n+ 2 - n+3 [tl. e escopmg senes ' ]<br />
TI ~ 3 l' l' (1 1 1 1 1 1 ) 1 1 11 c<br />
ms, 6 (<br />
n = l n n + n-oo n-oo n 1l n<br />
3 ) = tm Bn = 1m + 2 + 3 - - + 1 - - + 2 - ..,----+ 3<br />
= 1 + 2 + 3 = 6 . onverges<br />
© 2012 Cengagc Learning. All Righ!S Reserv<strong>ed</strong>. Muy not be scann<strong>ed</strong>. copi<strong>ed</strong>. or duplicotcd, or po>l<strong>ed</strong> lo n publicly occcs:Jiblc wobsitc. In whole or ip part.
SECTION 11.2 SERIES 0 55<br />
47. F~r the series .. ~ 1<br />
( e 1 1" - e 1 / (n+l)) ,<br />
S n = i~ ( el/i _ el/(i+I)) = (el _ el /2) + (el /2 _ el/ 3) + ... + ( el/n _ el / (n+l)) = e _ el /(n+l)<br />
[telescoping series]<br />
Thus, f: (el/n - el/ (n+l)) = lim S n = lim (e - e 1 / (n+l)) = e - e 0 = e- 1.<br />
n=l n-oo n--+oo<br />
Converges<br />
49. (a) Many people would guess that x < 1, but note that x consists of an infinite number of9s.<br />
9 9 9 9 ~ 9 l " h " . . .<br />
(b) x = 0.99999 .. . = + + 10 100 1000<br />
+ , 10 000<br />
+ ··· = .~<br />
1 10 .. , w 11c 1s a geo~etnc senes w1th a 1 = 0.9 and<br />
. 0.9 0.9 1 h . 1<br />
r = 0.1. Its sum IS 1<br />
_ 0. 1<br />
= 0. 9<br />
= , t at 1s, x = .<br />
(c) The number 1 has two decimal representations, 1.00000 . .. and 0.99999 . . . .<br />
(d) Except for 0, all rational nurnbe.rs that have a terminating decimal representation can be written in more than one way. For<br />
example, 0.5 can be written as 0.49999 . . . as well as 0.50000 ... .<br />
- 8 8 . . . .tl 8 d 1 l a 8/10 8<br />
51. 0.8 = 10<br />
+ 102<br />
+ · · · IS a geometriC senes WI 1 a = 10<br />
an r = 10<br />
. t converges to - - = / = -<br />
1 - 1" 1 - 1 10 9 :<br />
- 516 516 "516 516 . . . . 516 1<br />
53. 2.516 = 2 + 103<br />
+ 106<br />
+ · · · . Now 103<br />
+ 106<br />
+ · · · IS a geometnc senes With a= 103<br />
and r = 103<br />
. It converges to<br />
_a_ = 516/ 10 3 = 516/ 10 3 = 516 . Thus. 2 . 516 = 2 516 = 2514 = 838<br />
1 .:... r 1 - 1/ 103 999/ 103 999 ' + 999 999 333 ·<br />
- 42 42 42 42 . . . . 42 1<br />
55. 1.5342 = 1.53 + 104<br />
+ 106<br />
+ · · · . Now 104<br />
+ 106<br />
+ · · · 1s a geometnc senes w1th a = 104<br />
and r = 102<br />
.<br />
a 42/ 10 4 42/ 10 4 42<br />
It converges to 1 - 1· = 1 - 1/ 102 = 99/ 102 = 9900 ·<br />
- 42 153 42 15,147 42 15,189 5063<br />
Thus, 1. 5342 = 1. 53 + 9900 = 100 + 9900 = 9900 + 9900 = 9900 ~r 3300 ·<br />
00 00<br />
57. 2..: (-5)"xn = 2..: (- 5x )" is a geometric series with r = - 5x, so ilie series converges ¢:? lrl < 1 ~<br />
n = l<br />
n=l<br />
5 x<br />
. 1 - r 1 - - 5x = 1 + 5x ·<br />
J- 5xJ < 1 ~ JxJ < t, that is, -t < X< r In that case, the sum of the series is _ a_ = ( 5 X ) -<br />
00<br />
59. 'E<br />
n=O<br />
(x - 2)"<br />
00<br />
(X-<br />
2) " X-<br />
-,<br />
2<br />
so the series converges<br />
3 .. = E - 3 - is a geometric series with r = - 3<br />
n=O<br />
~ Jrl < 1 ~<br />
x - 2<br />
¢:? - 1 < - 3- < 1 - 3 < x- 2 < 3 ~ - 1 < x < 5. In that case, the sum of the series is<br />
a 1<br />
1- r = 1<br />
_ x -2<br />
3<br />
1<br />
3 - (x - 2)<br />
3<br />
3<br />
5- x ·<br />
' ~ 2" ~ (2)". . . . .... 2 h .<br />
61. L..- ---;; = L..- - IS a geometric senes Wlw r = :-• sot e senes converges<br />
n=O X n=O X _- X<br />
~ Jrl < 1<br />
2 < Jxl x > 2 or x < -2. In that case, the sum of the series is - a = __!__<br />
1 2 / = ~ 2 .<br />
1 - r 1 - X X -<br />
© 2012 Ccngage !.earning. All RighiS Rcscn "Cd. May nol be scaru>Cd. copic:d. or duplicat<strong>ed</strong>. or post<strong>ed</strong> lo a publicly aecC>Sible wcbsilc. in whole or in part.
56 0 CHAPTER 11 INFINITE SEQUENCES AND SERIES<br />
00 00<br />
63. :L: en:z' = :L: (e"T is a geometric series with r = e"', so the series converges {::} lrl < 1 {::} lex! < 1 {::}<br />
n =O n=O<br />
1<br />
- 1 < e"' < 1 0 < e"' < 1 {::} x < 0. In that case, the sum of the series is _a_ = - -.<br />
1 - r l -ex<br />
65. After defining J, We use convert (f, parfrac); in Maple, Apart in Mathematica, or Expand Rational and<br />
. . . 0 . fi d th h I . 3n2 + 3n + 1 1 1<br />
s~mpl~fy 10 enve to n at t e genera term IS (n 2 + n)3 = n 3<br />
( )3 . So the nth partial sum is<br />
. n+1<br />
n(1 1 ) ( 1) (1 1) (1 1 ) 1<br />
Sn = ~.:'f1<br />
k3 - (k + 1)3 =<br />
1 - 23 + 23 - 33 + " . + n3 - (n + 1)3 = . 1 - (n + 1)3<br />
The series converges to lim s,. = 1. This can be confirm<strong>ed</strong> by directly computing the sum using<br />
n-oo<br />
s um (f , n=l.. i nfinity) ; (in Maple), Sum [f, {n , 1 , Inf inity}] (in Mathematica), or Calculus Sum<br />
(from 1 to oo) and Simplify (in Derive).<br />
67. For n = 1, a1 = 0 since s1 = 0. For n > 1,<br />
. n- 1 (n - 1) - 1 (n - l)n- (n + l)(n- 2) 2<br />
an = Sn - Sn- 1 = _n _+ _l - (n- 1) + 1 = (n + l)n = -n '(n- +:-1=-)<br />
I ~ I' 1' 1 - 1/n 1<br />
A so, nL;:l a,.,. = n~~ Sn = n~ 1 + 1/n = .<br />
69. (a) The quantity of the drug in the body after the first tablet is 150 m g. After the second tablet, there is 150 mg plus 5%<br />
of the first 150-mg tablet, that is, (150 + 150(0.05)] mg. After the third tablet, the quantity is<br />
(150 + 150(0.05) + 150(0.05) 2 ] = 157.875 mg. After n tablets, the quantity (in mg) is<br />
= W 1 - 0.05 ..<br />
n 1 . . . 150(1 - 0.05") 3000 ( ")<br />
150 + 150 ( 0.05) + ... + 150 ( 0.05 ) - .·We can use Formula 3 to wnte this as<br />
1 _ 0.0 5<br />
_(b) The number of milligrams remaining in the body in the long run is lim ( 3 ~go (1 - 0.05")] = 3 ~go (1- 0) ~ 157.895,<br />
n.-.oo .<br />
only 0.02 mg more than the amount after 3 tablets.<br />
71. (a) The first step in the chain occurs when the local government spends D dollars. The people who receive it spend a<br />
fraction e of those D dollars, that is, D e dollars. Those who receive the De dollars spend a fraction c of it, that is,<br />
D c 2 dollars. Continuing in U1is way, we see U1at the total spending after n transactions is<br />
D(1 en)<br />
Sn = D + De + De 2 + · · · + Dc" - 1 = - by (3).<br />
1 -c<br />
(b) lim Sn = lim D( 1 - c" ) = __..!2_ lim (1 - c") = __..!2_ [since 0 < c < 1 => lim c" = o]<br />
n.-oo n.-+oo 1 - C 1 - C n -oo 1 - C n--.oo .<br />
= D [since c + s = 1] = kD [since k = 1/ s]<br />
8<br />
If c = 0.8, then s = 1 - c = 0.2 and the multiplier is k = 1/ s = 5.<br />
73. f (1 + c)- n is a geometric series with a ,;, (1 + c)- 2 and r = (1 + c)-t, so the series converges when<br />
n = 2<br />
IC1+ c)- 1 1< 1 {::} l1 +el > 1 {::} 1 +e>1or1+e< - 1 {::} c > Oore < 7 2.Wecalculatethesumofthe<br />
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. . (1+ c)- 2<br />
scn es and set 1t equal to 2: ) 1<br />
= 2
58 0 CHAPTER 11 INFINITE SEQUENCES AND SERIES<br />
83. Suppose on the contrary that l:(an + bn) converges. Then l:(an + bn) and I: an are convergent series. So by<br />
Theorem 8(iii), I: [(a,. + bn) - a,.] would also b~ convergent. But I: [(an + bn) -an] = I: bn, a contradiction, since<br />
I: bn is given to be divergent.<br />
85. The partial sums {sn} form an increasing sequence, since s,. - Sn- 1 =an > 0 for all n. Also, the sequence {sn} is bound<strong>ed</strong><br />
since Sn ::::; 1000 for all n. So by the Monotonic Sequence Theorem, the sequence of partial sums converges, that is, the series<br />
I: an is convergent.<br />
87. (a) At the first step, only the interval ( ~ , ~) (length ~) is remov<strong>ed</strong>. At the second step, we remove the intervals ( i, ~) and<br />
( ~ ' ~).which have a total length of2 · Ut At the third step 1 we remove 2 2 intervals, each of length (~ ) 3 • In general,<br />
at the nth step we remove 2n- 1 intervals, each oflength ( ~) n, for a length of 2"- 1 · ( ~)"' = ~ ( ~) n- 1 • Thus, the total<br />
length of all remov<strong>ed</strong> intervals is f ~ ( t) n- 1 = 1 :./; 13<br />
n=l<br />
the n th step, the leftmost interval that is remov<strong>ed</strong> is ( ( 1<br />
= 1 [geometric se~ies with a = t and 7' = t] . Notice. that at<br />
)" , ( j )"),so we never remov~ 0, and 0 is in the Cantor set. Also,<br />
the rightmost interval remov<strong>ed</strong> is (1 -<br />
12127 d B<br />
are 3• 3•·9• 9• 9• an 9·<br />
(j)", 1 - (tf'), so 1 is never remov<strong>ed</strong>. Some other numbers in the Cantor set<br />
(b) The area remov<strong>ed</strong> at the first step is ~; at the second step, 8 · ( ~) 2 ; at tbe third step, {8) 2 · ( i) 3 • In general, the area<br />
remov<strong>ed</strong> at t.he nth step is {8)"-1<br />
00<br />
1<br />
(8)"- 1 1 / 9<br />
n>;1 9 9 = 1 - 8/ 9 = 1.<br />
a r = H ~r-l. so the totat area of all remov<strong>ed</strong> squares is<br />
oo n 1 1 1 2 5 5 3 23<br />
89. (a) For n>;1 (n + 1)!' 81 = 1. 2 = 2' S2 = 2 + ~ = 6' sa = 6 + 1 . 2. 3. 4 24,<br />
23 4 119 . (n + 1}!-1<br />
1 ) 1<br />
s 4 = - + = -.The denommators are (n + 1}!, so a guess would be sn = (<br />
24 1 · 2 · 3 · 4 · 5 120 n + .<br />
1 21 - 1 (k + 1)! - 1<br />
(b) For n = 1, 8 1 =<br />
2 = ~·so the formula holds for n = 1. Assume 8k = (k + )! . Then<br />
1<br />
(k + 1}! - 1 k + 1 (k + 1}! - 1 k + 1 (k + 2)! - (k + 2) + k + 1<br />
Sk+l = (k + 1)! + (k + 2)! = (k + 1)! + (k + l)!(k + 2) = (k + 2)!<br />
- (k + 2)! -1<br />
- (It+ 2)!<br />
Thus, the formula is true for n = k + 1. So by induction, the guess is correct.<br />
. · . (n + 1)! - 1 . [ 1 ] oo n<br />
(c) lim 8n = lim ( )I = hm 1 - ( )' = 1 and so L: (<br />
n-oo ,._00 n + 1 . n-oo n + 1 . "=1 n + 1<br />
)! = 1.<br />
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SECTION 11.3 THE INTEGRAL TEST AND ESTIMATES OF SUMS 0 59<br />
11.3 The Integral :rest and Estimates of Sums<br />
1 r 2 ·1<br />
1. The picture sh?WS that a 2 = 2<br />
1. 3 < 1<br />
1<br />
x 1 . 3 dx,<br />
y<br />
1 131 00 1 [ 00 1<br />
a 3 = "1.3 < "1.3 dx, and so on, so 2:: 1':3 < "1.3 dx. The<br />
. 3 . 2 X n = 2 n . 1 X<br />
integral converges by (7.8.2) withp = 1.3 > 1, so the series converges.<br />
0 2 3 4<br />
X<br />
3. The function f(x) = 1/ ~ = x- 1 / 5 is continuous, positive, and decreasing on [1 , oo ), so the Integral Test applies.<br />
f oo x- 1 1 5 dx = lim<br />
1<br />
1 . . t-.oo 1 t-.oo<br />
J' x- 1 / 5 dx = lim [2x 4 1 5 ] t = lim (~t 4 1 5 - .§.) = oo, so E. 1/ ifrf diverges.<br />
4<br />
1 t--.oo ·. 4 n = 1<br />
5. The function f(x) = ( 2<br />
x ~ 1<br />
) 3<br />
is continuous, positi~e , and decreasing on [1, oo), so the lf1tegral Test applies .<br />
. roo<br />
1<br />
dx ~ lim r 1 dx = lim [-.!<br />
1<br />
] t = lim ( -<br />
1<br />
+ _.!_) = _.!__<br />
1 1<br />
(2x + 1)3 t-+oo 1 1 (2x + 1) 3 t-.oo. 4 (2x + 1)2<br />
1<br />
t- oo 4(2t + 1)2 36 36<br />
Since this improper integral is-convergent, the series E ( 2<br />
1<br />
1 ) 3<br />
. n=l n +<br />
is also convergent by the Integral Test.<br />
7. The function f(x) =-/-- is continuous, positive, and decreasing on [1, oo), so the futegral Test applies.<br />
X + 1 . .<br />
roo ~ 'dx = lim r -/-- 1<br />
dx = lim [- 2<br />
1 ln( x<br />
2<br />
+ 1 )] t = - 2<br />
1 lim [ln( e + 1) - ln 2] = oo. Sine~ this improper<br />
} 1<br />
X + 1 t-.oo } 1<br />
X + t-->oo .<br />
1<br />
. t--+oo ·<br />
integral is divergent, the series E ~ is also divergent by the Integral Test.<br />
n=1 .n + 1<br />
9. E ~ is a p-series with p = -/2 > 1, so it converges by (1 ).<br />
n=1 n . .<br />
1 1 1 1 ~ 1 Thi . . . 'th 3 1 . b ( )<br />
11.1+-+-+ - + - + ···= 6 3 · s tsap-seneswL p = > ,so1tconverges y I.<br />
8 27 64 125 n=l n<br />
1 1 1 1<br />
00<br />
1 1<br />
13. 1 +- +- + - + - + · · · = L: --. The function f(x) = 2<br />
x _- 1· is<br />
3 5 7 9 n =l 2n - 1<br />
continuous, positive, and decreasing on [1, oo ), so the· Integral Test applies.<br />
1 -·- dx = lim ft - 2<br />
1<br />
dx = lim a 1n 12x - 111 ~ = ~ lim (1n(2t - 1) - 0) = oo: so the series r: - 2<br />
1<br />
r oo -<br />
} 1 2x- 1 t -+oo 1 X- 1 t -+oo t-->oo n=l n - 1<br />
diverges.<br />
oo vfn+ 4 'oo ( vfn 4 ) oo 1 oo 4<br />
15. L: --2- = L: - 2 +2 = L: 3 / 2 + L: 2•<br />
n =1 n n = l n n n=l n n =1 n<br />
~ 1 . . 'th 3 .<br />
6 3!2 IS a convergent p-senes w1 p = 2 > 1.<br />
n = l n<br />
E 4 = 4 E ~ is a constant multiple of a convergent p-series with p = 2 > 1, so it converges. The sum of two<br />
2<br />
n=l n . n = l n<br />
convergent series_ is-convergent, so the original series is conv~rgent.<br />
© 201 2 Ccognse Learning. All Righls RCSCJV
60 0 CI:IAPTER 11 INFINITE SEQUENCES AND SERIES<br />
17. The function f(x) = ~<br />
4<br />
is continuous, positive, and decreasing on [1, oo), so· we can apply the Integral Test.<br />
X + .<br />
l<br />
oo 1 · . ~t 1 · . [1 _1 x]t · ·1 . [ · _1 (t) _1 (1)]<br />
--dx = lim - - dx = lim -tan - = - hm tan - - tan -<br />
1 x2 + 4 t --+00 1 x 2 + 4 t --+oo 2 2 1 2 t--+oa 2 2<br />
=.!. [~ - tan- 1 (.!.)]<br />
. 2 2 . 2<br />
Therefore, the series f: ~ converges.<br />
n=l n +<br />
19 ill n . ln n . ln 1 0 Tb fu . f( ) lnx . . d · · [ 2 )<br />
. ~ - = ~ -<br />
3 3<br />
smce - = . e nchon . x = - 3<br />
1s contmuous an postttve on , oo .<br />
n = l n n=t n 1 X<br />
I<br />
f'(x) = x 3 (1/x)-(ln x)(3x 2 )=x 2 -3x 2 lnx=1 - 3lnx - 1 {::}<br />
. (x3)2 x6 x4<br />
3<br />
x > e 1 1 3 ~ 1.4,' so f is decre~ing on [2, oo), and the Integral Test applies.<br />
lnx dx ~ lim [- lnx - - 1 1<br />
-] t = lim [-_!._(2 lnt + 1) + .!.] (~) - , so the series 4<br />
f: In~<br />
100<br />
Inx dx = lim 1t<br />
2<br />
x3 t -oo 2<br />
x3 t-+oo 2x 2 4x 2 1<br />
converges.<br />
t-+oo 4t 2 4 n= 2 n<br />
(*): u = hl x, dv = x - 3 dx => du = (1/x) dx, v = -~x - 2 , so<br />
I ln x d 1 -2 1 . ;· 1 -2 ( 1 / ) d 1 -21n · 1 ;· -3d 1 -2 1<br />
1 -2 . C<br />
~ x = - 2 x n x - - 2 x x x = - 2 x x + 2 x x = - 2 x nx- 4 x + .<br />
(**):lim (~2lnt+ 1 ) J:!:_lim 2/ t =-l lim ]:_ =0.<br />
t--+oo 4t 2 t --+oa 8t<br />
4 t--+oo t2<br />
21. f (x) = +is continuous and positive on [2, oo), and also decreasing since J'(x) =- 1 2 Tlnln) 2<br />
< 0 for x > 2, so we can'<br />
x = x . x x<br />
use the lntegra1Test.1 ."" ln 1 1<br />
dx= lim [ln(lnx)J; = lim [ln( lnt)~ln(ln2)]= oo,s otheseries f - -diverges.<br />
. 2 X X t--+oo t--+oo . n=2 n 1 n n<br />
23. The function f(x)'= e 1 1"jx 2 is continuous, positive, and decreasing on [1, oo), so the Integral Test applies.<br />
[g(x) = e 1 1~ is decreasing and dividing by x 2 do es~'t change that fact.]<br />
l<br />
oo ~ t e1frz; · t · . 00 el/n<br />
f(x) dx = lim - 2 - dx = lim [-e11 x] = - lim (e 1 / t - e) = -(1 - e) = e- 1, so the series 2:: - 2 -<br />
1 t-oo 1 X t --t-oo . 1 t --+CXJ n=1 n<br />
converges.<br />
. 1 1 1 1 -<br />
25. The funct10n f(x) = - - - 2<br />
= - - -<br />
3 2<br />
+ -- [by partial fractions] is continuous, positive and decreasing on [1, oo),<br />
q; +x x x x+ l<br />
so the Integral Test applies.<br />
00<br />
1 It<br />
f(x)dx = lim ( 21 -- 1 +-- 1 ) dx = lim [- -1 - ln x +ln(x +:t) . J t<br />
1 t --+oo 1 X X X + 1 t--+oo X l<br />
= hm . [-- 1 + ln-- t + 1 + 1-ln2 ] =0 + 0 + 1-ln2<br />
t--+oo t t<br />
Th e . mtegra ' I converges, so th e senes . 1 ~ - -- I<br />
2 3<br />
converges. .<br />
n=l n +n<br />
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,<br />
SECTION 11.3 THE INTEGRAL TEST AND ESTIMATES OF SUMS 0 61<br />
27. T he function f(x) = coFxx is neither positive nor decreasing on (1, oo), so the hypotheses of the Integral Test are not<br />
. · oo COS7Tn<br />
satisfi<strong>ed</strong> for the senes L: ~ .<br />
n = l y n<br />
00 1<br />
29. We have already shown (in Exercise 21) that when p = 1 the series E (l ) diverges, so assume that p =1- L<br />
. n=2 n n n p<br />
f (x) = ( 1 ) is continuous and positive on [2, oo), and f'(x) = - ~~ ~ : 1 < 0 ifx > e - P, so that f is eventually<br />
X ln x P · X X 'P<br />
decreasing and we can use the Integral Test.<br />
roo ---:-:--1-:-- d l' [ (ln X) 1-p] t<br />
} 2 x(ln x)P x=t~ 1 - p<br />
2<br />
[forp :f- 1] = lim [ (ln t)1- p - (ln 2)1 - p]<br />
h oo · 1-p 1-p<br />
This limit exists whenever 1 - p < 0 ¢:? p > 1, so the series converges for p > L<br />
31 . Clearly the series cannot converge if p 2': -~ . becau se then lim n(1 + n 2 )P =1- 0. So assume p. < -~ . Then ·<br />
n-- 1. Unless specifi<strong>ed</strong> otherwise, the domain of a function f is the<br />
set of real numbers x such that the expression for f ( x) makes sense and defines a real number. So, in the case of a series, it's<br />
the set of real numbers x such that the series is convergent.<br />
00 (3)4<br />
35. (a) n~l ;;:<br />
E 4 = 81 I: 4 = 81 90 = -1o<br />
00 81 00 1 (7T4) 97T4<br />
n=l n<br />
n = 1 n<br />
00<br />
1 1 1 1<br />
00<br />
1 . 7T 4 ( 1 1 ) 7r 4 17<br />
(b) k~6 (k - 2)" = 34 + 44 + 54 + ... = k~3 k4 = 90 - 14 + 24 [subtract a I and a2] = 90 - 16<br />
37. (a) j (x) = ...;- is positive and continuous and f'(x ) =- ~ is negative for x > 0, and so the Integral Test applies.<br />
X X -<br />
00 . 1 1 1 1 1<br />
n~l n2 ~ sw = I2 + 22 + 32 + .. : + 102 ~ 1.549768.<br />
Rw :::; roo ...;- dx = lim [- 1 ] t = lim (-.!. + 1<br />
1<br />
} 10<br />
X t --.oo X<br />
10<br />
00<br />
1 100<br />
11 X 10 X<br />
t --.oo<br />
t<br />
0 ) = 1<br />
, so the error is at most 0.1.<br />
10 (b) 8 10 + 2 dx :::; S :::; SlO + 21<br />
dx =? SlQ + 1<br />
\ :::; S :::; 8 10 + fa<br />
1.549768 + 0 .090909 = 1.640677 :::; s :::; 1.549768 + 0.1 = 1.649768, so we gets ~ 1.64522 (the average of 1.640677<br />
and 1.649768) with error :::; 0.005 (the maximum of 1.649768 - 1.64522 and 1.64522- 1.640677, round<strong>ed</strong> up).<br />
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62 0 CHAPTER 11 INFINITE SEQUENCES 1,\ND SERIES<br />
(c) The estimate in part (b) iss ~ 1.64522 with error ~ 0.005. The exact value given in Exercise ~4 is 1r 2 / 6 ~ 1.644934.<br />
The difference is less than 0.0003.<br />
(d) Rn ~ /.~ --;. dx = .!.. So Rn < 0.001 if.!. < - 1 - {::} n > 1000.<br />
,. x n n 1000<br />
r R,. ~ 1 oo (2x + 1) - 6 dx = t~ [ 10( ; 2<br />
~ ) 1 5 l0(<br />
39. f(x) = 1/{2x + 1) 6 is continuous, positive, and decreasing on [1, oo), so the Integral Test applies. Using (3),<br />
2<br />
n \<br />
1 ) 5<br />
. To be correct to five decimal places, we want<br />
10( 2<br />
n 1 + 1<br />
) 5<br />
~ 1 ~ 6<br />
¢:} (2n + 1) 5 ~ 20,000 {::} n ~ ~ ( .e-'20,000- 1) ~ 3.12, ;muse n = 4.<br />
4<br />
1 1 1 1 1<br />
S4 = n~ 1<br />
(2n + 1)6 = 36 + S6 + 7<br />
6 + ge ~ 0.001446 ~ 0.00145.<br />
41. f: n-1.oo 1 = f: 1 ~ 01<br />
is a convergent p-series with p = 1.001 > 1. Using (2), we get<br />
n = 1<br />
n=1 n<br />
Rn < !.oo x-l.001 dx = lim [ x - o.oo1 ] t = - 1000 lim [-1-]t = - 1000(--1-) = 1000.<br />
- n t->oo - 0.001 " t->oo x0.001 n n0.001 n0.001<br />
1000 - 1000<br />
We want Rn < 0.000 000 005 {::} n 0<br />
. 001 < 5 x 10 9 {::} n°·001 > 5<br />
x 10 9<br />
n > ( 2 X 10ll) 1000 = 2 1000 X 1011,000 ~ l.07 X 10301 X 1011.000 = l.07 X 1011.301.<br />
{::}<br />
43. (a) From the figure, a2 +as+···+ an ~ Jt f(x) dx, so with<br />
y<br />
1 1 1 1 1 1" 1<br />
f(x)=-,-+-+-+ .. ·+ - ~ -dx= ln n.<br />
x2 3 4 n 1 x<br />
1 1 1 1<br />
Thus, Bn = 1 + z + '3 + 4 + .. · + ;:;: ~ 1 + ln n.<br />
(b) By part (a), s 10 o ~ 1 + ln 10 6 ~ 14.82 < 15 and<br />
0 2 3 4·" n<br />
8109 ~ 1 + ln 10 9 ~ 21.72 < 22.<br />
45. bin" = ( e 1 " b) Inn = ( e 1 " ")In b = nln b = n _ 1 n b. This is a p-series, which converges for all b such that - ln b > 1 ¢:}<br />
1<br />
lnb < - 1 ¢:} b < e- 1 {::} b < 1/e [with b > 0].<br />
11.4 The Comparison Tests<br />
1. (a) We cannot say anything about I: an. If an > bn for all nand :L b,. is convergent, then :L an could be convergent or<br />
divergent. (See the note after Example 2.)<br />
(b) If an < bn for all n, then I; an is convergent. [This is part (i) of the Comparison.Test.)<br />
n n 1 1 " II ~ n b . 'th ~ 1 , w h' 1c I 1 converges<br />
n 2n n n=l n + 1 n = l n<br />
3. 2<br />
n 3 + 1<br />
< - 2 3<br />
= - 2<br />
< 2 .or a n ~ 1, so L..- -<br />
2<br />
- 3<br />
-- converges y companson WI L..-<br />
2<br />
because it is a p-series with p = 2 > 1.<br />
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SECTION 11.4 THE COMPARISON TESTS D 63<br />
n + 1 n 1 ~ n + 1 d' b · ··h ~ 1<br />
5. -- > - - = - for all n 2: 1, soL- 1verges y companson w1t L- h' h d' b · ·<br />
1<br />
, w 1c Jverges ecause 1t 1s a<br />
1<br />
nVn nVn Jn n=l n vn n=l vn<br />
p-series with p = ! ~ 1.<br />
00<br />
9" 9" ( 9 )"<br />
oo '9n<br />
3 + 10n 10" 10 - . n=l n=l 3 + 10<br />
7. < - = - for all n > 1. L:: (for is a convergent geometric series (lrl'= fo < 1 ), so L:: n<br />
converges by the Comparison Test.<br />
lnk 1 [' l k "k .>3] ~ lnkd. b · 'h~ 1 h'h d' b ·<br />
9. - > - for all k 2: 3 smce n • > 1 .or _ , so LJ - k 1verges y com pan son w1t LJ -k, w 1c 1verges ecause 1t<br />
k k k = 3 k=3 •<br />
is a p-series with p =: 1 ~ 1 (the harmonic series). Thus, f lnkk diverges since a finite number of terms doesn't affect the<br />
convergence or divergence of a series.<br />
I<br />
k=l<br />
{fk ifk kl/3 1 00 • {fk . . 00 1<br />
11. < ~ = k 312<br />
= k 7 16<br />
for all k 2: 1, so L:: converges by compar1son w1th L::<br />
716 ,<br />
Jk3 + 4k + 3 .v k3 k=l ,jk 3 + 4k + 3 k=l k<br />
· which converges because it is a p-series with p = ~ > 1.<br />
arctann 7r/2 fi II ~ arctann b . . h 1r ~ 1 hi h .<br />
13. 1. 2<br />
< ""1.2 or a n 2: 1, so LJ ~, 2 converges y comparison w1t - 2<br />
LJ ""1.2• w c converges because 1t is a<br />
n . n n =l n n=l n<br />
constant times ap-series with p = _1.2 > 1.<br />
4''+ 1 15.<br />
4 . 4" (4)"<br />
n _<br />
3 2<br />
00<br />
(4)"<br />
00<br />
(4)"<br />
> 3n"" = 4 3 for all n 2: 1. n~l 4 3 = 4 n~l 3 is a divergent geometric series (lrl = ~ > 1), so<br />
00 4n+l<br />
L:: - -- diverges by the Comparison Test.<br />
n=l 3"- 2<br />
1 1<br />
17. Use the Lirni~ Comparison Test with an = JTi2'+T and bn = n:<br />
1<br />
lim an = lim n = lim ' = 1 > 0. Since the harmonic series f .!. diverges, so does<br />
n-oo bn n-oo Jn 2 + 1 n- --- > - ( - ) = - - or use the Test for Divergence.<br />
1 + 3" 3n + 3" 2 3" 2 3<br />
© 2012 Cengoge Le:uning. All R;ghL< Resel'\'<strong>ed</strong>, May not be SCIUUl<strong>ed</strong>, copi<strong>ed</strong>, or duplicat<strong>ed</strong>, or post<strong>ed</strong> to a publicly acc'cssihlc website, in whole or in pM.
64 D CHAPTER 11 INFINITE SEQUENCES AND SERIES<br />
21 . Use the Limit Comparison Test with an = 2<br />
f+"2 and bn = ; :<br />
n +n+ 1 n 12<br />
. an . n 3 1 2 Jn+2 . (n 3 1 2 Jn+2)l(n 3 1 2 vn) . ) 1 +2ln /1 1<br />
hm - = hm = lim = lim = - = - > 0<br />
n--.oo bn n--.oo 2n 2 + n + 1 n--.oo (2n 2 + n + 1)ln 2 n--.oo 2 + 1ln + 1l n 2 2 2 ·<br />
Since f: ; 12<br />
is a convergent p-series [P = ~ > 1], the series f: 2<br />
f+"2 also converges.<br />
n=l n . n= l n + n + 1<br />
5+ 2n 1<br />
23. Use the Limit Compa'rison Test with a,. = ( 1<br />
+ n 2<br />
)2 and bn = n 3<br />
:<br />
lim an li n 3 (5 + 2n) l' 5n 3 + 2n 4 1ln 4 l' * + 2 . 2 0 s· ~ 1 .<br />
n --.oo bn. = n--.~ (1 + n2)2 = n~ ( 1 + n2)2 · l l (n2 )2 = n!...~ (.l;,. + l )2 = > . mce n'S::l n3 IS a convergent<br />
...<br />
p-senes . [ p = 3 > 1 ], t h e senes . ~ 5 + 2n l<br />
L.J (<br />
2 ) a so converges.<br />
2<br />
n.=l 1 + n<br />
n 2 1 oo ../n 4 + 1<br />
2 ( ) = -- for all n ~ 1, so I: 3 2<br />
diverges by comparison with<br />
n n + 1 n + 1 · n=l n + n<br />
f: - 1 - = f: ~.which diverges because it is a p-series with p = 1 ~ 1.<br />
n.=ln+1 n=2n<br />
27. Use the Limit Comparison Test with an = (1 + ~) 2 e- n and bn = e-": lim abn = lim .(1 + ~) 2 = 1 > 0. Since<br />
n n-oo n n-oo n<br />
f: e-n = f: · :. is a convergent geometric series [lrl = ~ < 1], the series f: (1 + ~) 2 e-n also converges.<br />
n=l n=l e n=l n<br />
29. Clearly n! = n(n - 1)(n - 2) · · · (3)(2) > 2 · 2 · 2 · · .. · 2 · 2 = 2''- 1 so..!_ < - 1 -. E - 1 - is a convergent geometric<br />
. - ' n! - 2n-l n =1 2 n -1<br />
series [lr I = ~ < 1], so E --\ converges by the Comparison Test.<br />
n=l n .<br />
31. Use the Limit Comparison Test with an == sin (;) and bn = ; . Then I: an and 2: bn are series with positive terms and<br />
I . an . sin(1ln) . sinO . ~ . . . .<br />
1m - = 1 rm _ = hm - - = 1 > 0. Smce L.J bn IS the divergent harmontc senes,<br />
n--.oo bn n--.oo 1ln 0--+D 0 n=l<br />
00<br />
2: sin (1l n) also diverges. [Note that we could also use !'Hospital's Rule to ev~l uate the limit:<br />
n = 1 ·<br />
. sin(ll x) H • cos(l/x) · ( -l/x 2 ) . 1<br />
hm I = lim I 2<br />
= hm cos - =cosO= 1.)<br />
x - oo 1 X %-oo -1 X x_____.oo X<br />
lO 1 1 1 1 1 1 1 1 .<br />
33. n~l ~ = .J2 + ..ff7 + v'82 + · · · + .JI0,05I ~ 1.24856. Now ~ < ..JTiJ = n 2 , so the error IS<br />
00<br />
1<br />
Rw ~ Tw ~ 21 dx = lim [-- 1 ] t = lim ( -- 1 +- 1 ) = - 1 = 0.1.<br />
t --.oo t 10 10<br />
10 X t --.oo X<br />
10<br />
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SECTION 11.5 ALTERNATING SERIES 0 65<br />
10 cos 2 1 cos 2 2 cos 2 3 cos 2 10 cos 2 n 1 .<br />
35. I; s-n cos 2 n = ~ + --+ --+ .. · + - - - ~ 0.07393. Now -- < - so the error IS<br />
n =l 5 5 2 5 3 5 10 5" - 5n'<br />
l<br />
oo 1 . j·t -x . [ 5- "' ] ' . ( 5 - t 5-1o) 1 _8<br />
R10 ~ T10 ~ -dx= lim 5 dx= lim - ln = lim - - 5 1 5<br />
+ ln S = 510 ln < 6.4 x 10 .<br />
10 5"' t~oo 10 t~oo 10 t~ oo n<br />
5<br />
d,. 9 d . ~ 9 . . . . (I I 1 1) o d d d ~ dn<br />
37. Since - < - for each n, an smce L.., - 1s a convergent geometnc senes r = 10 < , . 1 2 3 . .. = L.., -<br />
10" - 10" n=1 10n n = l 10"<br />
will always converge by the Comparison Test.<br />
39. Since I: a .. converges, lim an = 0, so there exists N such that ian - 01 < 1 for all n > N => . 0 ~ ar,. < 1 for<br />
n->oo<br />
all n > N => 0 ~ a~, ~ an. Since I; an converges, so does I; a~ by the Comparison Test.<br />
41 . (a) S ince lim an = oo, there is an integer N such that abn > 1 whenever n > N. (Take M = 1 in Definition 11.1.5.)<br />
-oo~ n .<br />
Then an > bn whenever n > Nand since I: b,. is divergent, I; a" is also divergent by the Comparison Test.<br />
. 1 d b 1 fi > 2 h I' an lim. n lim X II li 1 lim<br />
(b) (1) If an = - an n = - or n _ . , t en 1m -b = - 1<br />
. - = -ln = m - 1 = x = oo,<br />
Inn n n-oo n n ........ oo n n X--+00 X x-oo 1 X x-oo<br />
~ 1 . .<br />
so by part (a), L.., . -ln IS d1vergent.<br />
n=2 n<br />
.. ln n d b<br />
1 th ~ b · h d' h · · d I' an<br />
n n n=l n--+oo n.<br />
( 11) If an = -- an n = -, en L.., n 1s t e 1vergent armomc senes an 1m -b<br />
lim Inn = lim lnx = oo,<br />
,,_ oo x-oo<br />
00<br />
so I: an. diverges by part (a).<br />
n=l<br />
43. lim nan = lim a/,. , so we apply the Limit Comparison Test with bn = .! . Since lim nan > 0 we know that either both<br />
n--+oo n - oo 1 n n ,,_00<br />
series converge or both series diverge, and we also know that f .! diverges [p-serics with p = 1]. Therefore, I; an must be<br />
. n =l n<br />
divergent.<br />
45. Yes. Since I: an is a convergent series with positive terms, lim an= 0 by Theorem 11.2.6, and I: bn =I: sin( an) is a<br />
series with positive terms (for large enough n). We have lim bn = lim sin( an) = 1 > 0 by Tbeor~m 2.4.2<br />
n-oo an n--+oo an<br />
n~oo<br />
[ET Theorem 3.3.2]. Thus, l::bn is also convergent by the Limit Comparison Test.<br />
11.5 Alternating Series<br />
1. (a) An alternating series is a series whose terms are alternately positive and negative.<br />
(b) An alternating series f; an= f; ( -l)"- 1 bn, where bn =ian I, converges if O < b,.+l ~ b,. for all nand lim b,, = 0.<br />
n=l n=l n-+oo<br />
(This is the Alternating Series Test.)<br />
(c) The error in<strong>vol</strong>v<strong>ed</strong> in using the partial sum s.,. as an approximation to the total sum s is the remainder R ,. = s - sn and the<br />
size of the error is smaller than bn+l; that is, IR~,I ~ bn+l· (This is the Alternating Series Estimation Theor~m.)<br />
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66 0 CHAPTER 11 INFINITE SEQUENCES AND SERIES<br />
00<br />
2 4 6 8 10 n 2n 2n . 2 2 . -<br />
3. -- +- - - +---+···= :E{-1) --.Now limb,. = lim-- = lun / = - =fO.Smce<br />
5 6 7 8 9 n=l n + 4 n-oo n-oo n + 4 n-too 1 + 4 n 1<br />
lim an =I 0 (in fact the limit does not exist), the series diverges by the Test for Divergence.<br />
n-+oo<br />
00 00 1 00 1<br />
5. :E a,. = :E ( -1f- 1 2 + 1 = :E ( -1)"- 1 b,.. Now b,. = - 2 1<br />
> 0, {bn} is decreasing, and lim b,. = 0, so the<br />
n=l n=l n n=l n + n-oo<br />
series converges by the Alternating Series Test.<br />
-1)"b,.. Now lim b,. = lim ~- 1 3<br />
jn = - =f 0. Since lim a,. =f 0<br />
n=l n=l n + n=l n-+oo n-+oo + 1 n 2 n-+00<br />
7. E a,.= E (<br />
- 1)" 2<br />
3 n -<br />
1<br />
1 = f (<br />
(in fact the limit does not exist), the series diverges by the Test for Divergence.<br />
00 00 00 1<br />
9. :E a,.= :E ( - 1)"e-" = :E ( -1)"bn. Now b,. = n > 0, {bn} is decreasing, and lim b,. = 0, so the series converges<br />
n = l n=l n =l e n-+oo<br />
by the Alternating Series Test.<br />
n2<br />
11. b,. = n 3<br />
+ 4<br />
> 0 for n 2:: 1. {bn} is decreasing for n 2:: 2 since<br />
lim b,. = lim 1<br />
1 /4 n/<br />
(x 3 + 4)(2x) - x 2 (3x 2 ) _ x(2x 3 + 8- 3x 3 ) _ x(8- x 3 )<br />
(x3 + 4 )2 - (x3 + 4 )2 - (x3 + 4 )2 < 0 for x > 2. Also,<br />
3<br />
= 0. Thus, the series E (-1)"+1 ~converges by the Alternating Series Test.<br />
n -+00 n-op + n n=l n 3 + 4<br />
13. lim b~ = lim e 2 fn = e 0 = 1, so lim (-l)n- 1 e 2 1" does not exist. Thus, the series f: (-1)"- 1 e 2 1" diverges by the<br />
n-+oo ra.-oo n-+oo n=l<br />
Test for Divergence.<br />
sin(n + ~)1r (-1)" 1<br />
15. a,. =<br />
1 + .,fti = 1 + .,fii," Now b,. = + Vn > 0 for n 2:: q, {b,.} is decreasing, and .. ~~~ b,. = 0, so the series<br />
1<br />
oo sin(n +.! )1r<br />
:E fo converges by the Alternating Series Test.<br />
n=O 1 + n<br />
17. n~l ( - 1)" sin(~J bn =sin(~) > 0 for n 2:: 2 and sin(~) 2:: sin( n: 1<br />
). and ,.~ sin(~) = sinO = 0, so the<br />
series converges by the AJternating Series Test.<br />
n" n·n···· ·n<br />
19. - = > n =><br />
n! 1· 2 · ·· · · n -<br />
by the Test for Divergence.<br />
n"<br />
lim- =oo =><br />
n.-+oo n!<br />
( 1)" " · oo n<br />
lim - n does not exist. So the series :E (-1}" ~, diverges<br />
n ..... oo n. 1 n=l n.<br />
21. The graph gives us an estimate for the sum of the series<br />
E (-O.~) " of -0.55.<br />
n=l n.<br />
(0 8)"<br />
bs = ~ :::::: 0.000 004, so<br />
-1<br />
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SECTION 11.5 ALTERNATING SERIES D 67<br />
~ ( -0.8t :.._ - ,;.., ( - 0.8t<br />
LJ ' ~57- LJ '<br />
n=1 n. n=1 n.<br />
~ -0.8 + o.32 - o.oss3 + o.oi 706- o.oo2 731 + o.ooo 364- o.ooo 042 ~ -o.sso7<br />
Adding b 8 to s 7 does t:~ot change the fourth decimal place of 57, so the sum_ of the series, correct to four decimal places,<br />
is - 0.5507.<br />
23. The series f: ( - 1 );+ 1 satisfies (i) of the Alternating Series Test because (<br />
1<br />
)6 < -i and (ii)' lim -i = 0, so the<br />
n=l n · n + 1 n n -+oo n<br />
1 1<br />
series is convergent. Now bs = 56<br />
= 0.000064 > 0.00005 and b6' = 66<br />
~ 0.00002 < 0.00005, so by the Alternating Series<br />
Estimation Theorem, n = 5. (That is, since the 6th term is less than the desir<strong>ed</strong> error, we ne<strong>ed</strong> to afld the first 5 terms to get the<br />
sum to the desir<strong>ed</strong> accuracy.)<br />
25. The series f: ( - 1 )",· satisfies (i) of the Alternating Series Test because 10<br />
. +lt 1<br />
)' < - 0<br />
1<br />
1 and (ii) lim - 1 - = 0,<br />
n =O 10" n. " n + . 1 " n n-ioo 10" n !<br />
so the series !s convergent. Now bs = 10<br />
! 31<br />
~ 0.000 167 > 0.000 005 and b4 = 10<br />
! 41<br />
= 0.000 004 < 0.000 005, so by<br />
the Alternating Series Estimation Theorem, n = 4 (since the series starts with n = 0, -not n = 1). (That is, since the 5th tenn<br />
is less than the desir<strong>ed</strong> error, we ne<strong>ed</strong> to add the first 4 terms to get the sum to the desir<strong>ed</strong> accuracy.)<br />
1 1<br />
27. b4 = I = -- :::; o.ooo 025, so<br />
8. 40,320<br />
00<br />
(- 1)" 3 (-1)" ' 1 1 1<br />
n~1 (2n)! ~ 53 = n~l (2n)! = -2 + 24 - . 720 ~ ~ 0 .4 59722<br />
Adding b 4 to S3 does not change the fourth decinlal place of 5s, so by th~ Alternating Series Estimation Theorem, the sum of<br />
. the series, correct to four decimal places, is - 0.4597.<br />
72 -<br />
29. b7 = 107 = 0.000 004 9, so<br />
00 (- 1 )n- ln2 6 (-1)n- ln2<br />
"' "' 1 4 9 16 2s 36 ·0 067<br />
n~<br />
1<br />
10" :::; 56 = n-7;:<br />
1<br />
10" = 10 - 100 + 1000 - 10,000 + 100,000 - 1,000,000 = · 614<br />
Adding b 7 to 56 does not change the fourth decimal place of 56, so by the Alternating Series Estimation Theorem, the sum of<br />
the series, correct to four decimal places, is 0.0676.<br />
I<br />
00 ( -1)n-l 1 1 1 1 1 1 1<br />
31. I: = 1- - + - - - + · · · + - - - + --- + · · · . The 50th partial sum of this series is an<br />
n = 1 n 2 3 4 49 50 51 52<br />
underestimate, since "~ 1<br />
( - 1 ~n - l = 5so + ( 5<br />
1<br />
The result can be seen geometrically in Figure 1.<br />
1 - 1<br />
52 ) + ( ; 3 - 1<br />
) + ; · ·, and the te~s in parentheses are all positive. '<br />
54 1<br />
33. Clearly bn = - - is decreasing and eventually positive and lim bn = 0 f~r any p. So the Series converges (by the<br />
n+p<br />
n -+oo<br />
Alternating Series Test) for any P. for which every bn is defin<strong>ed</strong>, that is, n + p :/:- 0 for n 2 1, or p is not a negative integer.<br />
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68 D CHAPTER 11 INFINITE SEQUENCES AND SERIES<br />
35. 2: b 2 n = 2: 1/(2n) 2 clearly converges (by comparison with ~e p-series for p = 2). So suppose that 2: (-1r-I bn<br />
. .<br />
converges. Then by Th~orem 11.2.8(ii), so does 2: [(-1 t- 1 1<br />
bn + bn] ::: 2 ( 1 + l + i + · · · ) = 22: - . But 2<br />
thi~<br />
. n -1<br />
diverges by comparison with the harmonic series, a contradiction. Therefore, 2: (....:1)"- 1 b,. must diverge. The Alternating<br />
Series Test does not apply since {bn} is not. decreasing.<br />
11.6 Absolute Convergence and the Ratio and Root Tests<br />
1. (a) Siilce lim I an+l I = 8 > 1, part (b) of the Ratio Test tells us that the series 2: an is divergent.<br />
n-oo an. . .<br />
(b) Since lim I an+l I = 0.8 < 1: part (a) of the Ratio Test tells us that the series 2: an is absolutely convergent (and<br />
n.-CX> an<br />
therefore convergent).<br />
(c) Since lim I an+l I = 1, the Ratio Test fails and the series 2: an might converge or it might diverge.<br />
-~ an . .<br />
3. lim lan+l l = lim In+ 1 · 5 " 1 = lim 1.!. · n+ 1 1 =.!.lim 1 + 1 /n = .!. (1) = .!. < 1,sotheseries E ~is<br />
n-too an n-oo 5n+l n n-too 5 n 5 n-too 1 5 5 n=l 5n<br />
absolutely co.nvergent by the Ratio Test.<br />
1<br />
5. bn = - 5 n + n-->oo . n=O 5n + 1<br />
1 > 0 for n 2::' 0, { bn} is decreasing for n ~ 0, and lim bn = 0, so E ( -1 t converges by the Alternating<br />
Series Test. To determine absolute convergence, choose an = ~ to get<br />
- n<br />
li m -b<br />
an<br />
=<br />
1'<br />
tm<br />
1/n l' ' 5n + 1 5 0 1 d. . b .h L ' . C . ,.. . h th<br />
1 / ( 5 1 ) = rm --= > , so ~ -- 1verges y t e mut ompar1son . est w1t e<br />
n~~ n n-~ n + n-+oo n n=l 5n + 1<br />
' .<br />
hannonic series. Thus, the series E 5<br />
( - 1 )" 1<br />
is conditionally convergent.<br />
n=O n+<br />
7. lim lak+Il = lim [(k-+: 1 ) (~_)'"+ 1 ] = lim k +<br />
1 (~)<br />
1<br />
= ~ lim (1<br />
. k~oo a k k-oo k(~)k k -too k 3 3 k~ oo ' k<br />
+ .!..) = ~'( 1 ) = 1 < ·1,sotheseries<br />
E k ( ~) k is apsolutely convergent by the Ratio Test. since the terms of this series are positive, absolute convergence is the<br />
n=1<br />
same as convergence.<br />
9. lim 1 an+ I I = lim [ (l.l)n+l . ~J = lim (1.1)n 4 1<br />
·= (1.1) lim<br />
n -- 1,<br />
n = l<br />
- 1)n (l.~)n diverges by the Ratio Test.<br />
n<br />
3<br />
' 3<br />
· = (1.1) lim ( ~ ~ )4<br />
el/n e ( 1 ) oo 1 · ' 00 e l /n<br />
11. Since 0 ~ - 3 - ~ 3 = e 3 and 2: 3 is a convergent p-series [p = 3 > 1 ], L: - 3<br />
- converges, and so<br />
n n n n = l n n = l n<br />
oo (- 1)"el/n.<br />
2::<br />
3<br />
is absolutely convergent.<br />
n=l n<br />
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SECTION 11.6 ABSOLUTE CONVERGENCE AND THE RATIO AND ROOT TESTS D 69<br />
. lan+l l . [ w n+l {n+1)4 2 "+ 1 ] . (10 n + 1) 5 . . 00 10"<br />
13. Ji_~ ~ = J2...~ (n + 2) 42n+3 . . 10" = n~ 42 . n + 2 = 8 < 1, so the senes n~l (n + 1)42 ~+1<br />
,<br />
is absolutely convergent by the Ratio Test. Since the. terms of this series are positive, absolute convergence is the same as<br />
convergence.<br />
00<br />
(-1)" arctann\ rr/ 2 . oo rr/ 2 rr 1 . . oo (-1)" arctann<br />
15.<br />
2<br />
< - 2<br />
, so smce 2::: - 2 = -<br />
\<br />
2<br />
2::: 2 converges (p = 2 > 1), the giVen senes 2:::<br />
2<br />
n n n = l n n =l n n=l n<br />
converges absolutely by the Comparison Test.<br />
= 0 and {-ln 1 } is decreasing. Now Inn < n, so<br />
.. = 2 Inn n -oo nn n<br />
17. f: ( - 1 )" converges by the Alte~nating Series Test s.ince lim - 1<br />
1<br />
diverges by the Comparison Test. Thus,<br />
Inn n n=2.n n = 2 nn<br />
- 1 - > .!, and since f: .! is the divergent (partial) harmonic series, f: - 1<br />
1<br />
~ (-1)". di' II<br />
w -In IS con twna y convergent.<br />
n=2 n<br />
00 00<br />
icos{nrr/ 3)1 1 1 . . cos{nrr/ 3)<br />
19.<br />
:5<br />
1 1<br />
and 2::: 1<br />
converges (use the Ratto Test), so the senes 2::: converges absolutely by the<br />
n. n. n=l n. n=l n.<br />
1<br />
Comparison Test.<br />
. . n 2 + 1 . 1 + 1/n 2 1 .<br />
21 . lim y/i(lJ = lim - 2 2 1 = hm 2 + 1 / 2 = - 2 < 1, so the senes 2::: 2 2 n-oo n-+oo n + n-oo n n =l n + 1<br />
Root Test.<br />
00<br />
( n 2 + 1 ) " .<br />
IS absolutely convergent by the<br />
23. lim y'jaJ = lim " (1 + .!)" 2 = lim (1 + .!)" = e > 1 [by Equation 7.4.9 (or 7.4* .9) [ ET 3.6.6] ],<br />
n--..oo n-... oo n n~ oo n<br />
1)"<br />
2<br />
00 (<br />
so the series L;: 1 + - diverges by the Root Test.<br />
n =l n<br />
. .<br />
. I an+ll . I (n + 1) 10 C!1QO"+l n! I . 100 (n + 1) 100 . 100 ( 1 )100<br />
25. hm -- - Inn · . - lim -- -- - lim -- 1 + -<br />
n-oo a,. -n- co (n + 1)! n 100100" .-n-oo n+ 1 n - n-co n + 1 n<br />
= 0 · 1 = 0 < 1<br />
oo nl00100" .<br />
so the series 2::: is absolutely convergent by the Ratio Test.<br />
n=l n 1<br />
27. Use the Ratio Test with the series<br />
1 · 3 1 · 3 · 5 1 · 3 · 5 · 7 n - 11 · 3 · 5 · · · · · (2n - 1) _ co ,._11 · 3 · 5 · · · · · (2n - 1)<br />
1 -3!+ - 5-!- - 7! + .. · +(- 1 ) {2n - 1)! +···-n~l( - 1<br />
) · (2n - 1)!<br />
lim lan+ll = lim ~ ( - 1)"~1·3·5 .... ·(2n-1)[2(n+1)-l]_ (2n-1)! I<br />
n-oo a,. n-oo (2(n+1) - 1]! (- 1)"- 1 ·1·3·5·····(2n-1)<br />
=lim ~ (-1)(2n+l)(2n- 1)! 1 = lim _.!_=0 < 1,<br />
n -oo (2n + 1)(2n)(2n - 1)! n-oo 2n..<br />
so the given series is ahsolutely convergent and therefore convergent.<br />
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70 D CHAPTER 11 INFINITE SEQUENCES AND SERIES<br />
00<br />
2 · 4 · 6 · · · · · (2n)<br />
00<br />
(2 · 1) · (2 · 2) · (2 · 3) · · · ·. (2. n) oo .2nn! oo . .<br />
29· n~l n! = n~l nl = .~<br />
1<br />
-;;;[ = n~l 2n, wh1ch diverges by the Test for<br />
Divergence since lim 2n = oo.<br />
n -oo<br />
31 B h . d fi . . lim I an+l I li 15n + 11 5 .<br />
. y t e recurs1ve e mt1on, -- = m --- = - > 1, so the series diverges by the Ratio Test.<br />
n-oo a,. n--400 4n + 3 4<br />
00<br />
b" cosn1r<br />
00<br />
bn 1<br />
•<br />
33. The series I: " = I: (-1)" 2':., where bn > 0 for n ~ 1 and lim bn = - .<br />
n=l n n=l n n-oo 2<br />
I a n+l l 1 . I ( - l )n+lb~+l n I 1<br />
li<br />
. b · n 1 ( ) 1 . ~ b~ cos n 1r .<br />
rn -- = 1m + · (<br />
. 1<br />
< 1, so the senes LJ iS<br />
n-+oo an n-oo n - n ~ n -~o oo n + n = l n<br />
absolutely convergent by the Ratio Test.<br />
1 ) b = 1m n -- 1 = - 2 1 = - 2<br />
35. (a) lim 1 1 / (n + 1 3<br />
3<br />
1<br />
) 1 = lim n = lim = 1. Inconclusive<br />
3<br />
n _,oo 1/n3 n -+oo (n + 1) n ->oo (1 + 1/ n)<br />
3<br />
(b) lim I (n 2<br />
++ 1<br />
1 ) ·<br />
2 " I = lim n 2<br />
+ 1 = lim ( - 2<br />
1 + -2<br />
1 ) = -2<br />
1 . Conclusive (convergent)<br />
n - oo n n n -too n n --+oo n<br />
(c) lim I ~ · ( V:<br />
1<br />
1 = 3 lim ·; n = 3 lim ~/ = 3. Conclusive (divergent)<br />
n .... oo v n + 1 - 3 n - n->oo n + 1 n -+oo v<br />
(d) li<br />
I<br />
I+Vn<br />
.Jn + 1 1 + n21 lim. [R1 1/n2 + 1 ] 1 r l .<br />
m ·--- = + - · = . nconc us1ve<br />
n .... oo 1 + (n + 1) 2 Vn n->oo n 1jn2 + (1 + 1/n) 2<br />
1 - = lxl · 0 = 0 < 1, so by the Ratio Test the<br />
n ->oo a,. n ->oo n + . X" n->oo n + 1 n ->oo n + 1<br />
37. (a) lim \ an+l I = lim I ( xn+~)' · n! I = lim 1-x -1 = lx l lim -<br />
00 x n<br />
series I; I converges for all x .<br />
n = O n.<br />
(b) Since the series of part (a) always convergt
SECTION 11.6 ABSOLUTE CONVERGENCE AND THE RATIO AND ROOTTESTS 0 71<br />
41. (i) Following the hint, we get that I ani < rn for n 2: N, and ~o since the geometric series 2:::'= 1<br />
r" converges [0 < r < 1],<br />
the series 2::=N lanl converges as well by the Comparison Test, and hence so does 2:::'= 1 lanl, so 2::_:'= 1 an is absolutely<br />
convergent.<br />
(ii) If lim . y'j'aJ = L > 1, then there is an integer N such that y'j'aJ >.1 for all n 2: N, so lanl > 1 for n 2: N. Thus,<br />
n-too -<br />
lim an "# 0, so 2:::'= 1<br />
an diverges by the Test for Divergence:<br />
n~oo<br />
(iii) Consider f ~ [diverges] and f ~ [converges]. For each sum, lim y'j'aJ = 1, so the Root Test is inconclusive·.<br />
n =l n n=l n n--+oo<br />
43. (a) Since 2:: an is absolutely convergent, and since la;t I ~ !ani and Ia ~ I ~. I ani (because a;t' and a~ each equal<br />
either an or 0), we conclude by the Comparison Test that both 2:: a;t and 2:: a;;- must be absolutely convergent.<br />
Or: Use Theorem 11 .2.8.<br />
(b) We will show by contradiction that both 2:: a;t and 2:: a ~ must diverge. For suppose that 2:: a;t converg<strong>ed</strong>. Then so<br />
'<br />
would l::(a;t'- ~an) by Theorem 11.2.8. But l::(a;t - ~an) = 2:: [ ~(an + !ani)- ~an ] = ~ 2:: I an i, which<br />
diverges because 2: a,. is only conditionally convergent. Hence, 2: a;t can't converge. Similarly, neither can 2:: a~.<br />
45. Suppose that 2:: an is conditionally convergent.<br />
(a) 2:: n 2 an is divergent: Suppose 2:: n 2 a,. converges. Then lim n 2 an = 0 by Theorem 6 in Section 11.2, so there is an<br />
• '1. ----+ 00 ;<br />
' 1<br />
integer N > 0 such that n > N =} n 2 lanl < 1. For n > N, we have !ani < 2 , so 2: !ani converges by<br />
. n n>N<br />
comparison with the convergent p-series I;: -.;.. In other words, 2:: an converges absoluiely, contradicting the<br />
. n>N n<br />
assumption that 2: an is conditionally convergent. This contradiction shows that 2:: n 2 an diverges.<br />
Remark: The same argument shows that 2: n'~' a .. diverges for any p > 1.<br />
(b) f (- lnl)n is conditionally convergent. It converges by the Alternating Series Test, but does not converge absolutely<br />
n =2 n n .<br />
·[by the Integral Test, since the function f ( x) = - ln 1 is c~ntinuous , positive, and decreasing on [2, oo) and<br />
. X X<br />
00<br />
1<br />
dx 1t dx [ t ] (-l}n<br />
- 1<br />
- = lim - 1<br />
- = lim ln{ln x )] = oo . Setting an = -In for n ~ 2, we find that<br />
2<br />
x n x t ~oo 2 x n x t ~ oo 2 n n<br />
00 00 ( -1)"<br />
n~<br />
2 nan = n~ ln n<br />
2<br />
converges by the Alternating Series Test.<br />
It is easy to find conditionally convergent series 2:: an such·that ~na n diverges. Tw~ examples are 2::<br />
00 ( -1)"-1<br />
"~<br />
1<br />
00 (-1t-1<br />
and<br />
n<br />
fo , both of which converge by the Alternating Series Test and fai l to converge absolutely because 2:: I ani is a<br />
p-series with p ~ 1. In both cases, 2:: nan diverges by the Test for Divergence.<br />
n=l<br />
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72 D CHAPTER 11 INFINITE SEQUENCES AND SERIES<br />
11.7 Strategy for Testing Series<br />
1. -- 1 < - 1 = (1)n - for all n > 1.<br />
n+3n<br />
f (~)n is a convergent geometric series [lrl = t < 1), so f ~<br />
3n 3 -<br />
n = l . n=l n + 3<br />
converges by the Comparison Test.<br />
3. lim la..l = lim _2:_ 2<br />
= 1, so lim an= lim (-1}"_2:_ does not exist. Thus, the series f (- 1)n __.!:_diverges by<br />
n - oo n-oo n + n--
17<br />
lim I an+l I=<br />
SECTION 11.7 STRATEGY FOR TESTING SERIES 0 73<br />
lim 11 · 3 · 5 .. · · · (2n- 1)(2n 1) . 2 · 5 · 8 · .... (3n - 1) I = lim 2n 1<br />
· n-oo a n n--+oo 2 · 5 · 8 · · · · · (3n- 1)(3n + 2) 1 · 3 · 5 · · · · · (2n- 1) n--+oo 3n + 2<br />
= lim 2 + 1/ n = ~ < 1 ,<br />
n--+oo 3 + 2/ n 3<br />
oo 1 . 3. 5 .. · .. (2n - 1)<br />
so the series 2::: ( 3<br />
) converges by the Ratio Test.<br />
n=l 2 · 5 · 8 · · · · · n - 1 .<br />
ln x<br />
1<br />
2 - ln x ? lrin. .<br />
2<br />
19. Let f(x) = r . Then f (x) = 2<br />
x 312<br />
< 0 when lnx > 2 or x > e-, so r ts decreasmg for n > e .<br />
vx . . vn<br />
. I R I lim In n I' 1/n 1' 2 0 h . ~ ( )"Inn b<br />
By I'Hosptta 's u e, r = tm ( ) = tm r = , sot e senes n6-<br />
1<br />
- 1 r;;, converges y the<br />
n--+oo vn n--+oo 1/ 2 Vn n - oo vn - vn<br />
Alternating Series Test.<br />
21 . lim lan. l = lim IC- l )ncos(l/n 2 )1 = lim icos(1/n 2 ).1 =cos O= 1, so the series E (-l)'' cos(l/ n 2 ) diverges by the<br />
n-oo n-oo n -'oo . n =l<br />
Test for Divergence.<br />
23. Using the Limit Comparison Test with a n = tan(.!.) and b,. = .!., we have<br />
n . n<br />
lim an = lim tan(1/ n) = lim tan(l/ x) M: lim sec2(1/x). (- 1/x2) = lim sec2(1/ x) = 12 = 1 > 0. Since<br />
n-oo bn n --+oo 1/ n :z:--+oo 1/x :r-oo -1/x 2 :z:--+oo<br />
00 00<br />
2::: bn is the divergent harmonic series, 2::: a,.. is also divergent.<br />
n = l<br />
n.=l<br />
U<br />
I<br />
. n + 1 oo n!<br />
f"L-H>O an .,._.00 e n n. ft. - 00 en + n+ln! r&.-oc e n n= l en<br />
· ~ li I an+l I . 1' I (n + 1)! c" 2 1<br />
1 . (n + 1)n! · e"2 1<br />
25. se t 1e Ratto . est. m - - = un < +t) ~ · -<br />
1<br />
= rm 2 2 = 1m ~+t = 0 < 1, so :L ---:r<br />
converges.<br />
27. roo ln: dx = lim [- ln X - .!.] t . [using integration by parts] M: 1. So E ln: converges by the Integral Test, and since<br />
./ 2 X t-oo X X 1 n=l n<br />
k ln k k ln k ln k . . oo k In A: •<br />
-----,-<br />
3<br />
< - k 3<br />
= -k?, the gtven senes 2:::<br />
3<br />
converges by the Companson Test.<br />
(k + l) .. ·- k=t(k+ l)<br />
29. f a n = f (-1)"___!_h = f (<br />
- 1)" b,,. Now b,. = - 1 - 1<br />
- > 0, {b,..} is decreasing, and lim b, = 0, so the series<br />
n =l n=l cos n "·=1 cos 1 n n -oo<br />
converges by the Alternating Series Test.<br />
0 W . 1 2 2 d ~ 1 '. . . ~ 1 . b h<br />
r: rtte co~ lt n = < - an 6 - ts a convergent geometrtc senes, so 6 --<br />
1<br />
- ts convergent y t e<br />
~ e" + e-" e" n = l e" n = l cos 1 n<br />
Comparison Test. So E (<br />
n=l<br />
- 1)" ___!_h is absolutely convergent and therefore convergent.<br />
cos n<br />
31. lim a~.; = lim k 5 4<br />
k--+oo k-oo 3 +<br />
k<br />
k = (divide by 4")<br />
oo s"<br />
Thus, 2::: ~ diverges by the T
74 0 CHAPTER 11 INFINITE SEQUENCES AND SERIES<br />
"2 jn 1 1 1 oo n2<br />
33. lim ~ = lim (~1 ) = lim [( 1 ) I t = li ( 1· 11<br />
r = - < 1, so the series I: (·-n-)<br />
n-+oo n-+00 n + n-+00 n + n m + n e . n=1 n + 1<br />
converges by the Root Test.<br />
35. an = n 1 ; 1 /n = n . ~ 1 /n, so let bn = ; and use the Limit Comparison Test. l . a, lim 1 1 0<br />
n-->oo<br />
un - = . --= ><br />
n-+oo bn n-+oo n1/n<br />
[see Exercise 4.4.61 ], so th~ series f: 1<br />
1 11<br />
,. diverges by comparison with the divergent harmonic series.<br />
. n=l n<br />
. 00<br />
37. lim ~ = lim (2 1 /" - 1) = 1 - 1 = 0 < 1, so the series I: ( y/2 - 1)"' converges by the Root Test.<br />
n-~oo n-too n-=1 .<br />
11.8 Power Series<br />
1. A power series is a series of the form :L:::'=o enx" = co + c1x + c2x 2 + c3x 3 + · ··,where x is a variable and the en's are<br />
constants call<strong>ed</strong> the coefficients of the series.<br />
More generally, a series of the form :L:::'=o c,,(x - a)" =eo+ c 1 (x- a)+ c2(x- a) 2 + · · · is call<strong>ed</strong> a power series in<br />
( x - a) or a power series center<strong>ed</strong> at a or a power series about a, where a is a constant.<br />
I I a ~ (-1) "+ 1<br />
lim n+ 1 = lim ( (n+1)x"-H ) I = lim I (-1)--x n+1 I = lim [( 1 + -1) JxJ ] = JxJ. By the Ratio Test, the<br />
n--+00 Un n-tOO - 1 71. nxn n-H)O n n-00 n '<br />
00 .<br />
series I: ( -1)"nxn converges when !x i < 1, so the radius of convergence R = 1. Now we' II check !he endpoints, that is,<br />
n=l<br />
x = ± 1. Both series f: (- 1)"n(±1)" = f: (=F1)"n diverge by the Test for Divergence since lim J(=F1)"nl = oo. Thus,<br />
n=l · n=l TI.--+OO<br />
the interval of convergence is I = ( -1, 1).<br />
x" ' . I a,.+1 1 . I x"+l 2n - 11 . (2n - 1 ) . (2- 1/n / lxJ ) = JxJ. By<br />
n- 1 n-+oo an n-+oo n + x" n-+oo n + n-+oo 2 + 1 n<br />
5. If an = - 2<br />
--, then lim -- = hm - 2 1 · -- = !1m - 2 1<br />
Jxl = hm<br />
·oo ~ oo 1 ·<br />
diverges by<br />
n=l n - 1 n=l n - 1<br />
the Ratio Test, the series I: - 2<br />
--converges when Jx! < 1, so R = 1. When x = 1, the series I: - 2<br />
. . I ~ 1 . 1 1 d 1 ~ 1 d' . . . I . I fth h . .<br />
companson Wit 1 6 - srnce --- > - an - 6 - 1verges smce It ts a constant mu tip e o e armoruc senes.<br />
n=1 2n 2n - 1 . 2n 2 n=l n<br />
When x = - 1, the series f: ( - 1 ) n converges by t~e Alternating Series Test. Thus, the interval of converge~ce is [ -1, 1).<br />
n=1 2n -1<br />
7. If an= x",, then lim lan+l l = lim I ( x"+\; ·. n ! I= lim 1- x -1 = !xi lim -<br />
n. n -+ oo an n-+oo n + 1 . X" n -+oo n + 1 n-+oo n + 1<br />
So, by the Ratio Test, R = oo and I= ( - oo, oo).<br />
1 - = lxl· 0 = 0 < 1 for all real x.<br />
lim I an+1 I = lim J (n + 1)2 x"+l . ~~ = lim I x(n + 1)2 1 = lim [JxJ (1 + .!.)2] = EJ.(1)2 = ~ JxJ. By the<br />
n - oo . an n-CXJ 2n+l n 2 x"· n.-oo 2n2 n-oo 2 n 2<br />
. .<br />
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SECTION 11.8 POWER SERIES 0 75<br />
l<br />
00 n2xn<br />
Ratio Test, the series 2:: ( -1)'" .- ,.- converges when t lx l < 1 ¢> lx l < 2, so the radius of convergence is R = 2.<br />
n = l 2<br />
lim I (:fl)n n 2 1 = oo. Thus, the interval of convergence is I = ( -2, 2).<br />
n -oo<br />
( - 3)"x"<br />
11. If an = n 312<br />
, then<br />
00<br />
2:: (=F1)"'n 2 diverge by the Test for Divergence since<br />
n=l<br />
( 3)n+l n+l 3/2 I I (. )3/21 ( )3/2<br />
I I I<br />
11m . -- a,.+t = 1' lffi - x · n = 1' Im - 3 X --<br />
n = 3 1 X<br />
I 1' tm . 1<br />
n-•oo a., n~oo (n + 1)3/2 ( - 3)nxn n~oo n + 1 n-+oo 1 + 1/n .<br />
= 3lx l (1) = 3 lxl<br />
00 ( -3)"<br />
By the Ratio Test, the series 2:: '- x "' converges when 3 lxl < 1 ¢> lx l < %, so R = %- When x = S· the series<br />
,.= 1 nyn<br />
· I: ~ converges by the Alternating Series Test. When x = - %, the series 2::<br />
00 (- 1)" . 00 1<br />
n=l n<br />
( p = ~ > 1). Thus, the interval of convergence is [-S, t].<br />
312<br />
n=l n<br />
is a convergent p-series<br />
I<br />
13.I f an = ( - 1 ) "~. then lim lan+t l = lim I xn+l . _4" lnn i= E11im 1n n =1.::1 · 1<br />
4" ln n n~oo an n-+oo 4n+l ln(n + 1) x" 4 n~oo ln(n + 1) 4<br />
[by !' Hospital's Rule] = 1: 1. By the Ratio Test, the series converges when 1:1 < 1 ¢:> lx l < 4, so R = 4. When<br />
00 00 00<br />
xn [(-1)(- 4)t 1 . 1 1<br />
00<br />
1<br />
- > - an d 2:: - is tbe<br />
n=2 n n = 2 n n =2 n n n n n n = 2 n<br />
x = - 4, 2:(-1)" 4<br />
n ln = 2::<br />
4 " ln = 2: - 1 - . Smce lnn < n for n ~2 . - 1<br />
divergent harmonic series (without then = 1 term), f - 1<br />
1 ' is divergent by the Comparison Test. When x = 4,<br />
n=2 n n .<br />
00 xn 00 1<br />
2:: ( - 1)n 4<br />
n ln = 2:: ( -1)"-ln , which converges by the Alternating Series Test. Thus, I = ( - 4, 4] .<br />
n =2 n n=2 n<br />
(x - 2)"' . I an+t I . ·1(x - 2)"+1 n 2 + 1 I . n 2 + 1<br />
15. If an = 2 1 , then lim - - = lim ( 1 ) 2 1 · ( 2 ) = lx- 21 lim ( 1 ) 2 = lx - 21. By the<br />
n + n ~oo an n-+oo n + + X - " n~ oo n + + 1<br />
oo (x- 2)"<br />
Ratio Test, the series 2::<br />
2<br />
converges when lx- 21 < 1 [R = 1] ¢:> -1 < x- 2 < 1 ¢:> 1 < x < 3. When<br />
n=O n + 1<br />
x = 1, the series f: (- 1)" + converges by the Alternating Series Test; when x = 3, the series 1<br />
f: - 1 - 2<br />
- converges by<br />
n = O n + n=O n + 1<br />
comparison with the p -series f: ~ [p = 2 > 1]. Thus, the interval of convergence is I = [1, 3]. ·<br />
n = l n .<br />
00<br />
3"'(x + 4)"<br />
By the Ratio Test, the series .~ Vn converges when 3 lx + 41 < 1 ¢:> lx + 41 <<br />
1<br />
1 [ R = t ] ¢:><br />
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76 0 CHAPTER 11 INFINITE SEQUENCES AND SERIES<br />
-t < x + 4 < t
SECTION 11.8 POWER SERIES 0 n<br />
(b) It does not follow that I:::'=o en( - 4)" is necessarily convergent. [See the comments after Theorem 3 about convergence at<br />
the endpoint of an interval. An example is Cn = ( - 1 t / ( n4").]<br />
(n!)k n<br />
31. If a,.=-( )' x , then<br />
kn.<br />
lim I an+ll = lim [(n + 1)!]k (kn)! lxl = lim - (n + 1)k lxl<br />
....... oo a., n ..... ex> (n!)k [k(n + 1)]! ,.....,CX> (kn + k)(kn + k- 1) · · · (kn + 2)(kn + 1)<br />
. [ (n+1) (n+1) (n+1)]<br />
= J~ (kn + 1) (kn + 2) ... (kn + k) lxl<br />
- l. t.m [--- n + 1 ] lim [--- n + 1 ] · · · 1. liD [--- n + 1 ] I X I<br />
- n-+ex> kn + 1 ,...... ex> kn + 2 n-oo kn + k<br />
= ( ~ J lxl < 1 ¢:} lxl < kk for convergence, and the radius of convergence is R = kk.<br />
33. No. If a power series is center<strong>ed</strong> at a, its interval of convergence is symmetric about a. lf a power series has an infinite radius<br />
of convergence, then its interval of convergence must be (- oo, oo), not [0, oo).<br />
(-1t X2n+l<br />
35. (a) If a,. = I(<br />
1 ) 122n+l, then<br />
n. n+ .<br />
I<br />
a I I<br />
r n+l li<br />
x2n+3 . nl(n<br />
.<br />
+ 1)'22n+l<br />
.<br />
l (x)2<br />
lim<br />
1<br />
,.:.,~ ---a,: = n ..... ~ (n + 1)!(n + 2)! 22n+3 . x2n+l = 2 ,..._<br />
So J 1 ( x) converges for all x and its domain is (- oo, oo).<br />
(b), (c) The initial terms of J1(x) upton = 5 are ao = ~·<br />
..... oo (n + 1)(n + 2) = 0 for all x.<br />
xu<br />
and a(j = - 176<br />
, 94 7<br />
, 200<br />
. The partial sums seem to<br />
approximate J1 ( x) well near the origin, but as lx l increases,<br />
we ne<strong>ed</strong> to take a large number of terms to get a good<br />
approximation.<br />
37. S2n-l = 1 + 2x + x 2 + 2x 3 + x 4 + 2x 5 + · · · + x 2 "- 2 + 2x 2 "- 1<br />
= 1(1 + 2x) + x 2 (1 + 2x) + x 4 (1 + 2x) + · · · + x 2 " - 2 (1 + 2x) = (1 + 2x)(1 + x 2 + x 4 + ... + x 2 ~ - 2 )<br />
1 - x 2 " 1 + 2x<br />
= (1 + 2:c)- 1<br />
-- 2 [by (11.2.3) with r = x 2 ] --+ -<br />
1<br />
-- 2 as n--+ oo by (11.2.4), when lx l < 1.<br />
- X - X<br />
I<br />
2n 1 +2x. 2n 0" I I 1Th &:<br />
A so<br />
1+2x. .<br />
S2n = S2n-1 + x --+ _ x 1 2 smce x --+ •Or x < . ere,ore, Sn --+ _ x 1 2<br />
smce S2n and S2n-1 both<br />
1 + 2x . . 1 + 2x<br />
approach --- 2 as n --+ oo. Thus, the mterval of convergence IS ( - 1, 1) and f ( x) = - - 2 .<br />
. 1-x 1 - x<br />
© 2012 Cengagc Lcoming. All Rights Reserv<strong>ed</strong>. May not be scnnncd, copi<strong>ed</strong>, or dupliCD.tcd: or post<strong>ed</strong> to a publicly acccs!tiblc website. in whole or in part.
78 D CHAPTER 11. INFINITE SEQUENCES AND SERIES<br />
39o We use the Root Test on the series I:; c..x" 0<br />
lxl < 1/c, so R = 1/c.<br />
We ne<strong>ed</strong> lim \/lc,x"l =I xi lim y'jCnT = c !x! < 1 for convergence, or<br />
n.- oo n--too<br />
41 . For 2 < x < 3, I:; cnx" diverges and I:; dnx".convergeso By Exercise 11.2.69, I.:;( en+ dn) x" diverges. Since both series<br />
converge for I xi < 2, the radius of convergence ofi.:;(en + dn) x" is 2.<br />
11.9 Representations of Functions as Power Series<br />
00 00<br />
10 If f(x) = I:; c..xn has radius of convergence 10, then!' (x) = I:; nc..x"- 1 also has radius of convergence 10 by<br />
n = O n = l ·<br />
Theorem 20<br />
3. Our goal is to write the function in the form - 1· 1 , and then use Equation ( I) to represent the function as a sum of a power<br />
- r<br />
1 1 00 00 0<br />
series. f(x) = -- = ( ) = I:; ( -xr' = I:; ( -1)"x" with 1-xl < 1 ¢:? lxl < 1, so R = 1 and I= ( - 1, 1).<br />
1 +X 1 - -X n =O n= O<br />
2 2( 1) 200<br />
(x)"<br />
oo 1<br />
5o f(x) = 3<br />
_ x = 3 1<br />
_ x/ 3<br />
= 3 n~o 3 or, equivalently, 2 fo 3<br />
,.+1 x" 0<br />
that is, when I xi < 3, so R = 3 and I = ( -3, 3).<br />
The lxl<br />
series converges when<br />
3 < 1,<br />
The geometric series ... ~o [- (~) 2 ] " conver~es when 1- (~) 2 1 < 1 ¢:? ~ ~ 2 1 < 1 ¢:? lxl 2 < 9 ¢:? lxl < 3, so<br />
R = 3and I= (-3,3).<br />
The series converges when lxl < 1, so R = 1 and I = ( - 1, 1).<br />
· ( ) 1 +X -(1 - x) + 2 ( 1 ) ~ n 2 ~ n<br />
A second approach: f x :::; -- = = - 1 + 2 -- = -1 + 2 LJ x = 1 + LJ x .<br />
1 - X 1 - X 1 - X n = O n=l<br />
A third approach:<br />
f(x)= 1 +x =(l +x)(- 1 -) = (l +x)(l +x+x 2 +x 3 + ·o·)<br />
1 - x 1-x<br />
= (1 + x + x 2 + x 3 + .. o) + (x + x 2 + x 3 + x 4 + o o o) = I+ 2x + 2x 2 + 2x 3 + o .. = 1 + 2 f: Xno<br />
n=l<br />
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SECTION 11.9 REPRESENTATIONS OF FUNCTIONS AS POWER SERIES D 79<br />
3<br />
11. f (x) =<br />
2 2 x - x-<br />
3 A B<br />
..,....------,-,-,---"7 = --+ -- => 3 = A(x + 1) + B(x - 2). Let x = 2 to get A = 1 and<br />
(x- 2)(x + 1) x - 2 x + 1 ·<br />
x = - 1 to get B = - 1. Thus<br />
3 __ 1<br />
x2- x - 2 - X- 2 X+ 1 - 2 1- (x/2) 1 - (- x) 2 n=O 2 n=O<br />
_ _ _ 1_=~( 1 )- 1 =-~ E (=)" _ E(-x)n<br />
= f: [-~ (1:)" -1(-1)"']xn = E [(- 1t+l- n~l]x"<br />
n=O 2 2 n=O 2<br />
We represent<strong>ed</strong> f as the sum of two geometric series; the first converges for x E (-2, 2) and the second converges for ( -1, ~) .<br />
Thus, the sum converges for x E ( - 1, 1) =I.<br />
13. (a) f(x ) = 1 = ·-d (-- -1 ) = , --d [ L..... ~ (- 1)n x "]<br />
(1+x) 2 dx 1+x dx n=O<br />
[from Exercise 3]<br />
~ ~<br />
= L (-1)"+ 1 nx"- 1 [from Theorem 2(i)] = L (-1)"(n + l)x" with R = ~ .<br />
n=l<br />
n=O<br />
In the last step, note that we decreas<strong>ed</strong> the initial value of the summation variable n by 1, and then increas<strong>ed</strong> each<br />
occurrence ofn in the term by 1 [also note that (-1)"+ 2 = (- 1)" ].<br />
(b)f(x)=<br />
1<br />
1<br />
=-1:j_[ ] =-1:j_[E(-1)"(n+ 1)x"] [frompart(a)]<br />
(1+x) 3 2dx (1+x) 2 2dx n=O<br />
~ ~<br />
= -~ L (-1)"(n + 1)nxn- 1 = ~ L (- 1)n(n + 2)(n + 1)x" with R = 1.<br />
n=l<br />
n=O<br />
x 2 1 1 ~<br />
(c)f(x) = ( 1<br />
+x) 3 =x 2 · (l+x) 3 = x 2 · 2nJ;<br />
0<br />
{- 1)''(n+2)(n+1)x" [frompart(b)]<br />
= 1: E (-1)"{n+2)(n+ l)x"+~<br />
2 n=O<br />
To write the power series with x " rather ~an ·x"+ 2 , we will decrease each occurrence of n in the term by 2 and increase<br />
1 . CX><br />
the initial value of the summation variable by 2. This gives us 2 .. (-1)"(n)(n- 1)x,. with R = 1.<br />
~2<br />
15. f(x) = ln(5 - x) = -~~ = _1: ~~ = -1:1 [f: (::\"] dx = C- ~ E xn+l = ·C..:... f: x"<br />
5-x 5 1 - x/5 5 n=O 5} 5n=o5"(n+ 1) n=tn5n<br />
Putting x = 0, we get C = ln 5. The series converges for Jx/51 < 1 Jx J < 5, so R = 5.<br />
1 1<br />
00<br />
17. We know that - 1 4<br />
= ( 4<br />
) = L ( -4x)". Differentiating, we get<br />
+ X 1- - X n=O<br />
f(x) = {1+x4x)2 = · -4x.<br />
for J-4xJ < 1 Jxl < ;}, so R = ;} .<br />
-4 2 =-X f;{- 4)n+1 (n+1)xn = f;(-1)n4n(n+1)xn+l<br />
(1 + 4x) 4 n=O n=O<br />
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80 0 CHAPTER 11 INFINITE SEQUENCES AND SERIES<br />
1 00<br />
19. By Example 5, ( )2 = L: (n + 1}xn. Thus,<br />
1- X n =O<br />
!( ) _ 1 + X _ 1 X _ ~ ( } ,n ~ ( n+l<br />
x - (1- x)2 - (1-x)2 + (1- x)2 - ~o n + 1 x + n~o n + 1}x<br />
00 00<br />
= 2::(n+1}xn+ E nxn [make the starting values equal]<br />
n=O<br />
n=l<br />
= 1 + f; [(n + 1) + n]xn = 1 + f; (2n + 1}xn = f; (2n + 1}xn with R = 1.<br />
n=l n=l n = O<br />
The series converges when l-x 2 / 16l < 1 x 2 < 16 lxl < 4, so R = 4. The partial sums are 9 1 = ~.<br />
16<br />
82 = s1 -<br />
x 3 x 5 x 1 x 9<br />
, .... Note that 8 1 corresponds to the first term of the infinite<br />
162 , 83 = 82 + 163 , 84 = 83 - 164 , 8s = S4 + 165<br />
sum, regardless of the value of the summation variable and the value of the exponent.<br />
As n increases, 8n(x) approximates f better on the interval of convergence, which is ( - 4, 4}.<br />
1 + x) I dx I dx I dx I dx<br />
=I L~<br />
0 (-1)nxn + n~o xn] dx = 1 [(1- x + x2 - x 3 +x 4 - • .. ) + (1 + x + x 2 + x 3 +x 4 + ... )] dx<br />
23. f(x) = In ( - - = ln(1 + x) - ln(1 - x) = - - + -- = ( ) + - -<br />
1 - X 1 +X 1 - X 1 - -X 1 - X<br />
= 1 (2 + 2x 2 + 2x 4 + · · ·) dx =I f; 2x 2 " dx = C + f; 2 2 X 2n + l<br />
n=O n=O n + 1<br />
, oo zx2n+l oo 1<br />
But /(0) = ln t = 0, soC = 0 and we have f(x) = E --with R = 1. Ifx = ±1, then f(x) = ± 2 E - 2 1<br />
,<br />
n = O 2n + 1 n =O n +<br />
which both diverge by the Limit Comparison Test with b,. = .!. .<br />
n<br />
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SECTION 11.9 REPRESENTATIONS OF FUNCTIONS AS POWER SERIES 0 81<br />
2x 2x 3 2x 5<br />
The partial sums are s1 = l' s2 = s1 + 3' S3 = s2 + 5 ' · · · ·<br />
As n increases, s,.(x) approximates f better on the interval of<br />
convergence, which is {-1, 1).<br />
25. _ t_ B = t . _1_ 8 = t f: ( tB )" = f: t8n+l<br />
1 - t 1 - t n = O n =O<br />
=> j . t 00 tB·n+2 1<br />
--dt = C + I: ---. The series for -- 8<br />
converges<br />
1 - t 8 n =O 8n + 2 1 - t<br />
when lt 8 l < 1 ¢} It I < 1, so R = 1 for that series and also the series for t/{1 - t 8 ). By Theorem 2, the series for<br />
I<br />
_!_a dt also has R = 1.<br />
1 - t<br />
oo x" oo x"+2<br />
27. From Example 6,ln{1 + x) = E ( - 1)"- 1 - for lx l < 1, so x 2 ln{1 + x) = I: {- 1t- 1 - -and<br />
n =1 n n = 1 n<br />
I<br />
x 2 ln(1 + x) dx = C + I: ( -1)"' (<br />
n=1 n n+ 3<br />
oo x"+3 .<br />
)' R = 1 for the series for ln{1 + x), so R = 1 for the series representing<br />
x 2 ln{1 + x) as well. By Theorem 2, the series for I x 2 ln{l + x) dx also has R = 1.<br />
29<br />
1 1<br />
f;(-x 5 )"=f:{-1)"xu" =><br />
· 1 + x 5 = 1 - (- x·5 ) = o<br />
I<br />
n =O n =<br />
1 ;· oo oo ,&n+l<br />
--.. dx = 2:{- l )"'x 5 " dx = C + 2:(-1)"-x<br />
5 1 . Thus,<br />
1 + x~ ,.=o n=O n +<br />
0.2 1 [ x6 x lt . ] 0.2 {0.2)6 {0.2) u . .<br />
I= - - dx = x- - + - - ··· = 0.2---+ - ---· ··. The senes IS alternating so if we use<br />
. £ 0 1 + x 5 6 11 0 6 11 . '<br />
the first two terms, the error is at most (0.2) 11 /11 ~ 1.9 x 10- 9 . So I ~ 0.2 - {0.2) 6 /6 ~ 0.199. 989 to six decimal places.<br />
31. We substitute 3x for x in Example 7, and find that<br />
j . I 00 "' (3x?"+l I 00 ".32.. +1 x2n+2 00 . .. 32n+l x2n+3<br />
xarctan(3:c)d:v = x 2:(-1) dx= 2:: (- 1)<br />
dx = C+ E(- 1) ( )( )<br />
n = O 2 n -1- 1<br />
n = O 2 n + 1<br />
n =O 2n -1- 1 2n -j- 3<br />
1<br />
0 .1 [<br />
3<br />
So x arctan(3x) dx = .....:::.__ 3 33 G 35 7 37 !) ] 0 .1<br />
-~ +~-__E._+ ···<br />
0 1 ·3 3·5 5·7 7 · 9 0<br />
1 9 243 2187<br />
= 103 - 5 X 10° + 35 X 107 - 63 X 1Q9 + .. ..<br />
• • 1 . 'f h 1 •<br />
2187<br />
3 8<br />
The senes IS a ternatmg, so 1 we use t ree terms, t 1e error IS at most x ~ .5 x 10- . s o<br />
63 109<br />
r ·l 1 9 243<br />
J<br />
0<br />
x arctan(3x) dx ~ 103<br />
-<br />
5 x 105 + 35 x ~ 0.000 983 to six decimal places.<br />
107<br />
82 0 CHAPTER 11 INFINITE SEQUENCES AND SERIES<br />
x 3 x 5 x 1 (0.2? (0.2) 5 (0.2) 7<br />
33 By Example 7 arctan x = x - - + - - - + · · · so arctan 0 2 = 0 2 - --+ --- --+ ...<br />
. ' 3 57' .. 3 5 7 .<br />
Th . . I . 'f t1 1 · (0.2) 7<br />
e senes 1s a ternatmg, so 1 we use 1ree terms, t 1e error ts at most - - ~ 0.000 002.<br />
7<br />
(0 2? (0 2?<br />
Thus, to five decimal places, arctan0.2 ~ 0.2- - ·3-<br />
+ -·5-<br />
~ 0.197 40.<br />
2 , 1 2<br />
00<br />
(- 1)"2n(2n- 1)x 2 "<br />
00<br />
(-1t2nx 2 "'<br />
00<br />
(- 1)"x 2 "+2<br />
X Jo (x) + XJo(X) +X Jo(x) = n~l 22n(n!)2 + 1 ~ 1<br />
22"(n!)2 + n~O 22" (n!)2<br />
00<br />
(- 1)"2n(2n-1)x 2 " oo (- 1)"2nx 2 '' oo (- 1t- 1 x 2 "<br />
= n~l 2 2 "(n!) 2 + ,~1 2 2 "(n!J2 + ,~1 22 n- 2 [(n- 1)!] 2<br />
_<br />
00<br />
(- 1)"2n(2n-1)x 2 "<br />
00<br />
(- 1t2nx 2 "<br />
00<br />
(-1)"(-1)- 1 2 2 n 2 x 2 "<br />
- r~l 2 2 "(nl)2 + n~l 2 2 "(n!)2 + ,~1 2 2 "(n!)2<br />
= ~ (- 1 )" [2n(2n - 1) + 2n - 2 2 n 2 ] x2"<br />
n~l 2 2 "(n!)2 ,<br />
= ~ (- 1 )" [4n 2 - 2n + 2n- 4n 2 ] 2n = 0<br />
L- 22" ( 1)2 X<br />
n=l n.<br />
Since 16<br />
.i 28<br />
~ 0.000062, it follows from The Alternating Series Estimation Theorem that, correct to three decimal places,<br />
J 0<br />
1<br />
Jo (x) dx ~ 1 - 1 ; + 3 ~ 0 ~ 0.920.<br />
00 Xn I 00 nxn- l 00 Xn-1<br />
37. (a) f(x) = n~O nf ::} f (x) = •~1 ~ = •~1 (n- 1)!<br />
oo x.,."<br />
I: - 1<br />
= f(x)<br />
n = O n<br />
(b) By Theorem 9.4.2, the only solution to the differential equation df(x)/d.'C = f(:c) is f(x) = Ke", but f(O) = 1, so<br />
K = 1 and f(x) = e".<br />
Or: We could solve the equation df(x)/ dx = f(x.) as a separable differential equation.<br />
n I I I n+l 21 . ( ) 2 .<br />
39. If a,.. = X 2' then by the Ratio Test, lim an+ I = lim (X )2 . ~ = lxl lim ....2::_1 = lx l < 1 for<br />
n - n.-oo an n-oo n + 1 xn n----+oo n +<br />
convergence, so R = 1. When x = ± 1, f I x: I = f ~which is a convergentp-series (p = 2 > 1), so the interval of<br />
n = l n " =1 n<br />
convergence for f is [-1, 1]. By Theorem 2, the radii of convergence off' and f" are both 1, so we ne<strong>ed</strong> only check the<br />
00 x" 00 nxn- 1 00 x"<br />
endpoints. f(x) = I; 2 => . f'(x ) = I: - - 2 - = I; --,and this series diverges for x = 1 (harmonic series)<br />
n=l n n=l n n=O n + 1<br />
@ 2012 Cengage Lc:tminiJ. All Rights Reserv<strong>ed</strong>. May not be sc:~.Mcd, copi<strong>ed</strong>, or duplicat<strong>ed</strong>, or post<strong>ed</strong> w n publicly accessible wcbsile, in whole or in part.
SECTION 11.10 TAYLOR AND MACLAURIN SERIES D 83<br />
00 n.-1<br />
and converges for x = - 1 (Alternating Series Test), so the interval of convergence is [-1, 1). !" (x) = I: ~ diverges<br />
n=l n+ 1<br />
I<br />
at both 1 and - 1 (Test for Divergence) since lim ~ = 1 =/= 0, so its interval ofc~nvergence is ( - 1, 1).<br />
n--.oo n+ 1<br />
• 00 ·X2n+ l 1<br />
41. By Example 7, tan- 1 x = L: ( -1)"-- for lxl < 1. In particular, for x = r.;• we<br />
n=O 2n+ 1 y3<br />
00<br />
7r - 1 ( 1 ) n (1/ v'3) 2 n +l ~ ( )" (1)" 1 1<br />
have 6 = tan y3 = n~O 1<br />
( - ) 2n + 1 = n~O 1<br />
- 3 y3 2n + 1' SO<br />
7r=~f (-1)" =2v'3f (-1)" .<br />
y'3 n = O (2n + 1)3" n=O (2n + 1)3"<br />
11.10 Taylor and Maclaurin Series<br />
oo J
84 0 CHAPTER 11 INFINITE SEQUENCES AND SERIES<br />
9.<br />
11.<br />
13.<br />
15.<br />
n<br />
0<br />
1<br />
2<br />
3<br />
4<br />
n<br />
0<br />
1<br />
2<br />
3<br />
4<br />
n<br />
0<br />
1<br />
2<br />
3<br />
4<br />
5<br />
6<br />
n<br />
0<br />
1<br />
2<br />
3<br />
(o) =<br />
f (n)(x) J (n) (0)<br />
{ 1 so sinhx= I: ( ifn is odd 2<br />
_ ) 1<br />
•<br />
n=o n+1.<br />
sinh x 0<br />
x2n+l<br />
Use the Ratio Test to find R. If an = ( )I, then<br />
cosh x 1<br />
2n + 1 .<br />
sinhx 0<br />
r I<br />
cosh x 1<br />
sinhx 0<br />
= 0 < 1 for all x, so R = oo.<br />
J Cnl(x) J(nl (l)<br />
x 4 - 3x 2 4 t :!012 Cengnge Lc:unin~ . All Rights Rcscn1:d. ~ tay not be SCnMCd. copk-d. or duplicat<strong>ed</strong>, or postct.l loa publicly ncccs.~i blc wcbsilt:, in whole or in part.
SECTION 11.10 TAYLOR AND MACLAURIN SERIES 0 85<br />
. lan+l l r l( - lt+2(x~2)"+ 1 n2" 1- li l(- l)(x-2)n,_ r . ( n) lx-21<br />
J~ ~ = n~ (n + 1) 2n+l . ( - 1)"+ 1 (x- 2)" - ,._n;., (n + 1)2 - ":~ n + 1 - 2-<br />
= lx - 2 1 < 1 for convergence, so lx - 21 < 2 and R = 2.<br />
2<br />
17.<br />
n t
86 0 CHAPTER 11 INFINITE SEQUENCES AND SERIES<br />
25. {/1 -x = [1 +(-x)FI 4 = f ( 1 / 4 ) ·(-x)"= 1+t(-x)+ H-, ~ )( -x) 2 + H-~),(-D( -x? + · ..<br />
n=O n 2 3.<br />
= 1<br />
_ !x+ ~ (- 1)"- 1 (-1)" · [3 · 7 · · · · · (4n - 5)]x"<br />
4 n=2 4n · n!<br />
1<br />
00<br />
3 · 7 .. · .. (4n - 5)<br />
= 1- - x- I: xn<br />
4 n=2 4n. n !<br />
and 1-xl < 1 ¢} lxl < 1, so R = 1.<br />
27.<br />
1<br />
1<br />
3<br />
=<br />
(2 + x) [2(1 + x/2)] n=O n<br />
(<br />
3<br />
= - 8<br />
1 ( 1 + ~ 2 ) - 3 = - 8<br />
1<br />
f: (- 3 ) (~ 2<br />
) ". The binomial coefficient is<br />
- 3) = (-3)(-4)(- 5) ..... (-3 - n + 1) = ·c-3)(- 4)( -5) ..... [- (n + 2)]<br />
n n! n!<br />
_ ( - 1)" · 2 · 3 · 4 · 5 · · .. · (n + l )(n + 2) _ ( - l)"(n + l )(n + 2)<br />
- 2 ·n! - 2<br />
I<br />
Thus 1 = ! ~ ( -l)"(n + 1)(n + 2) x" = ~ ( -1)"(n + 1)(n + 2)x" for ~ ~~ < 1 ¢} lxl < 2, so R = 2.<br />
' (2 + x) 3 8 n=O 2 2n n=O 2"+ 4 2 .<br />
00 X2n+ l 00 ( 1fX) 2n+ 1 oo 7r2n+ 1<br />
29. sin x= L::(-1)" ( 2 1<br />
) 1<br />
~ f(x) =sin(1rx) = L::(- 1)" ( 2 1<br />
) 1<br />
= L::(-1)" ( 2 1<br />
) 1<br />
x 2 ~+ 1 ,R = oo.<br />
n=O n + · n = O n + · n =O n + ·<br />
oo x"<br />
31. ex = E I<br />
n =O n.<br />
R =oo.<br />
2x ~ (2x)" ~ 2" x" J( ) x + 2o: ~ 1 n ~ 2" n ~ 2" + 1 n<br />
=> e = L.J -<br />
n=O n. n=O n n=O n. n =O n. n = O n.<br />
1 - = L.J - 1 - ,so x =e e = L.J 1 x + L.J 1 x = L.J - - 1 -x,<br />
00 x2n<br />
33. cosx = I: (-1)" (2 ·),<br />
n = O n ·<br />
35. We must write the binomial in the form (1 +expression), so we' ll factor out a 4.<br />
X X :i; X ( • X2)-l/ 2 X oo (-1)<br />
(X2)"<br />
J4 + x 2 = y'4(1 + x 2 /4) = 2 y'1 + x2 /4 = 2 1 + 4 = 2 ,~0 n 2 4<br />
X X ~ ( l )"' 1 · 3 · 5 · · · · · (2n- 1) 2n<br />
= - + - ~ - X<br />
2 2 n=l 2" . 4" . n!<br />
_<br />
X<br />
00<br />
n 1 · 3 · 5 · · · · · (2n- 1) 2n+l X 2 !xj<br />
- 2 + .. ~ 1 {-1) n! 23 ,.+ 1 x and 4 < 1 ¢> 2<br />
< 1 ¢> lxl
SECTION 11.10 TAYLOR AND MACLAURIN SERIES D 87<br />
1.5<br />
= 1- ~x '' + fixs- 7~oxl2 + ...<br />
The series for cos x converges for all x , so the same is true of the series for<br />
f(x), that is, R = c;o· Notice that, as n increases, Tn(x) becomes a better<br />
approximation to f(x).<br />
- 1.5 t--r.'-----+---~ ..:--11.5<br />
- 1.5<br />
\ . T 8 = T 9 = T 10<br />
= T 11<br />
•<br />
\ 't<br />
\<br />
.<br />
oo n oo ( )" oo n<br />
:r;<br />
41 . e<br />
(11)<br />
=<br />
""<br />
L- I,<br />
X<br />
so e<br />
- x<br />
=<br />
""<br />
L-<br />
....=::___1 ·= ""{-1)"<br />
L-<br />
.:._,SO<br />
n =O n. n =O n. n =O n 1<br />
f (x) = xe- "' = f: (-1)" ~ x"+l ·<br />
n=O n.<br />
The series for e"' converges for all x, so the same is true of the series<br />
for f(x); that is, R = oo. From the graphs off and the first few Taylor<br />
polynomials, we see that T,. ( x) provides a closer fit to f ( x) near 0 a!; n increases.<br />
43. 5° = 5°<br />
(<br />
7T ) 7r . oo x2" x2 x4 x6<br />
1800<br />
= 36<br />
radians and cos x = .. ~o ( -1t ( 2<br />
n)! = 1 - 2f + 41<br />
- 6f + ·· ·,so<br />
7T (7r/ 36) 2 (7r/ 36) 4 (7r/ 36) 6 (7r/ 36) 2 . (7r/ 36)~<br />
cos - = 1 - --- + - + ... Now 1 - ~ 0 99619 and addmg 2 4 X w-G<br />
36 2! 4! 6! . 21 . 4! ~ .<br />
does not affect the fifth decimal place, so cos 5° ~ 0.99619 by the Alternating Series Estimation Theorem.<br />
(b)sin- lx= l 1 dx=C + x + f: 1·3 · 5 ·· .. · (2n-1)x 2 n+l<br />
V1- x 2 n = l (2n + 1)2n · n !<br />
= x + f; 1 · 3 · 5 · · · · · { 2 n - 1 ) x 2 n+l since 0 = s in- 1 0 = C.<br />
n=l (2n + 1)2". nl<br />
(16) 00 x 2n<br />
47. COSX = "~ 0 (- 1 )" { 2 n)! I<br />
oo<br />
X Gn+2<br />
x cos(x 3 ) dx = C + .~o ( - 1)" ( 6<br />
n + 2<br />
)( 2<br />
n)! , with R = oo.<br />
(16) oo n x2" oo x2"<br />
49. cosx = .~o ( -1) ( 2 n)! => cosx - 1 = .~<br />
1 ( - 1)" { 2 n)! . 1 oo 2n- 1<br />
cos x - = 2: {-1)" .::_____ =><br />
X n=l (2n)!<br />
I<br />
COS X - 1 oo n X 2n •<br />
dx = C + 2:::{-1) 2<br />
· ( 2<br />
) 1 , wtth R = oo.<br />
X n =l n n .<br />
@ 20 12 Ccngagc Lcunling. All Rights Ri:st:rvc:J. May not be scwmct.l. cupicd, or duplicat<strong>ed</strong>, or pO!'Itr.:d to a publicly IU: cc~ s i blc website, in whole or in pan.
88 0 CHAPTER 11 INFINITE SEQUENCES AND SERIES<br />
• oo x2n+l 00 x2n+ 4<br />
51 . arctanx = 2: (- 1)" - 2 1<br />
for lxl < 1, so x 3 arctanx = 2: ( -1)"- - for Jxl < 1 and<br />
n = O n + n =O 2n+ 1<br />
I<br />
oo<br />
x2n+5<br />
x 3 arctanx dx = C + ,..;;:o(-1)"' ( 2<br />
n + 1<br />
)( 2<br />
n + )' 5<br />
Since ~ < 1, we have<br />
(1/2) 5 (1/2r (1/ 2) 9 · · {1/ 2) 11<br />
1.5 - ~ + ~ ~ 0.0059 and subtracting Til ~ 6.3 x 10- 6 does not affect the fourth decimal place,<br />
so J112 0 x<br />
3 arctan x dx ~ 0.0059 by the Alternating Series Estimation Theorem.'<br />
(1/ 2) I oo ( 1/ 2) x4n+1<br />
n =O n , n =O n 4n + .1<br />
53. v'1 + x4 = (1 +.x 4 ) 1 1 2 00<br />
= 2: (x 4 )", so }1 + x 4 dx = C + L: --and hence, since 0.4 < 1,<br />
we have<br />
1<br />
I= } 1 +x 4 00<br />
0.4<br />
(1/ 2) (0.4) 4n+l<br />
dx = E<br />
o n = O n 4n + 1<br />
(o.4) 1 ~ (o.4) 5 H-4) (o.4) 9 H-t)(-~) (o.4) 13 H-4)(-~)(-~) (o.4) 17<br />
=(1)or+l!-5-+-2-,---9-+ 3! ~ + 4! ---rr- +· ··<br />
(0.4) 5 (0.4) 0 (0.4) 13 5(0.4) 17<br />
= 0.4 + ----w- - ----;:;2 + 208 - 2176 + ...<br />
Now (O~~)o ~ 3.6 x 10- 6 < 5 x 10- 6 , so by the Alternating Series Esti~ation Theorem, I ~ 0.4 + (0~~) 5 ~ 0.40102<br />
(correct to five decimal places).<br />
x -ln(1 + x) x- (x - l x 2 + l x 3 - lx 4 + lx 5 - • • ·) lx 2 - lx 3 + l x 4 - l x 5 + · · ·<br />
55. lim = lim 2 3 ., 4 fi = lim 2 :l 4 5<br />
x-0 x2 x-.0 x· .,_,o X 2<br />
= lim(! - l x + lx 2 - ix3 +: .. ) = !<br />
m-+0 2 3 4 5 2<br />
since power series are continuous functions.<br />
sinx- x + l x 3<br />
6<br />
57. lim<br />
x-o x5<br />
since power series are continuous functions.<br />
. 2 ~ ~ ~ ~ ~<br />
_59. From Equation 1 I, we have e-x = 1 - I + - 21<br />
- I + · · · and we know that cos x . = 1 - - 21<br />
+ - 41<br />
- · · · from<br />
1. . 3. . .<br />
Equation 16. Therefore, e-"' 2 cosx = (1 - x 2 + ~x 4 - ·- ·) (1 - ~x 2 + i4x 4 - - - • ). Writing only the tenns with -<br />
degree < 4 we get e-"' 2 cos x = 1 - lx 2 + .l.x 4 - x 2 + lx" + lx 4 + . ·. = 1 - !!.x 2 + ~x 4 +- ·.<br />
- ' 2 24 2 2 2 24 .<br />
© 2012 Cc.ngace Learning. All Rights Rcscn·cd. May not be SCOJ:mcd, oo picd~ or duplicat<strong>ed</strong>, or po.stcd too publiCly accessible website, in whole or in part.
61 _ x_ x<br />
· sin x - x- tx 3 + 1~ 0 x5 - · · · ·<br />
x- *x3 + l ~o xs - .. . j x<br />
SECTION 11.10 TAYLOR AND MACLAURIN SERIES 0 89<br />
1 I 2 7 4<br />
+ 6x +aGo X + · · ·<br />
x - ~ x3 + rlo xs - .. .<br />
l x3 _ ....Lxs + .. .<br />
6 120<br />
1 3 1 5<br />
oX - 36X + .. .<br />
From the long division above, ~ = 1 + ~x 2 + 3 ~x4 + · · ·.<br />
· I SlllX ·<br />
I<br />
oo 4n oo ( - x ~~) n a<br />
63. L; (- 1) "~ = L; - - 1<br />
- = e- x· , by (I I).<br />
n=O n. n =O n.<br />
3<br />
65. ~(-1) " - J nn = ~(- 1 )" _ 1 ( 3 / 5 )" = ln( 1 + ~) [fromTable l ] = ln~<br />
n=l n5 n = 1 n 5<br />
5<br />
n. (i ) ( )<br />
71. lfp is an n th-degree polynomial, then p
90 0 CHAPTER 11 INFINITE SEQUENCES AND SERIES<br />
00<br />
75. (a) g(x) = :L; 00 (k) x"' => g'(x) = :L; (k) nx"- 1 , so<br />
n~ n n~ n<br />
00<br />
00<br />
(1 + x)g'(x) = (1 + x) :L; 00 (k) nxn- (k) (k)<br />
1 = :L; nx"'- 1 + :L; nxn<br />
n=1 n n=1 n n = 1 n<br />
. .<br />
I<br />
Replace n with n<br />
= f: ( k ) (n 1<br />
+ 1)xn + f: (k)nxn<br />
+ 1 ]<br />
n=O n+ n = O n [ in the first series<br />
00 00<br />
_ k(k-1)(k-2) .. ·(k-n+1)(k-n) n<br />
[ k(k- 1)(k - 2)···(k-n+ 1)] ,,<br />
- n~o (n + 1 ) (n + 1)! x + n~O (n) nl x<br />
00<br />
(n + l )k(k - '1)(k - 2) ... (k- n + 1) n<br />
= n~O (n+ 1)! . ((k -n)_+n]X<br />
= k n~o k(k - 1)(k- 2l! ... (k- n + 1) x" = k fo ( ~) x" = kg(x)<br />
Thus, g'(x) = k<br />
1<br />
g(x).<br />
+x<br />
(b) h(x) = (1 + x)-k g(x) =><br />
h'(x) = - k(l + x)- k- 1 g{x) + (1 + x)-k g'(x)<br />
= -k(l + x)-i.:- 1 g(x) + (1 + x)- k kg(x)<br />
1+x<br />
= -k(l + x)-k- 1 g(x) + k{1 + x)-"'- 1 g(x) = 0<br />
[Product Rule]<br />
[from part (a)]<br />
(c) From part (b) we see that h(x) must be constant for x E ( -1, 1), so h(x) = h(O) = 1 for x E ( -1, 1).<br />
Thus, h{x) = 1 = (1 + x)-k g{x) g(x) = {1 + x)k for x E ( -1, 1).<br />
11.11 Applications of Taylor Polynomials<br />
1. (a)<br />
J (x) t
SECTION 11.11 APPLICATIONS OF TAYLOR POLYNOMIALS D 91<br />
I<br />
(b)<br />
X f To =T1 T2 = Ts T4 =Ta Te<br />
1r<br />
4<br />
7f<br />
2<br />
0.7071 1 0.6916 0.7074 0.7071<br />
0 1 - 0.2337 0.0200 - 0.0009<br />
1T - 1 1 -3.9348 0.1239 -1.2114<br />
3.<br />
(c) As n increases, Tn ( x ) is a good approximation to f ( x) on a larger and larger interval.<br />
n f(nl(x) j(n)(2)<br />
0 1/x<br />
1<br />
2<br />
1 - 1/ x 2 1<br />
- 4<br />
2 2/x 3 1<br />
4<br />
3 · - 6/x 4 3<br />
- 8<br />
3 J (n)(2)<br />
Ts(x) = 2::::: -- 1<br />
- (x- 2)"<br />
·n=O n.<br />
1 1 1 3<br />
= 2 - l (x - 2) + 4 (x- 2? - j_ (x- 2?<br />
0! 1! 2! 3!<br />
= t- H x - 2) + ~(x - 2) 2 - fa(x- 2) 3<br />
5.<br />
n J
92 0 CHAPTER 11 INFINITE SEQUENCES AND SERIES<br />
9.<br />
n J
SECTION 11.11 APPLICATIONS OF TAYLOR POLYNOMIALS 0 93<br />
15.<br />
n f (n) (x) J (n){1)<br />
0 x2/3 1<br />
1 1x- l /3 2<br />
3 3<br />
2 _1x- 4/3 2<br />
9<br />
- g-<br />
3 .§..x- 7/3 8<br />
27 27<br />
4<br />
- ~ x- 10 / 3<br />
81<br />
(c) 0.00006<br />
. 2/9 8/27<br />
(a) f(x) = x 2 1 3 ~ Ta(x) = 1 + ~(x- 1) - 2!(x - 1) 2 + 3!(x - 1) 3<br />
= 1 + ~(x- 1) - · ~(x-I?+ 8<br />
4 1 (x - 1) 3<br />
(b) IR3(x)l5 ~ lx - 11 4 , where J j < 4 >(~)~5 M. Now 0.8 5 x 5 1.2 =?<br />
lx- l l 5 0.2 ::}<br />
lx - 11 4 5 0.0016. Since jJ < 4 >(x) l is decreasing<br />
on [0.8, 1.2], we can take M =I j < 4 >(o.s)l = ~(0 . 8) - 10 1 3 , so<br />
IR3(x)l 5<br />
~(0 8)-10/3<br />
81<br />
' (0.0016) ~ 0.000096 97.<br />
24<br />
From the graph of IRa(x)l = Jx 2 / 3 -<br />
Ta(x) I• it seems that the<br />
error is less than 0.000 053 3 on [0.8, 1.2).<br />
17.<br />
n<br />
f(n) (x)<br />
J (n){O)<br />
(a) f(x) = secx ~ T2(x) = 1 + ~x 2<br />
0 secx<br />
1<br />
1 secx tanx<br />
2 secx (2sec 2 x- 1)<br />
0<br />
1<br />
3 sec x tan x ( 6 sec 2 x - 1)<br />
(b) IR2(x)l 5 ~ lx l 3 , where I t < 3 > (x) l 5 M . Now - 0.2 5 x 5 ~ . 2 =:> lx l 5 0.2 =? lxl 3 5 (0.2) 3 .<br />
f(3) (x) is an odd function and it is increasing on (0, 0.2) since secx and tanx are i"ncreasing on [0, 0.2) ,<br />
so I J C3>(x) l 5 J < 3 >(o.2) ~ 1.085 158892. Thus, IR2(x)l 5 J < 3 >i~· 2 ) (0.2) 3 ~ 0.001447.<br />
(c)<br />
0.0004<br />
· From the graph ofiR2(x)l = isecx - T2(x)l, it seems that the<br />
error is less than 0.000 339 on [-0.2, 0.2).<br />
19.<br />
n·<br />
0<br />
J(x)J 5 M. Now 0 5 x $ 0.1 =?<br />
x 4 5 (0.1) 4 , and letting x = 0.1 gives<br />
3<br />
ex 2 (12x + 8x 3 )<br />
0<br />
IR3(x)l $ eo.Dl (12 + ~!8 + 0.0016) (0.1)4 ~ 0.00006.<br />
4<br />
e"' 2 (12 + 48x 2 + 16x 4 )<br />
® 2012 Ccngngc Lc:uning. AU Ri~tsRc.scn·cd. M.D.)' not be .scrutncd. cori<strong>ed</strong>, orduplicattd. or post<strong>ed</strong> to a publicly acc
94 D . CHAPTER 11 INFINITE SEQUENCES AND SERIES<br />
(c) 0.00008<br />
~------------~<br />
y = IR,(x)l<br />
From the graph of jR3(x)l = le~ 2<br />
error is less than 0.000 051 on [0, 0.1].<br />
- T3(x) l, it appears that the<br />
21.<br />
n J
SECTION 11.11 APPLICATIONS OF TAYLOR POLYNOMIALS D 95 ·<br />
1 3 1 s Bth I · s ·<br />
27. sinx = x- 3!x +Six - · · ·. y e A ternatmg enes<br />
Estimation Theorem, the error in the approximation<br />
sin x = x - ~x 3 is less than I ;,x 5 1 < 0.01 ¢:}<br />
lx 5 l < 120(0.01) ¢:} lxl < (1.2) 115 ~ 1.037. The curves<br />
y = x - ix 3 andy= sinx- 0.01 intersect at x ~ 1.043, so<br />
the graph confirms our estimate. Since both the sine function<br />
0.9<br />
r-------------~~~~--~<br />
y=sinx+ o.oV /<br />
/ / y=x-kx 3<br />
/ ~ 'fooox-O.QJ<br />
0.9 "-L,__L,.:...__ _.__________./ 1.2<br />
0.8<br />
and the given approximation are odd functions, we ne<strong>ed</strong> to check the esti~ate only for x > 0. Thus, the desir<strong>ed</strong> range of<br />
. values for xis - 1.037 < x < 1.037.<br />
x 3 x 5 x 7<br />
29. arctan x = x - 3<br />
+<br />
5 - 7<br />
+ · · · . By the Alternating Series<br />
Estimation Theorem, the error is less than l -~ x 7 1 < 0.05 ¢:}<br />
y =arctan x + 0.05<br />
jx 7 j < 0.35 ¢:} lxl < (0.35? 17 ~ 0.8607. The curves<br />
y = x- tx 3 + kx 5 andy= arctanx + 0.05 intersect at<br />
x ~ 0.9245, so the graph confirms our estimate. Since both the<br />
arctangent function and the given approximation are odd functions,<br />
we ne<strong>ed</strong> to check the estimate only for x > 0. Thus, the desir<strong>ed</strong><br />
range of values for xis - 0.86 < x < 0.86.<br />
- 1<br />
y = x - .!. x 3 + .!. xs<br />
3 5<br />
31. Let s(t) be the position function of the car, and for convenience set s(O) = 0. The velocity of the car is v(t) = s' (t) and th~<br />
acceleration is a(t) = s" (t), so the second degree Taylor Polynomial is T2(t) = s(O) + v(O)t + a~O) t 2 = 20t + t 2 . We<br />
estimate the distance travel<strong>ed</strong> during the next second to be s(1) ~ T2(1) = 20 + 1 =21m. The function T 2 (t) would not be<br />
accurate over a full minute, since the car could not possibly maintain an acceleration of 2 m/s 2 for that long (if it did, its final<br />
spe<strong>ed</strong> would be 140 m/ s ~ 313 mi/ hl).<br />
33. E = ~2 - (D!d)2 = ~2- D2(1:d/ D)2 = ; 2 [1 - (1+ ~r2l<br />
We use the Binomial Series to expand (1 + d/ D ) - 2 :.<br />
when D is much larger than d; that is, when P is far away from the dipole.<br />
35. (a) If the water is deep, then 27rd/ L is large, and we know that tanh x -+ 1 as x -+ oo. So we can approximate<br />
tanh(27rd/ L) ~ 1, andsov 2 ~ f!L/ (27r) ¢:} v ~ ..jgLj (21r).<br />
© 2012 Cengage Learning. All Rights Reserve-d. May not be SC41lncd, copi<strong>ed</strong>, or duplical<strong>ed</strong>, or poSi<strong>ed</strong> 10 a publicly accessible website. in whole o r in part.
96 D CHAPTER 11 INFINITE SEQUENCES AND SERIES<br />
(b) From the table, the first term in the Maclaurin series of<br />
tanhx is x, so ifthe water is shallow, we can approximate<br />
n<br />
0<br />
1<br />
j (n) (x)<br />
tanhx<br />
sech 2 x<br />
f (nl(o)<br />
0<br />
1<br />
2<br />
-2sech 2 xtanhx<br />
0<br />
3<br />
2sech 2 x (3tanh 2 x- 1)<br />
- 2<br />
(c) Since tanh x is an odd function, its Maclaurin series is alternating, so the error in the approximation<br />
2trd 2trd. . . IJ"'(O)I (2trd) 3 1 (2trd) 3<br />
tanh L ~ L IS less than the first neglect<strong>ed</strong> term, wh1ch IS --<br />
3<br />
,- L =<br />
3 L .<br />
If L > lOd, then ~ ( 2 ~d) 3 < ~ ( 2tr · 1<br />
1<br />
0 ) 3<br />
= ;<br />
3<br />
5 , so the error in the approximation v 2 = gd is less<br />
. .<br />
gL tr3<br />
than 7r • ~ 0.0132gL.<br />
2 375<br />
37. (a) Lis the length of the arc subtend<strong>ed</strong> by the angle B, soL = RB =><br />
B=L/R.NowsecB=(R+C)/R. => RsecB=R+C =><br />
C = RsecB- R = Rsec(L/R)- R.<br />
(b) First we'll find a Taylor polynomial n(x) for f(x) = secx at X = 0.<br />
n<br />
/(nl(x)<br />
0 secx<br />
1 secx tanx<br />
2 secx(2tan 2 x + 1)<br />
3 secxtanx(6tan 2 x + 5)<br />
4 secx(24tan 4 x + 28tan 2 x + 5)<br />
/(n) (0)<br />
1<br />
0<br />
1<br />
0<br />
5<br />
Thus, f(x) = secx ~ T4(x) = 1 + ~(x- o? + -fi(x-W = 1 + ~x 2<br />
+ f4x 4 • By part (a),<br />
(c) Taking L = 100 km and R = 6370 km, the formula in part (a) says that<br />
C = Rsec(L/ R)- R = 6370 sec(100/6370)- 6370 ~ 0.785 009 96544 km.<br />
. . £ 2 5£ 4 100 2 5. 100 4 .<br />
The formula m part (b) says that C ~ 2<br />
R + 24<br />
R 3<br />
= 2<br />
. 6370<br />
+ 24<br />
. 63703<br />
~ 0.785 009 957 36 km.<br />
The difference between these two results is only 0.000 000 008 08 km, or 0.000 008 08 m!<br />
39. Using f(x) = Tn(x) + Rn(x) with n = 1 and x = r, we have J(r) = T1(r) + R 1(r), where T1 is the first-degree Taylor ,<br />
polynomial off at a. Because a= Xn, J(r) = f(xn) + f'(xn)(r- x,.. ) + R1(r). But r is a root off, so f(r) = 0<br />
and we have 0 = f ( Xn) + J' ( Xn) ( r -<br />
Xn) + R1 ( r). Taking the first two terms to the left side gives us<br />
© 2012 Ccngagc Learning. AU RighlS Rcscr.·cd. May not be scann<strong>ed</strong>, copi<strong>ed</strong>. or duplicat<strong>ed</strong>, or poslci.lto u publicly accessible website, in whole or in pan.
CHAPTER 11 REVIEW 0 97<br />
J'(xn)(xn - r) - f (xn) = R 1 (r). Dividing by f'(xn), we get Xn - r - ;,~:~) = :(~::).By the formula for Newton's<br />
method, the left side of the prec<strong>ed</strong>ing equation is Xn+I - r, so lxn+l - r l = I ;(~:}) I· Taylor's Inequality gives us<br />
IR 1 (r)l ::=; I J'~~r)llr - xnl 2 • Comb i nin~ this inequality with the facts lf"(x)l ::=; M and IJ'(x)l ~/(gives us<br />
lxn+l - rl ::=;<br />
M \2<br />
/( \xn - r ·<br />
2<br />
11 Review<br />
CONCEPT CHECK<br />
1. (a) See Definition 11.1.1.<br />
(b) See Definition 11.2.2.<br />
(c) The terms ofthe sequence {an} approach 3 as n becomes large.<br />
(d) By adding sufficiently many terms of the series, we can make the partia l sums as close to 3 as we like.<br />
2. (a) Sec the definition on page 721 [ET page 697].<br />
(b) A sequence is monotonic if it is either increasing or decreasing.<br />
(c) By Theorem 11.1.1 2, every bound<strong>ed</strong>, monotonic sequence is convergent.<br />
3. (a) See (4) in Section 11.2.<br />
(b) The p-se~ies f: J.. is converge~ ! ifp > 1.<br />
n=l 77,1'<br />
4. If 2: a ,. = 3, then lim an = 0 and lim S n = 3.<br />
n - oo<br />
n ---t ex><br />
5. (a) Test for Divergence: If lim a,, does not exist or if lim an =I 0, then the series 2:"':.. 1<br />
an is divergent.<br />
n---+oo n --+ oo n-<br />
(b) Imegral Test: Suppose f is a continuous, positive, decreasing function on [1, oo) and let an= f(n). Then the series<br />
:2:;::'= 1<br />
an is convergent if and only if the improper integral f 1<br />
00<br />
(i) If f 1<br />
00<br />
f(x) dx is convergent, then 2:;::'= 1<br />
a, is convergent.<br />
(ii) If f 1<br />
00<br />
f(x) dx is divergent, then I::;"= 1 a,. is divergent.<br />
(c) Comparison Test: Suppose that E an and 2: b,. are series with positive terms.<br />
(i) If E b,. is convergent and an ::=; bn for all n, then E a,. is also convergent.<br />
(ii) If E br. is divergent and an ~ bn for a ll n, then 2: an is also divergent.<br />
f(x) dx is convergent. In other words:<br />
(d) Limit Comparison Test: Suppose that E a,. and E bn are series with positive tenns. If lim (a,/bn) = c, where c is a<br />
n.- oo<br />
finite number and c > 0, then either both series converge or both diverge.<br />
(e) Alternating Series Test: I fthe alternating series :2:;::'= 1 ( - 1 )n- 1 bn = b1 - b2 + b:3 - b4 + bo - ba + · · · [b,. > OJ<br />
satisfies (i) bn+l :::; bn for all n and (ii) lim b,. = 0, then the series is convergent.<br />
n-
98 0 CHAPTER 11 INFINITE SEQUENCES AND SERIES<br />
(f) Ratio Test:<br />
(i) If lim I an+l ) = L < 1, then the series f an is absolutely convergent (and therefore convergent).<br />
n-oo an no=l<br />
(. ") If 1" I an+1 I L 1 1" I O.n+l I h h . ~ . d"<br />
ll un - - = > or 1m -- = oo, t en t e senes L- a n IS Ivergent.<br />
n-oo an n--+oo an n=l<br />
(iii) If lim I an+l I = 1, the Ratio Test is inconclusive; that is, no c~nclusion can be drawn about the convergence or<br />
n-+co an<br />
(g) Root Test:<br />
divergence of 2:: an.<br />
(i) If lim \li(i:j = L < 1, then the series 2::;:'= 1<br />
a,. is absolutely convergent (and therefore ~nvergent).<br />
n-co<br />
(ii) If lim \li(i:j = L > 1 ~r<br />
n-.oo<br />
(iii) If lim<br />
11.--+00<br />
lim ~ = oo, then the series 2::;:'= 1<br />
an is divergent.<br />
n--+oo<br />
~ = 1, the Root Test is inconclusive.<br />
6. (a) A series L an is call<strong>ed</strong>ab~olutely convergent if the series of absolute values 2:: I ani is convergent.<br />
(b) If a series L an is absolutely convergent, then it is convergent.<br />
(c) A series L:al' is call<strong>ed</strong> conditionally convergent if it is convergent but not absolutely convergent.<br />
7. (a) Use (3) in Section 11.3.<br />
(b) See Example 5 in Section 11.4.<br />
(c) By adding terms until you reach the desir<strong>ed</strong> accuracy given by the Alternating Series Estimation Theorem.<br />
00<br />
8. (a) 2:: en(x - a)"<br />
n=O<br />
00<br />
(b) Given the power series L en(x- a)", the radius of convergence is:<br />
n=O<br />
(i) 0 if the series converges only when x = a<br />
(ii) oo if the series converges (or all x, or<br />
(iii) a positive number R such that the series converges if lx - al < Rand diverges if lx- al > R.<br />
(c) The interval of convergence of a power series is the interval that consists of all values of x for which the series converges.<br />
Corresponding to the cases in part (b), the interval of convergence is: (i) the single point {a}, (ii) all real numbers, that is,<br />
the reaL number line ( -oo, oo ), or (iii) an interval with endpoints a - R and a + R which can contain neither, either, or<br />
both of the endpoints. In this case, we must test the series for ·convergence at each endpoint to determine the interval of<br />
convergence.<br />
9. (a), (b) See Theorem 11 .9.2.<br />
n<br />
f(i)(a)<br />
10. (a) Tn(x) = L - .- 1<br />
- (x - a)'<br />
i =O ~.<br />
oo j (n){ )<br />
(b) I: __ a_ (x - a)"<br />
n=O n l<br />
® 2012 Cengnge Learning. All Rights Reserv<strong>ed</strong>. May not be scann<strong>ed</strong>, copi<strong>ed</strong>, or duplicat<strong>ed</strong>, or post<strong>ed</strong> ton publicly ncccssiblc website, in wbole or in part.
CHAPTER 11 REVIEW 0 99<br />
oo f (n) (O)<br />
xn [a= 0 in part (b)]<br />
(c) L -- 1<br />
-<br />
n=O n .<br />
(d) See Theorem 11.10.8.<br />
(e) See Tay.lor's Inequality ( 11.10.9).<br />
11. (a)-( f) See Table I on page 786 [ ET 762].<br />
12. See the binomial series ( 11 .10.17) for the expansion. The radius of convergence for the binomial series is 1.<br />
TRUE-FALSE QUIZ<br />
1. False. See Note 2 after Theorem I 1.2.6.<br />
3. True. If lim an = L, then as n -7+ 00, 2n + 1 -+ oo, so a2n + l -+ L.<br />
n - oo<br />
5. False. For example, take en = ( - 1)" j (n6n).<br />
7. False, smce . hm . I _+_ an 1 I = lim . I 1 · n3 ~ 1 = lun .· I n3 . · -1/n31 - = hm . 1 = 1.<br />
n-oo a,. n - oo (n + 1) 3 1 n--+oo (n + 1) 3 1j n 3 n -oo (1 + 1/ n) 3<br />
9. False. See the note after Example 2 in Section 11.4.<br />
11. True. See (9) in Section II. I .<br />
13. True.<br />
111<br />
/ (0) 1<br />
By Theorem 11.10.5 the coefficient of x 3 is ~ = 3<br />
=> !'"(0) = 2.<br />
Or: Use Theorem 11 .9.2 to differentiate f three times.<br />
15. False. For example, let a.. = bn = ( -1 r. Then {an} and { bn} are divergent, but a,.b,. = 1, so {a,. b .. } is convergent.<br />
17. True by Theorem 11.6.3. [2:: ( -1)" a,. is absolutely convergent and hence convergent.]<br />
19. True. 0.99999 . . . = 0.9 + 0.9(0.1) 1 + 0.9(0.1? + 0.9(0.1) 3 + · · · = f: (0.9)(0.1)n-l = __2.:Q_ = 1 by the formula<br />
n=l 1 - 0.1<br />
for the sum of a geometric series [S = a 1 / (1 -;- r)] with ratio r satisfying lrl < 1.<br />
21 . True. A finite number of terms doesn't affect convergence or divergence of a series.<br />
EXERCISES<br />
1.<br />
{ 2 + n 3 } . . 2 + n 3 · . 2/ n 3 + 1 1<br />
1 2 3 converges smce lim<br />
+ n = 1 2 3 lim 11 3 2 = - 2 .<br />
n --+00 + n n--+00 n +<br />
3<br />
3. lim a,. = lim ~ = lim / ~ = oo, so the sequence diverges.<br />
n--+OC n-+oo 1 + n n-+oo 1 n + 1 .<br />
© 2012 C
100 0 CHAPTER 11 INFINITE SEQUENCES AND SERIES<br />
l nsinnl n 1 · .<br />
5. !ani = - 2<br />
- -<br />
1<br />
::; ~ 1<br />
{ (<br />
< - ,so Jan!-> 0 as n-> oo. Thus, lim a,.= 0. The sequence.{an} is convergent.<br />
n + n + n n --+oo •<br />
3)4"} ( 3)4:t<br />
7. 1 + ~ is convergent. Let y ~ 1 +; . Then<br />
lim lny = lim 4xln(1 + 3/x) = lim ln( 1 + 3 /x) ~ lim ~ ( - ~)<br />
:r:-oo :z:--+oo x --+ oo 1/ (4x} x--+oo - 1/ (4x2)<br />
limy= lim<br />
( 1+ - 3)"" =e 12 .<br />
.:c-oo n---.oo n<br />
I . 12 2<br />
liD = 1 ,so<br />
x - oo 1 + 3/x<br />
9. We use induction, hypothesizing that an- 1 < an < 2. Note first that 1 < a2 = t (1 + 4) = 1 < 2, so the hypothesis holds<br />
fo r n = 2. Now assume that ak-1 < ak < 2. Then ak = ~(ak - 1 + 4) < ~(a~.:+ 4) < ~(2 + 4) = 2. Soak < ak+l < 2,<br />
and the induction is complete. To find the limit of the sequence, we note that L = lim an = lim an+l =><br />
'ft.-oo n-oo<br />
L = HL + 4) => L = 2.<br />
n n 1<br />
00<br />
n · ·<br />
00<br />
1<br />
11. --g--- 1<br />
< 3<br />
= 2 , so L; --g--- 1<br />
converges by the Comparison Test with the convergent p-series L; 2<br />
[ p = 2 > 1].<br />
n + n n n=l n + n=l n<br />
13 ). lan+ll 1' [ (n+1)3 5n ] l' (1 1)3 1 1 1 ~ na .b h R . .,.,<br />
. 1m -- = 1m<br />
Tl.___.OO an n - 00 n n n-oo n 5 5 n=l 511.<br />
5 + 1 · 3 = rm +- ·- =- < , so LJ - converges y t e at1o ,est. ·<br />
15. Let f ( x) = ~. Then f is continuous, positive, and decreasing on [2, oo), so the Integral Test applies.<br />
xvlnx<br />
00<br />
1<br />
1t<br />
1 !.In L !n t<br />
f (x)dx= lim ~ dx [u = ln x. du=_: 0, {bn} is decreasing, and lim bn = 0, so the series 1<br />
E (- 1)n- 1 Vn converges by the Alternating<br />
n + n--+oo n=l n + 1<br />
Series Test.<br />
@ 2012 O:ng•gc U:;uning. All RighlS Reserv<strong>ed</strong>. Moy not be seaM<strong>ed</strong>, copi<strong>ed</strong>, or duplicot<strong>ed</strong>, or post<strong>ed</strong> to D publicly occcssiblc website, in whole or in p;u1.
CHAPTER 11 REVIEW 0 101 l<br />
23. Consider the series of absolute values: E n -I/ 3 is a p-series with p = t ~ 1 and is therefore divergent. But if we apply the<br />
1'1=1<br />
Alternating Series Test, we see that bn = 3~ > 0, {bn} is decreasing, and lim bn = 0, so the series E (- 1)"- 1 n - 1 / 3<br />
yn n-Hxl n=l<br />
converges. Thus, E (-1)"- 1 n- 1 1 3 is conditionally convergent.<br />
tt= l<br />
an+ll ~ ( - 1t+ 1 (n+2) 3n+l . 2 2 "+1 l - n+2 3 _ 1+(2/n) 3 3 .<br />
/ n) · 4 --+ 4 < 1. asn--+ oo, so by the Ratro<br />
l<br />
25. ~ ·~ 22n+3 · (- 1 )"(n + 1 ) 3 " .- n+ 1 · 4 - 1 + ( 1<br />
oo ( -1)"(n + 1)3"<br />
Test, I:<br />
2 "+ 1 is absolutely convergent.<br />
n=l 2<br />
oo (- 3)n- 1 oo (- 3)n-l oo (- 3)"-1 1 oo (- 3)n- l 1 oo ( 3 ) n-l 1 ( · 1 )<br />
27· I: 23n = I: (23)n = L 8" = -8 I: 8"- 1 = -8 L - -8 = -8 1 ( 3/8)<br />
n=I n=l n=l n=l n=l - -<br />
1 8 1<br />
=s· u = u<br />
00<br />
29. :L[tan- 1 (n+ 1)-tan- 1 n) = lim Bn<br />
n = l<br />
n-oo<br />
31. 1 - e + - 21<br />
. -<br />
= lim [(tan- 1 2- tan- 1 1) + (tan- 1 3- tan- 1 2) + · · · + (tan- 1 (n + 1)- tan- 1 n)J<br />
n-oo<br />
= lim [tan- 1 (n + 1)- tan- 1 1) = 1!. - 1!. = 1!.<br />
n-oo 2 4 4<br />
e2 e3 e4 oo e" oo ( -e)n oo x"<br />
31 . + - 41 . - .. · = L (-1)"1 = E -- 1- = e- e since e"' = I: I for all x.<br />
1 · 1 ( oo x" oo (-x)" )<br />
33. cosh x = -(e"' +e-"') = - I: 1 + L -- 1<br />
-<br />
2 2 n=O n. n=O n.<br />
n=O n. n=O n. . n=O n .<br />
oo ( -1) n+l · 1 1 1 1 1 1 1<br />
' 35· n~l n5 = 1 - 32 + 243 - 1024 + 3125 - 7776 + 16,807- 32,768 + ... ·<br />
. 1 1 oo (-1)"+1 7 (- 1)"+1<br />
~ E 5 ~ 0.9721.<br />
' n=l n n=l n<br />
Srnce ba = 85<br />
= 32 768<br />
< 0.000031, I:<br />
5<br />
00<br />
1 8 1 1 1<br />
37. I: - 2 5<br />
" ~ E - 2 5<br />
" ~ 0.18976224. To estimate the error, note that - 2<br />
- ·- < -,so the remainder term is<br />
n=l + n=l + + 5" 5n .<br />
00<br />
1<br />
00<br />
1 1/5 9<br />
Ra = n~o<br />
2 + 5 " < n~o S" = 1 _ 115 = 6.4 ~ 10,- 7 [geometric series with a= -b and r = i ].<br />
© 20 I 2 Ceng~ge lc::tming. All Rights Resc:r•cd. May not be sc:umcd, copi<strong>ed</strong>, or duplicat<strong>ed</strong>. or post<strong>ed</strong> io n publicly ncccs.siblc website, in ,whole or in part.
102 0 ~HAPTER 11 INFINITE SEQUENCES AND SERIES<br />
39. Use the Limi~ Comparison Test. lim I(~ )a,, I = lim n + 1 = Jim (1 + .!) = 1 > 0.<br />
n--+00 an n - oo n n-+oo n<br />
Since E Ia,. I is convergent, so is I: \ ( n : 1 ) a,. I, by the Limit Comparison Test.<br />
41. lim I a,.+ll = lim [ lx + 21"+1 . n4" ,.] = lim [__2!:___ lx + 2[] = [x + 21 < 1 ¢:} lx + 21 < 4, so R = 4.<br />
n - oo an !'-oo (n + 1) 4"+ 1 lx + 21 n-oo n + 1 4 4<br />
oo (x + 2)n<br />
lx + 21 < 4 ¢:? -4 < x + 2 < 4 ¢:? - 6 < x < 2. lfx = - 6, then the series E becomes<br />
. • n =l n 4"<br />
f: (- 4<br />
4r = f: (- l )", the alternating harmonic series, which converges by the Alternating Series Test. When x = 2, the<br />
n=l 71. n = l n<br />
series becomes the harmonic series f: .!, which diverges. Thus, I= [- 6, 2).<br />
n=l n<br />
43. lim I Un+l l = lim 1 2 n+ 1 (x- 3 )"+1 · y'n+3 ,., = 2 jx- 31 lim Jn + 3 = 2lx- 31 < 1 ¢:} jx- 31 < ~.<br />
n-oo a,. n-oo ../n + 4 2"(x - 3) n-oo n + 4<br />
soR = ~- lx- 31 < ~ ¢:?<br />
· · oo 2"{x - 3}"<br />
-~ < x- 3 < ~ ¢:} ~ < x < ~ · For x =~. the series I: .;n+3 becomes<br />
· n =l n +3<br />
45.<br />
I: ~ = I: ;/ 2<br />
, which diverges [P = ~ $ 1), but for X = ~' we get I: ~ .which is a convergent<br />
n=O y n + 3 n=3 n n=O v n + 3<br />
alternating series, so I = [ ~ , ~).<br />
n /("l(x) / (n)(:g)<br />
1<br />
0 sinx 2<br />
1 cosx .il<br />
2 - sin x<br />
3 - cosx<br />
4 sin x<br />
•.<br />
2<br />
1<br />
- 2<br />
_ _il<br />
2<br />
1<br />
2<br />
"C) {3) (7r) {4}C)<br />
sinx = f(~)+f'(~)(x-~)+!.____i_(x-~) 2 + / 6 (x-~) 3 +/ 6 (x - ~) 4 + ...<br />
6 6 . 6 2! 6 3! 6 4! 6<br />
47. - 1<br />
+-<br />
1 -X = 1 _ ~- X) = n~O (- x)" = n~O {-1)" x" for lxl < 1<br />
2 00<br />
=> ~ =I: {-1f'x"+ 2 withR= 1.<br />
1+x n = O<br />
© 2012 Ccngnge Learning. All RighiS R=n-.:d. May not be S
49./- 1 - dx = -ln{4- x) + C and<br />
4-x<br />
I I I 1 1 1 1 00<br />
(X)" 1/ oo x" 1 oo xn+1<br />
4 - X . 4 1 - x/4 4 n=O 4 4 n=O 4" . 4 n=O 4"(n + 1} ·<br />
--dx = - - - dx = - L: - dx = - E - dx = - E + C So<br />
CHAPTER 11 REVIEW D 103<br />
1 00 xn+1 00 xn+1 00 x" .<br />
ln{4 - x)=-- 4<br />
E ,.( 4 1 ) +C= - E 4 ,.+ 1( + ) +C = - 1<br />
E n +C.Puttmg x=O,wegetC = ln4.<br />
n=O n+ n=O n n=1 n 4<br />
oo x"<br />
. Thus, f(x) = ln{4- x) = ln4- E<br />
rt=1 n 4<br />
Another solution:<br />
" . The series converges for Jx/ 41 < 1
104 0 CHAPTER 11 INFINITE SEQUENCES AND SERIES<br />
(b) 1.5<br />
(c) IR3 (x) l $ ~~ ix - 11 4 , where J! ( 4 ) (x) J $ M with<br />
j ( 4 >(x) = - ~~x- 7 1 2 . Now 0.9 ~ x $ 1.1 =><br />
- 0.1 ~ X- 1 ~ 0.1 => (x- 1) 4 $ ·(0.1) 4 ,<br />
d I . 0 . . 15<br />
an etting x = .9 gtves M = ( _ )7 ,<br />
16 0 9 12<br />
so<br />
0<br />
2 IRa(x) l $<br />
16 (0. ~~ 712 41 (0.1)4 :::::: 0.000005648<br />
:::::: 0.000 006 = 6 X lO- G<br />
(d) 5 X 10- 6 From the graph of IR3(x)l = I JX - T3(x)l. it appears that<br />
the error is less than 5 x w- 6 on [0.9, 1.1].<br />
oo • x2n+l x3 x6 x7 . x3 xs x7<br />
59.sin x = 2:(- 1t ( 2 1<br />
) 1<br />
= x- - +---+· ··,sosmx - x=--+---+··· and<br />
n=O n + . 3! 5! 7! 3! 5! 7!<br />
00 00 00<br />
61. f(x ) = 2: Cn x" => f(-x)= l: cn(-xt = 2:(- 1)"c.,x"<br />
n=O n =O n=O<br />
(a) Iff is an odd function, then f( - x) = - f(x) =><br />
00 00<br />
2: (- 1)"cnx" = 2: - c,,x". The coefficients of any power series<br />
u=O<br />
are uniquely determin<strong>ed</strong> (by Theorem 11.10.5), so ( -1)" Cn = - en.<br />
If n is even, then ( - 1)" = 1, so en = -c,. => 2en = 0 => c,. = 0. Thus, all even coefficients are 0, that is,<br />
Co = C2 = C4 = · · · = 0.<br />
00 00<br />
(b) Iff is even, then f (-x) = f(x) => 2: {-lt c,. xn = 2: Cn x" => {-1)" Cn = Cn.<br />
n=O<br />
n = O<br />
If n is odd, then ( - 1 )" = -1, so -en = Cn => 2c,.. = 0 => Cn = 0. Thus, all odd coefficients are 0,<br />
that is, Cl = C3 = C5 = · · · = 0.<br />
n=O<br />
'<br />
@) 2011 Ccngoge Lcorning. All Rigbls Reserv<strong>ed</strong>. Moy not be scann<strong>ed</strong>. copi<strong>ed</strong>, or
D<br />
PROBLEMS PLUS<br />
1. It would be far too much work to compute 15 derivatives of f. The key idea is to remember that j 1, Pn -> oo as n-> oo.<br />
so= 3 eo = 1<br />
81 = 3 · 1 e1 = 1 / 3<br />
S2 = 3 · 4 2 e2 = 1/ 3 2<br />
S 3 = 3 · 4 3 e3 = 1/ 3 3<br />
(c) The area of each of the small triangles add<strong>ed</strong> at a given stage is one-ninth of the area of the triangle add<strong>ed</strong> at the prec<strong>ed</strong>ing<br />
stage. Let a. be the area of the original triangle. Then the area an of each of the small triangles add<strong>ed</strong> at stage n is<br />
e 2012 CcngUl_;C Lcamin&- All Rig)tts RL"SCr\'cd. Ma)' nol be scann<strong>ed</strong>. copi<strong>ed</strong>. or duplic31Cd. or pos1l-d to a publicly accessible websile, in whole or in part. 105
106 0 CHAPTER 11 PROBLEMS PLUS<br />
an = a · _.!:_ = .!::... Since a small triangle is add<strong>ed</strong> to each side at every stage, it follows that the total area An add<strong>ed</strong> to the<br />
9n 9n<br />
figure at the ntl1 stage is An = Sn-1 ·an = 3 · 4n- 1 · 9~,<br />
4n-1<br />
= a· 32 n_ 1<br />
. Then the total area enclos<strong>ed</strong> by the snowflake<br />
. 2 3<br />
curve is A = a + A1 + A2 + As + · · · = a + a · ~ + a · 3<br />
; + a · : 5<br />
+ a · : 7<br />
+ · · · . After the first term, this is a<br />
. . . h . 4 A a/3 a 9 8a B I f h . . I 'I I<br />
geometric senes w1t common ratlo !) , so. = a'+ 1<br />
_ .1 = a + 3 · 5<br />
= S. ut t 1e area o .t e ongma eqUI atera<br />
9 .<br />
. I . h 'd 1 . 1 1 . 7r J3 S h I d b h wfl k . 8 J3 2<br />
tnang e w1t s1 e IS a = '2 · · sm J3<br />
3 = 4 . o t e area enc ose y t e sno a e curve IS 5 · =<br />
4<br />
T .<br />
7. (a) Let a= arctanx and b =arctan y. Then, from Formula 14b in Appendix D,<br />
(b) tana-tanb tan(arctan x)-tan(arctany) x-y<br />
tan a - = 1 +tan a tan b = 1 + tan(arctanx) tan(arctan y) = 1 + xy<br />
. x-y<br />
Now arctan x - arctan y = a - b = arctan( tan( a - b)) = arctan - 1<br />
-- since --t < a - b < ¥.<br />
+xy<br />
(b) From part (a) we have<br />
120 1 28,561<br />
120 t 1 - t Wi - 239 t 28,441 t 1 7r<br />
arc t an 119<br />
- arc an 2311<br />
-.arc .an<br />
120 1<br />
= arc an 28<br />
_ 561<br />
= arc an = 4<br />
1 + Wi . 239 28,441<br />
(c) Replacing y by-yin the formula of part (a), we get arctanx +arctan y = arctan 1<br />
x + y . So<br />
-xy<br />
1 + 1<br />
4 arctan 1 2( 1 1) 2 5 5 2 t 5 t 5 t 5<br />
5 = arctan 5 + arctan 5 = arctan<br />
1 1<br />
- arc an 12 - arc an 12 + arc an 12<br />
1-5. 5<br />
..!!... +..!!...<br />
12 12<br />
= arctan<br />
5 5<br />
= arctan g~<br />
1 -12'12<br />
Thus, from part (b), we have 4 arctan t - arctan 2 i 9<br />
= arctan ~ i~ - arctan 2<br />
; 9<br />
= "i.<br />
x 3 · x 5 x 7 x 9 x 11<br />
(d) From Example 7 in Section 11.9 we have arctanx = x- 3 + 5 - 7+ 9 -~ +· ··,so<br />
11 1 1 1 1 1<br />
arctan 5 = 5 - 3 · 53 + 5 · 55 - 7 · 57 + 9 · 59 - 11 · 511 + .. .<br />
This is an alternating series and the size of the terms decreases to 0, so by the Alternating Series Estimation Theorem,<br />
the sum lies between ss and s6, that is, 0.197395560
CHAPTER 11 PROBLEMS PLUS D 107<br />
l<br />
9. We start with the geometric series f; x" = - 1 - , lxl < 1, and differentiate:<br />
n.=O 1- X<br />
}: nxn- l = -d }: xn = -d - 1<br />
_ = ( 1<br />
- ) 2<br />
for lxl < 1<br />
oo d(<br />
00<br />
) d(1) 1<br />
n=l X n=O X X X<br />
for lxl < 1. Differentiate again:<br />
oo oo X<br />
=> L nxn = X L nxn-l = ..,.---~<br />
n =1 n =l (1 - x)2<br />
00<br />
_ d x (1 - x? - x · 2(1 - x)( - 1) x + 1 ~ 2 ,.. x 2 + x<br />
}: n 2 x" 1 - - - = => L- n x = =><br />
n= 1 - dx (1 - x)2 - (1 - x) 4 (1 - x)3 n =l (1 - x)S<br />
f; n3xn- 1 = !!:.._ x 2 + x = (1 - x?(2x + 1) - (x 2 + x)3(1 - x) 2 ( - 1) = x 2 + 4x + 1 ==><br />
n=l dx (1 - x) 3 (1 - x ) 6 (1 - x)~<br />
f; n 3 x" = x3<br />
t<br />
n = l<br />
4 2<br />
x<br />
1- X<br />
)~ x , lx l < 1. The radius of convergence is 1 because that is the radius of convergence for the<br />
geometric series we start<strong>ed</strong> with. If x = ± 1, the series is }:n 3 (±1)n, which diverges by the Test For Divergence, so the<br />
interval of convergence is (-1, 1).<br />
11. ln(1- : 2<br />
) = ln( n~~ 1 ) ,;, In (~+ ll~n - 1 ) = ·in[(n +1)(n- 1)]- lnn 2<br />
= ln(n + 1) + ln(n - 1) - 2ln n = ln(n - 1) - Inn- ln n + In(n + 1)<br />
n- 1 n - 1 n<br />
= In - n - - [lnn - ln(n+ 1)] = In - n - -In n+ 1<br />
.<br />
k ( 1) k ( n-1 n )<br />
Let Sk = E In 1 - 2 = E In - - - In - - 1<br />
fork ~ 2. Then<br />
n = 2 n n = 2 n n+<br />
1 2) ( 2 3) ( k - 1 k ) 1 k<br />
. . ( 1 k ) 1<br />
Sk =<br />
(<br />
ln - - In - + In - - In - + · · · + In - - - In - - = In- - In - - so<br />
2 3 3 4 k k+1 2 k+ 1 '<br />
00 (<br />
}: In 1 - - 1) = lim Sk = hm In- -In-- = In - -ln 1 = ln 1 -In 2 - In 1 = - In 2.<br />
n = 2 n 2 k-oo k-oo 2 k + 1 2<br />
·I<br />
13. (a) The x-intercepts of the curve occur where sin x = 0 x = mr,<br />
nan integer. So using the formula for disks (and either .a CAS or<br />
l-+-l-~~+-''r7"""'-'...,.._---t 40 sin 2 x = ~ (1 - cos 2x) and Formula 99 to evaluate the integral),<br />
the <strong>vol</strong>ume of the nth bead is<br />
- I v. f"'' ( - :r:/10 . )2 dx rmr -:r:/ 5 . 2 d<br />
n = rr J (,.-l) .. e sm x = rr J (n- 1 ) .. e sm x x<br />
= 2;g,rr (e- (n-l)lf/5- e- mr/5)<br />
(b) The total <strong>vol</strong>ume is<br />
00<br />
rr .{ 0<br />
e- :r:/S sin 2 x dx = f V,, = 2 ~it f; [e- (n- 1 )"/ 5 - e-n"/ 5 ] = ;g; 2 [telescoping sum].<br />
n =l n=l<br />
Another method: If the <strong>vol</strong>ume in part (a) has been written as Vn = 2 ~g; e-n"/ 5 (e"/ 5 -<br />
as a geometric series with a= 2 ~g; (1 - e-"1 5 ) and r = e-"/ 5 .<br />
1), then we recognize f; Vn<br />
· n=l<br />
© 2012 Ccngo.ge Learning. All Rights Rcsr.n·<strong>ed</strong>. l\.Jay nol be settnr~-d, copiW, or dupl icuiL-d, or post<strong>ed</strong> to a publicly accessible website. in whole or in part.
108 D CHAPTER 11 PROBLEMS PLUS<br />
15. If Lis the length of a side of the equilateral triangle, then the area is A= ~L · 4L = 4£ 2 and so £ 2 =faA.<br />
Let r be the radius of one of the circles. When there are n rows of circles, the figure shows that<br />
L<br />
L = v'3r+r+(n-2)(2r)+r+v'3r=r(2n-2+2../3),sor = ( y'3 )'<br />
2 n+ 3-1<br />
The number of circles is 1 + 2 + · · · + n = n( n 2<br />
+ 1 ) , and so the total area of the circles is<br />
n(n + 1) 2 n(n + 1) £ 2<br />
An = 1fT = · 7r 2<br />
2 2 4(n+v'3- 1)<br />
n(n+ 1) 4A/J3 n(n+1) 1rA<br />
= 2 1f 4(n+v'3 - 1) 2 = {n+../3-1) 2 2../3 =><br />
An n(n + 1) 1r<br />
A = (n+../3-1) 2 2.;3<br />
1 + 1/n 1r 1r<br />
= -> -- asn ->oo<br />
[1+(J3-1)/n) 2 2v'3 2../3<br />
r 2r 2r<br />
~-----L------~<br />
17. As in Section 11.9 we have to integrate the function x"' by integrating series. Writing x"' = ( e 1 " "')"' = e"' 1 ""' and using the<br />
Maclaurin series for e"', we have x"' = ( eln "')"' = e"' 1 " "' = E ( x In x) n 1<br />
= E x" (~ x) n . As with power series, we can<br />
n = O n. n=O n.<br />
1<br />
·1 oo 11<br />
x"(ln xt oo 1 1 1<br />
dx = 2: I<br />
integrate this series term-by-term: x"' dx = 2:<br />
1<br />
0 n=O 0 n. n=O n. 0<br />
n(ln x)"- 1 x"+ 1<br />
withu = (lnx)",dv =x" dx,sodu= · dxand v = --: .<br />
x n+ 1<br />
f \"(lnx)"dx = lim /. 1 x"(ln xfdx= lim [x"+l (lnx)"]<br />
} 0 t-o+ t . t-o+ n + 1 t t-o+ it n +<br />
= o- _ n _ t x"(Inxf- 1 dx<br />
n+ 1 } 0<br />
(where !'Hospital's Rule was us<strong>ed</strong> to help evaluate the first limit). Further integration by parts gives<br />
1<br />
x"(lnxf dx. We integrate by parts<br />
- lim ( 1 ___!!:__ 1<br />
x"(1nxf- 1 dx<br />
(<br />
1<br />
x"(ln x)k dx = -~ 1<br />
} 0 n + 1 0<br />
1<br />
x"(ln x)k-<br />
1<br />
dx and, combining these steps, we get<br />
1x"(ln x)"dx = (-1)"n!11 x"dx= (-1)"n! =?<br />
0<br />
(n+1)" 0<br />
· (n+ 1)n+l<br />
1 "' d = ~ ..!_ 1 1 "(In ·)"' d = ~ ..!_ (- 1t n! = ~ (-1)" = ~ (- 1)"-1<br />
X X L... I X X X L... I ( 1 ) +1 , 0 n=O n. 0 . n=O n. n + n n=O L... ( n + 1 " n=l n"<br />
) +1 L... ·<br />
00 x2n+l 1<br />
19. By Table 1 in Section 11. 10, tan- 1 x = E ( - lt- 2<br />
' --for JxJ < 1. In particular, for x = r.;• we<br />
n=O n+ 1 v3<br />
7r ( 1 )<br />
00<br />
(1/../3) 2 "+ 1 00<br />
(1)" i 1<br />
have 6 = tan-1 y'3 = n~O ( - .1t 2n + 1 = n~O ( -1)" 3 y'3 2n + 1' SO<br />
6 00 (-1)" 00 (-1)" ( 00 ' (-1)" ) .<br />
·rr - - - 2 3 - 2 3 1 =><br />
- ../3 n~O (2n + 1)3" - y'3 n~O (2n + 1)3" - y'3 + f1 (2n + 1)3n<br />
00 (-1)" 7r<br />
E =--1.<br />
n=1 (2n + 1)3" 2 y'3<br />
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CHAPTER 11 PROBLEMS PLUS 0 109<br />
x x 2 · x 3 x 4<br />
21. Let f (x) denote the left-hand side of the equation l + 2T + 4<br />
! + 6f + 8f + .. · = 0. lfx;:::: ~· then f(x) ;:::: 1 and t~ere are<br />
no solutions oftheequation. Note that / (- x 2 ) = 1 - 2! + 4<br />
! -<br />
x2 x4 x s xs<br />
6<br />
! + 8f - · · · = cosx. The solutions of cosx = 0 for<br />
x < 0 are given by x = i - 1rk, where k is a positive integer. Thus, tl1e solutions off ( x) = 0 are x = - (i - 1r k) 2 , where<br />
k is a positive integer.<br />
23. Call the series S. We group the terms according to the number of digits in their denominators:<br />
Now in the group gn , since we have 9 choices for each of then digits in the denominator, there are 9n terms.<br />
Furthermore, each term in 9n is less than<br />
10 ~_ 1 [except for the first term in gl]. So 9n < gn ·<br />
10 ~_ 1 = 9( { 0<br />
)"'- 1 •<br />
00<br />
Now L: 9( 1<br />
9<br />
n = l<br />
0<br />
) n - l is a geometric series witl1 a = 9 and r = fa < 1. Therefore, by the Comparison Test,<br />
oo oo 9 n - 1 9<br />
S = L: 9n < L: 9(w) = 1 _ 9110<br />
= 90.<br />
n = l n=l<br />
Use the Ratio Test to show that the series for u, v, and w have positive radii of convergence (oo in each case), so<br />
Theorem 11'.9.2 applies, and hence, we may differentiate each of these series:<br />
du 3x 2 6x 5 9x 8 x 2 x 5 x 8<br />
- = -+-+-+··· = - +-+ - +· ·· = w<br />
dx 3! 6! 9! 2! 5! 8!<br />
. . dv x 3 x 6 x 9 dw x 4 x 7 x 10<br />
Similarly, dx = 1 + 3! + 6f + 9! + · · · = u, and dx = x + 4<br />
! + 7T + lO! + · · · = v.<br />
So u' = w, v' = u, and w' = v. Now differentiate the left-hand side of the desir<strong>ed</strong> equation:<br />
d<br />
dx (u 3 + v 3 + w 3 -<br />
3uvw) = 3u 2 u' + 3v 2 v' + 3w 2 w' - 3(u'vw + uv'w + uvw')<br />
= 3u 2 w + 3v 2 u + 3w 2 v - 3(vw 2 + u 2 w + u~ 2 ) = 0 =><br />
u 3 + v 3 + wa - 3uvw = C. To find the value of the constant C, we put x = 0 in the last equation and get<br />
1 3 + 0 3 + 0 3 - 3(1 · 0 · 0) = C => C = 1, so u 3 + •v 3 + w 3 - 3uvw = 1.<br />
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12 D VECTORS AND THE GEOMETRY OF SPACE<br />
12.1 Three-Dimensional Coordinate Systems<br />
1. We start at the origin, which has coordinates (0, 0,. 0). First we move 4 units along the positive x-axis, affecting only the<br />
x-coordinate, bringing us to the point ( 4, 0, 0). We then move 3 units straight downward, in the negative z-direction. Thus<br />
only the z-coordinate is affect<strong>ed</strong>, and we arrive at {4, 0, - 3).<br />
3. The distance from a point to the yz-plane is the absolute value of the x-coordinate of the point. G(2, 4, 6) has the x -coordinate<br />
with the smallest absolute value, soC is the point closest to the ·yz-plane. A( - 4, 0, -1) must lie in the xz-plane since the<br />
distance from A to the xz-plane, given by the y-coordinate of A, is 0.<br />
5. The equation x + y = 2 represents the set of all points in<br />
JR 3 whose x- andy-coordinates have a sum of2, or<br />
equivalently where y = 2 - x. This is the set<br />
{(x, 2 - x , z) I x E JR, z E R } which is a vertical plane<br />
y=2 - x<br />
y=2-x,z= O<br />
that intersects the xy-plane in the line y = 2 - x, z = 0.<br />
X<br />
7. We can find the lengths of the sides of the triangle by using the distance formula between pairs of vertices:<br />
IPQI = )(7 - 3)2 + [0 - (-2)]2 + [1 - (- 3)] ~ = yf16 + 4 + 16 = 6<br />
IQRI = j(l- 7)2 + {2 - 0) 2 + {1 - 1) 2 = vf36 + 4 + o = .J40 = 2 v'15<br />
IRPI = )(3 - 1)2 + (- 2 - 2)2 + (- 3 - 1) 2 = vf4+ 16+ 16 = 6<br />
The longest side is QR, but the Pythagorean Theorem is not satisfi<strong>ed</strong>: IPQI 2 + IRPI 2 =f:. IQRI 2 . Thus PQR is not a right<br />
triangle. PQR is isosceles, as two sides have the same length.<br />
9. (a) First we find the distances between points:<br />
IABI = ){3- 2)2 + {7- 4)2 + (-2 - 2)2 = v'26<br />
IBGI = ){1- 3) 2 + (3 - 7) 2 + [3- (- 2)]2 = v'45 = 3 v'5<br />
IACI = )(1- 2)2 + (3 - 4) 2 + (3- 2) 2 = J3<br />
In order for the points to lie on a straight line, the sum of the two shortest distances, must be equal to the longest distance.<br />
Sin
112 0 CHAPTER 12 VECTORS AND THE GEOMETRY OF SPACE<br />
(b) First we find the distances between points:<br />
I DEl = J(l- 0) 2 + [- 2- ( - 5)]2 + (4 - 5) 2 ::::: VIT<br />
IEFI = )(3 - 1) 2 + [4 -<br />
( - 2)]2 + (2 - 4) 2 = v'44 = 2 VIT<br />
IDFI = )(3 - 0) 2 + [4- (-5)]2 + (2 - 5) 2 = v'99 = 3Vll<br />
Since I DEl + IEFI = ID F I, the three points lie on a straight line.<br />
11. An equation of the sphere with center (-3, 2, 5) and radius 4 is [x - (- 3)] 2 + (y - 2) 2 + (z - 5) 2 = 4 2 or<br />
(x + 3) 2 + {y - 2) 2 + (z- 5) 2 = 16. The intersection of this sphere with the yz-plane is the set of points on the spher~<br />
whose x-coordinate is 0. Putting x = 0 into the equation, we have 9 + (y - 2) 2 + (z- 5) 2 = 16, x = 0 or<br />
(y- 2) 2 + (z- 5) 2 = 7, x = 0, which represents a circle in the yz-plane with center (0, 2, 5) and radius -/7.<br />
13. The radiusofthesphere is the distance between (4,3, - 1) and (3,8, 1): r = )(3 - 4)2 + (8 - 3)2 + [1 - (-1)]2 = V3o.<br />
Thus, an equation of the sphere is (x - 3) 2 + (y - 8) 2 + (z- 1) 2 = 30.<br />
15. Completing squares in the equation x 2 + y 2 + z 2 - 2x - 4y + 8z = 15 gives<br />
recognize as an equation of a sphere with center (1, 2, - 4) and radius 6.<br />
17. Completing squares in the equation 2x 2 - 8x + 2y 2 + 2z 2 + 24z = 1 gives<br />
2(x 2 - 4x + 4) + 2y 2 + 2~z 2 + 12z + 36) = 1 + 8 + 72 :::} 2(x - 2) 2 + 2y 2 + 2(z + 6? = 81 :::}<br />
(x - 2) 2 + y 2 + (z + 6) 2 = 8 , which we recognize as an equation of a sphere with center (2, 0, -6) and<br />
21<br />
radius j¥ = 9/ v'2.<br />
19. (a) lfthe midpointofthe line segment from P 1(x 1 ,y1,z1) to P 2 (x 2 ,y 2 ,z 2 ) is Q = (x 1 ;x 2 , Y 1 ;v 2 , z 1 ; z 2 ) ,<br />
then the distances IPl Q l and IQP21 are equal, and each is half of IPt P21· We verify that this is the case:<br />
IP1QI = v [~ (x1 + X2) - X1] 2 + a
!QP2\ = J [x2 - ~(X 1 + X2)] 2 + [y2- ~(Y l + Y2)] 2 + [z2 - ~ (z1 + z2)) 2<br />
SECTION 12.1 THREE-DIMENSIONAL COORDINATE SYSTEMS 0 113<br />
= (~x2- ~ x1) 2 + ( h2- ~ ·yt) 2 + (tz2 - ~z1 ) 2 = J G) 2 [(x2- xd + (y2 - yl) 2 + (z2 - z1) 2 )<br />
.= ~J(x2 - xd + (y2- y1) 2 + (z2- zd = ~ \P1P2\<br />
So Q is inde<strong>ed</strong> the midpoint of P 1P2.<br />
(b) By part (a), the midpoints of sides AB, B C and C A are P1 ( -~, 1, 4), P2 (1, ~. 5) and P3 (~ , ~, 4). (Re ~all that a m<strong>ed</strong>ian<br />
of a triangle is a line segment from a vertex to the midpoint of the opposite side.) Then the lengths of the m<strong>ed</strong>ians are:<br />
IAP2\ = Jo 2 + (~ - 2) 2 + (5 - 3) 2 = jf+4 = -fj = ~<br />
\BP3\ = J (~ + 2) 2 + (~) 2 + (4- 5) 2 = J¥ + * + 1 = J2j = ~JW<br />
\CP1\ = J (-~- 4) 2 + (1 :__ 1} 2 + (4- 5? = J¥ + 1 = ~.J85<br />
21 . (a) Since the sphere touches the xy-plan.e, its radius is the distance from its center, (2, -3, 6), to the xy-plane, namely 6.<br />
Therefore r = 6 and an equation of the sphere is (x- 2? + (y + 3? + (z - 6? = 6 2 = 36.<br />
(b) The radius of this sphere is the distance from its center (2, - 3, 6) to the yz-plane, which is 2. Therefore, an equation is<br />
(~- 2) 2 + (y + 3) 2 + (z - 6) 2 = 4.<br />
(c) Here the radius is the distance from the center (2, - 3, G) to the xz-plane, which is 3. Therefore, an equation is<br />
(x- 2) 2 + (y + 3) 2 + (z- 6} 2 = 9.<br />
23. The equation x = 5 represents a plane parallel to the yz-plane and 5 units in front of it.<br />
25. The inequality y < 8 represents a half-space consisting of all points to the left of the plane y = 8.<br />
27. The inequality 0 ~ z ~ 6 represents all points on or between the horizontal planes z = 0 (the xy-plane) and z = 6.<br />
29. Because z = - 1, all points in the region must lie in the horizontal plane ~ = -1. In addition, x 2 + y 2 = 4, so the region<br />
consists of all points that lie on a circle with radius 2 and center on the z-axis that is contain<strong>ed</strong> in the plane z = -1.<br />
31. The inequality x 2 + y 2 + z 2 ~ 3 is equivalent to jx 2 + y 2 + z 2 ~ J3, so the region consists of those points whose distance<br />
from the origin is at most V3. This is the set of all points on or inside the sphere with radius ../3 and center (0, 0 , 0).<br />
33. Here x 2 + z 2 ~ 9 or equiva lently Jx2 + z 2 ~ 3 which describes the set of all points in JR 3 whose distance from they-axis is<br />
at most 3. Thus, the inequality represents the region consisting of all points on or inside a circular cylinder of radius 3 with<br />
axis the y-axis.<br />
35. This describes all points whose x-coordinate is between 0 and 5, that is, 0 < x < 5.<br />
37. This describes a region all of whose points have a distance to the origin which is greater than r, but smaller than R. So<br />
inequalities describing the region are r < jx 2 + y 2 + z 2 < R, or r 2 < x 2 -1- y 2 + z 2 < R 2 .<br />
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114 0 CHAPTER 12 VECTORS AND THE GEOMETRY OF SPACE<br />
39. (a) To find the x- andy-coordinates of the point P , we project it onto £2<br />
I<br />
and project the resulting point Q onto the x- and y-axes. To find the<br />
z-coordinate, we project P onto either the xz-plane or the yz-plane<br />
(using our knowl<strong>ed</strong>ge of its x- or y-coordinate) and then project the<br />
resulting point onto the z-axis. (Or, we could draw a line parallel to<br />
QO from P to the z-axis.) The coordinates of Pare (2, 1, 4).<br />
(b) A is the intersection of £ 1 and £2, B is directly below the<br />
y-intercept of L2, and 0 is directly above the x-intercept of £2.<br />
41. We ne<strong>ed</strong> to find a set of points { P(x, y , z) II API = I BPI}.<br />
J(x + 1) 2 + (y - 5)2 + (z- 3) 2 = J(x - 6)2 + (y - 2)2 + (z + 2)2 ~<br />
(x + 1) 2 + (y- 5) + (z - 3) 2 = (x - 6) 2 + (y - 2) 2 + (z + 2) 2 =><br />
x 2 + 2x + 1 + y 2 - lOy + 25 + z 2 - 6z + 9 = x 2 - 12x + 36 + y 2 - 4y + 4 + z 2 + 4z + 4 '=> 14x - 6y - lOz = 9.<br />
Thus the set of points is a plane perpendicular to the line segment joining A and B (since this plane must contain the<br />
perpendicular bisector of the line segment AB).<br />
43. The sphere x 2 + y 2 + z 2 = 4 has center (0, 0, 0) and radius 2. Completing squares in x 2 - 4x + y 2 - 4y + z 2 - 4z = -11<br />
gives (x 2 - 4x + 4) + (y 2 - 4y + 4) + (z 2 - 4z + 4) = - 11 + 4 + 4 + 4 => (x - 2) 2 + (y - 2? + (z- 2) 2 = 1,<br />
so this is the sphere with center (2, 2, 2) and radius 1. The (shortest) distance between the spheres is measur<strong>ed</strong> along<br />
the line segment connecting their centers. The distance between (0, 0, 0) and (2, 2, 2) is<br />
.J(2 - 0)2 + (2- 0)2 + (2 - 0)2 = ffi = 2 J3, and subtracting the radius of each circle, the distance between the<br />
spheres is 2 v'3 - 2 - 1 = 2 .J3 - 3.<br />
12.2 Vectors<br />
1. (a) The cost of a theater ticket is a scalar, because it has only magnitude.<br />
(b) The current in a river is a vector, because it has both magnitude (the spe<strong>ed</strong> of the current) arid direction at any given<br />
location.<br />
(c) If we assume that the initial path is linear, the initial flight path from Houston to Dallas is a vector, because it has both<br />
magnitude (distance) and direction.<br />
(d) The population of the world is a scalar, because it has only magnitude.<br />
3. Vectors are equal when they share the same length and direction (but not necessarily location). Using the symmetry, of the<br />
. -----+ --+ ---+ ------+ --+ --+ __. -----+<br />
parallelog~am as a guide, we see that AB = D C, DA = CB, DE = EB, and EA = CE.<br />
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5. (a)<br />
un<br />
(d)<br />
(b)<br />
(c)<br />
u<br />
uNw<br />
SECTION 12.2 VECTORS 0<br />
w~<br />
u<br />
v+w<br />
(e)<br />
(f)<br />
u<br />
u<br />
u -:Gv<br />
~ v Dw<br />
v+u+w u-w-v<br />
115<br />
---><br />
7. Because the taif of dis the midpoint of QR we have QR = 2d, and by .the Triangle Law,<br />
a + 2d = b =? 2d=b-a =? d=!{b-a) = th - ~a. AgainbytheTriangleLawwehavec + d = bso<br />
c = b - d = b - ( ~ b - ~a) = ~ ~ + ~b .<br />
9. a= (3 - (- 1), 2 - 1}=(4, 1} 11.a=(2 - (-1), 2 - 3} = (3,-1}<br />
y<br />
B(3, 2) A(-1, 3)<br />
y<br />
0 X<br />
X<br />
13. a = ·(2 - 0, 3 - .3,-1 - 1} = (2, 0,-2} 15. (-1, 4} + (6, - 2} = (-1 + 6, 4 + ( - 2)} = (5, 2)<br />
A(0, 3, 1)<br />
y<br />
X<br />
X<br />
17. (:~, 0, 1) + (0, 8, 0) = (3 + 0, 0 + 8, 1 + 0)<br />
= (3, 8, 1)<br />
(3, 0, 1)<br />
X<br />
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116 0 CHAPTER 12 VECTORS AND THE GEOMETRY OF SPACE<br />
19. a + b = (5 + (-3) ,-12 + (-6)) = (2, - 18)<br />
2a + 3b = (10, -24) + (-9, -18) = (1, -42)<br />
lal = y'5 2 + ( - 12) 2 = .jl69 = 13<br />
Ia- b l = 1(5- ( -3), - 12- ( -6))1 = 1(8, -6)1 = y'8 2 + ( - 6)2 = v'100 = 10<br />
21. a+ b = (i'+ 2j - 3 k) + ( -2 i - j + 5 k) = - i + j + 2k<br />
2a + 3b = 2 (i + 2j - 3 k) + 3 ( -2 i - j + 5 k) = 2 i + 4j - 6 k - 6 i - 3j + 15 k = - 4 i + j + 9k<br />
lal = y'12 + 22 + ( -3)2 = v'i4<br />
Ia - b l = l(i+ 2j - 3 k) - (-2 i -j +5 k)l = l3 i + 3 j - 8k l = y'3 2 +3 2 + (-8)2 = v'82<br />
23. The vector - 3 i + 7 j has length l-3 i + ~ j I = y' ( -3)2 + 7 2 = v'58, so by Equation 4 the unit vector with the same<br />
d . . . 1 ( 3. 7 ' ) 3 . 7 .<br />
1rect1on IS r,:n - 1 + J =- r.;o 1 + r.;oJ·<br />
v58 v58 v58<br />
25. The vector 8 i - j + 4 k has length 18 i - j + 4 k l = y'8 2 + ( - 1) 2 + 4 2 = J8I = 9, so by Equation 4 the unit vector with<br />
the same direction is ~ (8 i - j + 4 k ) = ~ i - ~ j + ~ k.<br />
27.<br />
y<br />
. .j3<br />
From the figure, we sec that tan 0 = - 1<br />
- = J3 => B = 60°.<br />
X<br />
29. From the figure, we see that the x-component ofv is<br />
y<br />
v1 = I vi cos(1r / 3) = 4 ·. ~ = 2 and they-component is<br />
V2 = I vi sin( 1r / 3) = 4 · .,;; = 2 v'3. Thus<br />
V = (th, V2) = (2, 2 .j3 ).<br />
v<br />
Vz<br />
v,<br />
X<br />
31. The velocity vector v makes an angle of 40° with the horizontal and<br />
has magnitude equal to the spe<strong>ed</strong> at which the footba ll was thrown.<br />
From the figure, we see that the horizontal component ofv is<br />
I vi cos 40° = 60 cos 40° ~ 45.96 ft/s and the vertical component<br />
is I vi sin 40° = 60 sin 40° ~ 38.57 fils.<br />
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SECTION 12.2 VECTORS 0 117<br />
33. The given force vectors can be express<strong>ed</strong> in terms of their horizontal and vertical components as -300 i and<br />
~00 cos 6~ 0 i + 200 sin 60°j = 200 ( ~) i + 200 ( ~) j = 100 i + 100 v'3 j . The resultant force F is the sum of<br />
these two vectors: F = ( -300 + 100) i + (0 + 100 v'3) j = - 200 i + 100J3j. Then we have<br />
IFI ~ V(-200) 2 + (100v'3) 2 = .j70,000 = 100 V7 ~ 264.6 N. Let() be the angle F makes with the<br />
positive · · x-ax1s. · Th en tan () = lOO _ J3 = - . J3 an d t h e termma . I pomt . o f F 1· 1es m . t h e secon d qua d rant, so<br />
200 2<br />
() = tan- 1 ( - ~) + 180° ~ -40.9° f. 180° = 139.1°.<br />
35. With respect to the water's surface, the woman's velocity is the vector sum of the velocity of the ship with respect<br />
to the water, and the woman's velocity with respect to the ship. If we let north be the positive y-direction, then<br />
v = (0, 22) + (-3, 0) = (-3, 22). The woman's spe<strong>ed</strong> is lvl = .jg + 484 ~ 22.2 mi/ h. The vector v makes an angle()<br />
with the east, where()= tan- 1 ( ~;) ~ 98°. Therefore, the woman's direction is about N(98- 90) 0 W = N8°W.<br />
37. Let T 1 and T 2 represent the tension vectors in each side of the<br />
clothesline as shown in the figure. T 1 and T 2 have equal vertical<br />
components and opposite horizontal components, so we can write<br />
T 1 = - a i + b j and T 2 = a i + b j [a, b > 0]. By similar triangles, !!_ = 0·08 => a = 50b. The force due to gravity<br />
a 4<br />
acting on .Ule shirt has magnitude 0.8g ~ (0.8)(9.8) = 7.84 N, hence we have w = - 7.84j. The resultant T 1 + T 2<br />
of the tensile forces counterbalances w , so T 1 + T 2 = -w => (-a i + bj ) + (a i + bj ) = 7.84j =><br />
( -50b i + bj ) + (50b i + bj ) = 2bj = 7.84j => b = 7·: 4 = 3.92 and a= 50b = 196. Thus the tensions are<br />
T 1 = - a i + bj = - 196i + 3.92j and T 2 = ai + bj = 196i + 3.92j .<br />
Alternatively, we can find the value of() and proce<strong>ed</strong> as in Example 7.<br />
39 .. (a) Set up coordinate axes so that the boatman is ·at the origin,- the canal is<br />
y<br />
border<strong>ed</strong> by the y-axis and the line x = 3, and the current flows in the<br />
negative y-direction. The boatman wants to reach the point (3, 2). Let() be<br />
the angle, measur<strong>ed</strong> from the positive y-axis, in the direction he should<br />
steer. (See the figure.)<br />
X<br />
In still water, the boat has velocity v b = (13 sjn B, 13 cos()) and the velocity of the current is V c (0, -3.5), so the true path<br />
of the boat is determin<strong>ed</strong> by the velocity vector v = v b + V c = (13 sin B, 13 cos() -<br />
3.5). Let t be the time (in hours)<br />
after the boat departs; then the position of the boat at time t is given by tv and the boat crosses the canal when<br />
tv = (13sin0, 13cos0- 3.5) t = (3, 2). Thus 13(sinO)t = 3 =><br />
3<br />
t = - 13<br />
. ()and (13cosB- 3.5) t = 2.<br />
sm ·<br />
118 0 CHAPTER 12 VECTORS AND THE GEOMETRY OF SPACE<br />
Substituting gives {13 cos (} - 3.5) 13<br />
:in(} = 2 => 39 cos 8- 10.5 = 26 sin(} (1). Squaring both sides, we have<br />
1521 cos 2 8 - 819 cos 8 + 110.25 = 676 sin 2 () = 676 (1 - cos 2 9)<br />
2197 cos 2 (} - 819 cos(} - 565.75 = 0<br />
The quadratic formula gives<br />
_ 819 ± v'C--819)2- 4{2197)( - 565.75)<br />
cos<br />
8<br />
. - ·2{2197)<br />
· = 819 ± ~ 5 g~ 42 ' 572 :::::: 0. 72699 or - 0.35421<br />
The acute value for(} is approximately cos-- 1 (0.72699) :::::: 43.4°. Thus the boatman should steer in the direction that is<br />
43.4° from the bank, toward upstream.<br />
Alternate solution: We could solve ( 1) graphically by plotting y = 39 cos 8- 10.5 andy = 26 sin 8 on a graphing device<br />
and finding the appoximate intersection point (0.757, 17.85). Thus 0:::::: 0.757 radians or equivalently 43.4°.<br />
(b) From part (a) we know the trip is complet<strong>ed</strong> when t = ·- 3<br />
~<br />
(} .· But 8 :::::: 43.4°, so the time requir<strong>ed</strong> is approximately<br />
1 sm<br />
. 3 13sm 43 . 4<br />
0<br />
:::::: 0.336 hours or 20.2 minutes.<br />
41 . The slope of the tangent line to the graph ofy = x 2 at,the point {2, 4) is<br />
dy I = 2xl = 4<br />
· dx x=2 x=2<br />
and a parallel vector is i + 4j which has length Ji + 4j J = .)1 2 + 42 = ..JPi, so unit vectors parallel to the tangent line<br />
are ±~ (i + 4 j ).<br />
~ _.. --t ,_.. -+ ------+- ------+- ----+- ._. ---+ ---+ ( --+)<br />
43. By the Triangle Law, AB + BC = AC. Then AB + BC + CA = AC + CA, but AC + CA = AC + -AC = 0.<br />
--+ --+ --+<br />
SoAB+BC+CA = 0.<br />
45. (a), (b) (c) From the sketch, we estimate that s :::::: 1.3 and t:::::: 1.6.<br />
(d) c = sa + t b # 7 = 3s + 2t and 1 = 2s - t.<br />
Solving these equations gives s = t and t = V.<br />
47. Jr- roJ is the distance between the points (x, y , z) and (xo, y 0 , z 0 ), so the set of points is a sphere with radius 1 and<br />
center (xo, Yo, zo).<br />
Alternate method: Jr - ro J = 1 # J(x- xo) 2 + (y- vo)2 + (z- zo) 2 = 1 #<br />
(x - xo? + (y - yo) 2 + (z - zo? = 1, which is the equation of a sphere with radius 1 and center (xo, yo, zo).<br />
© 2012 Ccngagc Learning. All Rights Reserv<strong>ed</strong>. Moy not be scann<strong>ed</strong>. copi<strong>ed</strong>. orduplicotcd, or J>
SECTION 12.3 THE DOT PRODUCT 0 1-19<br />
49. a+ (b + c) = (at, a2} + ((bt , b2} + (c1, c2}) = (at , a2} + (b1 + c1, ~ + c2)<br />
= (a1 + b1 + c1, a2 + ~ + c2} = ((a1 + bt) + c1, (a2 + ~) + c2)<br />
= (a1 + b1 ,a2 + b2} + (ct, c2} = ((a1,a2} + (b1 , b2}) + (c1,c2}<br />
.= (a+ b} + c<br />
_____. -+ -+<br />
51 . Consider triangle ABC, where D and E are the midpoints of AB and BC. We know that AB. + BC = AC (1) and<br />
---+ ---+ _. I .-;. ---+ ---+ -+ . . ----+ ---t<br />
DB +BE= DE (2). However, DB = ~AB, and BE = ~BC. Substituting these expressions for DB and BE into<br />
---+ ---+ -+ ---t ----+ . .-. --+<br />
· (2) gives ~AB + ~BC = DE. Comparing this with (1) gives DE = ~AC. Therefore·AC and DE are parallel and<br />
12.3 The Dot Product<br />
1. (a) a · b is a scalar, and the dot product is defin<strong>ed</strong> only for vectors, so (a · b) · c has no meaning.<br />
(b) (a . b) c is a scalar multiple of a vector, so it does have meaning.<br />
(c) Both [a[ and b · care scalars, so [al (b ·c) is an ordinary product of real numbers, and has meaning.<br />
(d) Both a and b +care vectors, so th~ dot product a · (b +c) has meaning.<br />
(e) a · b is a scalar, but cis a vector, and so the two quantities cannot be add<strong>ed</strong> and a · b -1:' c has no meaning.<br />
(f) Ia ! is a scalar, and the dot product is defin<strong>ed</strong> only for vectors, so lal · (b + c ) has no meaning.<br />
3. a· b = ( - 2, ~) · (- 5,1 2} = (-2)(-5} + (~)(12) = 10 + 4 = 14<br />
5. a · b = (4, 1, i ) · (6, -3, - 8) = (4)(6) + (1)(- 3) + (i) (- 8) = 19<br />
7. a· b = (2 i + j ) · (i - j + k ) = (2)(1) + (1)( - 1) + (0)(1) = 1<br />
'-1<br />
' '• \<br />
9. By Theorem 3, a · b = lal lb1 cos 0 = (6)(5) cos 2 ; = 30 ( - ~) = -15.<br />
11. u , v , aJ)d ware all unit vectors, so the triangle is an equilateral triangle. Thus the angle between u and v is 60° and<br />
u · v = lullvl cos60° = (1 )( 1) (~) = ~. lfw is mov<strong>ed</strong> so it has the same initial point as u , we can see that the angle<br />
between them is 120° and we have u · w = lul lw l cos 120° = (1)(1) ( -~) = - ~.<br />
13. (a) i · j = (1, 0, 0) · (0, 1, 0) = (1)(0) + (0)(1) + (0)(0) = 0. Similarly, j · k = (0)(0) + (1)(0) + (0)(1) ,;, 0 and<br />
k . i = {0){1) + {0){0) + (1)(0) = 0.<br />
Another method: Because i, j , and k are mutually perpendicular, the cosine factor in each dot product (see Theorem 3)<br />
is cos~= 0.<br />
(b) By Pr~perty I of the dot product, i · i = lil 2 = 1 2 = 1 since i is a unit vector. Similarly, j · j = lj l 2 = 1 and<br />
k . k = lkl 2 = 1.<br />
© 2012 Ccngage Le:sming. All Ri!lhiS Reservod. May no1 be scann<strong>ed</strong>. copi<strong>ed</strong>, ordupl.ical
120 D CHAPTER 12 VECTORS AND THE GEOMETRY OF SPACE<br />
15. ja j = vW + 3 2 = 5, jbj = }2 2 + (- 1)2 = v'5, and a· b = {4){2) + {3)(- 1) = 5. ~rom Corollary 6, we have<br />
a· b 5 1 · . ( 1 ) ·<br />
cos(}= jajjb/ = 5<br />
. v'5 = vg· So the angle between a and b IS(}= cos- 1 v'5 ~ 63°.<br />
17. jaj = .,j3 2 + ( -1)2 + 5 2 = .;35, jbj = .,j( - 2) 2 + 4 2 + 3 2 = v'29, and a · b ~ {3){ -2) + ( -1)(4) + {5)(3) = 5. Then<br />
cos B = l:l · l~ l = .;35 ~ v'29 = v' 1 ~ 15<br />
and the angle between a and b is. B = cos- 1 ( v'1~15 ) ~ 81°.<br />
19. jaj = .J42 + ( -3)2 + 12 = .;26, jbj = .,j22 + 02 + ( -1)2 = v'5, and a. b = {4){2) + ( - 3)(0) + (1)( -1) = 7.<br />
h (} a· b · 7 7 d 8<br />
. _1 ( 7 )<br />
T en cos = jajjbj = v'26 . v'5 = v'130 an = cos v'130 ~ .<br />
52 o<br />
21. Let p, q, and r be the angles at vertices P, Q, and R respectively.<br />
--> -'-+<br />
Then p is the angle between vectors PQ and P R , q is the angle<br />
--> --><br />
between vectors QP and QR, and r is the angle between vectors<br />
--> . ---+<br />
RPaildRQ.<br />
--> --> .<br />
Thuscos = PQ·PR = (- 2 • 3 ) · ( 1 • 4 ) = - 2 + 12 =___!Q_and =cos- 1 (___!Q_) ~48° . Sim i larl,<br />
p !?O!!nl .,j(- 2)2+32 .j12+ 42 y'l3ffi v'ffi p y'ffi y<br />
---+ ---+<br />
QP . QR . (2, -3) · (3, 1) 6 - 3 3 - 1 ( 3 )<br />
75 o d<br />
cosq = IQ?IIQRI = v'4 + 9 v'9 + 1 = v'13 v'f5 = v'13Q so q =cos v'l3Q ~ an<br />
r ~ 180° - {48°+ 75°) = 57°.<br />
. ,___.,2 ,___.,2 1--,-+12<br />
,__., ,__.<br />
. ~-~-n<br />
Alternate solution: Apply the Law of Cosines three times as follows: cos p =<br />
2 PQ PR<br />
1<br />
_ i?R( -I?QI 2 .- IQJil. 2 _ j?Q( -lnl 2 -I Q'RI 2<br />
cos q - ,___.,, __., , and cosr - ---. .-.<br />
· 2 PQ QR 2 PR QR<br />
1 11 1<br />
23. (a) a . b = ( -5){6) + (3)( - 8) + {7){2) = -40 -:/= 0, so a and bare not orthogonal. Also, since a is not a scalar multiple<br />
ofb, a and bare not parallel.<br />
(b) a · b = ( 4) ( -3) + (6)(2) = 0, so a and bare orthogonal (and not parallel).<br />
(c) a· b = (-1){3) + (2){4) + {5)(-1) = 0, soaand bare orthogonal (and not parallel).<br />
(d) Because a = - ~ b, a and b ~e parallel.<br />
--+ ---+ ----+ --+ ----+ ----+<br />
25. QP = (-1, - 3, 2), QR = (4, - 2, - 1), and QP · QR = -4 + 6-2 = 0. Thus QP and QRare orthogonal, so the angle of<br />
the triangle at vertex Q is a right angle.<br />
© 2012 Cengogc Learning. All RighiS Resen'
SECTION 12.3 THE DOT PRODUCT 0 121<br />
21. Let a = a 1 i + a 2 j + a 3 k be a vector orthogonal to both i + j and i + k, Then a · (i + j ) = 0 a1 + a2 = 0 and<br />
a . (i + k ) = 0 a 1 + a 3 = 0, so a 1 = - a2 ~ -a3. Furthermore a is to be a unit vector, so 1 = ai + a~+ a§ = 3ai<br />
. . 1 h l . 1. l k d 1 '+ 1 ' + 1 k tw . h .<br />
1mphes a 1 = ± 73<br />
. T us a= 73<br />
1- 7aJ- 73 .an a = - 73 1 7aJ 73 are o sue umt vectors.<br />
29. The line 2x - y = 3 y ·= 2x - 3 has slope 2, so a vector parallel to the line is a = (1, 2}. The line 3x + y = 7 <br />
y = - 3x + 7 has slope - 3, so a vector parallel to the line is b = (1, - 3). The angle between the lines is the same as the<br />
angle 8 between the vectors. Here we have a · b = (1)(1) + (2)( - 3) = - 5, lal = v'P + 2 2 = v'5, and<br />
a · b -5 - 5 1 v'2<br />
[hi = J1 2 + ( -3)2 = v'lQ, so cos 8 = I alibi = y'5. v'1Q = 5<br />
J2 =- J2 or ""'2· Thus 8 = 135°, and the<br />
acute angle between the lines is 180° - 135° = 45°.<br />
31. The curves y = x 2 andy= x 3 meet when x 2 = x 3 x 3 - x 2 = 0 x 2 (x - 1) = 0 x = 0, x = 1. We have<br />
i:-x 2 = 2x and .!£x 3 = 3x 2 , so the tangent lines of both curv~s have slope 0 at x = 0. Thus the angle between the curves is<br />
dx<br />
dx<br />
0° at th~ point (0, 0). For x = 1, ! x 2 1:c=l = 2 and d~ xt:=l = 3 so the tangent lines at the point (1, 1) have slopes 2 and<br />
3. Vectors parallel to the tangent lines are (1, 2} and (1, 3}, and the angle (} between them is given by<br />
cos(} = (1, 2} . (1, 3} = 1 + 6 = _ 7_<br />
1(1, 2} 1 1(1,3}1 v'5v'W 5v'2<br />
~3 . Since 1(2, 1, 2)1 = v'4 + 1 + 4 = V9 = 3, using Equations 8 and 9 we have cos a=~ . cos/3 =~. and cos-y = ~· The<br />
direction angles are given by a = cos- 1 ( i) ~ 48°, /3 = cos- 1 (~) ~ 71°, and 1 = cos- 1 (~) = 48°.<br />
35. Since I i- 2j-3kl = v'1 + 4 + 9 = v'l4, Equations 8 and 9 give cos a = 7IT• cos/3 =*' and cos-y = 7&, while<br />
a = cos - 1 (~ ) ~74° , {3 = cos - 1 (-~) ~ 122 °, and /=cos - 1 (- vih) ~ 143°.<br />
37. l(c, c, c) I = v'c 2 + c 2 + c 2 = v'3c [since c > 0], so .cos a = cos/3 = cos:-y = ~ = ~ and<br />
v3c v3<br />
r;an a · b -5 · 4 + 12 · 6<br />
39. Ia I = J( -5)2 + 12 2 = v w:.r = 13. The scalar projection of b onto a is comp 11<br />
b = -<br />
1<br />
-<br />
1<br />
= = 4 and the<br />
a 13<br />
vector projection ofb onto a is, proj 11 b = ( j~~)<br />
: = 4 · f3 (-5, 12} = (-~, - ~ ).<br />
1 1<br />
41 . la l = v'9 + 36 + 4 = 7 so the scalar projection ofb onto a is compab = j~~ = ~ (3 + 12..:. 6) = l The vector<br />
. . f b . . . b 9 a 9 l (3 6 2} o (3 6 2} ( 27 54 18 )<br />
proJeCtiOn o onto a IS, prOJa = 7j;f = 7 · 7 , , - = 4o , , - = 49 • 49• -49 ·<br />
® 2012 Ccngage t eaming. All Rights Rcscn·cd. May not be scann<strong>ed</strong>, copi<strong>ed</strong>, or dupljeatcd, or post<strong>ed</strong> ton publicly accessible \\"Cbsilc, in whole or in part.
122 0 CHAPTER 12 VECTORS AND THE GEOMETRY OF SPACE<br />
43. lal = J4 + 1 + 16 = J2I so the scalar projection ofb onto a is compa b = a<br />
1a 21 J2I<br />
'<br />
· bl = 0 -;; 2 = - 1 - while the vector<br />
. . fb . . b 1 a 1 2i-j + 4 k 1 ( • • ) 2 1 4<br />
prOJeCtion 0 Onto a IS prOJa = J2T jaj = J2T · J2T = 2l 2 1 - J + 4 k = 21 i - 2T j + 2T k.<br />
a · b a·b<br />
45. ( ortha b ) · a = (b - proja b ) · a = b · a - (proja b } · a = b · a - --<br />
2<br />
2 a· a = b · a - - - 2<br />
Ia I = b · a - a· b = 0.<br />
Ja l<br />
lal<br />
So they are orthogonal by (7).<br />
a·b<br />
47. compa b = j;f = 2 ~ a · b = 2lal = 2 v'IO. lfb = (b1, b:l, b3) , then we ne<strong>ed</strong> 3bl + Ob2- 1b3 = 2 v'IO.<br />
One possible solution is obtain<strong>ed</strong> by taking b1 = 0, b2 = 0, bs = - 2 v'IO. In general, b = ( s, t, 3s - 2 v'10 ), s, t E llt<br />
49. The displacement vector is D = (6 - 0) i + (12 - 10} j + (20- 8) k = 6 i + 2j + 12 k so, by Equation 12, the work done is<br />
W = F · D = (8 i - 6j + 9 k ) · (6i + 2j + 12k} = 48 - 12 + 108 = 144joules.<br />
51. Here ID I = 80 ft, IF I = 30 lb, and B = 40°. Thus<br />
w =F . D = IF IIDI cos B = (30)(80) cos 40° = 2400 cos 40° ~ 1839 ft-lb.<br />
53. First note that n = (a, b) is perpendicular to the line, because if Q1 = (a1, bt) and Q2 = (a2 , b2) lie on the line, then<br />
-----+<br />
n · Q1 Q2 = aa2 -<br />
aa1 + ~ - bb1 = 0, since aa2 + bb:l = - c = aa1 + bb1 from the equation of the line.<br />
I<br />
Let P2 = (x2, y2) lie on the line. Then the distance from P1 to the line is tJ:te absolute value of the scalar projection<br />
o fp ----+p.<br />
(P-----t<br />
1 2 onto n . compn 1 2 = I I = r:?'7l:? = r::?7'l:?<br />
P. ) In · (x2- x1, Y2 - Yl )l lax2- ax1 + by2 -lnJ1I lax1 + by1 + cl<br />
n v a2 + b2 v a2 + b2<br />
since ax2 + by2 =-c. The requir<strong>ed</strong> distance is 1( 3 )(- 2 ) + (- 4 )( 3 ) + 5 1 = 1 3 .<br />
. )32 + (- 4)2 5<br />
,I<br />
55. For convenience, consider the unit cube position<strong>ed</strong> so that its back left corner is at the origin, and its <strong>ed</strong>ges lie along the<br />
coordinate axes. The diagonal of the cube that begins at the origin and ends at (1, 1, 1} has vector representation (1, 1, 1).<br />
The angle B between this vector and the vector of the <strong>ed</strong>ge which also begins at the ori~i n and runs along the x-liXis [that is,<br />
. . b B (1, 1, 1) · (1, 0, 0) 1<br />
( )]<br />
1,0,0 1sg1ven ycos = I( 1 , 1 , 1 }I I( 1 , 0, 0)I = J3<br />
57. Consider the H- C-H combination consisting of the sole carbon atom and the two hydrogen atoms that are at (1, 0, 0) and<br />
(0, 1, 0) (or any H-C-H combination, for that matter). Vector representations of the line segments emanating from the<br />
carbon atom and extending to these two hydrogen atoms are ( 1 - t, 0 - t, 0 - t ) = ( t, - t, -t) and<br />
(0- t. 1-t. 0- t) ~ (- t, t, - t ). The bond angle, B, is therefore given by<br />
® 2012 Cengage Lcoming. All Righrs Reser\'cd. Me~y no1 be scann<strong>ed</strong>, copi<strong>ed</strong>. or duplicat<strong>ed</strong>, or post<strong>ed</strong> to lt publicly accessible wcbshc, in whole or in p:.trt.
SECTION 12.4 THE CROSS PRODUCT , 0 123<br />
59. Let a = (a1, a2, a3) and= (b1, b2, b3).<br />
Property 2:<br />
a· b = (a1, a2, a3 ) · (b1, b2, bs) = a1b1 + a2b2 + asbs<br />
= b1a1 + ~a2 + b3a3 = (b1, ~. bs ) · (a1 , a2, as) = b ·a<br />
Property 4:<br />
(ca) · b = (ca1, ca2, ca3) · (b1, b2, bs) = (ca1)b1 + (ca2)b2 + (ca3)b3<br />
= c(a1b1 + a2b2 + asbs) = c (a· b) = a1(cb1) + a2(~) + a 3(cbs)<br />
= (a1 , a2, as)· (cbt, c~, cbs) = a· (c b)<br />
Property 5: 0 ·a= (0, 0, 0) · (a1, a2, a3) = {O)(al) + (O)(a2) + (O)(a3) = 0<br />
61 . Ia · b l = l lal lbl cos8l = lallbl lcos 81. Since lcos81 ~ 1, Ia · bl = lal lbl lcos 81 ~ lal lbl.<br />
Note: We have equality in the case ofcos8 = · ± 1, so 8 = 0 or 8 = 1r, thus equality when a and bare parallel.<br />
63. (a) t he Parallelogram Law states that the sum of the squares of the<br />
lengths of the diagonals of a parallelogram equals the sum of the<br />
squares of its (four) sides.<br />
(b) Ia + b l 2 =(a + b) · (a + b)= lal 2 + 2(a ·b)+ lbl 2 and Ia - b l 2 = (a- b)· (a- b)= lal 2 -<br />
2(a · b)+ lb l 2 •<br />
Adding these two equations gives Ia + b l 2 + Ia- b l 2 = 2 lal 2 + 2 lbl 2 .<br />
12.4 The Cross Product<br />
1. a x b = 6 0<br />
j<br />
0 8<br />
k<br />
-2 = I~<br />
0<br />
-2 . 6 - 2 . 6 o I<br />
1 - J + k<br />
0 0 0 0 8<br />
1<br />
= [0 - (- 16)]i - (0 - O)j + (48 - O) k = 16 i + 48k<br />
1<br />
1<br />
1<br />
Now (a x b)· a = (16, 0, 48) · (6, 0, -2) = 96 + 0 - 96 = 0 and (a x b)· b = (16, 0, 48) · (0, 8, 0) = 0 + 0 + 0 = 0, so<br />
a x b is orthogonal to both a and b .<br />
j<br />
k<br />
3. a x b = 1 3 - 2 = I ~ ~~I i - ~-~ - ~ lj + l ~~ ~I k<br />
-1 0 5<br />
·= (15 - O)i - (5 - 2) j + [0 - (- 3))k = 15 i -3j +3k<br />
Since (a x b) · a= {15i - 3j + 3k) · (i + 3j - 2k) = 15 - 9-6 = O, a x bisorthogonal to a.<br />
Since (ax b)· b = (15 i - 3j + 3 k) · ( - i + 5 k) = - 15 + 0 + 15 = 0, a x b is orthogonal to b .<br />
® ::!012 CCngng.: Learning. All Rights Reserv<strong>ed</strong>. Mny not be scann<strong>ed</strong>. copi<strong>ed</strong>, ur duplicAt<strong>ed</strong>, or post<strong>ed</strong> to u publicly ncccs.c~iblc wcbsile, in whole or in pan.
124 0 CHAPTER 12 VECTORS AND THE GEOMETRY OF SPACE<br />
j k<br />
5. ax b = 1 - 1 -1<br />
= ,-~ -~ li 11 - 1, . 11<br />
1 1 J + 1 -~ lk<br />
1<br />
1 2 2 2<br />
2 1 2<br />
Now {ax b)· a= {~ i-j + ~ k ) · (i-j - k) = ~ + 1- ~<br />
~ 0 and<br />
(a x b ) · b = { ~ i - j + ~ k) · G i + j + ~ k) = ~ - 1 + ~ = 0, so a x b is orthogonal to both a and b .<br />
j k<br />
= 1 1/t I<br />
i -I t 1/t I· + I t<br />
7. a x b = t 1 1/t<br />
1 I k<br />
t21 t21 J t<br />
e 2 t2<br />
t 2 1<br />
= (1 - t) i - (t- t) j + (t 3 - t 2 ) k = (1 - t) i + (t 3 - t 2 ) k<br />
Since {a X b). a = (1 -t,O,t 3 - t2). (t, 1, 1/t) = t-e +0 +t 2 - t = 0, a X b is orthogonal to a.<br />
Since (a x b ) · b = ( 1 - t, 0, t 3 - t 2 ) • ( t 2 , t 2 , 1) = t 2 - t 3 + 0 + t 3 - t 2 = 0, a x b is orthogonal to b .<br />
9. According to the discussion prec<strong>ed</strong>in.g Theorem II, i x "j = k, so (i x j ) x k = k x k = 0 [by Example 2).<br />
11. (j - k) X (k- i) = (j - k) X k + (j - k) .X ( - i)<br />
= j X k + (- k) X k + j X (-i) + (-k) X (- i)<br />
= (j X k ) + (-1){k X k) + (- 1)(j Xi)+ (-1) 2 (k X i)<br />
= i + (-1) 0 + (-1)(- k) + J = i + j + k<br />
by Property 3 of Theorem I I<br />
by Property 4 of Theorem 11<br />
by Property 2 of Theorem I 1<br />
by Example 2 and<br />
the discussion prece<strong>ed</strong>ing Theorem II<br />
13. (a) Since b x cis a vector, the dot product a· (b x c) is meaningful and is a scalar.<br />
(b) b ·cis a scalar, so a x (b ·c) is mean<strong>ingles</strong>s, as the cross product is defin<strong>ed</strong> only for two vectors.<br />
(c) Since b x cis a vector, the cross'product a x (b x c) is meaningful and results in another vector.<br />
(d) b · cis a scalar, so the dot product a · (b ·c) is mean<strong>ingles</strong>s, as the dot product is defin<strong>ed</strong> only for two vectors.<br />
(e) Since (a . b ) and ( c . d) are both scalars, the cross product (a . b ) X ( c . d) is mean<strong>ingles</strong>s.<br />
(f) a x band c x d arc both vectors, so the dot product (ax b)· (c x d ) is meaningful and is a scalar.<br />
15. If we sketch u an~ v starting from the same initial point, we see.that the<br />
angle between them is 60°. Using Theorem 9, we have<br />
lu x vi = lullvl sin 0 = {12)(16) sin 60° = 192 · v; = 96 v'3.<br />
By the right-hand rule, u x v is direct<strong>ed</strong> into the page. .<br />
v<br />
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j k<br />
SECTION 12.4 THE CROSS PRODUCT . 0 125<br />
17. axb = 2 - 1 3 = ~-~ :Ii - I~ : lj + I~<br />
-1, k = ( -1-6) i - (2-12)j+ [4- (- 4)] k = - 7i+10j+8 k<br />
2 .<br />
4 2 1<br />
j k<br />
=1 - ~ ~ li -I:<br />
21 .<br />
b x a = 4 2 1 k = [6- (- 1)] i-{12 - 2)j + (-4 - 4) k = 7 i - lOj- 8 k<br />
~ l j + 1: -1<br />
2 -1 3<br />
Notice a x b = - b x a here, as we know is always true by Property I of Theorem II .<br />
19. By Theorem 8, the cross product of two vectors is orthogonal to both vectors. So we calculate<br />
i j k 12 11 I 3 11 I 3 21<br />
(3,2,1)x(-1, 1, 0) = 3 2 1 = i- · j + k= - i-j +5 k.<br />
1 0 -1 0 - 1 1<br />
- 1 1 0<br />
. I b h ± (-1, -1,5) ±(-1, -1,5) h . ( 1 1 5 )<br />
../3 , t at 1s, - M, -M ~ 37:3<br />
3<br />
So two umt vectors orthogona to ot are v' 1<br />
+ 1<br />
+ 25<br />
=<br />
21 . Let a= (a1, a2, as). Then<br />
j<br />
k<br />
O x a= 0 0 0 0 lk= O,<br />
= Ia: :31i - ~~ :sl j + 1:1 a2<br />
a1 a2 as<br />
j<br />
k<br />
axO = a1 a2 as<br />
0 0 0<br />
i _ I a1 as, . + I a1 a21 O<br />
0 0 0 0 J 0<br />
k = O.<br />
= I a~ as I<br />
25. ax (b+c) =a x (b1 +c1,b2 +c2,bs +c3)<br />
= (a2(bs + cs)- a3(b2 + c2), as(b1 + c1)- a1(b3 + cs) , a1(b2 + c2)- a2(b1 + q))<br />
= ((a2bs - asb2) + (a2cs - asc2), (asb1- a1bs) + (asc1 - a1c3), (a1b2 - a2b1) + (a1c2 - a2c1))<br />
= (a2bs- asb2 , aab1 - a1bs, a1b2- a2b1 ) + (a2cs - asc2, asc1- a1c3, a1c2 - a2c1)<br />
= (a x b) + (a X c)<br />
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126 0 CHAPTER 12 VECTORS AND THE GEOMETRY OF SPACE<br />
27. By plotting the vertices, we can see that the parallelogram is determin<strong>ed</strong> by the<br />
---+ ---+<br />
vectors AB = (2, 3) and AD= (4, -2}. We know that the area of the parallelogram<br />
determin<strong>ed</strong> by two vectors is equal to the length of the cross product of these vectors.<br />
. ---+<br />
In order to compute the cross product, we consider the vector AB as the three-<br />
---+<br />
dimensional vector (2, 3, 0) (and similarly for AD), and then the area of<br />
parallelogram ABCD is<br />
y<br />
j<br />
k<br />
IA13 x ml= 2 3 o =l(o)i- (o) j+ (-4 - 12)kl=l-16kl = 16<br />
4 - 2 0<br />
---+ ---+<br />
29. (a) Because the plane through P, Q, and R contains the vectors PQ and P R, a .vector orthogonal to both of these vectors<br />
---+ ---+<br />
(such as their cross product) is also orthogonal to the plane. Here PQ = ( - 3, 1, 2} and P R = (3, 2, 4}, so<br />
---+ ---+<br />
PQ x PR = ((1)(4)- (2)(2), (2)(3)- (-3)(4), (-3)(2)- (1)(3)) = (0, 18, - 9)<br />
Therefore, (0, 18, - 9) (or any nonzero scalar multiple thereof, such as (0, 2, - 1)) is orthogonal to the plane through P , Q,<br />
andR.<br />
(b) Note that the area of the tr ~angle determin<strong>ed</strong> by P, Q, and R ls equal to half of the area of the<br />
parallelogram determin<strong>ed</strong> by the three points. From part (a), the area of the parallelogram is<br />
I.PQ x PR I = I (0, 18, -9) I = y'O + 324 + 81 = v'4o5 = 9v'5, so the area of the triangle is ~ · 9v'5 = ~ .J5.<br />
---t ---+<br />
31 . (a) PQ = (4, 3, - 2) and P R = (5, 5, 1}, so a vector orthogonal to the plane through P, Q, and R is<br />
---+ ---+<br />
PQ x P R = ((3)(1) - ( -2)(5), ( -2)(5) - (4)(1), (4)(5) - (3) (5)) = (13, - 14, 5) [or any scalar mutiple thereof].<br />
---+ ---+<br />
(b) The area of the parallelogram determin<strong>ed</strong> by PQ and PR is<br />
IPQ x P'RI = 1(13,-14, 5)1 = yl13 2 + (- 14) 2 +5 2 = v'395,sotheareaoftriangle PQR is ~v'39Q.<br />
33. By Equation 14, the <strong>vol</strong>ume of the parallelepip<strong>ed</strong> detennin<strong>ed</strong> by a , b , and cis the magnitude of their scalar triple product,<br />
1 2 3 11 21 1- 1<br />
which is a · (b x c) = -1 1 2 = 1 - 2<br />
. 1 4 2<br />
2 1 4<br />
Thus the <strong>vol</strong>ume of the parallelepip<strong>ed</strong> is 9 cubic units.<br />
---t --+ --+<br />
35. a= PQ = (4, 2, 2}, b = PR = (3, 3, -1), and c = P S = (5,5, 1).<br />
'<br />
24 1 + 31-1 2 11<br />
= 1(4-2)-2(- 4-4) + 3(-1-2) = 9.<br />
so the <strong>vol</strong>ume of the parallelepip<strong>ed</strong> is 16 cubic units.<br />
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SECTION 12.4 THE CROSS PRODUCT D 127<br />
1 5 - 2<br />
37. u · (v x w)= 3-1 0 =1 1- 1·<br />
0 3 0 3<br />
1- 5 1 1+(- 2) 1 -ll = 4 + 60 - 64 = 0,whichsaysthatthe<strong>vol</strong>ume<br />
9 - 4 5 - 4 5 9 . .<br />
5 9 - 4<br />
of the parallelepip<strong>ed</strong> determin<strong>ed</strong> by u, v and w is 0, and jhus these three vectors are coplanar.<br />
39. The magnitude of the torque is Jr J = Jr x FJ = JrJJFJ sinO = (0.18 m)(60 N) sin(70 +lOt = 10.8 sin80° r::::: 10.6 N·m.<br />
41 . Using the notation of the text, r = {0, 0.3, 0) and F has direction (0, 3, -4). The angle(} between them can be determin<strong>ed</strong> by<br />
(0, 0.3, 0) . {0, 3, -4) . 9 0.9<br />
cos 9 = => cos = =><br />
J(o, o.3, o)JI{o,3, -4)1 (0.3)(5)<br />
cosO = 0.6 => (J r::::: 53.1°. Then Jr l = lri iFI sin(} =><br />
100 = 0.3IFisin53.1° => 1F i r:::::417N.<br />
43. From Theorem 9 we have Ia x bl = la l lbl sin 9, where 0 is the angle between a and b, and from Th,eorem 12.3.3 we have<br />
a· b = lal lb l cosO => Ja l lbl = a · be. Substituting the second equation into the first gives Ia x b l = a · b sin B, so<br />
cos cos (J<br />
Ia X b l = tan9. Here Ia X b l = 1(1, 2, 2)1 = V1 + 4 + 4 = 3, so tan() = Ia x bbl = ~ = J3 => () = 60° .<br />
. a· b a · v3<br />
45. (a)<br />
The distance between a point and a line is the length of the perpendicular<br />
from the point to th~ line, here In I =d. But referring to triangle PQS,<br />
a<br />
d = In I = IQP' sin (J = lbl sin 9. But 8 is the angle between QP = b<br />
--+ Ia x bl<br />
and QR = ~ ·Thus ~y Theorem 9, sinO = lai Jb l<br />
d d = lb1 . 9 = Jbi Ja x bl = Ia x b J<br />
an so 'I sm Jai Jbl laJ<br />
--+ ---+<br />
(b) a= QR = (- 1, - 2, - 1) and b = QP = (1, - 5, - 7}. Then<br />
a x b = (( -2)( -7) - ( -1){ 7 5), ( - 1)(1)- ( - 1)( - 7) , ( -1)( -5)- ( - 2)(1)} = (9, - 8, 7).<br />
Thus the distance is d = Ia [:I bl = ~ JBl + 64 + 49 = ji¥ = j¥.<br />
47. From Theorem 9 we have Ia x b l = [al[b[ sin 9 so<br />
Ja x b] 2 = [a[ 2 [b[ 2 sin 2 () = [a[ 2 lb[ 2 (1 - cos 2 B)<br />
= fa i 2 Jb[ 2 - ([ai Jb [ cos 9) 2 = [a[ 2 lb[ 2 - (a · b) 2<br />
by Theorem 12.3.3.<br />
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128 D CHAPTER 12 VECTORS AND THE GEOMETRY OF SPACE<br />
49. (a - b) x (a + b) = (a - b) x a + (a - b) x b<br />
=ax a+ (-b)x a +ax b+(- b)x b<br />
=(a X a)- (b x a)+ (a X b)- (b x b)<br />
= 0 - (b x a) + (a x b)- 0<br />
=(ax b)+ (a x b)<br />
= 2(a x b)<br />
by Property 3 of Theorem 11<br />
.<br />
by Property 4 ofTheorem I I<br />
by Property 2 ofTheorem II (with c =: -1)<br />
by Example2<br />
by Property I of Theorem ll<br />
51. a X (b X c) + b X (c X a) + c X (a X b)<br />
= [(a · c)b- (a · b)c] + [(b · a)c- (b · c)a] + [(c · b)a - (c: a)b]<br />
= (a · c)b - (a· b)c+ (a · b)c- (b · c)a+ (b · c)a- (a · c)b = 0<br />
by Exercise 50<br />
53. (a) No. If a · b = a · c, then ·a · (b - c) = 0, so a is perpendicular to b - c, which can happen if b -:f. c. For example,<br />
let a = (1, 1, 1), b = (1, 0, 0) and c = (0, 1, 0).<br />
(b) No. If a x b =ax c then a x (b- c) = 0, which implies that a is parallel to b- c; which of course can happen<br />
ifb -:f. c.<br />
(c) Yes. Since a· c = a· b , a is perpendicular to b - c, by part (a). From part (b), a is also parallel to b - c. Thus since<br />
a -:f. 0 but is both parallel and perpendicular to b - c, we have b - c = 0, so b = c.<br />
12.5 Equations of Lines and Planes<br />
1. (a) True; each of the first two lines has a direction vector parallel to the direction vector of the third line, so these vectors are<br />
each scalar multipl~s of the third direction vector. Then the first two direction vectors are also scalar multiples of each<br />
other, so these vectors, and hence the two lines, are parallel.<br />
(b) False; for example, the x- and y-axes are both perpendicular to the z-axis, yet the x- andy-axes are not parallel.<br />
(c) True; each of the first two planes has a normal vector parallel to ~he normal vector of the third plane, so these two normal<br />
vectors are parallel to each other and the planes are parallel.<br />
(d) False; for example, the xy- and yz-planes are not parallel, yet they are both perpendicular to the xz-plane.<br />
(e) False; the x- and y-axes are not parallel, yet they are both parallel to the plane z = 1.<br />
(f) True; if each line is perpendicular to a plane, then the lines' direction vectors are both parallel to a normal vector for the<br />
plane. Thus, the direction vectors arc parallel to each other and the lines are parallel.<br />
(g) False; the planes y = 1 and z = 1 are not parallel, yet they are both parallel to the x-axis.<br />
(h) True; if each plane is perpendicular to a line; then any normal vector for each plane is parallel to a direction vector for the<br />
line. Thus, the normal vectors are parallel to each other and the planes are parallel.<br />
(i) True; see Figure 9 and the accompanying discussion.<br />
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SECTION 12.5 EQUATIONS OF LINES AND PlANES 0 129<br />
(j) False; they can be skew, as in Example 3.<br />
(k) True. Consider any normal vector for the plane and any direction vector for the line. If the normal vector is perpendicular<br />
to the direction vector, the line and plane are parallel. Otherwise, the vectors meet at an angle B, 0° :S B < goo., and the<br />
line will intersect the plane at an angle goo -B.<br />
3. For this line, we have r 0 = 2 i + 2.4j + 3.5 k and v = 3 i + 2 j - k, so a vector equation is<br />
r = ro + t v = (2 i + 2.4 j + 3.5 k ) + t(3 i + 2j - k) = (2 + 3t) i + {2.4 + 2t) j + (3.5 - t) k and parametric equations are<br />
X = 2 + 3t, y = 2.4 + 2t, Z = 3.5 - t.<br />
5. A Line perpendicular to the given plane has the same direction as a normal vector to the plane, such as<br />
n = (1, 3, 1). So r 0 = i + 6 k, and we can take v = i + 3j + k. Then a vector equation is<br />
. '<br />
r = {i + 6 k) + t(i + 3j + k ) = (1 + t) i + 3tj + (6 + t) k, and parametric equations are x = 1 + t, y = 3t, z = 6 + t.<br />
7. The vector v = (2 - 0, 1 - ~ . - 3- 1) = (2, ~ . -4) is parallel to the line. Letting Po = (2, 1, -3), parametric equations<br />
2 1<br />
1 3 4 h'l · . X - 2 y - 1 Z + 3<br />
2 - = 112<br />
are x = 2 + t, y = + 2 t, z =- - .t, w 1 e symmetnc equat1ons are -<br />
X - 2 = 2 y _ 2 = Z + 3.<br />
2 - 4<br />
= ---=4 or<br />
9. v = (3- ( -8), - 2- 1,4- 4) = (11, - 3,·0), and letting Po= (- 8, 1, 4), parametric equations are x = -8 + llt,<br />
4 0 4 I ·1 . . x + 8 y - 1 4 N . h h h d' .<br />
y = 1 - 3t, z = + t = , w 11 e symmetnc equations are ----u- :;::<br />
_ 3<br />
, z = . ottcc ere t at t e trectton number<br />
c = 0, so rather than writing z ~ 4 in the symmetric equation we must write the equation z = 4 separately.<br />
11. The line has direction v = (1, 2, 1). Letting Po = (1, - 1, 1), parametric equations are x = 1 + t, y = -1 + 2t, z = 1 + t<br />
. . 1 y + 1 1<br />
an d symmetriC equatiOns are x - = - - = z - .<br />
2<br />
13. Direction vectors of the lines are v 1 = (- 2 - ( - 4), 0 - ( - 6), - 3 - 1) = (2, 6, - 4) and<br />
v 2 = (5 - 10,3 - 18, 14 - 4) = ( -5, - 15, 10), and since v 2 = -:-~ v1 ,<br />
the direction vectors and thus the lines are parallel.<br />
15. (a) The li ne passes through the point (1, - 5, 6) and a direction vector for the line is ( - 1, 2, - 3), so symmetric equations for<br />
. x- 1 y+5 z -6<br />
the line are ---=1 = - 2<br />
- = --=3.<br />
x- 1 y+5 0-6 x - 1<br />
(b) The line intersects the xy-plane when z = 0, so we ne<strong>ed</strong> -=1' = - 2<br />
- = --=3 or -=1' = 2 => x = - 1,<br />
Y ~ 5 = 2 => y = -1. Thus the point of intersection with the xy-plane is ( - 1, - 1, 0). Similarly for the yz-plane,<br />
we ne<strong>ed</strong> x = 0 => 1 = y ~ 5 = z ~ 3 6 => y = - 3, z = 3. Thus the line intersects the yz-plane at {0, - 3, 3). For<br />
the xz-plane, we ne<strong>ed</strong> y = 0 => => x = - ~, z = - ~ . So the line intersects the xz-plane<br />
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130 D CHAPTER 12 VECTORS AND THE GEOMETRY OF SPACE<br />
\~~ From Equation 4, the line segment. from ro ~ 2 i - j + .4 k to r 1 = 4 i + 6j + k is<br />
r (t) = (1 - t)ro +tr1 = (1 - t)(2 i - j +4 k) +t(4 i + 6j + k ) = (2 i -j +4k) +t(2 i + 7j- 3k), 0 ::=; t ::=; 1.<br />
l 19. Since the direction vectors (2, -1, 3} and (4, - 2, 5} are not scalar multiples of each other, the Lfnes aren't parallel. For the<br />
\ ' ' \- -<br />
lines to intersect, we must be able to find one value oft and one value of s that produce the same point from the respective<br />
..------______: ----·----- --~·-----------"'--<br />
parametric equations. Thus we ne<strong>ed</strong> to satisfy tlie following three equations: 3 + 2t = 1 + 48, 4 - t = 3 - 28,<br />
...-----<br />
1 + 3t = 4 + 5s. Solving the last two equations we get t = 1, 8 = 0 and checking, we see that these values don't satisfy the<br />
first equation. Thus the lines aren't parallel and don't intersect, so they must be skew lines.<br />
r<br />
·• 21. Since the direction vectors (1, - 2, - 3} and (1, 3, -7) aren't scalar multiples of each other, the lines aren't parallel. Parametric<br />
equations of the lines are L1: x = 2 + t, y = 3- 2t, z = 1 - 3t and L2: x ·= 3 + 8, y = -4 + 38, z = 2 -78. Thus, for the<br />
lines to intersect, the three equations 2 + t = 3 + 8, 3 - 2t = -4 + 3s, and 1-3t = 2-78 must be satisfi<strong>ed</strong> simultaneously.<br />
Solving the first two ~quations gives t = 2, 8 = 1 and checking,_ we see that these values do satisfy the third equation, so the<br />
lines intersect when t = 2 and 8 = 1, that is, at the point ( 4, -1, -5).<br />
23. Since the plane is perpendicular to the vector (1, -2, 5), we can take (1, - 2, 5} as a normal vector to the plane.<br />
(0, 0, 0) is a point on the plane, so setting a= 1, b = -2, c = 5 and xo = 0, yo = 0, z 0 = 0 in Equation 7 gives<br />
1(x - 0) + (-2)(y- 0) + 5(z - 0) = 0 or x- 2y + 5z = 0 as an equation of the plane.<br />
25. i + 4j + k = (1, 4, 1} is a normal vector to the plane and (- 1, ~, 3) is a point on the plane, so setting a= 1, b = 4, c = 1,<br />
xo = - 1, yo = ~, zo = 3 in Equation 7 gives 1 (x - (-1)] + 4 (11 - ~) + 1(z - 3) = 0 or x + 4y + z = 4 as an equation of<br />
the plane.<br />
27. Since the two planes are parallel, they will have the same normal vectors. So we can taken = (5, -1, - 1), and an equation of<br />
the plane is5(x-1) -1(y- (- 1)]-1[z- (-1)] = Oor5x -11- z = 7.<br />
29. Since the two planes are parallel, they will have the same normal vectors. So we can taken = (1, 1, 1), and an equation of the<br />
plane is 1(x- 1) + 1 (y - ~) + 1 (z - ~) = 0 or x + y + z = Jt or 6x + 6y + 6z = 11.<br />
31. Here the vectors a= (1- 0, 0 - 1, 1 1 1) = (1, -1, 0) and b = (1 - 0, 1- 1, 0- 1) = (1, 0, - 1) lie in the plane, so<br />
ax b is a normal vector to the plane. Thus, we can taken= ax b = (1 - 0, 0 + 1, 0 + 1) = (1, 1, 1}. If P o is the point<br />
(0, 1, 1), an equation of the plane is 1(x- 0) + 1(y - 1) + 1(z - 1) = 0 or x + y + z = 2.<br />
33. Here the vectors a = (8 - 3, 2 - ( - 1), 4- 2} = (5, 3, 2) and b = (-1 - 3; - 2 - ( -1), -3- 2} = (-4, - i , - 5) lie in<br />
the plane, so a normal vector to the plane is n = a x b = (- 15 + 2, -8 + 25, - 5 + 12) = (-13, 17, 7) and an equation of<br />
the plane is - 13(x - 3) + 17[y- (-1)) + 7(z - 2) = 0 or -13x + 17y + 7z = -42.<br />
35. If we first find two nonparallel vectors in the plane, their cross product wi ll be a normal vector to the plane. Since the given<br />
line lies in the plane, its direction vector a= (-2, 5, 4) is one vector in the plane. We can verify that the given point (6, 0, -2)<br />
© 2012
SECTION 12.5 EQUATIONS OF LINES AND PLANES 0 131<br />
does not lie on this line, so to find another nonparallel vector b which lies in the plane, we can pick any point on the line and<br />
find a vector connecting the points. If we putt = 0, we see that ( 4 1 3 1<br />
7) is ·on the line, so<br />
b = (6 - 4,0- 3, -2 -7) = (2, -3, -9) and n = a x b = (-45 + 12,8-18,6 -10) = (- 33, - 10, - 4). Thus, an<br />
equation of the plane is -33{x - 6)- 10(y- 0)- 4[z - ( -2)] = 0 or 33x +lOy + 4z = 190.<br />
37. A direction vector for the line of intersection is a= n1 x n 2 = {1, 1, - 1) x (2, -1, 3) = (2 1<br />
- 5 1<br />
- 3), and a is parallel to the<br />
desir<strong>ed</strong> plane. Another vector parallel to the plane is the vector .connecting any point on the line of intersection to the given<br />
point ( - 1 1<br />
2 1<br />
1) in the plane. Setting x = 0, the equations of the planes r<strong>ed</strong>uce toy- z = 2 and -y+ 3z = 1 with<br />
simultaneous solution y = ~ and z = ~. So a point on the line is ( 0 1 ~ 1 ~) and another vector parallel to the plane is<br />
( - 1 1 -~ , -~) .Then a normal vector to the plane is n = (2, - 5 1 -3) x (-1 1 -~, -~ ) = (-2,4, - 8) and an equation of<br />
the plane is -2{x + 1) + 4(y- 2) - B(z - 1) = 0 or x - 2y + 4z = --:1.<br />
39. If a plane is perpendicular to two other planes, its nonnal vector is perpendicular to the normal vectors of the other two planes.<br />
Thus (2, 1, -2) x (1, 0, 3) = (3- 0, -2- 6, 0 - 1) = (3, -8, - 1) is a normal vector to the desir<strong>ed</strong> plane. The point<br />
(1, 5, 1) lies on the plane, so an equation is 3(x - 1) - B(y - 5)-(z- 1) = 0 or 3x - By- z = - 38.<br />
41. To find the x-intercept we set y = --- z = 0 in the equation 2x + 5y + z = 10<br />
and obtain 2x = 10 => x = 5 so the x-intercept is (5, 0, 0). When<br />
x = z = 0 we get 5y = 10 => y = 2, so they-intercept is (0, 2, 0).<br />
Setting x = y = 0 gives z = 10, so the z-intercept is (0, 0, 10) and we<br />
graph the portion of the plane that lies in the first octant.<br />
X<br />
43. Setting y = z = 0 in the equation 6x - 3y + 4z = 6 gives 6x = 6 =><br />
x = 1, when x = z = 0 we have -3y = 6 => y = - 2, and x = y = 0<br />
implies 4z = 6 => z = ~.so the intercepts are (1, 0, 0), {0, -2, 0), and<br />
(0 1 '0, ~). The figure shows the portion of the plane cut off by the coordinate<br />
planes.<br />
X<br />
I<br />
@ ubstitute the parametric equations ofthe line into the equation of the plane: (3- t) - (2 + t) + 2{5t) = 9 =><br />
Bt = 8 => t = 1. Therefore, the point of intersection of the line and the plane is given by x = 3 - 1 = 2, y = 2 + 1 = 3,<br />
and z = 5(1) =5 1<br />
that is, the point (2, 3, 5).<br />
47. Parametric equationS for the line are x = t, y = 1 + t, z = ~t and substituting into the equation of the plane gives<br />
4{t)-{1+t)+3(~t)=8 => ~t=9 => t=2.Thusx = 2,y=1+2 .=3,z=H2)=1andthepointof<br />
intersection is (2, 3 1<br />
1).<br />
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132 D CHAPTER 12 VECTORS AND THE GEOMETRY OF SPACE<br />
49. Setting x = 0, we see that (0, 1, 0) satisfies the equatio~s of both planes, so that they do in fact have a line of intersection.<br />
v = o 1 x 0 2 = (1, 1, 1} x (1, 0, 1) = (1, 0, -1} is the direction of this line. Therefore, direction numbers of the intersecting<br />
line are 1, 0, - 1.<br />
51. Normal vectors for the planes are 01 = (1, 4, -3} and 0 2 = ( -3, 6, 7), so the normals (and thus the planes) aren't parallel.<br />
. .<br />
But 01 · 0 2 ,;, -3 + 24 - 21 = 0, so the normals (and thus the planes) are perpendicular.<br />
53. Normal vectors for the planes are 01 = (1, 1, 1) and 02 = (1, -1, 1). The normals are not parallel, so neither are tl1e planes.<br />
Furthe'rmore, o 1 · o 2 = 1 - 1 + 1 = 1 =F 0, so the planes aren't perpendicular. The angle between them is given by·<br />
= __ 1_ = ~ 1(1)<br />
lolllo21 .j3.j3 3 => O=cos- 3 ::::::70.5o.<br />
cos8 = O t . 0 2<br />
55. The normals are 0 1 = (1, -4, 2) and 0 2 = (2, -8, 4). Since 02 = 2o 1 , the normals (and tlms the planes) are parallel.<br />
57. (a) To find a point on the line of intersection, set one of the variables equal to a constant, say z = 0. (This will fail if the line .of<br />
intersection does not cross the xy-plane; in that case, try setting x or y equal to 0.) The equations of the two planes r<strong>ed</strong>uce<br />
to x + y = 1 and x + 2y = 1. Solving these two equations gives x = 1, y = 0. Thus a point on ilie line is (1, 0,·0).<br />
A vector v in the direction of this intersecting line is perpendicular to the normal vectors of both planes, so we can take<br />
v = o 1 x o 2 = (1, 1, 1) x (1, 2, 2) = (2- 2, 1 - 2, 2- 1) = (0, - 1, 1}. By Equations 2, parametric equations for tlle<br />
line are .'1: = 1, y = - t, z = t.<br />
. 0] . 0 2 1 + 2 + 2 5 h fi 0 -1 ( 5 ) 15 80<br />
(b) The angle between the planes satisfies cos 0 = I II I = r;; If\ = r;;. T ere ore = cos r;; :::::: . .<br />
0] 02 v 3 v 9. 3 v 3 3 v 3<br />
59. Setting z = 0, tlle equations of the two planes become 5x- 2y = 1 and 4x + y = 6. Solving tllese two equations gives<br />
x = 1, y = 2 so a point on the line of intersection is (1, 2, 0). A vector v in the direction of this intersecting line is<br />
perpendicular to the normal vectors ofbotll planes. So we can use v = o 1 x 0 2 = (5, - 2, -2) x (4, 1, 1} = (0, - 13, 13} or<br />
equivalently we can take v = (0, -1, 1}, and symmetric equations for the line are x = 1, y _=- 1<br />
2 = I or x = 1, y-2 = -z.<br />
61. The distance from a point (x, y , z) to (1, 0, - 2) is d 1 = ..j(x- 1)2 + y 2 + (z + 2)2 and the distance from (x, y, z) to<br />
(3,4, 0) is d 2 = ..j(x- 3) 2 + (y- 4) 2 + z 2 . The plane consists of all points (x, y, z) where d1 = d2 => d{ = di
SECTION 12.5 EQUATIONS OF LINES AND PLANES 0 133<br />
therefore bcx + aetJ + abz = abc + 0 + 0 or bcx + acy + abz = abc. Notice that if a ol 0, b ol 0 and c ol 0 then we can<br />
' . X '1j Z 1 Th' . d . be I<br />
rewnte the equat1on as - + - + - = . ts 1s a goo equat1on to remem r.<br />
a b c<br />
65. Two vectors which are perpendicular to the·requir<strong>ed</strong> line are the normal of the given plane, (1, 1, 1), and a direction vector for<br />
the given line, (1, - 1, 2). So a direction vector for the requir<strong>ed</strong> line is (1, 1, 1} x (1, -1, 2} = (3, - 1, -2}. Thus L is given<br />
by (x, y, z} = (0, 1, 2} + t(3, -1, - 2}, or in parametric form, x = 3t, y = 1- t, z = 2- 2t.<br />
67. Let Pi have normal vector D i. Then n1 = (3, 6, -3}, n 2 = (4, -12, 8), n a = (3, -9, 6), ll4 = (1, 2, - 1}. Now n 1 = 3D4,<br />
so n1 and n4 are parallel, and hence P1 and P4 are parallel; similarly P 2 and P3 are parallel because n2 = ! n 3. However, n 1<br />
and n 2 are not parallel (so not all four planes are parallel). Notice that the point (2, 0, 0) lies on both P 1 and P 4 , so these two<br />
planes are identical. The point ( ~ , 0, 0) lies on P2 but not on P3, so these are different planes.<br />
69. Let Q = (1, 3, 4) and R = (2, 1, 1), points on the line corresponding to t = 0 and t = 1. Let<br />
. .<br />
---> --+<br />
P = ( 4, 1, - 2). Then a = QR = (1, - 2, -3), b = QP = (3, - 2, - 6) . The distance is<br />
d = Ia x bl = 1(1, -2,-3} x (3,-2,-6}/ = 1(6, - 3,4)1 = J6 2 + (-3)2 + 42 = J6I = {61<br />
/al 1(1, -2, -3}/ 1(1, - 2, - 3}1 JP+(- 2)2 + (-3)2 v'I4 V14'<br />
1 BE<br />
. tl<br />
9<br />
d'ta . D iax1+by1+cz1 +d/ /3(1) +2(- 2)+6(4)-5/ /181 18<br />
7 . y quat1on , 1e IS nee IS = = = r.n = - .<br />
Ja2+b2+ c2 . J 32 + 22 + 62 v49 7<br />
73. Put y = z = 0 in the equation of the first plane to get the point (2, 0, 0) on the plane. Because the planes are parallel, the<br />
distance D between them is the distance from (2, 0, 0) to the second plane. By Equation 9,<br />
D = /4(2) - 6(0) + 2(0) - 31 = _ 5_ = _5_ or 5 VT4.<br />
J42 + ( - 6)2 + (2)2 V56 2 VI4 28<br />
75. The distance between two parallel planes is the same as the distance between a point on on~ ofilie planes and the oilier plane.<br />
Let Po = (xo, yo, zo) be a point on the plane given by ax+ by + cz + dt = 0. Then axo + byo + czo + d 1 = 0 and the<br />
distance between Po and ilie plane given by ax + by + cz + d2 = 0 is, from Equation 9,<br />
D = /ax o + byo + czo + d2/ = /-dl + d2 / , /d1 - d2 /<br />
Ja2 +b2+c2 Ja2 + b2+ c2 Ja2+ b2+c2<br />
77. £ 1: x = y = z =? x = y (1). £2: x + 1 = y/2 = z/3 =? x + 1 = y/2 (2). The solution of (l ) and (2) is<br />
x = y = - 2. However, when x = -2, x = z =? z = - 2, but x + 1 = z/3 =? z = - 3, a contradiction. Hence the<br />
lines do not intersect. For L1. v1 = (1, 1, 1), and for L2, v2 = (1, 2, 3}, so the lines are not parallel. Thus ilie lines are skew<br />
lines. If two lines are skew, they can be view<strong>ed</strong> as lying in two parallel planes and so ilie distance between the skew lines<br />
would be the same as the distance between these parallel planes. The common normal vector to the planes must be<br />
perpendicular to boili (1, 1, 1) and (1, 2, 3), the direction vectors of the two lines. So set<br />
® 2~ 12 Ccogogc Lc;uning.. All Rights R.esen·cd. Mny not be SC4M<strong>ed</strong>, copi<strong>ed</strong>, or duplicat<strong>ed</strong>, or post<strong>ed</strong> to a publicly accessible website, in whole or in pan.
134 D CHAPTER 12 VECTORS AND THE GEOMETRY OF SPACE<br />
n = {1, 1, 1) x {1, 2, 3) = (3- 2, - 3 + 1, 2- 1) = {1, - 2, 1). From above, we know that ( - 2, -2, - 2) and ( -2, - 2, -3)<br />
are points of £1 and~ respectively. So in the notation of Equation 8, 1{ - 2)- 2( - 2) + 1( - 2) + d 1 = 0 ~ d 1 = 0 and<br />
1( -2) - 2( -2) + 1( -3) + d2 = 0 ~ d2 = 1.<br />
By Exerci~e 75, the distance between these two skew lines is D =<br />
IO - II<br />
. v 1 +4 + 1<br />
Alternate solution (without reference to planes): A vector which is perpendicular to both of the lines is<br />
n = (1, 1,1) x (1, 2, 3) = (1, - 2, 1). Pick any point on each of the lines, say ( - 2, -2, -2) and ( -2, - 2, - 3), and form the<br />
vector b = (0, 0, 1) connecting the two points. The distance between the two skew lines is the absolute value of the 'sca,lar<br />
. . . . ln·bl 11 · 0-2·0+1·11 1<br />
proJection ofb along n , that IS, D = - - - = = u;·<br />
. 1n 1 v 1 +4+1 v6<br />
79. A direction vector for £1 is v1 = (2, 0, -1) and a direction vector for £ 2 is v 2 = (3, 2, 2). These vectors are not parallel so<br />
neither are the lines. Parametric equations for the lines are £1 : x = 2t, y = 0, z = - t, and £2: x = 1 + 3s, y = - 1 + 2s,<br />
z = 1 + 2s. No values oft and s satisfy·these equations simultaneously, so the lines don't intersect and hence are skew . . we<br />
can view the lines as lying in two parallel planes; a common normal vector to the planes is n = v 1 X v2 = (2, - 7, 4). Line<br />
£ 1 passes through the origin, so (0, 0, 0) lies on one of the planes, and. (1, - 1, 1) is a point on £2 and therefore on the other<br />
plane. Equations of the planes then are 2x- 7y + 4z = 0 and 2x- 7y + 4z- 13 = 0, and by Exercise 75, the distance<br />
I0-(-13)1 13<br />
between the two skew lines is D = = rcn.<br />
v4+49 +16 v69<br />
Alternate solution (without reference to planes): Direction vectors of the two lines are v1 = (2, 0, -1) and v 2 = (3, 2, 2).<br />
Then n = v 1 x v 2 = (2, - 7, 4} is perpendicular to both lines. Pick any point on each of the lines, say (0, 0, 0) and (1, -1, 1),<br />
and form the vector b = (1, -1, 1) connecting the two points. Then the distance between the two skew lines is the absolute<br />
value of the scalar projection ofb along n , that is, D = ln 1·1bl = 1 2 + 7 + 4 1 = ~ 6 3 9 .<br />
D V4 + 49+16 YO::!<br />
81 . If a =F 0, then ax + by+ cz + d = 0 =? a(x + dja) + b(y - 0) + c(z - 0) = 0 which by (7) is the scalar equation of the<br />
plane through the point ( -d/ a, 0, 0) with normal vector (a , b, c). Similarly, if b =F 0 (or if c =F 0) the equation of the plane can<br />
be rewritten as a(x - 0) + b(y + djb) + c(z - 0) = 0 [or as a(x- 0) + b(y- 0) + c(z + djc) = 0] which by (7) is the<br />
scalar equation of a plane through the point (0, - d/b, 0) [or the point (0, 0, -d/c)] with normal vector (a, b, c).<br />
SECTION 12.6 CYLINDERS AND QUADRIC SURFACES 0 135<br />
12.6 Cylinders and Quadric Surfaces<br />
1. (a) In JR 2 , the equation y = x 2 represents a parabola.<br />
X<br />
(b) In R 3 , the equation y = x 2 doesn't in<strong>vol</strong>ve z, so any<br />
horizontal plane with equation z = k intersects the graph<br />
z<br />
in a curve with equation y = x 2 • Thus, th~ surface is a<br />
parabolic cylinder, made up of infinitely many shift<strong>ed</strong><br />
copies of the same parabola. The rulings are parallel to<br />
the z-axis.<br />
X<br />
(c) In R 3 , the equation z = y 2 also represents a parabolic<br />
z<br />
cylinder. Since x doesn't appear, the graph is form<strong>ed</strong> by<br />
moving the parabola z = y 2 in the direction of the x-axis.<br />
Thus, the rulings of the cylinder are parallel to the x-ax.is.<br />
3. Since y is missing from the equation, the vertical traces<br />
x 2 + z 2 = 1, y = k, are copies of the same circle in<br />
the plane y = k. Thus the surface x 2 + z 2 = 1 is a<br />
circular cylinder with rulings parallel to they-axis.<br />
5. Since x is missing, each vertical trace z = 1 - y 2 ,<br />
x = k, is a copy of the same parabola in the plane<br />
x = k. Thus the surface z = 1 - y 2 is a parabolic<br />
cylinder with rulings parallel to the x-axis.<br />
.1'<br />
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136 D CHA~TER 12 VECTORS AND THE GEOMETRY OF SPACE<br />
7. Since z is missing, each horizontal trace xy = 1,<br />
z = k, is a copy of the same hyperbola in the plane<br />
z = k. Thus the surface xy = 1 is a hype~bolic<br />
cylinder with rulings parallel to the z-axis.<br />
9. (a) The traces ofx 2 + y 2 - z 2 = 1 in x = k are y 2 - z 2 = 1 - k 2 , a family of hyperbolas. (Note that the hyp!!rbolas are<br />
orient<strong>ed</strong> differently for -1 < k < 1 than for k < - 1 or k > 1.) The traces in y = k are x 2 - z 2 = 1 - k 2 , a similar<br />
family of hyperbolas. The traces in z = k are x 2 + y 2 = 1 + k 2 , a family of circles. Fork = 0, the trace in the<br />
xy-plane, the circle is of radius 1. As I ki increases, so does the radius of the circle. This behavior, combin<strong>ed</strong> with the<br />
hyperbolic vertical traces, gives the graph of the hyperboloid of one sheet in Table 1.<br />
(b) The shape of the surface is unchang<strong>ed</strong>, but the hyperboloid is<br />
rotat<strong>ed</strong> so that its axis is the y-axis. Traces in y = k are circles,<br />
while traces in x =. k and z = k are hyperbolas.<br />
(c) Completing the square in y gives x 2 + (y + 1) 2 -<br />
z 2 = 1. The<br />
surface is a hyperboloid identical to the one in part (a) but shift<strong>ed</strong><br />
one unit in the negative y-direction.<br />
11. For x = y 2 + 4z 2 , the traces in x = k are y 2 + 4z 2 = k. When k >. 0 we<br />
have a family of ellipses. When k = 0 we have just a point at the origin, and<br />
the trace is empty for k < 0. The traces in y = k are x = 4z 2 + k 2 , a<br />
family of parabolas opening in the positive x-direction. Similarly, the traces<br />
in z = k are x = y 2 + 4k 2 , a family of parabolas opening in the positive<br />
x-direction. We recognize the graph as an elliptic parabofoid with axis the<br />
x-axis and vertex the origin.<br />
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0<br />
-<br />
SECTION 12.6 CYLINDERS AND QUADRIC SURFACES 0 137<br />
13. x 2 = y 2 + 4z 2 . The traces in x = k are the ellipses y 2 + 4z 2 = k 2 . The<br />
traces in y = k are x 2 -<br />
4z 2 = k 2 , hyperbolas for k =/; 0 and two<br />
intersecting lines if k = 0. Similarly, the traces in 'z = k are<br />
x 2 - y 2 = 4k 2 , hyperbolas for k =/; 0 and two intersecting lines if k = 0.<br />
We recognize the graph as an elliptic cone with axis the x-axis and vertex<br />
the origin.<br />
15. - x 2 + 4y 2 - z 2 = 4. The traces in x = k are the hyperbolas<br />
4y 2 - z 2 = 4 + k 2 . The traces in y = k are x 2 + z 2 = 4k 2 - 4, a family of<br />
circles for lkl > 1, and the traces in z ·= k are 4y 2 -<br />
x 2 = 4 + k 2 , a family<br />
of hyperbolas. Thus the surface is a hyperboloid of two sheets with<br />
axis the y-axis.<br />
17. 36x 2 + y 2 + 36z 2 = 36. The traces in x = k are y 2 + 36z 2 = 36(1 - k 2 ),<br />
a family of ellipses for lkl < 1. (The traces are a single point for lkl = 1<br />
and are empty for lkl > 1.) The traces in y = k' are the circles<br />
36x 2 + 36z 2 = 36- k 2 {"} x 2 + z 2 = 1 - tok 2 , lkl < 6, and the<br />
traces in z = k are the ellipses 36x 2 + y 2 = 36(1 - k 2 ), 1~ 1 < 1. The<br />
graph is an ellipsoid center<strong>ed</strong> at the origin with intercepts x = ± 1, y = ± 6,<br />
z = ±1.<br />
19. y = z 2 - x 2 • The traces in x = k are the parabolas y = z 2 -:- k 2 ;<br />
the traces in y = k are k = z 2 -<br />
x 2 , which are hyperbolas (note the hyperbolas<br />
are orient<strong>ed</strong> differently fork > 0 than fork < 0); and the traces in z = k are<br />
the parabolas y = k 2 .-<br />
y z 2 x2<br />
x 2 • Thus, l = 12<br />
- ]2 is a hyperbolic paraboloid.<br />
X<br />
2 2<br />
21. This is the equation of an ellipsoid: x 2 + 4y 2 + Dz 2 = x 2 + ~ + ___!___, = 1, with x-intercepts ±1, y-intercepts ±i<br />
(1/2) (1/3t<br />
and z-intercepts ±~. So the major axis is the x-axis and the only possible graph is Vll.<br />
23. This is the equation of a hyperboloid of one sheet, with a = b = c = 1. Since the coefficient of y 2 is negative, the axis of the<br />
hyperboloid is the y-axis, hence the correct graph is II.<br />
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138 D CHAPTER 12 VECTORS AND THE GEOMETRY OF SPACE<br />
25. There are no real values of x and z that satisfy this equation for y < 0, so this surface does not extend to the left of the<br />
. .<br />
xz-plane. The surface intersects the plane_y = k > 0 in an ellipse. Notice that y occurs to the first power whereas x and z<br />
occur to the second power. So the surface is an elliptic paraboloid with axis they-axis. Its graph is VI.<br />
27. This surface is a cylinder because the variable y is missing from the equation.' The intersection of the surface and the xz-plane<br />
is an ellipse. So the graph is VIII .<br />
. z2 .<br />
29. y 2 = x 2 + ~z 2 or y 2 = x 2 + g represents an elliptic<br />
cone with vertex (0, 0, 0) and axis the y-axis.<br />
• 2<br />
31. x 2 + 2y - 2z 2 = 0 or 2y = 2z 2 - x 2 or y = z 2 - ~<br />
represents a hyperbolic paraboloid with center (0, 0, 0).<br />
z<br />
33. Completing squares in y and z gives<br />
4x 2 + (y - 2) 2 + 4(z - 3) 2 = 4 or<br />
xz + (y ~ 2 2<br />
) + (z - 3) 2 = 1, an ellipsoid with<br />
center (0, 2, 3).<br />
z<br />
35. Completing squares in all three variables gives<br />
(x - 2) 2 - (y + 1? + (z- 1) 2 = 0 or<br />
(y + 1? = (x - 2) 2 + (z -1?, a circular cone with<br />
center (2, -1, 1) and axis the horizontal line x = 2,<br />
z =l.<br />
y<br />
37. Solving the equation for z we get z. = ±)1 + 4x 2 + y 2 , so we plot separately z = ) 1 + 4x 2 + y 2 and<br />
z = - ) 1 + 4x 2 +y 2 .<br />
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SECTION 12.6 CYLINDERS AND QUADRIC SURFACES 0 139<br />
1<br />
To restrict the z-range as in the second graph, we can use the option view= - 4 . . 4 in Maple's p l"ot3d command, or<br />
Plo.t Range - > {- 4 , 4} in Mathematica's Pl ot 3D command.<br />
39. Solving the equation for z we get z = ±y' 4x 2 + y 2 , so we plot separately z = y' 4x 2 + y2 and z = - y' 4x2 + y2.<br />
41.<br />
z z=2 43. The surface is a paraboloid of re<strong>vol</strong>ution (circular paraboloid) with vertex at<br />
the origin, axis the y-axis and opens to the right. Thus the trace in the<br />
yz-plane is also a parabola: y = z 2 , x = 0. The equation is y = x 2 + z 2 .<br />
The parabola<br />
y=x2<br />
X<br />
45. Let P = (x, y, z) be an arbitrary point equidistant from (-1, 0, 0) and the plane x = 1. Then the distance from P to<br />
(-1, 0, 0) is y'(x + 1) 2 + y 2 + z 2 and the distance from P to the plane x = 1 is lx- 11 /Vf'i = lx - 11<br />
(by Equation 12.5.9). So lx- 11 = y'(x + 1)2 + y 2 + z 2 ¢::> (x- 1) 2 = (x + 1) 2 + v. 2 + z 2 ¢:><br />
x 2 - 2x + 1 = x 2 + 2x + 1 +. y 2 + z 2 ¢:> - 4x = y 2 + z 2 • Thus the collection of all such points P is a circular<br />
paraboloid with vertex at the origin, axis the x-axis, which opens in the negative direction.<br />
2 2 2<br />
47. (a) An equation for an ellipsoid center<strong>ed</strong> at the origin with intercepts x = ±a, y = ± b, and z = ±cis x 2<br />
+ Y 2<br />
+ .=.._ = 1.<br />
a b c 2<br />
Here the poles of the model intersect the z-axis at z = ± 6356.523 and the 'equator intersects the x- andy-axes at<br />
x = ±6378.137, y = ±6378.137, so an equation is<br />
x2 y2 z2 .<br />
(6378.137) 2 + (6378.137) 2 + (6356.523)2 = 1<br />
• • ~ ~ I ~<br />
(b) Traces to z = k are the Circles (6378.137)2 + (6378.137)2 = 1 - (6356.523)2<br />
2 2 = (6378.137)2- (6378.137)2<br />
k2<br />
x + y ~6.~ .<br />
® 2012 C.ngagc Lc:lming. All Rights Re
140 D CHAPTER 12 VECTORS AND THE GEOMETRY OF SPACE<br />
(c) To identify the traces in y = mx we substitute y = mx into the equation of the ellipsoid:<br />
As expect<strong>ed</strong>, this is a family of ellipses.<br />
x 2 (mx? . z 2<br />
(6378.137) 2 + (6378.137)2 + (6356.523) 2 = 1<br />
(1 + m 2 )x 2 z 2<br />
(6378.137) 2 + (6356.523)2 = 1<br />
x2<br />
(6378.137) 2 /(1 + m2) + (6356.523)2 = 1<br />
z2<br />
49. If (a, b, c) satisfies z = y 2 - x 2 , then c = b 2 - a 2 . L 1 : x = a + t, y = b + t, z = c + 2( b - a )t,<br />
£2: x =a+ t, y = b- t, z = c- 2(b + a)t. Substitute the parametric equations of £1 into the equation<br />
of the hyperbolic paraboloid in order to find the points of intersection: z = y 2 - x 2 =><br />
. '<br />
I<br />
c + 2(b - a)t = (b + t) 2 - (a+ t) 2 = b 2 - a 2 + 2(b- a)t => c = b 2 - a 2 . As this is true for· all values oft,<br />
L 1 lies on z = y 2 - x 2 . Performing similar operations with £2 gives: z = y 2 - x 2 =><br />
c- 2(b + a)t = (b- t) 2 - (a+ t? = b 2 - a 2 - 2(b + a)t => c = b 2 - a 2 . This tells us that all of £2 also lies on<br />
51.<br />
The curve of intersection looks like a bent ellipse. The projection<br />
of this curve onto the xy-plane is the set of points (x, y, 0) which<br />
satisfy x 2 + y 2 = 1 - y 2 x 2 + 2y 2 = 1 <br />
y2<br />
x 2 + = 1. This is an equation of an ellipse.<br />
{1//2)2<br />
12 Review<br />
CONCEPT CH ECK<br />
1. A scalar is a real number, while a vector is a quantity that has both a real-valu<strong>ed</strong> magnitude and a direction.<br />
2. To add two vectors geometrically, we can use either.the Triangle Law or the Parallelogram Law, as illustrat<strong>ed</strong> in Figures 3<br />
and 4 in ·section 12.2. Algebraically, we add the corresponding components of the vectors.<br />
3. For c > 0, c a is a vector with. the same direction as a and length c times the length of a. If c < 0, ca points in the opposite<br />
direction as a and has length JcJ times the length of a. (See Figures 7 and IS in Section 12.2.) Algebraically, to find c a we<br />
multiply each component of a by c.<br />
4. See (I) in Section 12.2.<br />
5. See Theorem 12.3.3 and Definition 12.3.1.<br />
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CHAPTER 12 REVIEW 0 141<br />
6. The dot product can be us<strong>ed</strong> to find the angle between two vectors and the scalar projection of one vector onto another. In<br />
particular, the dot product can determine if two· vectors are orthogonal. Also, the dot product can be us<strong>ed</strong> to determine the<br />
work done moving an object given the force and di~placement vectors.<br />
7. See the box<strong>ed</strong> equations as well as Figures 4 and 5 and the accompanying discussion on page 828 [ET 804].<br />
B. See Theorem 12.4.9 and the prec<strong>ed</strong>ing discussion; use either (4) or (7) in Section 12.4.<br />
9. The cross product can be us<strong>ed</strong> to create a vector orthogonal to two given vectors as well as to determine if two vectors are<br />
parallel. The cross product can also be us<strong>ed</strong> to find the area of a parallelogram determin<strong>ed</strong> by two vectors. In addition, the<br />
cross product can be us<strong>ed</strong> to determine torque if the force and position vectors are known .<br />
•<br />
10. (a) The area of the parallelogram determin<strong>ed</strong> by a and b is the length of the cross product: Ia x bl.<br />
(b) The <strong>vol</strong>ume of the parallelepip<strong>ed</strong> determin<strong>ed</strong> by a, b, and cis the magnitude of their scalar triple product: Ia . (b x c )l.<br />
11. If an equation of the plane is known, it can be written as ax + by + cz + d = 0. A normal vector, which is' perpendicular to the<br />
plane, is (a , b, c) (or any scalar multiple of (a , b, c)). If an equation is not known, we can use points on the plane to find two<br />
non-parallel vectors which lie in the plane. The cross, product of these vectors is a vector perpendicular to the plane.<br />
12. The angle between two intersecting planes is defin<strong>ed</strong> as the acute angle between their normal vectors. We can find this angle<br />
using Corollary 12.3.6.<br />
13. See (1), (2), and (3) in Section 12.5.<br />
14. See (5), (6), and (7) in Section 12.5.<br />
15. (a) Two (nonzero) vectors are parallel if and only if one is a scalar multiple of the other. In addition, two nonzero vectors are<br />
parallel if and only if their cross product is 0.<br />
(b) Two vectors are perpendicular if and only if their dot product is 0.<br />
(c) Two planes are parallel if and only if their normal vectors are parallel.<br />
--+ --+<br />
16. (a) Determine the vectors PQ = (a1, a2 , a3) and P R = (b1, ~ . b3) . If there is a. scalar t such that .<br />
(a1, a 2, a3) = t (b1, ~ . b3 ), then the vectors are parallel and the points must all lie on the same line.<br />
~ ---+ -+ ---+<br />
Alternatively, if PQ x PR = 0, then PQ and PR are parallel, ~o P, Q, and Rare collinear.<br />
Thirdly, an algebraic method is to determine an equation of the line joining two o.f the points, and then check whether or<br />
not the third point satisfies this equation.<br />
--+ --+ --+<br />
(b) Find the vectors PQ =a, P R = b , PS = c . a x b is normal to the plane form<strong>ed</strong> by P, Q and R, and so S lies on this<br />
plane if a x band care orthogonal, that is, if(a x b)· c = 0. (Or use the reasoning in Example 5 in Section 12.4.)<br />
Alternatively, find an equation for the plane determin<strong>ed</strong> by three of the points and check whether or not the fourth point<br />
satisfies this equation.<br />
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142 D CHAPTER 12 VECTORS AND THE GEOMETRY OF SPACE<br />
17. (a) See Exercise 12.4.45.<br />
(b) See Example 8 in Section 12.5.<br />
(c) See Example 10 in Section 12.5.<br />
18. The traces of a surface are the curves of intersection of the surface with planes parallel to the coordinate planes. We can find<br />
the trace in the plane x = k (parallel to the yz-plane) by setting x = k and determining the curve represent<strong>ed</strong> by the resulting<br />
equation. Traces in the planes y = k (parallel to the xz-plane) and z = k {parallel to the xy-plane) are found similarly.<br />
19. See Table I in Section 12.6.<br />
TRUE-FALSE QUIZ<br />
1. This is false, as the dot product of two vectors is a scalar, not a vector.<br />
3. False. For example, ifu = i and ·v = j then iu · v i= 101 = 0 but iu l lvl = 1 · 1 = 1. In fact, by Theorem 12.3.3,<br />
lu · v i = liu llvl cos 91.<br />
5. True, by Theorem 12.3.2, property 2.<br />
7. True. If 9 is the angl~ between u and v , then by Theorem 12.4.9,lu x vi = iu l lvl sin 9 = lvllul sin 9 = 1':' x ui.<br />
(Or, by Theorem 12.4.1l,lu x v i = 1-v x ul = 1- lllv x ul = lv x ui.)<br />
9. Theorem 12.4.11 , property 2 tells us that this is true.<br />
11. This is true by Theorem 12.4.11, property 5.<br />
13. This is true because u x vis orthogonal to u (see Theorem 12.4.8), and the dot product of two orthogonal vectors is 0.<br />
15. This is false. A normal vector to the plane is n = (6, - 2, 4). Because (3, - 1, 2) = ~n , the vector is parallel ton and hence<br />
perpendicular to the plane.<br />
17. This is false. In IR 2 , x 2 + y 2 = 1 represents a circle, but { (x, y, z) I x 2 + y 2 = 1} represents a three-dimensional surface,<br />
namely, a circular cylinder with axis the z -axis.<br />
19. False. For example, i · j = 0 but i i- 0 and j =f. 0.<br />
21. This is true. Ifu and v are both nonzero, then by (7) in s~ c tion 12.3, u · v = 0 implies that u and v are orthogonal. But<br />
u x v = 0 implies that u and v are parallel (see Corollary 12.4.10). Two nonzero vectors can't be both parallel and<br />
orthogonal, so at least one of u, v must be 0.<br />
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CHAPTER 12 REVIEW 0 1~<br />
EX ERCISES<br />
1. (a) The radius of the sphere is the distance between the points ( - 1, 2, 1) and {6, -2, 3), namely,<br />
J [6 - ( -1)]2 + ( - 2 - 2) 2 + (3 - 1) 2 = 0f!}. By the formula for an equation of a sphere (see page 813 [ET 789]),<br />
an equation of the sphere with center ( -1, 2, 1) and radius J69 is (x + 1? + (y - 2) 2 + (z- 1) 2 = 69.<br />
(b) The intersection of this sphere with the yz -plane is the set of points on the sphere whose x-coord inate is 0. Putting x = 0<br />
into the equation, we have (y - 2) 2 + (z - 1) 2 = 68, x = 0 which represents a circle in the yz-plane with center (0, 2, 1)<br />
and radius .J68.<br />
(c) Completing squares gives (x - 4) 2 + (y + 1) 2 + (z + 3? = - 1 + 16 + 1 + 9 = 25. Thus the sphere is center<strong>ed</strong> at<br />
(4, -1, -3) and has radius 5.<br />
3. u . v = Ju J.JvJ c~s45° = (2)(3) '{!- = 3 .;2. Ju x vJ = JuJJvJ sin45° = (2)(3)~ = 3 .;2.<br />
By the right-hand rule, u x vis direct<strong>ed</strong> out of the page.<br />
5. Forthetwo vectors to be orthogonal, we ne<strong>ed</strong> (3, 2,x} · (2x,4,x} = 0 # (3)(2x) + (2){4) + (x)(x) = 0 #<br />
~ + 6x + 8 = 0
144 0 CHAPTER 12 VECTORS AND THE GEOMETRY OF SPACE<br />
15. The line has direction v = ( -3, 2, 3). Letting Po = ( 4, - 1, 2), parametric equation~ are<br />
X = 4 - 3t, y = -1 + 2t, Z = 2 + 3t.<br />
17. A direction vector for the line is a normal vector for the plane, n = (2, -1, 5), and parametric equations for the line are<br />
. X = -2 + 2t, y = 2 - t, Z = 4 + 5t.<br />
'<br />
19. Here the vectors a= (4-3,0- (-1) ,2 - 1) = (1,1,1} and b = (6 - 3,3- (-1), 1 - 1) = {3,4, 0) lie in the plane,<br />
so n = a x b = ( -4, 3, 1) is a normal vector to the plane and an equation of the plane is<br />
- 4(x- 3) +3.(y - (- 1)) + 1(z -1) = 0 or - 4x+3y +z = - 14.<br />
21 . Substitution of the parametric equations into the equation of the plane gives 2x- y + z = 2(2- t) - (1 + 3t) + 4t = 2 =?<br />
. .<br />
- t + 3 = 2 => t = 1. When t = 1, the parametric equations give x = 2 - 1 = 1, y = 1 + 3 = 4 and z = 4. Therefore,<br />
the point of intersection is (1,4,4).<br />
23. Since the direction vectors (2, 3, 4) and (6, --;1, 2) aren't parallel, neither are the lines. For the lines to intersect, the three<br />
equations 1 + 2t = -1 + 6s, 2 + 3t = 3 - s, 3 + 4t = -5 + 2s must be satisfi<strong>ed</strong> simultaneously. Solving the first two<br />
equations gives t = k. s = ~ and checking we see these values don't satisfy the third equation. Thus the lines aren't parallel<br />
and they don't intersect, so they must be skew.<br />
25. n 1 = (1, 0, -1) and n2 = (0, 1, 2). Setting z = 0, it is easy to see that (1, 3, 0) is a point on ,the line of intersection of<br />
x - z = 1 andy+ 2z = 3. The direction of this line is v 1 = n 1 x n 2 = (1, -2, 1). A second vector parallel to the desir<strong>ed</strong><br />
plane is v 2 = (1,1, -2), since it is perpendicular to x + y- 2z = 1. Therefore, the normal of the plane in question is<br />
n = v 1 x v 2 = (4 - 1, 1 + 2, 1 + 2) = 3 (1, 1, 1). Taking (x 0 , y 0 , z 0 ) = (1, 3, 0), the equation we are looking for is<br />
(x- 1) + (y- 3) + z = 0 ¢* x + y + z = 4.<br />
. l-2- (-24)1 22 .<br />
27. By Exercise 12.5.75, D - - ~·<br />
v26<br />
y'32 + I2 + ( -4)2<br />
29. The equation x = z represents a plane perpendicular to<br />
the xz-plane and intersecting the xz-plane in the line<br />
x = z,y = 0.<br />
31. The equation x 2 = y 2 + 4z 2 represents a (right elliptical)<br />
cone with vertex at the origin and axis the x -axis.<br />
X<br />
X<br />
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. . . y2 2<br />
33. An equivalent equation IS - x 2 + 4<br />
-;- z = 1, a 35. Completing the squate in y gives<br />
hyperboloid of two sheets with axis the y-ax.is. For<br />
IYI > 2, traces parallel to the xz-plane are circles.<br />
z<br />
CHAPTER 12 REVIEW 0 145<br />
z2<br />
4x 2 + 4(y - 1) 2 + z 2 = 4 or x 2 + (y - 1) 2 +- = 1,<br />
4 .<br />
an ellipsoid center<strong>ed</strong> at (0, 1, 0).<br />
(0, 1,2)<br />
(O, l ,-2)<br />
<br />
. x2 y2 2 2 2<br />
4 + 16 = l. The equation of the ellipsoid is ~ + i 6 + : 2<br />
= l, since the horizontal trace in the<br />
plane z = 0 must be the original ellipse. The traces of the ellipsoid in the yz -plane must be circles since the surface is obtain<strong>ed</strong><br />
2 2 2<br />
by rotation about the x-axis. Therefore, c 2 = 16 and the equation of the ellipsoid is ~ + i 6<br />
+ ~ 6<br />
= 1 <br />
4x 2 + y 2 + z 2 = 16.<br />
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D<br />
PROBLEMS PLUS<br />
1. Since three-dimensional situations are often difficult to visualize and work with, let<br />
us first try to find an analogous problem in two dimensions. The analogue of a cube<br />
is a square and the analogue of a sphere is a circle. Thus a similar problem in two<br />
dimensions is the following: if five circles with the same radius rare contain<strong>ed</strong> in a<br />
square of side 1 m so that the circles touch each other and four of the circles touch<br />
two sides ofthe square, find r.<br />
The diagonal of the square is .J2. The diagonal is a lso 4r + 2x . But x is the diagonal of a smaller square of side r . Therefore<br />
x = .J2r ~ .J2=4r+2x =4r+2.J2r= (4+2.J2)r ~ r = 4<br />
/{,/2.<br />
Let's use these ideas to solve the original three-dimensional problem. The diagonal of the cube is \/12 + 12 + 12 = v'a.<br />
The diagonal of the cube is also 4r + 2x where x is the diagonal of a smaller cube with <strong>ed</strong>ger. Therefore<br />
r;; r;; v'3 2 v'a - 3<br />
x = Jr 2 + 1·~ + r 2 = v'3 r ~ v'a = 4r+2x .=4r+2v3r = (4+2v3)r.Thus r= r;;=<br />
4 + 2v3 2<br />
The radius of each ball is ( v'3 - ~) m.<br />
3. (a) We find the line of intersection L as in Example 12.5. 7(b). Observe that the point ( - 1, c, c) lies on both planes. Now since<br />
L lies in both planes, it is perpendicular to both of the normal vectors n1 and n 2, and thus parallel to their cross product<br />
j<br />
k<br />
n 1 x n 2 = c 1 1 = (2c, -c 2 + 1, -c 2 - 1). So symmetric equations of L can be written as<br />
1 - c c<br />
-- x +1 = y - c z - c<br />
~<br />
1 =<br />
- 2c (..- - ~ c + 1<br />
, provid<strong>ed</strong> that c I 0, ±1.<br />
If c = 0, then the two planes are given by y + z = 0 and x = - 1, so symmetric· equations of L are x = - 1, y = - z. If<br />
c = - 1, then the two planes are given by -x + y + z = - 1 and x + y + z = -1, and they intersect in the line x = 0,<br />
y = - z - 1. If c = 1, then the two planes are given by x + y + z = 1 and x - y + z = 1, and they intersect in the line<br />
y = 0, X = 1- Z.<br />
(b) If we set z = t in the symmet~i c equations and solve for x andy separately, we get x + 1 = (t - c)( - 2 c)<br />
c 2 + 1<br />
(t- c)(c 2 - 1)<br />
y - c = c2 +1<br />
- 2ct + (c 2 - 1) (c 2 - 1)t + 2c El. . . fr th .<br />
~ x = c 2 + 1<br />
, Y = & + 1<br />
. unmatmg c om ese equattons, we<br />
have x 2 + y 2 = t 2 + 1. So the curve trac<strong>ed</strong> out by L in the plane z = tis a circle with center~~ (0, 0, t) and<br />
radius J fl + 1.<br />
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148 0 CHAPTER 12 PROBLEMS PLUS<br />
(c) The area of a horizontal cross-section of the solid is A ( z) = 1r(z 2 + 1 ), so V = J; A( z )dz = 1r ( ~ z 3 + z] ~ = 4 3 ,..<br />
53 53 54 50 5"- 2 -<br />
jv5 1 = 24<br />
. 32<br />
jv 1j = 23<br />
. 32<br />
. Similarly, jv5j = 24 . 33<br />
, jv.71 :;= 25 . 34<br />
, and in general, jv ,. j = 2 n_ 2 . 3 n_ 3 = 3(ir 2 .<br />
Thus<br />
f Jv,.J = lv1l + lv 2l + f 3(i)"- 2 = 2 + 3 + f 3(~ )"<br />
n = l n = 3 n=l<br />
00 ll<br />
= 5 + E .2.(!~)"- 1 = 5 + ___L_ [sum of a geometric series] = 5 + 15 = 20<br />
n=l<br />
2<br />
G 1 - ~<br />
7. (a) When 8 = 8., the block is not moving, so the sum of the forces on the block<br />
I<br />
must be 0, thus N + F + W = 0. This relationship is illustrat<strong>ed</strong><br />
geometrically in the figure. Since the vectors form a right triangle, we have<br />
- J!l_ 11-.• n <br />
tan(O.) - INI - n - fl-s·<br />
(b) We place the block at the origin and sketch the force vectors acting on the block, including the additional horizontal force<br />
H , with initial points at the origin. We then rotate this system so that Flies along the positive x-axis and the inclin<strong>ed</strong> plane<br />
is parallel to the x-axis. (See the follo~ing figure.)<br />
N<br />
F<br />
w<br />
IF! is maximal, so jFj = 11-. n for f) > B_,. Then the vectors, in terms of components parallel and perpendicular to the<br />
inclin<strong>ed</strong> plane, are<br />
N =n j<br />
W = ( - mg sin 0) i + ( -mg cos fJ) j<br />
F = (11-.n) i<br />
H = (hmin cos B) i + ( - hmin sin B) j<br />
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l<br />
CHAPTER 12 PROBLEMS PLUS 0 149<br />
Equ!lting components, we have<br />
JJ-,n- mg sin(}+ hm1n cosO= 0 ,=? hmin cos(}+ J-L •• n = mgsin(} (1)<br />
n- mg cos()- hrnin sinfJ = 0 =? hmin sin(}+ mgcos(} = n (2)<br />
(c) Since (2) is solv<strong>ed</strong> for n, we substitute into (1):<br />
hmin cos() + JJ-~ (hmin sin(}+ mg cos 9) = mg sin 9 =?<br />
hmin cos fJ + hmin/1-s sin fJ = mg sin fJ - mgJJ-. cos(}<br />
'*<br />
1<br />
(sin8 - JJ-.• cos'(} ) . ( tau8 - /1-., )<br />
tmin = mg cos(}+ 1-t. sin(} = mg 1 + JJ-. tau(}<br />
() kn (} th<br />
. b h ( tan8 - tanfJ8 ) d . . . . .<br />
From part a we ow JJ-~ = tan ., so IS ecomes min = mg +tan (}, tau(} an usmg a tngonomctnc tdentlty,<br />
1<br />
this is mg tau(8- 8. ) as desir<strong>ed</strong>.<br />
Note for fJ = 8., hmin = mgtau 0 = 0, which makes sense since the block is at rest for 8. , thus no additional force H<br />
is necessary to prevent it from moving. As(} increases, the factor tau(8- 8.), and hence the value of hmin, increases<br />
slowly for small values ofB- 8. but much more rapidly as 8 - e. becomes significant. This seems reasonable, as the<br />
steeper the inclin<strong>ed</strong> ~lane, the less the hori zont~l components of the various forces affect the movement of the block, so we<br />
would ne<strong>ed</strong> a much larger magnitude of horizontal force to keep the block motionless. If we allow f) --+ 90°, corresponding<br />
to the inclin<strong>ed</strong> plane being plac<strong>ed</strong> vertically, the value of hmin is quite large; this is to be expect<strong>ed</strong>, as it takes a great<br />
amount of horizontal force to keep an object from moving vertically. In fact, without friction (so (} 8 = 0), we would have<br />
(} --+ 90° :::? hmin --+ oo, and it would be impossible to keep the block from slipping.<br />
(d) Since hmo.x is the largest value of h that keeps the block from slipping, the force of friction is keeping the block from ·<br />
moving up the inclin<strong>ed</strong> plane; thus, F is direct<strong>ed</strong> down the plane. Our system of forces is similar to that in part {b), then,<br />
except that we have F = -(JJ-.n) i. (Note that IFI is again maximal.) Following our proc<strong>ed</strong>ure in parts (b) and (c), we<br />
equate components:<br />
- JJ-,n- mgsin(} + hmux cos(} = 0 :::? hme.x cos(}- /)- 8<br />
n = mgsin8<br />
n - mg cos (} - hm:l.X sin (} = 0<br />
'* hmax sin (} + mg cos e = n<br />
Then substituting,<br />
hmax cos 8 - /1-s ( hmax sin 8 + mg cos B) = mg sin 8 '*<br />
hmax cos(} -<br />
hmn.x/1-s sin(} = mg sin(} + mgJJ-. cos e '*<br />
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150 0 CHAPTER 12 PROBLEMS PLUS<br />
h _ (sin0+J.£.cos8) _ · ( tan8 +J.£.)<br />
max- mg - mg<br />
cos 0 - J.£ 8<br />
sin B 1 - J.£ 8<br />
tan B<br />
tanB +tan B. )<br />
== mg ( 1- tan B. tanB = mgtan(B +B .• )<br />
We would expect hmax to increase as 8 increases, with similar behavior as we establish<strong>ed</strong> for hmin, but with hmax values<br />
al~ays larger than hmin· We can see that this is. the case if we graph hmo.x as a function of 8, as the curve is the graph of<br />
hmin translat<strong>ed</strong> 20. to the left, so the equation does seem reasonable. Notice that the equation pr<strong>ed</strong>icts hmo.x --+ oo as<br />
. 8--+ (90° - 0., ). In fact, as hmnx increases, the nonnal force increases as well. When (90° - Bs) :50 :S: 90°, the<br />
horizontal force is completely counteract<strong>ed</strong> by the sum of the f!Orlnal and frictional forces, so no part of the horizontal<br />
force contributes to moving the block up the plane no matter how large its magnitude:<br />
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13 D VECTOR FUNCTIONS<br />
13.1 Vector Functions and Space Curves<br />
1. The component functions v'4- t 2 , e- 3 t, and ln(t + 1) are a ll defin<strong>ed</strong> wher4 - t 2 2: 0 => - 2 ~ t ~ 2 and<br />
t + 1 > 0 => t > - 1, so the domain ofr is ( -1, 2].<br />
1 1<br />
- =<br />
t-+O t--+o sin 2 t . t.-o sin 2 t lim sin 2 t<br />
~ t--+0 t2<br />
3. lim e- 31 = e 0 = 1, lim~= lim -<br />
and lim cos 2t = cos 0 = 1. Thus<br />
t-+0<br />
=<br />
1 _ I_ _ 1<br />
. t)2 - !2 - '<br />
(<br />
lim~<br />
t --+0 t<br />
lim (e-st i + 4 j +cos 2t k) = [lim e- 3 t] i + [urn 4] j + [rim cos 2t] k = i + j + k.<br />
t-+O sm t t-+O t-+O sm t t -+O<br />
] . 1 + t 2 1 . (1/ t2 ) + 1 0 + 1<br />
1 1 . _ 1 t ,.<br />
1 . 1 - e-2 t<br />
S. ~~ 1- t2 1 . 1 1 = t~ (1/t2)- 1 = 0 - 1 = - 't 2.~ tan = 2 • t~ t = t2.~ t - -te_2_t = 0 - 0 = 0· Thus<br />
(<br />
1 + e 1 - e- 2 t) .<br />
lim - 1<br />
2 ,tan-1 t, t =(-l,f,O).<br />
t--+oo - t<br />
7. The corresponding parametric equations for this curve are x = sin t, y = t.<br />
We can make a table of values, or we can eliminate the parameter: t = y =><br />
x = sin y, with y E JR. By comparing different values oft, we find tl)e direction in<br />
which t increases as indicat<strong>ed</strong> in the graph.<br />
X<br />
9. The corresponding parametric equations are x = t, y = 2 - t, z = 2t, which are<br />
parametric equations of a line through the point (0, 2, 0) and with direction vector<br />
(1, -1, 2).<br />
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152 D CHAPTER 13 VECTOR FUNCTIONS<br />
11. The corresponding parametric equations are x = 1, y = cos t, z = 2 sin t.<br />
Eliminating the parameter in y and z gives ~ 2 + (z/ 2) 2 = cos 2 t + sin 2 t = 1<br />
or y 2 + z 2 / 4 = 1. Since x = 1, the curve is an ellipse center<strong>ed</strong> at (1, 0, 0) in<br />
the plane x = 1.<br />
X<br />
13. The parametric equations are X = e. y = t 4 • z = t 6 . These are positive<br />
for t =I 0 and 0 when t = 0. So the curve lies entirely in the first octant.<br />
The projection of the graph onto the xy-plane is y = x 2 , y > 0, a half parabola.<br />
Onto the xz-plane z = x 3 , z > 0, a half cubic, and the yz-plane, y 3 = z 2 •<br />
I<br />
I<br />
I<br />
I<br />
I<br />
I<br />
I<br />
I<br />
I<br />
, I<br />
,'<br />
,,,- y=xl<br />
,,'<br />
X<br />
I<br />
15. Tbe projection of the curve onto the xy-plane is given by r (t) = (t , sin t, 0} [we use 0 for the z-component] whose graph<br />
is the curve y =sin x, z = 0. Similarly, the projection onto the xz-plane is r (t) = (t, 0, 2 cost), whose graph is the cosine<br />
wave z = 2 cos x, y = 0, and the projection onto the yz-plane is r (t) = (0, sin t, 2 cost) whose graph is the ellipse<br />
2<br />
- I<br />
y<br />
-2<br />
xy-plane<br />
xz-plane<br />
yz-plane<br />
From the projection onto the yz-plane we see that the curve lies on an elliptical<br />
cylinder with axis the x-axis. The other two projections show that the curve<br />
oscillates both vertically and horizontally as we move in the x-direction,<br />
suggesting that the curve is an elliptical helix that spirals along the cylinder.<br />
17. Taking ro = (2, 0, 0) and r1 = (6, 2, -2), we have from Equation 12.5.4<br />
r (t) = {1 - t)ro +tr1 = {1- t) {2,0, 0) +t{6, 2,-2), 0 ~ t ~ 1 or r {t) = (2 + 4t,2t, - 2t);O ~ t ~ 1.<br />
Parametric equations are x = 2 + 4t, y = 2t, z = -2t, 0 ~ t ~ 1.<br />
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SECTION 13.1 VECTOR FUNCTIONS AND SPACE CURVES 0 153<br />
19. Taking r o = (0, - 1, 1) and r 1 = ( ~ . ~. i), we have<br />
r (t) = (1-t) ro +tr1 = (1 - t) (0, - 1, 1) + tO,~.~), 0 ~ t ~ 1 or r (t) = (~ t ,- 1 + 1t,1 - tt), 0 ~ t ~ 1.<br />
. .<br />
Parametric equations are x = ~t, y = - 1 + ~t, z = 1 - tt. 0 ~ t ~ 1.<br />
21 . x = tcost, y = t, z = t ~in t, t ;:::: 0. At any point (x, y, z) on the curve, x 2 + z 2 = t 2 cos 2 t + t 2 sin 2 t = t 2 = y 2 so the<br />
curve Hes on the circular cone x 2 + z 2 = y 2 with axis they-axis. Also notice that y ;:::: 0; the graph is II.<br />
23. x = t, y = 1/ (1 + t 2 )., z = t 2 • At any point on the curve we have z = x 2 , so the curve lies on a parabolic cylinder parallel<br />
to they-axis. Notice that 0 < y ~ 1 and z ;:::: 0. Also the curve passes through (0, 1, 0) when t = 0 and y --+ 0, z --+ oo as<br />
t --+ ±oo, so the graph must be V.<br />
25. x = cos Bt, y = sin Bt, z = e 0 ·8 L, t ;:::: 0. x 2 + y 2 = cos 2 Bt + sin 2 Bt = 1, so the curve lie~ on a circular cylinder with<br />
axis the z-axis. A point (x, y, z) on the curve lies directly above the point (x, y, 0), which moves counterclockwise around the<br />
unit circle in the xy-plane as t increases. The curve starts at (1-, 0, 1), when t = 0, and z --+ oo (at an increasing rate) as<br />
t --+ oo, so the graph is IV.<br />
27. Ifx = tcost, y = tsint, z = t, then x 2 + y 2 = t 2 cos 2 t + t 2 sin 2 t = t2 = z 2 ,<br />
so the curve lies on the cone z 2 = x 2 + y 2 . Since z = t, the curve is a spiral on<br />
this cone.<br />
29. Parametric equations for the curve are x = t, y = 0, z = 2t - t 2 • Substituting into the equation of the paraboloid<br />
gives 2t- t 2 = t 2 => 2t = 2t 2 => t = 0, 1. Since r(O) = 0 and r (1) = i + k , the points of intersection<br />
are(O, O, O)and (1, 0, 1).<br />
31. r (t) =(cost sin 2t,sint sin2t,cos 2t).<br />
We include both a regular plot and a plot<br />
showing a tube of radius 0.08 around the ·<br />
·curve.<br />
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154 0 CHAPTER 13 VECTOR FUNCTIONS<br />
33. r(t) = (t, t sin t, t cost) 35. r (t) = (cos2t,cos3t,cos4t)<br />
X<br />
37.<br />
:li = (1 +.cos 16t) cost, y = (1 +cos 16t) sin t, z = 1 +cos 16t. At any<br />
point on the graph,<br />
X<br />
x 2 + y 2 = (1 +cos 16t) 2 cos 2 t + (1 +cos 16t) 2 sin 2 t<br />
= (1 + cos 16t ) 2 = z 2 , so the graph lies on the cone x 2 + y 2 = z 2 •<br />
From the graph at left, we see that this curve looks like the projection of a<br />
leav<strong>ed</strong> two-dimensional curve onto a cone.<br />
39. Ift = - 1, then x = 1, y = 4, z = 0, so the curve passes through the point (1,4, 0). 1ft= 3, then x = 9, y = - 8, z = 28,<br />
so the curve passes through the point {9, - 8, 28). For the point (4, 7, - 6) to be on the curve, we require y = 1 - 3t = 7 ==><br />
t = -2. But then z = 1 + ( - 2? = -7 # -6, so (4, 7, -6) is not on the curve.<br />
41. Both equations are solv<strong>ed</strong> for z, s~ we ~an substitute to eliminate z: J x2 + y2 = 1 + y ==> x 2 + y 2 = 1 + 2y + y 2 ==><br />
x 2 = 1 + 2y ==> y = ~(x 2 - 1). We can form parametric equations for the curve C of intersection by choosing a<br />
parameter x = t, then y = ~(t 2 - 1) and z = 1 + y = 1 + ~(t 2 - 1) = ~(t2 + 1). Thus a vector function representing C<br />
is r (t) = t i + 1 (t 2 - 1) j + ~(t 2 + 1) k. "·<br />
43. The projection of the curve C of intersection onto the xy-plane is the circle x 2 + y 2 = 1, z = 0, so we can write x = cost, ·<br />
y = sin t, 0 $ t $ 27!'. Since C also lies on the surface z = x 2 - y 2 , we have z = x 2 - y 2 = cos 2 t - sin 2 tor cos 2t.<br />
Thus parametric equations for Care x = cost, y = sin t, ; = cos 2t, 0 $ t $ 27l', and the corresponding vector function<br />
is r(t) = cost i + sin t j +cos 2t k, 0 $ t $ 27!'.<br />
45.<br />
The projection of the curve C of intersection onto the<br />
X<br />
z<br />
xy-plane is the circle x 2 + y 2 = 4, z = 0. Then we can write<br />
x = 2 cost, y = 2 sin t, 0 $ t $ 27l'. Since C also lies on<br />
the surface z = x 2 , we have z = x 2 = (2 cost)~ = 4 cos 2 t.<br />
Then parametric equations for C are x = 2 cost, y = 2 sin t,<br />
@) 20J2 Cengll£e Lcnming. All Right! Reserv<strong>ed</strong>. Mny not he sclllUl<strong>ed</strong>. copi<strong>ed</strong>, or dupliC3h!'d. or post<strong>ed</strong> 10 a publicly accessible website, in whole or in part.
SECTION 13.1 VECTOR FUNCTIONS AND SPACE CURVES D 155<br />
47. For the particles to collide, we require r1 ( t) = r 2 ( t) ¢:> ( t 2 , 7t - 12, t 2 ) = .( 4t - 3, t2, 5t - 6). Equating components<br />
gives t2 = 4t- 3, 7t - 12 = t2, and t 2 = 5t- 6. From the first equation, t 2 - 4t + 3 = 0 ¢:> (t- 3)(t -1) = 0 sot = 1<br />
or t = 3. t = 1 does not satisfy the other two equations, butt = 3 does. The piuticles collide when t = 3, at the<br />
point (9, 9, 9).<br />
49. Let u (t) = (u1(t), u2(t), u3(t)) and v (t) = (v1 (t), V2(t), va(t)). In each part of this problem the basic proc<strong>ed</strong>ure is to use<br />
Equation l and then analyze the individual component functions using the limit properties we have already develop<strong>ed</strong> for<br />
real-valu<strong>ed</strong> functions.<br />
(a) lim u (t) + lim v(t) = I lim u1 (t), lim u2(t), lim ua(t)) + I Jim V1 (t), lim v2(t), lim v3(t)) and the limits of these<br />
t-+a. t--+a \ t -+a t-+a t--+a \t--+a t--+a t-a.<br />
component functions must each exist since the vector functions both possess limits as. t -+ a. Then adding the two vectors<br />
and using the addition property oflimits for real-valu<strong>ed</strong> functions, we have that<br />
lim u(t) + lim v (t) = I Jim u1 (t) + lim v1 (t), lim u2(t) + lim v2(t), lim ua(t) + lim v3 (t))<br />
t --..o. t -+a \t--+a<br />
= lim (u1(t) +VI (t), U2(t) + V2(t), 'U3(t) + V3(t))<br />
t-+a .<br />
[using (I) backward]<br />
= lim (u(t ) + v(t)]<br />
t-a<br />
(b) lim cu(t) = lim (cu1 (t), cu2(t), C'U3(t)) = I lim cu1(t), lim cu2(t) , lim cua(t))<br />
t -+a t --+ a \t- a t -+a t -+a<br />
= I c lim u1(t), c lim u~ (t), c lim u3(t)) = c I lim u1 (t), lim u 2 (t), lim ua(t))<br />
\ t-a t -+ a t -+ a \t__,. a. t-+a t-+a<br />
= c lim (u1 (t), u2(t), u3(t)) = c lim u (t)<br />
t--+a<br />
t-a<br />
(c) lim u(t) · lim v(t) = I lim u1(t), lim u2(t), lim u3(t)) ·I lim v1(t), lim v 2 (t), lim v3(t))<br />
t-+a t--+a \t-a t -+a t-a \t--. n. · t-+a t - a.<br />
= [lim u1(t)] [u~ v1(t)] + [lim u2(t)] [lim v2(t)] _;_ .[lim us (t)] [urn v3(t)]<br />
t --+a t -+a t -+ a t --+a t -+a t-+a<br />
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156 0 CHAPTER 13 VECTOR FUNCTIONS<br />
(d) lim u (t) x ·lim v (t) = ( lim u1 {t), l.im u2(t), lim u3(t)) x ( lim v1(t), lim v2(t), lim va(t))<br />
t-a t-a t- a t--+a t.-a t-+o. t-+a t-ta.<br />
= ([lim u2(t)] [lim v3(t)] - [lim us(t)] [lim v2(t)] ,<br />
t-+a t-+a t-+n t -+a.<br />
[lim ua(t)] [lim v1(t)] -<br />
[lim u1(t)] [tim vs(t)],<br />
t-+a t-+a t -+a t-+a<br />
[p~ u1(t)) [~~ v2(t)) - [l~ u 2(t)] [E~ vl(t) ])<br />
= (J~ [u2(t)v3(t)- u3(t)v2(t)] , E~ [ua(t)v1(t) - u1(t)va(t)] ,<br />
~~ [u1(t)v2(t)- u2(t)v1(t)])<br />
= lim (·u2(t)va(t)- ua(t)v2(t), U3 (t) v1 (t) - u1(t)v3(t), u1(t)v2(t)- tt2(t)v1 (t))<br />
t-+a ·<br />
= lim (u (t) x v (t)]<br />
t-<br />
51 . Let r (t) = (! (t), g (t), h (t)) and b = {b1, b2, ba) . If lim r (t) = b, then lim r (t) exists, so by (1),<br />
t-a.<br />
t-+a<br />
b = lim r (t) ~ ( lim f(t), lim g(t), lim h(t)). By the definition of equal vectors we have lim f(t)·= b1, lim g(t) = b2<br />
t-+a t-a t-a t -+a t-a t -+a<br />
and lim h(t) = b 3 • But these are limits of real-valu<strong>ed</strong> functions, so by the definition of limits, for every c > 0 there exists<br />
t-Hl<br />
0, 02 > 0, oa > 0 so that ifO < It - a l < 01 then lf(t) - b1 l < c/3, ifO < it- al < 02 then lg(t) - b2l < c/ 3, and<br />
ifO < It- al 0 there exists o > 0 such that if 0 < It - a l < o then<br />
lr.(t)- b l ::; !f(t) - b1 l + lg(t) - b2 l + lh(t) - b3 l 0, there exists o > 0 such<br />
tharifO l(f(t)- b1 ,g(t) - b2,h(t) - ba) l < c *><br />
J [f(t) - b!) 2 + [g(t)- b2j2 + [h(t) - baj2 < c ¢:> (J(t) - b 1 ] 2 + [g(t)- b2] 2 + [h(t) - baf < c 2 . But each term<br />
on the left side of the last inequality is positive, so if 0 < It - al < o, then [! ( t) - b1] 2 < c 2 , [g( t) - b2] 2 < c 2 and<br />
[h(t) - b3] 2 < c 2 or, taking the square root of both sides in each ofthe above, lf(t) - b1 l < c, lg(t) - b2 l < . c and<br />
lh(t) - b3l
SECTION 13.2 DERIVATIVES AND INTEGRALS OF VECTOR FUNCTIONS 0 157<br />
13.2 Derivatives and Integrals of Vector Functions<br />
1. (a)<br />
r(4.5) - r(4)<br />
Q<br />
r(4.2)- r(4)<br />
)(.<br />
(b) r( 4·5)- r ( 4 ) = 2[r(4.5}- r(4}], so we draw a vector in the same<br />
0.5<br />
direction but with twice the length of the vector r(4.5}- r(4).<br />
r ( 4·2) - r ( 4 ) = 5[r( 4.2) - r ( 4)], so we draw a vector in the same<br />
0.2<br />
direction but with 5 times the (ength of the vector r( 4.2) - r( 4).<br />
(c) By Definition I, r'(4) = J..i~<br />
r( 4 + ~ - r ( 4 ). T (4) = ~ ~~!~I·<br />
(d) T( 4) is a unit vector .in the same direction as r' ( 4), that is, parallel to the<br />
tangent line to the curve at r(4) with length 1.<br />
3. Since (x + 2) 2 = t 2 = y - 1 =><br />
y = (x + 2? + 1, the curve is a<br />
parabola.<br />
(a), (c)<br />
(b) r'(t) = (1, 2t},<br />
r'(- 1) = (1,·-2)<br />
)(.<br />
5. x = sin t, y = 2 cos t so<br />
x 2 + (y/ 2) 2 = 1 and the curve is<br />
an ellipse.<br />
(a), (c)<br />
(b) r'(t) ~ cost i - 2sintj,<br />
r '(71")- 4 -21<br />
v'2.<br />
-y~<br />
'2<br />
J<br />
.<br />
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158 0 CHAPTER 13 VECTOR FUNCTIONS<br />
· 7. Since x = e 2 t = (et) 2 = y 2 , the<br />
(a), (c)<br />
y<br />
(b) r'(t) = 2e 2 t i + et j ,<br />
curve is part of a parabola. Note<br />
r '(O) = 2 i + j<br />
that here x > 0, y > 0.<br />
X<br />
9. r ' (t) = \! [t sin t], :t [e) ·, ! [t cos 2t]) = (t c~s t +sin t, 2t, t(- sin 2t) · 2 +cos 2t)<br />
= (t cost+ sin t , 2t, cos 2t - 2t sin 2t)<br />
11. r(t) =ti+j + 2Vt k => r'(t)=1 i + O j +2(~t - 1 1 2 )k = i+ ~ k<br />
13. r (t) = et 2 i - j + ln(1 + 3t) k => r'(t) = 2tet 2 3<br />
i + - - k<br />
1 +3t<br />
15. r ' (t) = 0 + b + 2t c = b + 2t c by Formulas 1 and 3 of Theorem 3.<br />
r'(O) = (1, 2, . 2). So lr'(O)I = .JP + . 2 2 + 2 2 = .J§ = 3 and<br />
19. r '(t) = - sin ti + 3j + 4cos 2tk => r'(O) = 3j + 4k. Thus<br />
T (~) = 1::~~~~ = .J02 +~2 +42 (3 j +4k) = t(3j +4k) = ~ j + ~ k.<br />
21. r(t) = (t, t 2 , t 3 ) => r' (t) = (1, 2t, 3t 2 ). Then r ' (1) = (1, 2, 3) and lr' (1)1 = \,/1 2 + 2 2 + 3 2 = v'l4, so<br />
( ) r ' ( 1) 1 ( ) 1 1 2 a ) " ( ) ( )<br />
T 1 = ir'( 1<br />
)i = V'i4 1,2,3 = \V'i4'V'i4 ' V'i4 . r t = 0,2,6t ,so<br />
j k<br />
r '(t) x r"(t) = 1 2t 3t2<br />
0 2 6t<br />
= 12t 3t21 i - 11 3t2 1· + 11 2t I k<br />
2 6t 0 6t J 0 2<br />
= {12f- 6t~ ) i - (6t- O) j + (2 - 0) k = (6f, -6t, 2)<br />
23. The vector equation for the curve is r(t) = ( 1 + 2 Jt, t 3 - t, t 3 + t ), so r '(t) = ( 1/ Vt, 3f - 1, 3t 2 + 1). The point<br />
(3, 0, 2) corresponds tot= 1, so the tangent vector there is r ' (1) = (1, 2, 4). Thus, the tangent line goes through the point<br />
(3, 0, 2) and is parallel to the vector (1, 2, 4). Parametric equations are x = 3 + t, y = 2t, z ,;,. 2 + 4t.<br />
25. The vector equation for the curve is r (t) = ( e- t cost, e- t sin t, e-t), so<br />
r' (t) = (e-t(-sin t) +(cost)( -e-t), e-t cost+ (sin t)( -e-t), (-e- t))<br />
= ( - e-t(cos t +sin t), e- t(cos t - sin t), -e-')<br />
The point (1, 0, 1) corresponds to t = 0, so the tangent vector there is<br />
r '(O) = (-e 0 (cos 0 +sin 0), e 0 (cos0- sin 0), -e 0 ) = (-1, 1, -1}. Thus, the tangent line is parallel to the. vector<br />
(- 1, 1, - 1} and parametric equations are x = 1 + ( -1)t = 1 - t, y = 0 + 1 · t = t, z = 1 + ( - 1)t = 1 - t.<br />
® 2012 Ccngogelcaming. All Rights Reserv<strong>ed</strong>. Mlly not be: scann<strong>ed</strong>, copi<strong>ed</strong>. or duplicat<strong>ed</strong>, or poste-d to a publicly accessible website, )n whole or In pan.
SECTION 13.2 DERIVATIVES AND INTEGRALS OF VECTOR FUNCTIONS 0 159<br />
27. First we parametrize the curve C of intersection. The projection of C onto the xy-plane is contain<strong>ed</strong> in the circle·<br />
x 2 + y 2 = 25, z = 0, so we can write x = 5 cost, y = 5 sin t. C also lies on the cylinder y 2 • + z 2 = 20, and z ~ 0<br />
near the point (3, 4, 2), so we can write z = )20 - y2 = V20 - 25 sin 2 t. A vector equation then for C is<br />
r (t) = (5c~st,5sint, )20- 25sin 2 t) =} r'(t) = ( -5sint,5cost, ~(20- 25sin 2 t)- 1 1 2 (-50sintcost)).<br />
The point (3, 4, 2) corresponds tot = cos- 1 (~),so the tangent vector there is<br />
The tangent line is parallel to this vector and passes through (3, 4, 2), so a vector equation for the line<br />
is r (t) = (3 - 4t)i + ( 4 + 3t)j + (2 - 6t)k.<br />
29. r(t) =(t,e- t, 2t - e ) =* r'(t)=(1,-e- t,2 - 2t). At(0,1,0),<br />
t = 0 and r '(O) = {1, - 1, 2). Thus, parametric equations of the tangent<br />
.line are x = t, y = 1 - t, z = 2t.<br />
31 . r(t) = (tcost,t,tsint) =* r'(t) = (cost-tsint, 1,tcost+sint).<br />
At ( -1r, 1r, 0), t = 1r and r ' (1r) = ( -1, 1, -1r). Thus, parametric equations<br />
of the tangent line are x = -1r -<br />
t, y = 1r + t, z = -1rt.<br />
33. The angle of intersection of the two curves is the angle between the two tangent vectors to the curves at the point of<br />
intersection. Since r; (t) = (1, 2t, 3t 2 ) and t = 0 at (0, 0, 0), r} (0) = (1, 0, 0) is a tangent vector to r 1 at (0, 0, 0). Similarly,<br />
r~(t) = (cost, 2cos2t, 1) and since r 2(0) = (0, 0, 0), r~ (0) = (1, 2, 1) is a tangent vectortor2 at (0, 0,0). IfO is the angle<br />
between these two tangent vectors, then cos 0 = 7f7r, (1, 0, 0 ~ · (1, 2, 1) = -js and 9 = col:l- 1 ( -js) ~ 66°.<br />
35. J; (t i - t 3 j + 3t 5 k ) dt = (J; t dt). i- (J; t 3 dt) j + (! 0<br />
2<br />
3t<br />
5<br />
dt) k<br />
= (~t 2 ]~ i - [tt 4 ]~ j + [~t 0 ] ~ k<br />
= H4- O)i - t{16 - O)j + ~(64- O) k = .2i -4j +32k<br />
® 2012 Ccngugc Lcnming. All Righls Reserv<strong>ed</strong>. Moy not he scann<strong>ed</strong>. copi<strong>ed</strong>. or duplicotcd. or post<strong>ed</strong> to a publicly acce.
160 . 0 CHAPTER 13 VECTOR FUNCTIONS<br />
37. f~"' 12 (3sin 2 t cost i + 3 sin t cos 2 t j + 2 sin t cost k ) dt<br />
= (I; 12 3 sin 2 t cost dt) i + (.ro" 12 3 sin t cos 2 t dt) j + (I; 12 2 sin t cost dt) k<br />
= [sin 3 t) ~ 12 i + [- cos 3 t] ~ 12 j+ [sin 2 t) ~ 12 k = ( 1 - 0) i + ( 0 + 1) j + ( 1 - 0) k = i + j + k<br />
39. f (sec 2 t i + t(e + 1) 3 j + t 2 ln t k ) dt = (f sec 2 t dt) i + (f t(t,2 + 1) 3 dt) j + (f t 2 ln t dt) k<br />
=tan t i + t(t 2 + 1) 4 j + (tt 3 ln t- ~t 3 )k + C,<br />
where C is a vector constant of integration. [For the z-cornponent,- integrate by pa~s with u = ln t, dv = t 2 dt.]<br />
41. r'(t) = 2t i + 3t 2 j + Vtk =} r (t) = t2.i + e j + tt 3 1 2 k + c , where cis a constant vector.<br />
But i + j = r (1) = i + j + i k + C. Thus C = -tk and r (t) = ·t 2 i + t 3 j + ( tt 3 1 2 - ~) k.<br />
Fo.r Exercises 43-46, 1et u(t) = (u 1 (t), u 2 (t), u 3 (t)) and v (t) = (v 1 ( t), v2(t), v 3 ( t)). In each of these exercises, the proc<strong>ed</strong>ure is to apply<br />
Theorem 2 so that the corresponding properties of derivatives of real-valu<strong>ed</strong> funciions can be us<strong>ed</strong>.<br />
d<br />
d<br />
43. -d [u (t) + v (t)] = -d (u1(t) +v1(t),u2(t) +v2(t),ua(t) +v 3 (t))<br />
t t .<br />
= (! [u1(t) +v1 (t)],! [u2(t) +v2(t)], ;t [ua(t) + va(t)])<br />
= M(t) + vW), u~(t) + vHt), u3(t) +vW))<br />
= M (t), u~ (t), u3(t)) + M(t), v~(t), v3(t)) = u' (t) + v' (t)<br />
d<br />
d<br />
45. dt [u(t) x v(t)] = dt (u2(t)va(t) - ua(t)v2(t),ua(t)v1(t) - u1(t)v3(t),u1(t)v2(t) - u2(t)v1(t))<br />
= (u~va(t) + u2(t)v3(t) - u3(t)v2(t) - u3(t)v2(t),<br />
u3(t)vl(t) + ua(t)v~ (t)- uW)va(t)- u1(t)vW),<br />
u~ (t)vz(t) .+ U1 (t)vW) - u2(t)vl (t) - uz(tM (t))<br />
= (uW)va(t) - u3(t)v2 (t), u3(t)vl (t)- u~(t)va(t), ui(t)'!!2(t)- u2(t)vl(t))<br />
+ (uz(t)vW) - ua(tM(t), u3(t)vi (t)- ul(t)v3(t), u1 (t)vW) :- uz(t)v~ (t))<br />
= u' (t) x v (t) + u (t) x v' (t)<br />
· Alternate solution: Let r(t) = u(t) x v(t). Then<br />
r(t +h)- r (t) = [u (t +h) X v(t +h)] - (u(t) x v(t)]<br />
= [u(t + h) x v(t + h)] - [u(t) x v(t)] + [u (t + h) x v(t)] - [u (t t h) X v (t)]<br />
= u (t + h) x [v (t +h) - v(t)] + [u(t +h) - u (t)] X v(t)<br />
(Be careful of the order of the cross product.) Dividing through by hand taking the limit as h--+ 0 we have<br />
r'(t) = lim u (t +h) x [v~ +h) - v (t)] + lim [u(t +h) - u (t)] X v(t) = u(t) x v'(t) + u'(t) X v (t)<br />
h- o . h- o h<br />
by Exercise 13.1.49(a) and Definition 1.<br />
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SECTION 13.3 ARC LENGTH AND CURVATURE D 161<br />
47. :t [u (t) · v(t)] = u'(t) · v (t) + u (t) ·.v'(t) [by Formula 4 of Theorem 3]<br />
= (cost, - sin t , 1) · (t, cost, sin t) + (sin t, cost, t) · (1,-sin t, cost)<br />
= t cos t - cos t sin t + sin t + sin t - cos t sin t + t cos t<br />
= 2tcost + 2sint - 2cost sint<br />
49. By Formula 4 of Theorem 3, f' (t) = u ' (t) · v (t) + u (t) · v' (t), and v' (t) = (1, 2t, 3t 2 ), so<br />
j'(2) = u'(2) · v{2) + u {2) · v'{2) = (3, 0, 4) · (2,4,8) + (1, 2, - 1) · (1, 4, 12) = 6 + 0 + 32 + 1 + 8- 12 = 35.<br />
d .<br />
51 . - [r(t) x r' (t)] = r' (t) ~ r' (t) + r (t) x r" (t) by Formula 5 of Theorem 3. But r' (t) x r' (t) = 0 (by Example 2 in<br />
dt<br />
Section 12.4). Thus,! [r (t) x r'(t)] = r(t) x r"(t).<br />
53. ~ Jr (t)l = dd [r(t) · r(t)] 112 = ~ [r (t) · r (t)]- 1 1 2 [2r(t) · r '(t)] = -<br />
1<br />
( 1 )I r(t) · r'(t)<br />
dt t · - r t<br />
55. Since u (t) = r (t) · [r'(t) x r"(t)),<br />
u' (t) = r 1 (t) · [r' (t) X r" (t)] + r (t) · ! [r' {t) X r 11 (t))<br />
= 0 + r(t). · (r"(t) x r "(t) + r'(t) X r 111 (t)]<br />
= r(t) · (r' (t) X r 111 (t)]<br />
[since r' (t) ..l r' (t) x r" (t)]<br />
[since r"(t) x r"(t) = 0]<br />
13.3 Arc Length and Curvature<br />
1. r (t) = (t,3cos t,3sint) =? r' (t) = (1,-3sint,3cost) =?<br />
Jr'(t)l = ) 1 2 + (- 3 sint) 2 + (3cost) 2 = ) 1 + 9(sin 2 t + cos 2 t) = v'IO.<br />
Then using Formula3, we have L = f~ 5 lr'(t)l dt = ts v'Wdt = .fi0t] ~ 5 ~ 10 .fiO.<br />
3. r(t) = v'2t i + etj + e- tk =? r'(t) = v'2i + etj - e-tk =?<br />
lr'{t)l = J ( v'2/ + (et) 2 + (-e- t)2 =:= J2 + e 2 t + e u = .J(et + e- t)2 = et + e- t [since et + e- t > OJ.<br />
Then L = }~ 1 lr' (t)i dt = J~( et + e- t) dt = [et - e- t] ~ = e- e- 1 .<br />
5. r(t) = i + t 2 j + t 3 k =? r'(t) = 2tj + 3t 2 k =? ir'(t)i = J4t 2 + 9t 4 = t J4 + 9t2 . [since t 2': OJ.<br />
Then L = J 1 0<br />
lr '(t)i dt = J; 2t tJ4 + 9t 2 dt = fa· t(4 + 9t 2 ) 3 1 = 2<br />
17 (133 1 2 - 4 3 1 2 ) = 2<br />
\ (13 3 / 2 - 8).<br />
7. r(t) = (t 2 , t 3 , t 4 ) =? r 1 (t) = (2t, 3t 2 , 4t 3 ) => lr' (t)i = V(2t? + (3t 2 )2 + (4t 3 ) 2 = J4t2 + 9t4 + 1fit6, so<br />
L = J~ lr'(t)l dt = J; J4t 2 + 9t 4 + 16t 6 dt ~ 18.6833.<br />
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162 0 CHAPTER 13 VECTOR FUNCTIONS<br />
9. r (t) = (sint,cost,tant) =} r'(t) = (cost, - sint,sec 2 t) =}<br />
lr'(t)l = ..jcos 2 t + (-sint)2 + (sec 2 t)2 = v'l + sec 4 t and L = J 0<br />
,. 14 lr'(t)l dt = J~"' 14 v'1 +sec 4 t dt ~ 1.2780.<br />
11. The projection of the curve C onto the xy-plane is the c6rve x 2 = 2y or y = ~ x 2 , z = 0. Then we can choose the parameter<br />
x = t =} y = ~t 2 . Since C also lies on the surface 3z = xy, we have z .= ~xy = ~(t)(~t2) = i t 3 . Then paramen:ic<br />
equations for C are x = t, y = ~ t2, z = *t 3 and the corresponding vector equation is r (t) = (t, ~ t 2 , ~ t 3 ). The origin<br />
corresponds to t = 0 and the point (6, 18,_36) corresponds tot = 6, so<br />
L = f 0<br />
6<br />
lr'(t)l dt = ]~ 6<br />
1(1, t , lt 2 )1 dt = J~ V1 2 + t 2 + (~t 2 ) 2 dt = f 0<br />
6<br />
J1 + t 2 + i t4. dt<br />
= f 0<br />
6<br />
J(1 + ~ t2 )2 dt = j~( 1 + tt 2 ) dt = [t + ~t 3 )~ = 6 + 36 = 42<br />
13. r (t) = 2t i + (1 - 3t) j + (5 + 4t) k =} r'(t) = 2i- 3j + 4 k and "* = lr'(t)l :::::: v'4 +9 + 16 = .;29. Then<br />
8 = 8( t) = J~ lr' ( u) I du = J; .;29 du = .;29 t. Therefore, t = ~ 8, and substituting for t in the original equation, we<br />
have r (t(s)) = ~8 i + (1- ~8)j + (5+ ~8) k.<br />
15. Here r (t) = (3 sin t , 4t, 3 cos t), so r' (t) = (3cos t, 4, -3 sin t) and lr'(t)l = ..)9 cos 2 t + 16 + 9 sin 2 t = v'25 = 5.<br />
The point (0, 0, 3) corresponds to t = 0, so the arc length function beginning at (0, 0, 3) and measuring in the positive<br />
direction is given by 8( t) = J; lr' ( u) I du = J; 5 du = 5t. 8( t) = 5 =} 5t = 5 . =} t = 1, thus your location after<br />
moving 5 units along the curve is (3 sin 1, 4, 3 cos 1).<br />
17. (a) r (t) = (t, 3 cost, 3 sin t) =} r' (t) = (1, - 3 sin t, 3 cost) =} lr' (t)l = ..)i + 9sin 2 t + 9 cos 2 t = v'IO.<br />
· ( ) r' (t) 1 ( . ) / 1 ;l • 3 )<br />
ThenT t = lr'{t)l = "710 1, - 3 sm t,3cost or \"Tlii,-'710smt,'711icost.<br />
T'(t) = fto (0, - 3cost, -3sint) =} IT '(t)l = 7io ..)o + 9cos 2 t + 9sin 2 t = #o· Thus<br />
N (t) = ~ ~:~~~ ~ = ~~~ (0,-3cos t,-3sin t) = (0,-cost,-sint).<br />
(b) ~t(t) = IT '(t)l = 3/v'W = ~<br />
lr'(t)l v'W 10<br />
19. (a)r(t) = (v'2t,et,e- t) =} r'(t) = (v'2,et,-e-t) =} Jr'(t)J=v'2 +e2L + e 2 t =..J(et+e-t)2= et+e-t.<br />
Then<br />
[after multiplying by ::]<br />
and<br />
T '(t) = _1_ (v'2e' 2e2t 0)- 2e2t (V'fe' e2t - 1)<br />
e 2 '+ 1 ' ' (e 2 ' + 1) 2 ''<br />
~ (e 2 ' ~ 1<br />
)2 ((e 2 ' + 1) (v'2e' ,2e 2 ',0) - 2e 2 ' ( .J2e',e 2 ', - 1)) = (e2' ~ 1<br />
) 2 (v'2e' (1 .:... e 2 ') ,2e 2 ',2e 2 ')<br />
® 2012 Cengagc Lc11nting. All Righls Reserv<strong>ed</strong>. Muy no1 be :scnnn
SECTION 13.3 ARC LENGTH AND CURVATURE d 163<br />
Then<br />
Therefore<br />
T '(t) e 2 t + 1 1 ( rn t( 2t) 2t 2t)<br />
N (t) = !T '(t)! = J2 et (e2t + 1)2 v2e 1 - e , 2e , 2e<br />
= 1 (J2et(1- e2t) 2e2t 2e2t) = _1_ (1- e2t J2et· J2et)<br />
J2et(e2t+1) ' , e2t+ 1 , ,<br />
!T'(t)! J2et 1 - J2et - J2e2t - .J2e2t<br />
(b) ~~:(t) = lr'(t)l = e2t + 1 . et + e t - eSt+ 2et + e-t - e4t + 2e2t + 1 -:- (e2t + 1)2<br />
21. r (t) = t 3 j + e k =} r'(t) = 3e j + 2t k, r"(t) = 6tj + 2 k, lr'(t)! = y'o 2 + (3t2)2 + (2t)2 = v'9t4 + 4t2,<br />
, " _ 2 _ lr'(t) X r"(t)i _ ·6t 2 _ 6t 2<br />
2 ) 312 .<br />
r' (t) x r"(t) = -6t2 i, Jr (t) X r (t)J - 6t . Then ~~:(t)- I () 3 -<br />
3 - ( 9 1 .<br />
r' t I ( v'9t4 + 4t2 ) t' + 4t<br />
23. r (t) = 3t i + 4sin t j + 4cost k =? r'(t) = 3 i + 4costj - 4sin tk, r"(t) = -4sintj - 4cos~k,<br />
Jr '(t)! = Jg + 16 cos 2 t + 16sin 2 t = .J9+T6 = 5, r '(t) x r"(t) = -16 i + 12cos t j - 12 sin t k,<br />
Jr' (t) x r " (t)! = }256 + 144 cos2 t + 144sin 2 t = J405 = 20. Then ~~:(t) = lr' (t) x r~ (t)J = 2 ~ = ..! .<br />
, Jr'(t)[ · 5 25<br />
25. r (t) = (t, t 2 , t 3 ) =? r' (t) = (1, 2t, 3t 2 ). The point (1, 1, 1) corresponds to t = 1, and 1) 1 (1) = (1, 2, 3) =?<br />
Jr '(1)1 = v'1 + 4 +9 = Jf4. r"(t) = (0, 2,6t ) =?<br />
r"(1) = (0, 2,6). r'(1) x r"(1) = (6, - 6,2),so<br />
·lr'(1) x r"(1)! = yf36 + 36 + 4 = J76. Then ~~:(1_} = lr'(~;,~ 1 ~'~( 1 )1 = ::;; = ~ VSJ.<br />
4 , 3 "( ) 2 ( ) lf"(x)! l12x 2 1 12x 2<br />
27. f (x) = x , f (x) = 4x , f x = 12x • II: x = [l + (J'(x))2)3/2 = [ 1 + ( 4 x3)2]3/ 2 = ( 1<br />
+ 16<br />
x6)3/ 2<br />
29. f(x) = xe"', J'(x) = xe"' + e"', f"(x) = xe"' + 2e"',<br />
lf"(x)i !xe"'+2e"'l !x+2Je"'<br />
~~:{x) = [1 + (/'(x))2J312 = [1 + (xe"' + e"')2)3/2 = [1 + (xe"' + e"')2 j3/2<br />
Jy"(x)J e"' _ :r 2x -3/2<br />
31. Since y' = y" = e"', the curvature is ~~:(x) = 312 = ( 1<br />
+ e 2 "' ) 312 - e (1 + e ) .<br />
. [1 + (y'(x))2]<br />
To find the maximum curvature, we first find the critical numbers of ~~:(x):<br />
1<br />
2:r<br />
+<br />
3 2x 1 2 2x<br />
'( ) _ "'( 2x)-3/2 "'(-;!)( 2x)- 5/2( 2 2x) _ x e - e _ x - e<br />
II: x - e 1 + e + e . 2 1 + e e - e (1 + e2"')5/2 - e (1 + e2x)5/2.<br />
~~:' (x) = 0 when 1 - 2e 2 "' = 0, so e 2 "' = ~ or x = - ~ ln 2. And since 1 - 2e 2 "' > 0 for x < - ~ ln 2 and 1 - 2e 2 "' < 0<br />
© 2012 Cengogc Learning. All Righls Reserv<strong>ed</strong>. Mny not be scn.nncd. copi<strong>ed</strong>. or dupljc:1tcd. or post<strong>ed</strong> to a publicly acces.coible website. in whole or in part
164 D CHAPTER 13 VECTOR FUNCTIONS<br />
for x > - ~ ln. 2, the maximum curvature is attain<strong>ed</strong> at the point (-~ ln 2, e(- In 2 ) 1 2 ) = (- ~ ln 2, ~).<br />
Since lim e"'(1 + e 2 x)- 3 1 2 = 0, ~~:(x) approaches 0 as x---+ oo.<br />
:.~: - oo<br />
33. (a) C appears to be changing direction more quickly at P than Q, so we would expect the curvature to be great~r<br />
,<br />
at P.<br />
(b) First we sketch approximate osculating circles at P and Q. Using the<br />
axes scale as a guide, we measure the radius of the osculating circle<br />
at P to be approximately 0.8 units, thus p = .!. =><br />
K,<br />
~ 1.3. Similar!y,'we estimate the radius of the<br />
p .8<br />
"'= .!. ~ - 0<br />
1<br />
c<br />
osculating circle at Q to be 1.4 units, son, = .!. ~ ....!...<br />
4 ~ p 1.<br />
0.7.<br />
X<br />
j6x-' 1 1 6<br />
[1 + (-2x-3)z]3/2 = :z;4 (1 + 4x- 6)3/ 2'<br />
The appearance of the two humps in this graph is perhaps a little surprising, but it is<br />
explain<strong>ed</strong> by the fact that y = x - 2 increases asymptotically at the origin from both<br />
-1<br />
directions, and so its graph has very little bend there. [Note th~t r;,(O) is undefin<strong>ed</strong>.]<br />
37. r(t) = (tet, e- 1 ,../2t) => .r '(t) = ((t+l)et,-e-'·,../2), r"(t) = ((t+2)et,e-t, O). Then<br />
r' (t) x r" (t) = ( -.J2e-t, ../2(t + 2)e 1 , 2t + 3), · Jr ' (t) x r " (t)l = y'2e- 21 + 2(t + 2) 2 e 2 t + (2t + 3)2,<br />
l r'(t)J = . l(t + 1 )2e2t + e zt + 2 , ( ) lr'(t) x r "(t)l y'2e- 2 t +'2(t + 2)2e 2 t + (2t + 3)2<br />
v and "' t = lr'(t)13 = ((t + 1)2e2t + e-2t + 2]3/2<br />
We plot the space curve and its curvature function for - 5 :::; t :::; 5 below.<br />
K(l)<br />
0.6<br />
y<br />
- 5<br />
5 I<br />
From the graph of r;,(t) we see that curvature is maximiz<strong>ed</strong> fort = 0, so the curve bends most sharply at the point (0, 1, 0).<br />
The curve bends more gradually as we move away from this point, becoming almost linear. This is reflect<strong>ed</strong> in the curvature<br />
graph, where n,(t) becomes nearly 0 as /tl increases.<br />
39. Notice that the curve b has two inflection points at which the graph appears alrnos,t straight. We would expect the curvature to<br />
be 0 or nearly 0 at these values, bu't the curve a isn't near 0 there. Th~ s, a must be the graph of y = f ( x) rather than the graph<br />
of curvature, and b is the graph .of y = K-( x).<br />
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SECTION 13.3 ARC LENGTH AND CURVATURE D 165<br />
41. Using a CAS, we find (after simplifYing)<br />
K(!)<br />
6 v'4cos2 t - 12cost + 13<br />
~~:(t) = ( 17<br />
_ 12<br />
cos t) 3 12 • (To compute cross<br />
products in Maple, use the VectorCa1cu1us or<br />
Li nearA1gebra package and the Cross Produc t (a, b)<br />
command; in Mathematica, use Cross (a, b].) Curvature is<br />
largest at integer multiples of2:rr.<br />
0 21T 41T 61r I<br />
43. X = t 2 => :i; = 2t => X = 2, y = t 3 => iJ = 3t 2 => ij = 6t.<br />
\xy- iJx\ j(2t)(6t) - (3t 2 )(2) j j12t2- 6t 2 1 6t2<br />
Then ~~:(t) = [:i;2 + y2)3/2 = [(2t)2 + (3t2)2j3/ 2 = (4t2 + W)3/2 = (4t2 + 9t4)3/2 ·<br />
45. x = et cos t => :i; = e 1 (cos t- sint) => x = e 1 ( - sint- cost)+ et(cost- sin t) = - 2et sin t,<br />
y = et sint => iJ = et (cos t + sin't) => y = e 1 (- sint +cost)+ et(cos t + sint) = 2et cost. Then<br />
\:i;ij - 1ixl jet(cos t- sin t)(2et cost) - e 1 (cos t + sin t)( -2et sin t}j<br />
x;( t) = . . = .:....__;..______;_;_---:-:_-:---:--------'-:-'-;;-;;;--~<br />
[x 2 + y 2 ] 3 1 2 ([et(cos t- sin t)J2 + [et(cos t +sin t)j2) 3 / 2<br />
j2e 21 (cos 2 t - sin t cost + sin t cos t + sin 2 t)j j2e 2 t(1) j 2e2t 1<br />
= [e21(cos2t - 2costsint +sin 2 t + cos2 t + 2cos t sint +sin 2 t)] 3 2<br />
' = [e2t(1 + 1)] 3 1 2 = e 31 (2) 3 / 2 = -..fiet<br />
2 ) _ _ r'(t) _ (2t,2t 2 ,1) _ (2t,2t\ 1) _ ( 2 2 1<br />
2 t 2 + 1<br />
,soT(1) - 3•3•3)·<br />
47. ( 1, 3 ,1 correspondstot - 1. T (t)-\r'(t)\ - v' 4<br />
t 2 + 4<br />
t 4<br />
+ - 1<br />
T '(t) = - 4t(2t 2 + 1)- 2 ( 2t, 2t 2 , 1) + (2t 2 + 1)- 1 (2,4t, 0) [by Formula 3 of Theorem 13.2.3]<br />
= (2t 2 + 1)- 2 ( -8t 2 + 4t 2 + 2, -8t 3 + 8t 3 + 4t, -4t) = 2(2t2 + 1)- 2 (1- 2t 2 , 2t, -2t)<br />
(1 - 2t 2 , 2t, -2t) - (1- 2t 2 , 2t, - 2t)<br />
v'1 - 4t2 + 4t4 + 8t2 - 1 + 2t 2<br />
49. (0,7r,-?) corresponds tot ='lr. r(t) = (2sin 3t,t,2cos3t) =><br />
T(t) = r' (t) = (6 cos 3t,1,-6sin3t) =: - 1- ( 6 cos 3 t, 1 ,- 6 sin 3 t ).<br />
\r'(t)\ } 36 cos2 3t + 1 + 36sin 2 3t V37<br />
T(1r) = ~ ( - 6, 1, 0) is a normal vector for the normal plane, and so (-6, 1, 0) is also normal. Thus an equation for the<br />
plane ·is - 6(x - 0) + 1(y -1r) + O(z + 2) = 0 ory·- 6x = ~ ·<br />
18<br />
ffi=><br />
N (t) = ~~:m , = (-sin3t, O, - cos 3t). So N(1r) = (0, 0, 1) and B (1r) = ~ (-6, 1, 0) x (0,0, 1) = ~ {1, 6, 0).<br />
Since B(1r) is a normal to the osculating plane, so is (1, 6, 0).<br />
An equation for the plane is 1(x- 0) + 6(y -1r) + O(z + 2) = 0 or x + 6y = 61r.<br />
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166 0 CHAPTER 13 VECTOR FUNCTIONS<br />
51. The ellipse is given by the parametric equations x = 2 cost, y = 3 sin t, so using the result from Exercise 42,<br />
~~:(t) = l:i:ii- xyl = l(-2sint}(-3sint)- (3cost)(- 2cost)1 = 6<br />
[:i: 2 + 1?] 3 / 2 ( 4 sin 2 t + 9 cos 2 t~ S/ 2 ( 4 sin 2 t + 9 cos 2 t)3/ 2 •<br />
At (2, 0}, t = 0. N.ow ~~:(0} = 2<br />
67<br />
= ~.so the radius of the osculating circle is<br />
1/ ~(0} = ~ and its center is ( -~, 0). Its equation is therefore (x + ~ ) 2 + y 2 = ¥·<br />
s<br />
At (0, 3}, t = ~.and~~:(~) = ~ = ~· So the radius of the osculating circle is~ and<br />
its center is ( 0, ~). Hence its equation is x 2 + (y - ~) 2 = lf.<br />
53. The tangent vector is normal to the normal plane, and the ve~ tor (6, 6, - 8) is normal to the given plane.<br />
But T (t) II r'(t) and (6, 6, -8) II (3, 3, -4), so we ne<strong>ed</strong> to find t such that r' (t) II (3, 3, - 4}.<br />
r (t) = (t 3 1 3t, t4 ) :::::? r ' (t) = (3t 2 , 3, 4t 3 ) II (3, 3, - 4) when t = - 1. So the planes are parallel at the point ( - 1, -3, 1).<br />
55. First we parametrize the curve of intersection. We can choose y = t; then x = y 2 = t 2 and z = x 2 = t 4 , and the curve is<br />
given by r (t) = (t 2 , t, t 4 ). r ' (t) = (2t, 1, 4t 3 ) and the point (1, 1, 1) corresponds tot= 1, so r ' (1) := (2, 1, 4) is a normal<br />
vector for the normal plane. Thus an equation of the normal plane is<br />
r' (t) 1<br />
2(x - 1) + 1(y - 1} + 4(z- 1) = 0 or 2x + y + 4z = 7. T (t) = -<br />
1<br />
'( )I = . ( 2t, 1, 4t 3 ) and<br />
· r t J 4t 2 + 1 + 16t 6<br />
T ' (t) = - H 4t2 + 1 + 1?t 6 ) - S/ 2 (8t + 96t 5 ) (2t, 1, 4t 3 ) + ( 4t 2 + 1 + 16t 6 ) - l/ 2 ( 2, 0, 12t 2 ). A normal vector for<br />
the osculating plane is B(l) = T (1) x N (1), but r'(1) = (2, 1, 4) is parallel to T (1) and<br />
T ' (1} = - ~ (21)- 3 : 2 ( 104} (2, 1, 4} + (21)- 1 1 2 (2, 0, 12) = 21<br />
3TI (-31, -26, 22) is parallel to N (1) as is (-31, .:...26, 22),<br />
so (2, 1, 4} x (- 31, - 26, 22} = (126, -168, -21) is normal to the osculating plane. Thus an equation for the osculating<br />
plane is 126(x - 1) - 168(y - 1) - 21(z- 1) = 0 or 6x- 8y - z = -3.<br />
dTI dT<br />
57 - ldTI_IdT/dtl_ ldT / dtl d N dT / dt -N ldt dt _dT /dt_dT b h Ch' R l<br />
. ~~:- ds- ds/dt - ds/dt an - l dT /dtl 'so~~: - ~dTids - ds/dt-ds yte am ue.<br />
dt dt<br />
59. (a) IB/ = 1 ~ B · B = 1<br />
d<br />
~ ds (B · B )= 0<br />
dB J. B<br />
ds<br />
(b) B = T x N :::::?<br />
: = ! (T x N) = :~ (T x N } ds~dt = :t (T x N } l r'~t)l = [(T ' x N ) + (T X N' ))lr' ~t )l<br />
= [( T X IT ' I + T X N lr'(t)l = Tr'Tt)'l<br />
1 T ' ) ( ')] 1 , T X N '<br />
dB<br />
:::::? -J. T<br />
ds<br />
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SECTION 13.3 ARC LENGTH AND CURVATURE 0 167<br />
(c) B = T x N => T _L N , B _L T and B _LN. SoB, T and N form an orthogonal set of vectors in the thre<strong>ed</strong>imensional<br />
space !R 3 . From parts (a) and (b), dBfds is perpendicular to both BandT, so dB/ ds is parallel to N .<br />
Therefore, dB/ ds = -r(s)N , where r(s) is a scalar.<br />
-<br />
(d) Since B = T x N , T _L N and both T and N are unit vectors, B is a unit vector mutually perpendicular to both T and<br />
N. For a plane curve, T and N always lie in the plane of the curve, so that B is a constant unit vector always<br />
perpendicular to the plane. Thus dB/ ds = 0, but dBfds = - r(s)N and N =/= 0, so r(s) = 0.<br />
61. (a) r ' = s' T => r" = s" T + s' T ' = s" T + s' ~~ s' = s" T + ~(s') 2 N by the first Serret-Frenet formula.<br />
(b) Using part (a), we have<br />
(c) Using part (a), we have<br />
r' X r" = (s' T ) X (s" T + ~cs') 2 N ]<br />
= ((si T ) x (s" T )] + ((s'T ) x (~(s'? N)) [by Property 3 of Theorem 12.4.11)<br />
= (s' s")(T x T ) + ~~;(s') 3 (T x N ) = 0 + ~~;(s'? B = ~~;(s') 3 B<br />
r 111 = [s" T + ~~;(s') 2 N ]' = s 111 T + s" T ' ;t- ~'(s'? N + 2~~;s' s" N + ~~;(s'? N '<br />
= s"' T + s"dT s' + ~'(s'? N + 2~s' s" N + ~(s') 2 dN s'<br />
ds<br />
ds<br />
= s"' T + s" s'~ N + K 1 (s') 2 N + 2~s' s" N + ~~;(s') 3 (-~ T + rB) [by the second formula]<br />
= [s"'- K 2 {s'?J T + (3~s' s" + ~~;'(s') 2 ] N + ~~;r(s') 3 B<br />
(d) Using parts (b) and (c) and the facts that B · T = 0, B · N = 0, and B · B = 1, we get<br />
(r ' x r ") . r 111 = ~~:(s') 3 B · { (s 111 - ~ 2 (s') 3 ] T + (3~s' s" + ~~ (s') 2 ] N + KT(s') 3 B} = ~~:(s') 3 ~r(s')3 = T<br />
Jr ' x r " J 2 · J~~:(s')3 B J 2 (~~;(s')3] 2 •<br />
63. r = (t, tt2, !t 3 ) => r' = (1, t, t 2 ), r" = (0, 1, 2t), r 111 = (0, 0, 2) => r ' x r" = (t 2 , - 2t, 1) =><br />
(r'x r")· r"' (t2, - 2t,1) ·(0,0,2) 2<br />
T - - - --,---....,..--<br />
- Jr' x r 11 J 2 - t 4 + 4t 2 + 1 - t 4 + 4t 2 + 1<br />
65. For one helix, the vector equation is r (t) = (10 cost, 10 sin t, 34t/(2rr)) (measuring in angstroms), because the radius of each<br />
helix is 10 angstroms, and z increases by 34 angstroms for each increase of 271' in t. Using the arc length formula, Jetting t go<br />
from 0 to 2.9 x 10 8 x 21r, we find the appmximate length of each helix to be<br />
B s . / ] 2.9xl0 8 x2rr<br />
L = J;·9xlO<br />
x zrr Jr'(t)J dt = j~ · 9 x 10 x zrr y (- 10 sint) 2 + {10cost) 2 + U!) 2 dt = ..j100 + (~!) 2 t<br />
0<br />
= 2.9 x 10 8 x 21r J10o + ( g! ) 2 ~ 2.07 x 10 10 A- more than two meters!<br />
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168 D CHAPTER 13 VECTOR FUNCTIONS<br />
13.4 Motion in Space: Velocity and Acceleration<br />
1: (a) Ifr(t) = x(t) i + y (t) j + z(t) k is the position vector of the particle at timet, then the average velocity over the time<br />
.interval [0, 1] is<br />
_ r(l)- r (O) _ (4.5 i + 6.0j + 3.0k)- (2.7 i + 9.8j + 3.7k) _ . 1 8 3 8 . 0 ? k s· . - . 1 I h th<br />
1 - . J - . . 1m1 ar y, over t e o er<br />
1-0 1<br />
•<br />
v ave - -<br />
intervals we have<br />
[0.5, 1] :<br />
[1, 2] :<br />
[1, 1.5]:<br />
_ r(1)- r (0.5) _ (4.5 i + 6.0j + 3.0 k)- (3.5 i + i2j + 3.3 k) _ 2 0<br />
. _ 2 4<br />
. _ 0 6<br />
k<br />
Vave- 1 - 0.5 - 0.5 - . 1 . J .<br />
V ave -<br />
_ r (2)- r (1) _ (7.3i+7.8j+2.7k) - (4.5i+6.0j +3.0k) _ . 28<br />
2 _ 1<br />
- .<br />
1<br />
1 8 ._ 03 k .<br />
- · 1 + · J ·<br />
_ r(1.5) -r(1) _ (5.9i+6.4j+2.8k)-(4.5 i +6.0j+3.0 k) _ 28<br />
. · 08<br />
._ 04<br />
k<br />
Yavo- - - · 1 + · J •<br />
1.5 -1 0.5 . .<br />
(b) We can estimate the velocity at t = 1 by averaging the average velocities over the time intervals [0.5, 1] and [1, 1.5]:<br />
v(1) ~ ~{(2 i - 2.4j- 0.6k) + (2.8 i +0.8j- 0.4k)] = 2.4i - 0.8j - o·.5k. Then the spe<strong>ed</strong> is<br />
Jv (1)\ ~ )(2.4) 2 + ( - 0.8)2 + ( -0.5)2 ~ 2.58.<br />
. 1 2 )<br />
3. r(t) = (- 2 t ,t =><br />
v{t) = r'(t) = (-t, i)<br />
a(t) = r"(t) = (-:- 1, 0)<br />
Att = 2:<br />
v{2) = ( -2, 1)<br />
a(2) .= (-1,0)<br />
jv(t)\ = Jt 2 + 1<br />
5.r(t)=3costi+2sintj =><br />
v (t) = -3sinti+2costj<br />
a(t) = -3costi- 2sintj<br />
Att = 71'/3:<br />
v(~) =-¥i+j<br />
a(i) = - ~ i - J3j<br />
(3,0)<br />
/<br />
X<br />
Jv (t)\ = )9sin 2 t + 4cos 2 t;:::: .) 4 + 5sin 2 t<br />
Notice that x 2 /9 + y 2 j4 = sin 2 t + cos 2 t = 1, so the path is an ellipse.<br />
Att = 1:<br />
z<br />
v(t) = i + 2tj<br />
a(t) = 2j<br />
v(1) = i + 2j<br />
a(1) = 2j<br />
Jv(t)l=~<br />
Here x = t, y = t2 => y = ~ 2 and z = 2, so the path of the particle is a<br />
parabola in the plane z = 2.<br />
X<br />
Y<br />
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SECTION 13.4 MOTION IN SPACE: VELOCITY AND ACCELERATION 0 169<br />
9. r (t) = (t2 + t, e - t , t 3 ) => v(t) = r' (t) = (2t + 1, 2t - 1, 3t 2 ), a(t) = v' (t) = (2, 2, 6t),<br />
lv (t)i = .j(2t + 1)2 + (2t- 1)2 + (3t 2 )2 = v'9t4 + 8t 2 + 2.<br />
11. r (t) = v'2t i + e' j + e-t k => v (t) = r'(t) = v'2i + e' j - e-' k, a(t) = v'(t) = e' j + e-' k,<br />
lv(t)l = v'2 + e 2 ' + e zt = J(e' + e-') 2 = e' + e-' .<br />
13. r (t) = e'(cost,sint, t) =><br />
v (t) = r'(t) = e'(cost,.sin t, t~ + et (-sint,cost, 1) = et (cost - sint,sint +cost, t + 1)<br />
a(t) = v'-(t) = 'e'(cost- sint - sin t - cost, sin t +cost+ cost - sin t , t + 1 + 1)<br />
= et(- 2sin t , 2 cost, t + 2)<br />
lv(t)i = etJcos 2 t + sin 2 t - 2costsint + sin 2 t + cos 2 t + 2sin tcost + t 2 + 2t + 1<br />
= e' .Jt 2 + 2t + 3<br />
15. a(t) = i + 2j => v(t) = J a(t) dt = J(i + 2j) dt = t i + 2tj + C and k = v {0) = C,<br />
.soC = k and v(t) = t i + 2tj + k. , r{t) = J v(t)"dt = J (t i + 2tj + k) dt = ~t 2 i + t2 j + t k + D .<br />
But i = r (0) = D, soD= i and r(t) = (tt 2 + 1) i + t 2 j + tk.<br />
17 .. (a) a(t) = 2t i + sintj +cos 2t k => ·<br />
v( t) = J (2t i + sin t j + cos 2t k) dt = t2 i - cost j + t sin 2t k + C<br />
and i = v (O) = -j + C, so C = i + j<br />
and v(t) = (t 2 + 1) i + (1- cost)j + t sin2tk.<br />
r (t) = J[(t 2 + 1) i + (1- cost)j + t sin2tk)dt<br />
= (tt 3 + t) i + (t- sin t)j- ~cos 2t k + D<br />
(b)<br />
0.6<br />
0.4<br />
% 0.2<br />
0<br />
Butj = r(O) = -i k+ D , soD = j + {kand r (t) = (tt 3 +t) i+ (t -sint + 1)j + (!- { cos2t) k.<br />
19. r (t) = (t 2 ' 5t, e- 16t) =} v(t) = (2t, 5, 2t - ,16), lv(t)i = v'4t 2 + 25 + 4t 2 - 64t + 256 = v'8t2 - 64t + 281<br />
and dd lv(t)i = t(se - 64t + 281)- 1 1 2 (16t - 64) . This is zero if and only if the numerator is zero, that is,<br />
t .<br />
16t - 64 = 0 or t = 4. Since :t lv(t)l < 0 fort < 4 a~d :t lv(t)l > 0 fort > 4, the minimum spe<strong>ed</strong> of -/153 is atta.in<strong>ed</strong><br />
at t = 4 units of time.<br />
21 . IF (t)i = 20 N in the direction of the positive z-axis, so F (t) = 20 k. Also m = 4 kg, r{O) = 0 and v {O) = i - j .<br />
Since 20k = F(t) = 4 a(t), a (t) = 5k. Then v (t) = 5t k + c1 where c1 = i-j so v(t) = i - j + 5tk and the<br />
spe<strong>ed</strong> is lv (t)l = v 1 + 1 + 25t 2 = v'25t 2 + 2. Also r (t) = t i - t j + ~t 2 k + cz and 0 = r (O), so C 2 = 0<br />
and r (t) = t i - t j + ~ t 2 k.<br />
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170 0 CHAPTER 13 VECTOR FUNCTIONS<br />
23. lv(O) I = 200 m/s and, since the angle of elevation is 60°, a unit vector in the _direction of the velocity is<br />
(cos60°)i+ (sin60°)j = ~ i + 4!j. Thusv(O) = 2oo(~i + 4j) = lOOi+ 100J3j and ifwesetuptheaxessothatthe<br />
projectile starts at the origin, then r(O) = 0. Ignoring air resistance, the only force is that due to gravity, so<br />
F(t) = ma(t) = -m9 j where 9 ~ 9.8 m/s 2 . Thus a(t) = -9.8j and, integrating, we have v (t) = - 9.8t j +C. But<br />
100 i + 100 V3 j = v(O) = C , so v( t) = 100 i + ( 100 V3 - 9.8t) j and then (integrating again)<br />
r (t) = lOOt i + (100 V3 t- 4.9t 2 ) j + D where·O = r(O) = D. Thus the position function of the projectile is<br />
'<br />
r(t) = lOO t i + (100 V3t- 4.9t 2 ) j .<br />
(a) Parametric equations for the projectile are x(t) = lOOt, y(t) = 100 V3 t- 4.9t2. The projectile reaches the ground when<br />
y(t) = O(and t > 0) => 100 V3t- 4.9t 2 = t(100 V3- 4.9t) = 0 => t = 10 4 ~f ~ 35.3 s. So the range is<br />
x( 10 4°;0) = 100e 0 4~f3) ~ 3535 rn.<br />
(b) The maximum height is reach<strong>ed</strong> when y(t) has a critical number (or equivalently, when the vertical component<br />
of velocity is 0): y' ( t) = 0 => 100 J3 - 9.8t = 0 => t = 10 0 ~{3 ~ 17.7 s. Thus the maximum height is<br />
v( 10~{!) = 100 V3 ( 10 ~{!) - 4.9 ( 10 9~{!r ~ 1531 m.<br />
(c) From part (a), impact occurs at t = 10 J.;f s. Thus, the velocity at impact is<br />
v ( 10 4°{!) = 1~0 i + [ 100 J3 - 9.8 ( 10 4°f)] j = 100 i- 100 J3j and the spe<strong>ed</strong> is<br />
lv ( 10 1.f) I= JIO,OOO + 30,000 = 200 mjs.<br />
25. As in Example 5, r (t) = (vo cos45°)t i + [(vo sin45°)t- t9t2] j = t[voJ2 t i + (voJ2 t- gt 2 ) j] . The ball lands when<br />
y = 0 (and t > .0) ::::>:<br />
voJ2 . . I . 1 rn2' vo J2 2 90<br />
9 an<br />
d<br />
t<br />
h<br />
e<br />
..<br />
m1t1a<br />
. I<br />
9 g '<br />
t = -- s. Now smce 1t ands 90 m away, 90 = x = - 2<br />
vo v L. --or v 0 =<br />
velocity is vo = J90g ~ 30 mjs.<br />
27. Let a be the angle ,of elevation. Then v 0 = 150m/sand from Example 5, the horizontal distance travel<strong>ed</strong> by the projectile is<br />
d = v5 s~ 2 a. Thus 1502 ;in 2 a = 800 => sin 2a = ~~~; ~ 0.3484 => 2a ~ 20.4° or 180 - 20.4 = 159.6°.<br />
Two angles of elevation then are a ~ 10.2° and a :=::: 79.8°.<br />
29. Place the catapult at the origin and assume the catapult is 100 meters from the city, so the city lies between (100, 0)<br />
and (600, 0). The initial spe<strong>ed</strong> is v 0 = 80 m/s and let 0 be the angle the catapult is set' at As in Example 5, the trajectory of<br />
the catapult<strong>ed</strong> rock is given byr(t) = (80cosO)ti + [(80sin0)t- 4.9t 2 ] j. The topofthe near city wall is at (100, 15),<br />
which the rock will hit when (80 cos 0) t = 100 => t = - 4<br />
5 O and (80 sin O)t - 4.9t<br />
2<br />
= 15 =><br />
cos<br />
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2<br />
SECTION 13.4 MOTION IN SPACE: VELOCin' AND ACCELERATION D 171<br />
80 sin 9 .--- 5 4.9 (-- 5 ) = 15 => 100 tan () - 7.65625 sec 2 fJ = 15. Replacing sec 2 9 with tan 2 8 + 1 gives<br />
4cos9 4 cosfJ<br />
7.65625 tan 2 9 - 100 tanfJ + 22.65625 = 0. Using the quadratic formula, we have tan 9 ~ 0.230635, 12.8306 =><br />
B ~ 13.0°, 85.5°. So for 13.0° < 9 < 85.5°, the rock will land beyond the near city wall. The base of the far wall is<br />
locat<strong>ed</strong> at (600, 0) which the rock hits if (80 cos fJ)t = 600 => t = 2<br />
15 n and (80sin 9)t - 4.9t<br />
2 = 0 =><br />
COSu<br />
. 15 ( 15 ) 2<br />
80sm0 · --n - 4.9 - (} = 0 => 600tan9- 275.625sec 2 fJ = 0 =><br />
2<br />
2cOS!7<br />
COS<br />
275.625 tan 2 9 - 600tan0 + 275.625 = 0. Solutions are tan(}~ 0.658678, 1.51819 => (} ~ 33.4°, 56.6°.· Thus the<br />
rock lands beyond the enclos<strong>ed</strong> city ground for 33.4° < (} < 56.6°, and the angles that allow the rock to land on city ground<br />
are 13.0° < () < 33.4~, 56.6° < 9 < 85.5°. If you consider t~at the rock can hit the far wall and bounce back into the city, we<br />
calculate the angles that cause the rock to hit the top of the wall at (600, 15): (80 cos 9)t = 600 => t = ~ and<br />
2 cos(}<br />
(80sinfJ)t- 4.9t 2 = 15 => 600tan0 -275.625sec 2 (} = 15 => 275.625 tan 2 (} - 600 tan(}+ 290.625 = o.<br />
Solutions are tan 9 :::::: 0. 727506, 1.44936 =><br />
13.0° < (} < 36.0°' 55.4° < (} < 85.5° .<br />
(} :::::: 36.0°, 55.4°, so the catapult should be set with angle fJ where<br />
31. Here a (t) = - 4 j - 32 k so v (t) = -4tj - 32t k + v o = - 4tj- 32tk + 50 i +80 k = 50i- 4tj + (80 - 32t) k and<br />
r (t) = 50t i- 2t 2 j + (BOt- 16e) k (note that r o = 0). The ball lands when the z-component ofr (t) is zero<br />
and t > 0: SOt - 16t 2 = 16t{5 - t) = 0 => t = 5. The position of the ball then is<br />
r (5) = 50(5) i - 2{5) 2 j + [80(5) - 16(5) 2 ] k = 250 i - 50 j or equivalently the point (250, - 50, 0). This is a distance of<br />
J250 2 + ( -50)2 + 0 2 = )65,000 :::::: 255ft from the origin at an angle oftan- 1 (~) :::::: 11.3° from the eastern direction<br />
toward the south. The spe<strong>ed</strong> of the ball is lv (5)l = I 50 i- 20j - 80 k l = J50 2 + ( - 20)2 + ( -80)2 = v'9300:::::: 96.4 ftls.<br />
33. (a) After t seconds, the boat will be 5t meters west of point A. The ve lo~ i ty<br />
20<br />
of the water at that location is 4 ~ 0 (5t)(40- 5t)j . The velocity of the<br />
boat in still water is. 5 i, so the resultant velocity of the boat is<br />
v(t) = 5 i + 4 ~ 0 (5t)( 40- 5t)j = 5i + ( ~t- fG-t2 ) j . Integrating, we obtain<br />
r (t) = 5t i + (~tl - -fGt 3 ) j +C. lfwe place the origin at A (and consider j<br />
to coincide with the northern direction) then r (O) = 0 => C = 0 and we have r (t) = 5t i + (~t 2 -<br />
reaches the east bank after 8 s, and it is locat<strong>ed</strong> at r(8) = 5(8)i + ( ~(8) 2 -<br />
downstream.<br />
1<br />
e) j . The boat<br />
16 -fG (8?) j = 40 i + 16j. Thus the boat is 16 m<br />
(b) Let Q be the angle north of east that the boat heads. Then the velocity of the boat in still water is given by<br />
5(cos Q) i + 5(sin Q) j . At t seconds, the boat is 5(cos Q)t meters from the west bank, at which point the velocity<br />
of the water is . 1 ~ 0 [5(cos Q)t][40- 5(cos Q)t] j . The resultant velocity of the boat is given by<br />
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172 D CHAPTER 13 VECTOR FUNCTIONS<br />
v(t) = 5(cos a:) i + [5 sin a:+ 4<br />
g 0<br />
(5t cos a:)( 40 - 5t cos a:)] j = (5 cos a:) i + (5 sin a:+ ~t cos a: - -fue cos 2 a:) j.<br />
Integrating, r (t) = (5t cos a:) i + (5tsin a:+ ~t 2 cos a:- -ftt 3 cos 2 a:) j (where we have again plac<strong>ed</strong><br />
the origin at A). The boat will reach the east bank ~hen 5t cos a: = 40 =? t = ~ = - 8 -.<br />
5 cos a: cos a:<br />
In order to land at point B ( 40, 0) we ne<strong>ed</strong> 5t sin a: + ~ e cos a: - ft t 3 cos 2 a: = 0 =?<br />
5(- 8 - ) sin a:+~ (- 8 2<br />
-)<br />
cos a: cos a: cos a:<br />
cos a:- -ft(- 8 3<br />
-) .cos 2 a:= 0 =?<br />
- 1 - (40sina: + 48 - 32) = 0 =?<br />
cos a:<br />
40 sin a: + 16 = 0 =? sin a: = - %. Thus a: = sin - l (-%) ~ -23.6°, so the boat should head 23.6° south of<br />
east (upstream). The path does seem realistic. The boat initially heads<br />
If<br />
upstream to counte~act the effect of the current. Near the center of the river,<br />
the current is stronger and the boat is push<strong>ed</strong> downstream. When the boat<br />
nears the eastern bank, the current is slower and the boat is able to progress<br />
upstream to arrive at point B.<br />
0 1--=:::==::::::::;::>o-f'...::::::::=:=::::~ 40<br />
- 12<br />
35. Ifr' (t) = c x r(t) then r' (t) is perpendicular to both c and r(t). Remember that r ' (t) points in the direction of motion, so if<br />
r' (t) is always perpendicular to c, the path of the particle must lie in a plane perpendicular to c. But r ' (t) is also perpendicular<br />
to the position vector r(t) which confines the path to a sphere center<strong>ed</strong> at the origin. Considering both restrictions, the path<br />
must be contain<strong>ed</strong> in a circle that lies in a plane perpendicular to c, and the circle is center<strong>ed</strong> on a line through the origin in the<br />
direction of c.<br />
37. r(t) = (3t - t 3 ) i + 3t 2 j =? r'(t) = (3 - 3t 2 ) i + 6tj,<br />
lr' (t)i = J(3- 3t 2 )2 + (6t)2 = v'9 + 18t 2 + 9t 4 = J(3 - 3t 2 ) 2 = 3 + 3t 2 ,<br />
r" (t) ·= -6t i + 6j, r' (t) x r" (t) = (18 + 18t 2 ) k. Then Equation 9 gives<br />
- r ' (t) . r" (t) - (3 - 3e)( -6t) + (6t)(6) - 18t + 18t 3 - 18t(1 + t 2 ) - 6t<br />
aT- lr '(t)/ - 3 +3t2 - 3 + 3t2 - 3(1 + t 2 ) -<br />
[or by Equation 8,<br />
I d [ 2] ] dE . LO . lr'(t) X r"(t)i 18 18e - 18(1 t 2 ) - 6<br />
aT = v = dt 3 + 3t = 6t an quat10n g1ves aN - ir'(t)i -<br />
3 + 3 t2 - 3 ( 1 + t 2) - ·<br />
39. r (t) = cost i +sin t j + t k =? r '(t) = - sint i + costj + k , ir'(t)i = Vsin 2 t + cos 2 t + 1 = V'2,<br />
r" (t) = - cost i- sin tj, r' (t) x r" (t) =sin t i - cos tj + k.<br />
_ r'(t) · r"(t) _ sintcost - sintcost _ 0<br />
d _lr'(t) x r"(t)l _ Vsin 2 t+cos 2 t+ 1 = .J2 = l<br />
Then D.T - lr'(t)i - . .j2 - an aN - lr'(t)i - -/2 -/2 ·<br />
41. r(t)=et i +-/2tj +e-t k =? r'(t)=et i+-/2j-e- 1 k, ir(t)i=v'e 21· +2+ e 2 t=J(e 1· +e t)2=e'·+e- t,<br />
e2t _ e- 2t (et + e- t)(et _ e- t) .<br />
r"(t) = et i + e-t k. Then aT = = = et - e:-t = 2sinht<br />
et + e t et + e t<br />
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CHAPTER 13 REVIEW D 173<br />
43. The tangential component of a is the length of the projection of a onto T, so we sketch<br />
the scalar projection of a in the tangential direction to the curve and estimate its length to<br />
be 4.5 (using the fact that a has length 10 as a guide). Similarly, the normal component of<br />
a is the length of the projection of a onto N, so we sketch the scalar projection of a in the<br />
normal direction to the curve and estimate its lengttJ to be 9.0. Thus aT ::::::: 4.5 cmf s 2 and<br />
X<br />
45. _If the engines are turn<strong>ed</strong> off at timet, then the spacecraft will continue to travel in the direction ofv(t), so we ne<strong>ed</strong> at such<br />
that for some scalars > 0, r(t) + s v (t) = (6, 4, 9). v(t) = r ' (t) = i + ~ j + (t 2<br />
~ 1<br />
) 2<br />
k =><br />
4 8(3-t)t<br />
so 7- t2 + 1 + (t2 + 1)2 = 9 ¢}<br />
24t- 12t 2 - 4<br />
- -;-:-;;:--:--::-.;--:- = 2 ¢} t 4 + 8t 2 - 12t + 3 = 0.<br />
(t2 + 1)2<br />
·a is easily seen that t = 1 is a root of this polynomial. Al~o 2 + ln 1 + 3 ~ 1 = 4, sot= 1 is the desir<strong>ed</strong> solution.<br />
13 Review<br />
CONCEPT CHECK<br />
1. A vector function is a function whose domain is a set of real numbers and whose range is a set of vectors. To find the derivative<br />
or integral, we can differentiate or integrate each component of the vector function.<br />
2. The tip of the moving vector r (t) of a continuous vector function traces out a space curve.<br />
3. The tangent vector to a smooth curve at a point P with position vector r(t) is the vector r ' (t). The tangent line at p ' is the line<br />
through P parnllel to the tangent vector r ' (t). The unit tangent vector is T (t) = ,:m,.<br />
4. (a) (a)- (f) See Theorem 13.2.3.<br />
5. Use Formula 1 3.3.2, or equivalently, 1 3.3.3.<br />
6. (a) TI1e curvature of a curve is"' = I~~ I where Tis the unit tangent vector.<br />
I I<br />
(b) K.(t) = . T r '(t)<br />
'(t)<br />
(c) K.(t) = lr' (t ) X r" (t)l<br />
[r'(t)[ 3 _ if"(x) l ·<br />
(d) K-(x)- [1 + (f'(x))2j3/2<br />
T'(t)<br />
7. (a) The unit normal vector: N (t) = IT'(t)l' The binormal vector: B (t) = T (t) x N (t).<br />
(b) See the discussion prec<strong>ed</strong>ing Example 7 in Section 13.3.<br />
8. (a) Ifr(t) is the position vector of the particle on the space curve, the velocity v (t) = r'(t), the spe<strong>ed</strong> is given by lv (t )i,<br />
and the acceleration a (t) = v ' (t) = r" (t).<br />
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174 D CHAPTER 13 VECTOR FUNCTIONS<br />
(b) a = a7:T +aNN where aT = v' and aN = 1w 2 .<br />
9. See the statement of Kepler's Laws on page 892 [ET 868].<br />
TRUE-FALSE QUIZ<br />
1. True. If we reparametrize the curve by replacing u = t 3 , we have r(u) = u i + 2u ~ + 3uk, which is a line through the origin<br />
with direction vector i + 2 j + 3 k.<br />
3. False. The vector function represents a line, but the line does not pass through the origin; the x-component is 0 only for t ;, 0<br />
which co~esponds to the point {0, 3, 0) not {0, 0, 0).<br />
d<br />
5. False. By Formula 5 ofTheorem 13.2.3, -d (u(t) x v(t)] = u '(t ) x v (t) + u (t) x v'(t).<br />
. t<br />
7. False. K- is the magnitude of the rate of change of the unit tangent vector T with respect to arc length s, not with respect tot.<br />
9. True. At an inflection point where f is twice continuously differentiable we must have !" ( x) = 0, and by Equation 13.3.11,<br />
the curvature is 0 there.<br />
11. False. If r(t) is the position of a moving particle at timet and lr(t)l = 1 then the particle lies on the unit circle or the unit<br />
sphere, but this does not mean that the spe<strong>ed</strong> lr'(t)l must be constant. As a counterexample, let r (t) = (t, Jl=t2), then<br />
r'(t) = ( 1, -t/J1 - t 2 ) and lr(t)l = Jt 2 + 1 - t 2 = 1 but lr'(t)l = yf1 + t 2 /(1- t 2 ) = 1 /~ which is not<br />
constant.<br />
13. True. See the discussion prec<strong>ed</strong>ing Example 7 in Section 13.3.<br />
EXERCISES<br />
1. (a) The corresponding parametric equations for the curve are x = t,<br />
y = cos 1rt, z = sin -rrt. Since y 2 + z 2 = 1, the curve is contain<strong>ed</strong> in a<br />
I<br />
circular cylinder with axis the x-axis. Since x = t, the curve is a helix.<br />
(b) r(t) = t i + cos rij +sin -rrt k =><br />
r'(t) = i--rrsin 1rt j + -rrcos -rrt k ~<br />
X<br />
r" ( t) = --rr 2 cos 1rt j - 1r 2 siri 7rt k<br />
3. The projection of the curve C of interSection onto the xy-plane is the circle x 2 + y 2 = 16, z = 0. So we can write<br />
x = 4 cost, y = 4 sin t, 0 S t S 21r. From the equation of the plane, w~ have z = 5 -<br />
x = 5 - 4 cos t, so parametric<br />
equations for Care x = 4 cost, y = 4 sin t, z = 5 - 4 cost, 0 S t S 2-rr, and the corresponding vector function is<br />
r(t) = 4 cost i + 4 sin t j + (5 - 4 cost) k , 0 S t S 21r.<br />
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5. f~(t 2 i +tcos 1rtj +sin 1rt k ) dt = (! 0<br />
1 t 2 dt) i + (I~ tcos 1rtdt) j +(I; sin 1rtdt) k<br />
where we integrat<strong>ed</strong> by parts in the y-component.<br />
I<br />
= [ ~t 3 ]~ i + (~sin 1rt]~ - f 0<br />
1 ~sin 1rtdt) j +[-~cos 1rt]~ k<br />
1 • [ 1 t]l . 2 k 1 • 2 • 2 k<br />
= ]j l + ;1fCOS1l" 0 J+ ;r ;= 3 1 - ;:lfJ+ ;r<br />
1. r(t) = (t 2 , t 3 , t 4 ) . =? r 1 (t) = ( 2t,3e,4t 3 ) =? Jr 1 (t) J = J 4t 2 + 9t4 + 16t 6 and<br />
CHAPTER 13 REVIEW· 0 175<br />
. L = J; Jr 1 (t)J dt = J; J4t2 + 9t 4 + 16t 6 dt. Using Simpson's Rule with f(t) = J4t 2 + 9t4 + 16t 6 and n = 6 we<br />
have tlt = 3 (i 0 = ~ and<br />
L ~ ';' [f(O) + 4f(~) + 2f{1) + 4Ja) + 2/(2) + 4f(~) + /{3)]<br />
= i [vo +o +o + 4 . ..j4(~/ + 9 (~) 4 + 16( ~) 6 + 2 . .J4(1)2 + 9{1) 4 + 16(1)6<br />
~ 86.631<br />
+ 4. J4G) 2 + 9G) 4 + 16Gt + 2 . .J4(2)2 + 9(2)4 + 16{2)6<br />
+4· 4(~) 2 +9(~) 4 +16(~t+.J4(3) 2 +9{3)4+ 16(3)6 ]<br />
9. The angle of intersection of the two curves, 8, is the angle between their respective tangents at the point of intersection.<br />
For both curves the point {1, 0, 0) occurs when t = 0.<br />
r W) = - sin t i +cos t j + k =? rHO) = j + k and r 2(t) = i + 2tj + 3t 2 k =} r~(O) = i.<br />
r i {0) · r~ (0) = (j + k ) · i = 0. Therefore, the curves intersect in a right angle, that is, 8 = ~.<br />
(b) T'(t) = -Ht 4 +t 2 + 1)- 3 1 2 (4t 3 +2t) (t 2 ,t, 1) + (t 4 + t 2 + 1)- 1 1 2 (2t, 1, 0)<br />
- 2t 3 - t 2 1<br />
(t4 + t2 + 1)3/2 (t , t , 1) + (t4 + t2 + 1)1/2 (2t, 1, 0)<br />
- ( -2t 5 - t 3 , - 2t 4 - t 2 , - 2t 3 - t) + (2t 5 + 2t 3 + 2t, t 4 + t 2 + 1, 0) - (t 3 + 2t, - t 4 + 1, -2t 3 - t)<br />
- (t4 + t2 + 1)3/2 - (t4 + t2 + 1)3/2<br />
1 Jt 6 + 4t 4 + 4t 2 + t 8 - 2t 4 + 1 + 4t6 + 4t 4 + t 2 JtB + 5t 6 + 6t 4 + 5t2 + 1<br />
IT (t)l = (t4 + t2 + 1)3/2 = (t4 + t2 + 1)3/2<br />
and<br />
(t 3 + 2t 1 - t 4 - 2t 3 - t)<br />
N (t) - ' '<br />
- Jt!!. + 5t6 + 6t4 + 5t2 + 1 .<br />
JT 1 (t)\ J t!!. + 5t 6 + 6t 4 + 5t2 + 1 Jt<br />
(c) "'(t) = Jrl(t)l = 4 + 4t 2 + 1<br />
(t4 + t2 + 1)2 or (t4 + t2 + 1)3/2<br />
I<br />
13 4 3 II 12 2 d ( ) lv"l<br />
. y = X , y = X an It X = ( 1 + (y')2)3/ 2<br />
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176 0 CHAPTER 13 VECTO'R FUNCTIONS<br />
15. r (t) = (sin2t, t,cos2t) => r '(t) = (2cos2t, 1, -2sin2t) => T (t) = -js (2 cos2t, 1, - 2sin2t) =><br />
T '(t) = ~ (- 4sin 2t, 0, -4cos 2t) =><br />
N (t) = (-sin 2t, 0,- cos2t). SoN = N ('rr) = (0, 0, - 1) and<br />
B = T x N = -js (-1, 2, 0}. So a normal to the osculating plane is (-1, 2, 0) and an equation is<br />
- l(x- 0) + 2(y- 1r) + O(z - 1) = 0 or x - 2y + 27r = 0.<br />
17. r (t) = tlnti +tj + e-t k, v (t) = r'(t) = (1 + lnt) i + j - e-~ k,<br />
lv (t)i = J(l + lnt)2 + 1 2 + (-e-:-t)2 = vf2 +2lnt + (ln t)2 +e- 2 t,<br />
a(t) = v'(t) = t i + e-t k<br />
19. We set up the axes so that the shot leaves the athlete's hand 7 ft above the origin. Then we are given r (O) = 7j,<br />
lv (O)I = 43 ft/ s, and v (O) has direction given by a 45° angle of elevation. Then a unit vector in the direction ofv(O) is<br />
~( i + j ) => v (O) = ~(i + j). Assuming air resistance is negligible, the only external force is due to gravity, so .as in<br />
Example 13.4.5 we have a = - g j where here g ~ 32 ft/s 2 . Since v' (t) = a (t), we integrate, giving v (t) = - gt j + C<br />
where C = v (O) = ~( i + j) => v (t) = ~ i + ( ~ - gt) j . Since r'(t) = v (t) we integrate again, SO·<br />
r(t) = ~t i + ( ~t - ~gt2) j + D. ButD = r (O) = 7j => r (t) = ~t i + ( ~t- ~gt2 + 7)j.<br />
(a) At 2 seconds, the shot is at r(2) = ~(2) i + ( ~ (2) - ~g(2) 2 + 7) j ~ 60.8 i + 3.8j, so the shot is about 3.8 ft. above<br />
the ground, at a horizontal distance of 60.8 ft from the athlete.<br />
(b) The shot reaches its maximum height when the vertical component of velocity is 0: ~ - gt = 0 =><br />
t =<br />
;: ~ 0.95 s. Then r (0.95) ~ 28.9 i + 21.4j , so the maximum height is approximately 21.4 ft.<br />
v2g .<br />
(c) The shot hits the ground when the vertical component ofr(t) is 0, so ~t - ~gt 2 + 7 = 0 =><br />
-16t2 + ~t + 7 = 0 => t ~ 2.11 s. r (2. 11) ~ 64.2 i - 0.08j , thus the shot lands approximately 64.2 ft from the<br />
athlete.<br />
21. (a) Instead of proce<strong>ed</strong>ing directly, we use formula 3 ofTheorem 13.2.3: r (t) = t R (t) =><br />
v = r'(t) = R (t) + t R'(t) = coswt i + sinwtj +tv a.<br />
(b) Using the same method as in part (a) and starting with v = R (t) + t R '(t), we have<br />
a= v' = R '(t) + R '(t) + t R "(t) = 2 R '(t) + t R "(t) = 2 vd + tad.<br />
(c) Here we have r(t) = e-' coswt i + e- t sin wtj ,; e-t R (t). So, as in parts (a) and (b),<br />
v = r'(t) = e-t R '(t)- e-t R (t) = e-t[R '(t) - R (t)) =><br />
a = v' = e-t[R"(t)- R '(t))- e-:t[R '(t) - R (t)) = e- ~ [R"(t)- 2 R ' (t) + R (t)J<br />
Thus, the Coriolis acceleration (the sum of the "extra" terms not in<strong>vol</strong>ving aa) is - 2e-~ v.i + e-t R.<br />
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CHAPTER 13 REVIEW 0 177<br />
23. (a) r (t) = R coswt i + R sin wtj => v = r '(t) = -wR sinwt i + wRcoswt j, so r = R(coswt i + sinwtj) and<br />
v = wR(- sinwti + coswtj). v · r = wR 2 ( -<br />
coswt sinwt + sinwtcoswt) = 0, so v ..L r . Since r points along a<br />
radius of the circle, and v ..L r , vis tangent to the circle. Because it is a velocity vector, v points in the direction of motion.<br />
(b) In (a), we wrote v in the form wR u, where u is the unit vector - sin wt i + cos wtj . Clearly I vi = wR lui = wR. At<br />
spe<strong>ed</strong> wR , the part1c . I e camp I etes one revo I ut10n, . a d. 1stance 2 R,. . T 2·nR 27r<br />
1r m t1me = - R = - .<br />
w . w<br />
(c) a = dv = - w 2 R coswti - w 2 Rsinwtj = - w 2 R(coswt i + sinwtj ), so a = - w 2 r. This shows that a is proportional<br />
dt<br />
to rand points in the opposite direction (toward the origin). Also, ja j = w 2 jr j = w 2 R .<br />
(d) By Newton's Second Law (see Section 13.4), F = ma, so jF j = m jaj = mRw 2 = m (~R) 2 = m ~~ 2<br />
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0 PROBLEMS PLUS<br />
1. (a) The projectile reaches maximum he~ght when 0 = ~~ = ! [(Vo sina)t- ~gt2] = vo sin a - gt; that is, when<br />
vosina .d ( . )(vosiiJ.a) 1 (vosina) 2<br />
t = ---an y = vosma --- - -g ---<br />
g g 2 g<br />
2 ° 2<br />
Vo sm a<br />
.<br />
Th'<br />
1s<br />
. h .<br />
1 'gh .<br />
IS t e max1mum 1e1 t attam<strong>ed</strong> when<br />
2 g .<br />
the projectile is fir<strong>ed</strong> with an angle of elevation a. This maximum height is largest when a = ~ . In that case, sin a = 1<br />
. . . vfi<br />
and the maximum he1ght IS-.<br />
2g<br />
(b) Let R = vfi /g. We are ask<strong>ed</strong> to consider the parabola x 2 + 2Ry - R 2 = 0 which can be rewritten as y = - __!._ x 2 + !!:. .<br />
2R 2<br />
The points on or inside this parabola are those for which - R ~ x ~ Rand 0 ~ y ~ ;~ x 2 + ~.<br />
When the projectile is<br />
fir<strong>ed</strong> at angle of elevation a, the points ( x, y) along its path satisfy the relations x = ( v 0 cos a) t and<br />
y = (v 0 sina)t- tyt 2 , whereO ~ t ~ (2v 0 sina)/g(as inExan1ple 13.4.5). Thus<br />
lxl ~ lvo cos a Cvo ;ina) I= I~ sin 2al ~ I~; I = IRI. This shows that -R ~ x ~ R.<br />
For t in the specifi<strong>ed</strong> range, we also have y = t ( vo sin a - ~ gt) = t gt Cvo ;in a - t) ;::: 0 and<br />
, X 9 X g 2<br />
2 2 x 2 =<br />
vo cos a 2 vo cos a v 0 cos a .<br />
0 (<br />
y = (vosma)----- - --)2<br />
= (tana)x -<br />
- 1 2 R) -1 2 1 2 ( R<br />
y- ( 2Rx + 2 = 2Rcos2 ax + 2Rx + tan a) x- 2<br />
1 2<br />
2 R 2 x +(tan a) x. Thus<br />
cos a<br />
= ~ ( 1 - _1_) + (tana)x _ R = x2 (1-sec 2 a) + 2R{tana) x - R 2<br />
2R cos 2 a 2 2R<br />
_ -{tan 2 a) x 2 + 2R (tan a) x - R 2 ·_ :- [{tana) x - R] 2 0<br />
- 2R - . 2R ~<br />
We have shown that every target that can be hit by the projecti le lies on or inside the parabola y = _ __!._ x 2 + R.<br />
2R 2<br />
Now let (a, b) be any point on or inside the parabola y = -<br />
2 ~ x 2 + ~· Then·-R ~a~ Rand 0 ~ b ~ -<br />
2 ~ a2 + ~-<br />
We seek an angle a such that (a, b) lies in the path of the projectile; that is, we wish to find an angle a such that<br />
b = -<br />
2 ~c~s 2 a a 2 +(tan a) a or equivalently b = ~ (tan 2 a ~1-<br />
1)a 2 +(tan a) a. Rearranging this equation we get<br />
;~ tan 2 a- a tan a+(;~+ b) = 0 or a 2 (tana? - 2aR(tana) + (a 2 + 2bR) = 0 (*) . This quadratic equation<br />
for tan a has real solutions exactly when the discriminant is nonnegative. N~w B 2 - 4AC ;::: 0 <br />
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180 0 CHAPTER 13 PROBLEMS PLUS<br />
(-2aR?- 4a 2 (a 2 + 2bR);:::: 0
CHAPTER 13 PROBLEMS PLUS 0 181<br />
The projectile hits the ground when (v sina)t- ~gt 2 = 0 => t = 2 ; sin a, so the distance travel<strong>ed</strong> by the projectile is<br />
(2v/g)sin o 1 (211/g)sin<br />
L(a) = lv(t)l dt = .9<br />
1 0 0<br />
v ) 2 v2<br />
( t - - sin a + 2 cos 2 a dt<br />
.9 9<br />
t - (v /g) sino<br />
=g 2<br />
[<br />
+ [(vl g) cosa]2 ln (t-~sino +<br />
2 g<br />
[using Formula 2 I in the Table of lntegrals]<br />
g<br />
=<br />
[v.<br />
2 (~.rna)' + (~ -gsma oo.o )' + (~'"'")' m ( ~ •ina+ (~ •mo)' + (~=a Y)<br />
+~sin o<br />
g (~ •ina )' + (~=o )'- (~oooo )' m( -~•ina+ (~•ina )' + (~=o Y)]<br />
2<br />
= g_ [~sino · ~+ v<br />
cos 2 2<br />
a ln(~ sino+~) +~ sino·~ - v cos 2 a ln(- ~ sino+~)]<br />
2 g g g2 g g g g g2 g g<br />
v 2 . + v 2 2 1n( (vl g)sina+v/g) v 2 . + v 2 2<br />
1 (1+sino)<br />
= - stno -cos a = -sma -cos an<br />
g 2f1 - (vI g) sin a +vI g g 2g 1 - sin a<br />
We want to maximize L( a) for 0 s; a s; 1r / 2.<br />
v<br />
2<br />
v<br />
2<br />
[ 2 1 - sin a 2 cos a . ( 1 + sin a)]<br />
L'(a) =-coso+ - cos a · . · . 2 1<br />
2 - 2cosa smo ln .<br />
g g +sma (1-smo) . 1 -sm o<br />
v 2 • v 2 [<br />
= - cos a + - cos 2 . ( 1 + sin<br />
2<br />
a)]<br />
2 a ·-- - 2 cos a sm a ln .<br />
g g coso , 1 -sma<br />
v 2 v 2 [ • (l+sina)] v 2 . = - COSO! [ 2- SIDQ . • ln (1+ sina . ) ]<br />
9 g 1 -SlDO! 9 1 -SJDO! ,<br />
= - COSO!+- COSO! 1-SlDO! ln<br />
L( a ) has critical points for 0 < a < 1r 12 when L' (a) = 0 => 2 - sin a ln ( ~ + s~ a) = 0 [since cos a =1= 0].<br />
- SIDQ<br />
Solving by graphing (or using a CAS) gives a~ 0.9855. Compare values at the critical point and the endpoints:<br />
L(O) = 0, L( 1r 12) = v 2 I g, and £(0.9855) ~ 1.20v 2 I g. Thus the distance travel<strong>ed</strong> by the projectile is maximiz<strong>ed</strong><br />
for a ~ 0.9855 or ~ 56°.<br />
7. We can write the vector equation as r (t) = ·at 2 + bt + c where a= (a1, a2, a a}, b = (b1 , b2 , ba }, and c = (c1, c2, c3).<br />
Then r ' (t) = 2t a+ b which says that each tangent vector is the sum of a scalar multiple of a and the vector b . Thus the<br />
tangent vectors are all parallel to the plane determin<strong>ed</strong> by a and b so the curve must be parallel to this plane. [Here we assume<br />
that a and bare nonparallel. Otherwise the tangent vectors are all parallel and the curve lies along a single line.] A normal<br />
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182 0 CHAPTER 13 PROBLEMS PLUS<br />
vector for the plane is a x b = (a2b3- a3~, a3b1 - a1b3, a1b2 - a2b1). The point (c1. c2, c3) lies on the plane (when<br />
t = 0), so an equation of the plane is<br />
or<br />
(a2b3- a31h)(x- c1) + (a3b1 - a 1 ~)(y - c2) + (a l~ - a2bl)(z - c3) = 0<br />
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14 0 PARTIAL DERIVATIVES<br />
14.1 Functions of Several Variables<br />
1. (a) From Table 1, f( - 15, 40) = -27, which means that if the temperature is - 15°C and the wind spe<strong>ed</strong> is 40 km/ h, then the<br />
air would feel equivalent to approximately -27°C without wind.<br />
(b) The question is asking: when the temperature is -2!tC, what wind spe<strong>ed</strong> gives a wind-chill index of - 30°C? From<br />
Table 1, the spe<strong>ed</strong> is 20 km/h.<br />
(c) The question is asking: when the wind spe<strong>ed</strong> is 20 km/ h, what tem·perature gives a wind-chill index of -49°C? From<br />
Table I, the temperature is -35°C.<br />
(d) The function W = j( -5, v) means that we fix T at -5 and allow v to vary, resulting in a function of one variable. In<br />
other wor~s, the function gives wind-chill index values ~o r different wind spe<strong>ed</strong>s when the temperature is -5° C. Fro111<br />
Table 1 (look at the row corresponding toT = - 5), the function decreases and appears to approach a constant value as v<br />
increases.<br />
(e)_The function W = j(T, 50) means that we fix vat 50 and allow T to vary, again giving a function of one variable. In<br />
other words, the function gives wind-chill index values for different temperatures when the wind spe<strong>ed</strong> is 50 km/h . From<br />
Table 1 (look at the column corresponding to v = 50), the function increases almost linearly as T increases.<br />
3. P(120, 20) = 1.47(120) 0 ' 65 (20) 0 ·35 ~ 94.2, so when the manufacturer invests $20 million in capital and 120,000 hours of<br />
labor are complet<strong>ed</strong> yearly, the monetary value of the production is about $94.2 million.<br />
5. (a) !(160, 70) = 0.1091(160) 0 .4 25 (70) 0 ·725 ~ 20.5, which means that the surface area of a person 70 inches (5 feet 10<br />
inches) tall who weighs 160 pounds is approximately 20.5 square feet.<br />
(b) Answers will vary depending on the height and weight of the reader.<br />
7. (a). According to Table 4, f( 4·0, 15) = 25, which means that if a 40-knot wind has been blowing in the open sea for 15 hours,<br />
it will create waves with estimat<strong>ed</strong> heights of25 feet.<br />
'<br />
(b) h = f(30, t) means we fix vat 30 and allow t to vary, resulting in a function of one variable: Thus here, h = f(30, t)<br />
gives the wave heights produc<strong>ed</strong> by 30-knot winds blowing for t.hours. From the table (look at the row corresponding to<br />
v = 30), the function increases but at a declining rate as t increases. In fact, the function values appear to be approaching a<br />
limiting value of approximately 19, which suggests that 30-knot winds cannot produce waves higher than about 19 feet.<br />
(c) h = f(v , 30) means we fix tat 30, again giving a function of one variable. So, h = f(v, 30) gives the wave heights<br />
produc<strong>ed</strong> by winds of spe<strong>ed</strong> "! blowing for 30 hours. From the table (look at the column corresponding to t = 30), the<br />
function appears to increase at an increasing rate, with no apparent limiting value. This suggests that faster winds (lasting<br />
30 hours) always create higher waves.<br />
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184 0 CHAPTER 14 PARTIAL DERIVATIVES<br />
9. (a) g(2, - 1) = cos(2 + 2( - 1)) = cos(O) = 1<br />
(b) x + 2y is defin<strong>ed</strong> for all choices of values for x andy and the cosine function is defin<strong>ed</strong> for all input values, so the domain<br />
of g is JR 2 .<br />
·(c) The range of the cosine function is [- 1, 1] and x + 2y generates all possible input values for the cosine function, so the<br />
range ofcos(x + 2y) is (-1, 1].<br />
(b) ft, .,fij, vz are defin<strong>ed</strong> o~ ly when x ~ 0, y ~ 0, z ~ 0, and ln(4 - x 2 - y 2 - z 2 ) is defin<strong>ed</strong> when<br />
4 - x 2 - y 2 - z 2 > 0 0, or ix 2 + y 2 < 1. So the domain off<br />
is { (x, y) j ix 2 + y 2 < 1 }, the interior of an ellipse.<br />
y<br />
, ,.,~,..- - - - - -- ..........,<br />
,,~ X<br />
-----<br />
17. J1 - x 2 is defin<strong>ed</strong> only when 1 - x 2 ~ 0, or<br />
x 2 ::; 1 ¢:}<br />
-1 ::; x ::; .1, and \./1 - y 2 is defin<strong>ed</strong><br />
only when 1 - y 2 ~ 0, or y 2 ::; 1 -¢* -1 ::; y::; 1.<br />
Thus the domain off is<br />
{(x,y)l-1::;x::;1, - 1 ::; y ::; 1}.<br />
19. Jy - x 2 is defi~<strong>ed</strong> only when y - x 2 ~ 0, or y ~ x 2 .<br />
ln addition, f is not defin<strong>ed</strong> if 1 - x 2 = 0
SECTION 14.1 FUNCTIONS OF SEVERAL VARIABLES 0 185<br />
21 . We ne<strong>ed</strong> 1 - x 2 - y 2 - z 2 ~ 0 or x 2 + y 2 + z 2 ~ 1,<br />
soD= { (x, y, z) I x 2 + y 2 + z 2 ~ 1} (the points inside<br />
or on the sphere of radius 1, center the origin).<br />
23. z = 1 + y, a plane which intersects the yz-plane in the<br />
line z = 1 + y, x = 0. The portion of this plane for<br />
x ~ 0, z ~ 0 is shown.<br />
X<br />
y<br />
25. z = 10 - 4x - Sy or 4x + Sy + z = 10, a plane with<br />
intercepts 2.5, 2, and 10.<br />
27. z = y 2 + 1, a parabolic cylinder<br />
(0, 0, 10)<br />
X<br />
29. z = 9 - x 2 - 9y 2 , an ell iptic paraboloid opening<br />
downward with vertex at (0, 0, 9) .<br />
31. z = )4- 4x 2 - y 2 so 4x 2 + y 2 + z 2 = 4 or<br />
y2<br />
z2<br />
x 2 + 4<br />
+ 4<br />
= 1 and z ~ 0, the top half of an<br />
ellipsoid .<br />
.r<br />
33. The point (- 3, 3) lies between the level curves with z-values 50 and 60. Since the point is a little closer to the level curve with<br />
z = 60, we estimate that f( -3, 3) ~56. The point (3, -2) appears to be just about halfway between the level curves with<br />
z-values 30 and 40, so we estimate !(3, - 2) ~ 35. The graph rises as we approach the origin, gradually from above, steeply<br />
from below.<br />
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186 0 CHAPTER 14 PARTIAL DERIVATIVES<br />
35. The point (160, 10), corresponding to day 160 and a depth of 10m, lies between the isotherrnals with temperature values<br />
of8 and 12°C. Since the point appears to be locat<strong>ed</strong> about three-fourths the distance from the 8°C isothermal to the l2°C<br />
isothermal, we estimate the temperature at. that p~int to be approximately 11° C. The point ( 180, 5) lies betwe~n the 16 and<br />
20°C isothermals, very close to the 20°C level curve, so we estimate the temperature there to be about 19.5°C.<br />
37. Near A, the level curves are very close together, indicating that the terrain is quite steep. At B, the level curves are much<br />
farther apart, so we would expect the terrain to be much tess steep than near A, perhaps almost flat.<br />
39.<br />
41.<br />
5<br />
X<br />
y<br />
X<br />
43. The level curves ~re (y - 2x) 2 = k or y = 2x ± ..Jk,<br />
k ~ 0, a fami ly of pairs of parallel lines.<br />
45. The level curves are vfx + y = k or y = -vlx + k, a'<br />
family of vertical translations of the graph of the root<br />
function y = -vx.<br />
y<br />
2<br />
0<br />
4321 0 1234<br />
-1<br />
-2<br />
47. The level curves are ye~ = k or y = ke-x, a family of<br />
exponential curves.<br />
49. The level curves are J y 2 - x 2 = k or y 2 - x 2 = k 2 ,<br />
k ~ 0. When k = 0 the level curve is the pair of lines<br />
y = ±x. Fork > 0, the level curves are hyperbolas<br />
with axis the y-axis.<br />
0<br />
X<br />
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SECTION 14.1 FUNCTIONS OF SEVERAL VARIABLES 0 187<br />
51. The contour map consists of the level curves k = x 2 + 9y 2 , a family of<br />
ellipses with major axis. the x -axis. (Or, if k = 0, the origin.)<br />
The graph of f(x, y) is the surface z = x 2 + 9y 2 , an elliptic paraboloid.<br />
X<br />
If we visualize lifting each eUipse k = x 2 + 9y 2 of the contour map to the plane<br />
z = k, we have horizontal traces that indicate the shape of the graph of f .<br />
X<br />
53. The isothermals arc given by k = 100/(1 + x 2 + 2y 2 ) or<br />
x 2 + 2y 2 = (100- k)/k [0 < k ~ 100], a family of ellipses.<br />
55. f(x, y) = xy 2 - x 3<br />
The traces parallel to the yz-plane (such as the left-front trace in the graph above) are parabolas; those parallel to the xz-plane<br />
(such as the right-front trace) are cubic curves. The surface is call<strong>ed</strong> a monkey saddle because a monkey sitting on the surface<br />
near the origin has places for both legs and tail to res ~.<br />
-4 4<br />
X<br />
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188 0 CHAPTER 14 PARTIAL DERIVATIVES<br />
59. z = sin(xy) (a) C (b) ll<br />
Reasons: This function is periodic in both x andy, and the function is the same when x is interchang<strong>ed</strong> withy, so its graph is<br />
symmetric about the plane y = x. ln addition, the function is 0 along tl1e x- andy-axes. These conditions are satisfi<strong>ed</strong> only by<br />
Can~ II.<br />
61. z = sin(x- y) (a) F (b) 1<br />
Reasons: This function is periodic in both X andy but is constant along the lines y = X + k, a condition satisfi<strong>ed</strong> only<br />
by F and I.<br />
(a) B<br />
(b) VI<br />
Reasons: This function is 0 along the lines x = ±1 andy = ±1. The only contour map in which this could occur is VI. Also<br />
note that the trace in the x z-plane is the parabola z = 1 - x 2 and the trace in the yz-plane is the parabola z = 1 - y 2 , so the<br />
graph is B.<br />
65. k = x + 3y + 5z is a family of parallel plane$ with normal vector (1, 3, 5).<br />
67. Equations for the level surfaces are k = y 2 + z 2 . For k > 0, we have a family of circular cylinders with axis the x-axis and<br />
radius ,fk. When k = 0 the level surface is the x-axis. (fhere are no level surfaces fork < 0.)<br />
69. (a) The graph of g is 'the graph off shift<strong>ed</strong> upward 2 units.<br />
(~)The graph of g is the graph off stretch<strong>ed</strong> vertically by a factor of2.<br />
(c) The graph of g i~ the graph off reflect<strong>ed</strong> about the xy-plane.<br />
(d) The graph of g(x, y) = - f(x, y) + 2 is the graph off reflect<strong>ed</strong> about the xy-plane and then shift<strong>ed</strong> upward 2 units.<br />
71. f(x,y) = 3x - x 4 - 4y 2 - lOxy<br />
20r-----------------.<br />
10<br />
0<br />
y<br />
Three-dimensional view<br />
y<br />
Front view<br />
It does appear that tlle function bas a maximum value, at the higher of the two "hilltops." From the front view graph, the<br />
maximum value appears to be approximately 15. Bolli h illtop~ could be consider<strong>ed</strong> local maximum points, as the values off<br />
there are larger than at the neighboring points. There doe~ not appear to be any local minimum point; although tlle valley sh(\pe<br />
between the two peaks looks like a minimum of some kind, some neighboring points have lower function values.<br />
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73.<br />
10-------------<br />
5<br />
0<br />
- 5<br />
y<br />
SECTION 14.1 FUNCTIONS OF SEVERAL VARIABLES D 189<br />
x+y<br />
f ( x, y) = - 2<br />
-- 2<br />
• As both x and y become large, the function values<br />
X +y<br />
appear to approach 0, regardless of which direction is consider<strong>ed</strong>. As<br />
(x, y) approaches the ~rigin, the graph exhibits asymptotic behavior.<br />
From some directions, f(x, y)--+ oo, while in others f(x, y) --+ -oo.<br />
(These are the vertical spikes visible in the graph.) If the graph is<br />
examin<strong>ed</strong> carefully, however, one can see that f(x, y) approaches 0<br />
along the line y = - x.<br />
75. f(x, y) = ecx 2 +v 2 • First, if c = 0, the graph is the cylindrical surface<br />
z = eY 2 (whose level curves are parallel lines). When c > 0, the vertical trace<br />
above they-axis remains fix<strong>ed</strong> whi.le the sides of the surface in the x-direction<br />
"curl" upward, giving the graph a shape resembling an elliptic paraboloid. The<br />
z<br />
level curves of the surface are ellipses center<strong>ed</strong> at the origin.<br />
c=O<br />
For 0 < c < 1, the ellipses have major axis the x-axis and the eccentricity increases as c --+ 0.<br />
y<br />
c = 0.5 (level curves in increments of 1)<br />
-1.2<br />
For c = 1 the level curves are circles center<strong>ed</strong> at the origin.<br />
-1.2<br />
c = 1 (level curves in increments of 1)<br />
[continu<strong>ed</strong>)<br />
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or post<strong>ed</strong> to o publicly oot
190 0 CHAPTER 14 PARTIAL DERIVATIVES<br />
When c > 1, the level curves are ellipses with major axis they-axis, and the eccentricity increases as c increases.<br />
-1.2<br />
c = 2 (level curves in increments of 4)<br />
For values of c < 0, the sides of the surface in the x-direction curl downward and approach the xy-plane (while the vertical<br />
trace x = 0 remains fix<strong>ed</strong>), giving a saddle-shap<strong>ed</strong> appearance to the graph near the point (0, 0, 1). The level curves consist of<br />
a family of hyperbolas. As c decreases, the surface becomes flatter in the x -direction and the surface's approach to the curve in<br />
the trace x = 0 becomes steeper, as the graphs demonstrate.<br />
c = - 0.5 (level curves in increments of0.25)<br />
-1.2<br />
c = -2 (level curves in increments of0.25)<br />
77. z = x 2 + y 2 + =Y· When c < -2, the surface intersects the plane z = k I= 0 in a hyperbola. (See the following graph.)<br />
-1.2<br />
It intersects the plane x =yin the parabola z = (2 + c)x 2 , and the plane x = - y in the parabola z = (2 -<br />
parabolas open in opposite directions, so the surface is a hyperbolic paraboloid.<br />
c)x 2 . These<br />
When c = - 2 the surface is z = x 2 + y 2 - 2xy = (x- y) 2 . So the surface is constant along each line x- y = k . That<br />
is, the surface is a cylinder with axis x - y = 0, z = 0. The shape of the cylinder is determin<strong>ed</strong> by its intersection with the<br />
plane x + y = 0, where z = 4x 2 , and hence the cylinder is parabolic with mini.ma ofO on the line y = x.<br />
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SECTION 14.1 FUNCTIONS OF SEVERAL VARIABLES 0 191<br />
c = - 5, z = 2 c = - 10 c= - 2<br />
When -2 < c:::; 0, z ~ 0 for all x andy. Ifx andy have the same sign, then<br />
x 2 + y 2 + cxy ~ x 2 + 1i - 2xy = (x-y) 2 ~ 0. IJthey have opposite signs, then cxy ~ 0. The intersection with the<br />
surface and the plane z = k > 0 is an ellipse (see graph below). The intersection with the s urface and the planes x = 0 and<br />
y = 0 are parabolas z = y 2 and z = x 2 respectively, so the surface is an elliptic paraboloid.<br />
When c > 0 the graphs have the same shape, but are reflect<strong>ed</strong> in the plane x = 0, because<br />
x 2 + y 2 + cxy = ( - x) 2 + y 2 + (- c)( - x)y. That is, the value of z is the same for c at (x, y) as it is for-cat ( -x, y).<br />
c = -1, z = 2' c = O c = 10<br />
So the surface is an elliptic paraboloid for 0 < c < 2, a parabolic cylinder for c = 2, and a-hyperbolic paraboloid for c > 2.<br />
79. (a) P = bLo: K 1 - o: =? p = bL"' J
192 0 CHAPTER 14 PARTIAL DERIVATIVES<br />
After entering the (x, y) pairs into a calculator or CAS, the resulting least squares regression line through the points is<br />
approximately y = 0.75136x + 0.01053, which we round toy= 0. 75x + 0.01.<br />
(c) Comparing the regression line from part (b) to the equation y = ln b +ax with x = ln(L/ I b = e 0 ·0 1 ~ 1.01. Thus, the Cobb-Douglas production function is<br />
p = bL"' ](1-a = 1.01£0.75 ](0.25.<br />
14.2 Limits and Continuity<br />
1. In general, we can't say anything about /(3, 1) I lim f(x, y) = 6 means that the values of f(x, y) approach 6 as<br />
{x,y)-•(3,1) ·<br />
(x, y) approaches, but is not equal to, (3, 1). Iff is continuous, we know that lim f(x, y) = f(a, b), so<br />
(x,y)--z<br />
-0.2<br />
-0.1<br />
-0.05<br />
- 0.2 -0.1<br />
-2.551 -2.525<br />
-2.525 -2.513<br />
-2.513 -2.506<br />
- 0.05 0 0.05 0.1 0.2<br />
-2.5 13 -2.500 -2.488 -2.475 -2.451<br />
- 2.506 -2.500 -2.494 -2.488 - 2.475<br />
-2.503 -2.500 -2.497 -2.494 -2.488<br />
0<br />
-2.500 -2.500<br />
- 2.500 -2.500 -2.500 - 2.500<br />
0.05<br />
-2.488 - 2.494<br />
- 2.497 -2.500 -2.503 - 2.506 - 2.5 13<br />
0.1<br />
-2.475 - 2.488<br />
- 2.494 -2.500 -2.506 -2.513 - 2.525<br />
0.2<br />
- 2.451 -2A75<br />
-2.488 - 2.500 - 2.513 - 2.525 - 2.551<br />
As the table shows, the values off ( x, y) seem to approach - 2.5 as ( x, y) approaches the origin from a variety of different<br />
directions. This suggests that lim f(x, y) = -2.5. Since f is a rational function, it is continuous on its domain. f is<br />
(:t:,!J) - {0,0)<br />
o b o o<br />
de fi ne d at<br />
bl' I th l' !( ) 0203 + 0302 - 5 5 'fy'<br />
( 0, 0 ), so we can use d1rect su stltutwn to esta IS 1 at 1m x, y =<br />
= --, ven mg<br />
0<br />
(:r.,.y)-->{0,0) 2 - 0 0 2<br />
our guess.<br />
5. f(x, y) = 5x 3 - x 2 y 2 is a polynomial, and hence continuous, so ' lim f(x, y) = f(l, 2) = 5{1) 3 - {1) 2 (2) 2 = 1.<br />
(x,y)-{1,2)<br />
7 . f( x, y ) . =<br />
4 - xy . . 1 .., . d h . . d .<br />
tS a ratwna ;unctiOn an encc contmuous on tts omam.<br />
X 2 +3y 2<br />
.<br />
(2, 1) is in the domain off, so f is continuous there and lim f(x, y) = !(2, 1) = (~)~ ( 2 ~~~~ 2<br />
{x,y) --< (2,1) +<br />
2<br />
7<br />
9. f(x, y) = (x 4 - 4y 2 )/(x 2 + 2y 2 ). First approach (0, 0) along the x-axis. Then.f(x, 0) = x 4 /x 2 = x 2 for x =f. 0, so<br />
f(x, y) --+ 0. Now approach (0, 0) along they-axis. For y =f. 0, f(O, y) = -4y 2 /2y 2 = -2, so f(x, y)--+ -2. Since f has<br />
two different limits along two different lines, the limit does not exist.<br />
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SECTION 14.2 LIMITS AND CONTINUITY 0 193<br />
11. j(x , y) = (y 2 sin 2 x)j(x 4 + y 4 ). On the x -axis, f(x, 0) = 0 for x '# 0, so f(x, y) --+ 0 as (x, y) --+ (0, 0) along the<br />
. x 2 sin 2 x sin 2 x 1 ( sinx) 2<br />
x-axis. Approaching (0, 0) along the hne y = x, f(x, x) = x 4 + x 4<br />
=<br />
2 x 2 = 2 --;-- for x :/: 0 and<br />
lim sin x = 1, so f(x, y)--+ ~·S ince f has two different limits along two different lines, the limit does not exist.<br />
:c- o x · ·<br />
13. f(x, y) = ~· We can see that the limit along any line through (0, 0) is 0, as well as along other paths through<br />
xz +yz<br />
(0, O) such as x = y 2 andy= x 2 . So we suspect that the limit exists and equals 0; we use the Squeeze Theorem to prove our<br />
assertion. 0 ~ I ~~~ lxl since lvl ~ )x 2 + y 2 , and lxl --+ 0 as (x, y)--+ (0, 0). So lim f(x, y) = o:<br />
x2 + y2 . (x,y)-(o,o)<br />
15. Let f(x, y) =<br />
x2yeY<br />
+ 4y<br />
X<br />
4 2<br />
. Then f(x, 0) = 0 for x '# 0, so f(x, y) --+ 0 as (x, y)--+ (0, 0) along the x -axis. Approaching<br />
2 2 ::r2 4 ::c2 a:2<br />
(0, 0) along they-axis or the line y = x also gives a limit ofO. But f (x, x 2 ) = x:: 4<br />
tx 2<br />
)2 = x 5<br />
: 4<br />
= T for x '# 0, so<br />
f(x, y) -+ e 0 /5 = t as (x,y)--+ (0, 0) along theparabola y = x 2 . .Thus the limit doesn't exist.<br />
19. eY 2 is a composition of continuous functions and hence continuous. xz is a continl.!ous function and tan t is continuous for<br />
t '# ~ + mr (nan integer), so the composition tan(a
194 0 CHAPTER 14 PARTIAL DERIVATIVES<br />
25. h(x, y) = g(f(x, y)) = (2x + 3y - 6) 2 + y'2x + 3y - 6. Since f is a. polynomial, it is continu o~s on JR 2 and g· is<br />
continuous on its domain { t I t ;::: 0}. Thus h is continuous on its domain.<br />
D = {(x, y) I 2x + 3y-6;::: 0} = { (x, y) I y;::: - ~x + 2}, which consists of all points on or above the line y = - ~x + 2.<br />
27.<br />
From the graph, it appears that f is discontinuous along the line y = x.<br />
3<br />
2<br />
z 1<br />
0<br />
-1<br />
2<br />
If we consider f(x, y) = e 1 /(a:-y) as a composition offun.ctions,<br />
g(x, y) = 1/(x- y) is a rational function and therefore continuous except<br />
where x- y = 0 =} y = x. Since the function h(t) = et is continuous<br />
everywhere, the composition h(g(x,y)) = e 1 1(x-v) = f(x,y) is<br />
continuous except along the line y = x, as we suspect<strong>ed</strong>.<br />
29. The functions xy and 1 + ex-v are continuous everywhere, and 1 + ex- v is never zero, so F(x, y) = xy is continuous<br />
. 1 + ex-y<br />
on its domain JR 2 .<br />
31. F(x, y) =<br />
l +x2 +y2<br />
1 - x 2 -y 2<br />
is a rational function and thus is continuous on its domain<br />
33. G(x, y) = ln(x 2 + y 2 - 4) = g(f(x, y)) where f(x, y) = x 2 + y 2 - 4, continuous on ]~_2, and g(t) = In t, continuous on its<br />
domain {t ['t > 0}. Thus G is continuous on its domain { (x,_y) I x 2 + y 2 -<br />
4 > 0} = { (x, y ) I x 2 + y 2 > 4 }, the_ exterior<br />
ofthe circle x 2 + y 2 = 4.<br />
35. f(x, y, z ) = h(g(x, y, z )) where g(x, y, z) = x 2 + y 2 + z 2 , a polynomial that is continuous<br />
everywhere, and h(t) = arcsint, continuous on [- 1, 1). Thus f is continuous on its domain<br />
{(x,y, z) l-1 S x 2 + y 2 + z 2 S 1} = {(x,y,z)! x 2 + y 2 + z 2 S 1 }, so f is continuous on the unit ball.<br />
if (x, y) =I= (0, 0)<br />
if (x,y) = (0,0)<br />
The first piece off is a rational function defin<strong>ed</strong> everywhere except at the<br />
origin, so f is continu~us on JR 2 except possibly at the origin. Since x 2 s 2x 2 + y 2 , we have jx 2 y 3 / (2x 2 + y 2 ) I. s jy 3 j. We<br />
. 2 3<br />
know that jy 3 \ --+ 0 as (x, y) --+ (0, 0). So, by the Squeeze Theorem, lim f(x, y) = lim<br />
2 ~ y 2 = 0.<br />
(a;,y)-(0,0) (x,y)- (0,0) X + Y<br />
But f(O, 0) = 1, so f is discontinuous at (0, 0). Therefore, f is continuous on the set { (x, y) I (x, y) =/= (0, 0) }.<br />
39 lim<br />
- x 3 + y 3 1 . (r cos 8)3 + (1' sin8) 3 lim ( 3 . 3 B)<br />
. - -- = tm = r· cos 8 + rsm =<br />
( x ,y)-(0,0) X 2 -f- y 2 r-->O+ 1' 2 r~o+<br />
0<br />
-x2 - y 2<br />
e - 1<br />
. 41. lim<br />
(a;,y)--> (0,0) x2 +y2<br />
lim e- r (- 2r·)<br />
2r<br />
r - o+<br />
= lim - e- r 2 = - e 0 = - 1<br />
r·-o+<br />
2<br />
[using !'Hospital's Rule]<br />
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SECTION 14.3 PARTIAL DERIVATIVES 0 195<br />
sin(xy)<br />
43. f(x, y) = xy<br />
{<br />
1<br />
if (x, y) :f. (0, 0)<br />
if (x, y) = (0, 0)<br />
From the graph, it appears that f is continuous everywhere. We know<br />
xy is continuous on R 2 .and sin t is continuous everywhere, so<br />
.<br />
( )<br />
R2 d sin(:cy) . . R2<br />
sin xy is contmuous on an --- IS contmuous on<br />
x y<br />
-1<br />
except possibly where x y = 0. To show that f is continuous at those points, consider any point (a, b) in R 2 where ab = 0.<br />
Because x y is continuous, x y --+ ab == 6 as {x, y ) --+ (a, b). If we lett= x y, then t--+ 0 as (x, y) --+ (a, b) and<br />
lim sin(xy). = Jim sin(t) = l by Equation 2.4.2 [ET 3.3.2). Thus lim f (x, y) = f(a, b) and f is continuous<br />
(x,y)-(a,b) x y t-o t (:z:,y)-(a,b)<br />
on R 2 •<br />
45. since Jx - aJ 2 = lxl 2 + Jal 2 . - 2JxJ ial cos O ~ Jxl 2 + Jal 2 - 2Jx.JJal = (lxJ -Ial) 2 , we have llxl - lall $ lx- aJ. Let<br />
t > 0 be given and set o = t. Then ifO < lx - al < o, llxJ- IaJI $ Jx - aJ < o = t . Hence limx-a Jxl = Jal and<br />
f (x) = Jxl is continuous on R".<br />
14.3 Partial Derivatives<br />
1. (a) 8T I 8x represents the rate of change ofT when we fi x y and t and consider T as a function of the single variable x, which<br />
describes how quickly the temperature changes when longitude changes but latitude and time are constant. 8T 18y<br />
represents the rate of change of T when we fix x and t and consider T as a function of.y, which describes how quickly the<br />
temperature changes when latitude changes but longitude and time are constant. 8T I 8t represents the rate of change ofT<br />
when we fix x andy and consider T as a function oft , which describes how quickly the temperature changes over time for<br />
a constant longitude and latitude.<br />
(b) f .,(158, 21, 9t represents the rate of change of temperature at longitude 158°W, latitude 21 °N at 9:00AM when only<br />
longitude varies. Since the air is warmer to the west than to the east, increasing longitude results in an increas<strong>ed</strong> air<br />
temperature, so we would expect .f:r: (158, 21, 9) to be positive. / 11 (158, 21 , 9) represents the rate of change oftemperature<br />
at the same time and location when only latitude varies. Since the air is warmer to the south and cooler to the north,<br />
increasing latitude results in a decreas<strong>ed</strong> air temperature, so we would expect / 11 (158, 21, 9) to be negative. ft(158, 21, 9)<br />
represents the rate of change of temperature at the same time and location when onJy time varies. Since typically air<br />
temperature increases from the morning to the afternoon as the sun warms it, we would expect f t(158, 21, 9) to be<br />
positive.<br />
3. a By De tlon<br />
. 4<br />
,<br />
f ( 15 30) lim f(- 15 + h,30) - f(- 15,30) h' h . b 'd . h<br />
T - , = h , w 1c we can approx1mate y cons1 ermg = 5<br />
() fini<br />
h-0<br />
and h = -5 and using the values given in the table:<br />
© 201 2 Ccngoge J.cnming. All Righ t" Rt:scr,.cd. May not~ .sc
196 D CHAPTER 14 PARTIAL DERIVATIVES<br />
7'(-15 30) ~ f (-10, 30) - f (- 15, 30)<br />
! = - 20 - (- 26) = ~ = 1 2<br />
' 5 5 5 . '<br />
" f{- 20, 30)- f (-15,30) - 33 - {- 26) - 7<br />
h( - 10, 30) ~ , _ 5<br />
= _ 5<br />
. = _ 5<br />
= 1.4. Averaging these values, we estimate<br />
fr( -15, 30) to be approximately 1.3. Thus, when tl1e actual temperature is - 15°C and the wind spe<strong>ed</strong> is 30 km/ h, the<br />
apparent temperature rises by about 1.3°C for every degree iliat the actual temperature rises.<br />
S . . 1 I f ( 15 30 ) . f (- 15, 30 +h) - f( -15, 30) . .<br />
tml ar y, v - , = 11m h wh1ch we can approxunate by considering h = 10<br />
l•- 0<br />
and h = - lO: f .., (- 15 , 30 ) ~ f (- 15,40) - f(- 15, 30) = - 27 - (-26) = -<br />
10 10<br />
1 = - 0. 1<br />
,<br />
10<br />
f(-15 20)-f(- 15 30) - 24 - (-26) 2 .<br />
f v{-15, 30) ~ ' _ 10<br />
' = _ 10<br />
= _ 10<br />
= - 0.2. Averaging these values, we estimate<br />
f..,( - 15, 30) to be appr~ximately - 0.15. Thus, when the actual temperature is - 15°C and ilie wind spe<strong>ed</strong> is 30 km/ h, ilie<br />
apparent temperature decreases by about 0.15°C for every km/ h that the wind spe<strong>ed</strong> increases.<br />
(b) For a fix<strong>ed</strong> wind spe<strong>ed</strong> v, the values of the wind-chill index W increase as temperature T increases (look at a column of<br />
the table), so a; is positive. For a fix<strong>ed</strong> temperature 1', the values of W decrease (or remain constant) as v increases<br />
(look at a row of the table), so ~: is negative (or perhaps 0).<br />
(c) For fix<strong>ed</strong> values ofT , the function values f (T, v) appear to become constant (or nearly constant) as v increases, so the<br />
corresponding rate of change is 0 or near 0 as v increases. This suggests that lim ( aw I ov) = 0.<br />
V->00<br />
5. (a) If we start at (1, 2) and mo~e in the positive x-direction, the graph off increases. Thus f,(1, 2) is positive.<br />
(b) If we start at (1, 2) and move in the positive y-direction, tbe graph off decreases. Thus fv(1, 2) is negative.<br />
7. (a) f zx = g, (!,), so f xx is the rate of change of f , in tl1e x -direction. fx is negative at ( - 1, 2) and if we move in the<br />
positive x-direction, the surface becomes less steep. Thus the values of f., are increasing and !:ex(- 1, 2) is positive.<br />
(b) f 1111 is the rate of change of f 11 in tlle y-direction. / 11 is negative at (- 1, 2) and if we move in the positive y-d.irection, the<br />
surface becomes steeper. Thus the values of f 11 are decreasing, and / 1111 ( - 1, 2) is negative.<br />
9. First of all, if we start at the point (3, - 3) and move in the positive y-direction, we see that both b and c decrease, while a<br />
increases. Both b and c have a low point at about (3, - 1.5), while a is 0 at this point. So a is defi nitely the graph of j 11<br />
, and<br />
one of b and c is the graph off. To see which is which, we start at the point (- 3, - 1.5) and move in the positive x-direction.<br />
b traces out a line with negative slope, while c traces out a parabola opening downward. This tells us that b is tlle x-derivative<br />
of c. Soc is the graph of f, b is the, graph of f ,, and a is tbe graph of f 11<br />
•<br />
© 20 12 Ccngage Lt4ming. All Rights RCSCTVcd. May not be scann<strong>ed</strong>. coplcd, or tluplica1cd, or pustl'd to n puhlicly accessible w~bsitc, in whole or in ~-
SECTION 14.3- PARTIAL DERIVATIVES 0 197<br />
11. f(x, y) = 16 - 4x 2 - y 2 => f ,., (x , y) = - 8x and fv(x, y) = -2y => f ,., (1, 2) = - 8 and fv (1, 2) = - 4. The graph<br />
off is the paraboloid z = 16 - 4x 2 - y 2 and the vertical plane y = 2 intersects it in the parabola z = 12 - 4x 2 , y = 2<br />
(the curve C 1 in the first figure). The slope of the tangent line<br />
to this parabola at (1, 2, 8) is f x(1, 2) = - 8. Similarly the<br />
plane x = 1 intersects the paraboloid in the parabola<br />
z ~ 12 - y 2 , X = 1 (the curve C2 in the second figure) and<br />
the slope of the tangent line at (1, 2, 8) is f v(1, 2) = - 4.<br />
JC<br />
X<br />
20<br />
0<br />
- 20<br />
2<br />
2<br />
Note that traces off in planes parallel to·the xz-plane. are parabolas which open downwru:d for y < 0 and upward for y > O,<br />
and the traces of ! r:: in these planes are straight lines, which have negative slopes for y < 0 and positive slopes for y > 0. The<br />
traces off in planes parallel to the yz-plane are cubi_c curves, and the traces of fu in these planes are parabolas.<br />
15. f(x, y) = y 5 ...:.. 3xy => f x(x, y) = 0 - 3y = - 3y, / y(x, y) = 5y 4 - 3x<br />
19. z = (2x + 3y) 10 => aaz = 10{2a: + 3y) 0 0<br />
2 = 20{2x + 3y) 9 , az = 10(2x + 3y) 0 0<br />
3<br />
= 30(2x + 3y) 0<br />
X y .<br />
21. f(x, y) = x f y = x y- 1 => fx(x, y) = y - 1 = 1/ y, fv (x, y) = -xy- 2 = - x/ y 2<br />
ax+ by<br />
23· f (x,y) = ex+dy => - , y (cx +dy)2 (ex + dy )2<br />
f - (x • ) = (ex + dy)(a) - (ax+ by)(c) = (ad·- bc)y'<br />
f ( x ) = (ex+ dy)(b} - (ax+ by)(d} = (be - ad)x<br />
1<br />
Y ' Y (ex+ dy) 2 (ex + dy} 2<br />
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198 D CHAPTER 14 PARTIAL DERIVATIVES<br />
25. g(u,v) = (u 2 v- v 3 ) 5 =? gu(u,v) = 5(u 2 v - v 3 ) 4 • 2uv = 10u·u(u 2 v - v 3 )\<br />
gu(u, v) = 5(u 2 v- v 3 ) 4 (u 2 - 3v 2 ) = 5(u 2 - 3v 2 )(u 2 v- v 3 ) 4<br />
. '<br />
29. F(x,y) = l " cos(et)dt =? Fx(x,y)= :xl" cos (e 1 ) dt=cos(e"' ) by theFundamentaiTheoremofCalculus, Partl;<br />
F 11 (x,y) = o·y<br />
0 1"' . t<br />
11<br />
cos(e) dt = oy - I. cos(e) dt = - oy f .. cos(et) dt = - cos(e 11 ).<br />
0 [ ( 11 . t ] 0 rv<br />
33. w = ln(x + 2y + 3z)<br />
ow 1 ow 2 ow 3<br />
=> ox = x + 2y + 3z' oy = x + 2y + 3z' oz = x + 2y + 3z<br />
35. u = xysin- ~(yz) =><br />
ou 1 xy 2<br />
- = xy . (y) = --;:::.:=:::::::::==:::<br />
az J 1- (yz)2 J 1- y2z2<br />
37. h(x, y, z, t) = x 2 y cos(z/t) => h.,(x; y, z, t) = 2xy cos(z/t), h 11 (x, y, z, t) = x 2 cos(z/t),<br />
hz(x, y, z, t) = -x 2 y sin(z/t)(1/ t) = ( - x 2 y/t) sin(z/t), ht(x, y, z, t) = -x 2 y sin(z/t)( -zC 2 ) = (x 2 yz/t 2 ) sin.(z/t)<br />
39 V 2 2 2 F h · 1<br />
1 ( 2 2 2) -1/2 ( )<br />
. u = x<br />
Xi<br />
1 + x 2 + ··· + Xn· or eac t = , ..., n, u,, = 2 x 1 + X2 + ·· · + Xn 2x; = .<br />
y X~ + X~ + ··· + x?,<br />
41. f(x, y) = 1n( x + Jx2 + y2) '*<br />
( ) _ 1 [ 1 ( 2 2)- 1/ 2( )] _ 1 ( X )<br />
f., x,y - ~ 1+2 x +y 2x - ~ 1+ ~ ,<br />
x + Y x2 + y2 x + Y x2 + y2 Y x2 + y2<br />
. 1 ( 3 ) .<br />
so f.,(, 34= ) 1 =11 !!=l<br />
3<br />
+~ + ...f32+42 s ( + s) s· .<br />
y l(x + y + z)- y(l ) x + z<br />
43. f(x,y, z) = x+y + z =? fv(x,y,z) = (x + y + z )2 = (x+y+ z )2'<br />
2+(-1) 1<br />
soj11(2,1, - 1) = ( 2 + 1 +(- 1 )) 2 4·<br />
45. f(x, y) = x1l - x 3 y =><br />
f ( ) _ 1 . f(x + h,y) - f(x,y) _ 1 . (x +h)y2 - (x + h) 3 y- (xy 2 - x 3 y)<br />
"' x, y - liD - liD<br />
,...... o h h-0 h .<br />
h(y 2 - 3x 2 y - 3xyh- yh 2 )<br />
= lim = lim (y 2 - 3x 2 y- 3xyh- yh 2 ) = y 2 - 3x 2 y<br />
,. ..... o h h ..... o<br />
f ( ) _ li f(x, y + h) - f(x;y) _ U:U x(y + h) 2 - x 3 (y + h)- (xy 2 - x 3 y) _ 1' h(2xy + xh- x 3 )<br />
11 x , y - h5 h - h ->0 h - "~ h<br />
· = lim {2xy + xh- x 3 ) = 2xy - x 3<br />
/a.-+0 .<br />
© 2012 Ccngnge Learning. All Rights Reserv<strong>ed</strong>. Mny not be scann<strong>ed</strong>, copi<strong>ed</strong>, orduplicot<strong>ed</strong>, or post<strong>ed</strong> to • publicly ncc=iblc website, in whole or in P."t.
8 8<br />
47. x 2 + 2y 2 + 3z 2 = 1 =? - (x 2 + 2y 2 + 3z 2 ) = - (1)<br />
ax<br />
ox<br />
8z - 2x x 8 ( 2 2 2) 8 ( )<br />
- 8<br />
= - = - - 3<br />
, and - 8<br />
x + 2y + 3z = - 0<br />
1<br />
x 6z z y y<br />
SECTION 14.3 PARTIAL DERIVATIVES D 199<br />
8z<br />
8z<br />
2x + 0 + 6z ox = 0 =? 6z ax = - 2x =?<br />
8z<br />
0 + 4y + 6z 8y = 0 =?<br />
8z<br />
6z - 0<br />
= -4y =?<br />
y<br />
49. ez = xyz => ez '- 8z = y (<br />
X -<br />
8z + Z · 1 )<br />
=?<br />
ax ax<br />
• 8z 8z yz<br />
(e- - xy) - 8<br />
= yz, so - 8<br />
= -_--.<br />
x x e- - x:y<br />
_ 8z 8z<br />
e- - - xy - = yz =><br />
OX ox<br />
8z xz<br />
ay = e=- xy<br />
=><br />
• 8z<br />
8z<br />
e- - xy - = xz , so<br />
8y<br />
e· oy - xy oy = xz => ( - ) 8z<br />
51. (a) z = f(x ) + g(y) => ~: = /'(x), ~ = ,q'(y)<br />
. 8z df QU df I 1<br />
(b) z = f(x + y). Let u = x + y. Then ox = du ox = du (1) = f (u) = f (x + y),<br />
. az = df au= df (1) = J'(u) = f'(x + y).<br />
8y du 8y du<br />
53. f(x , y) = x 3 y 5 + 2x 4 y =? f,(x, y) = 3x 2 y 5 + Bx 3 y, fu(x, y) = 5x 3 y 4 + 2x 4 . Then l rr::c (x, y) = 6xy 5 + 24x 2 y,<br />
f xu(x, y) = 15x 2 y 4 + 8x 3 , fux(x, y) = 15x 2 y 4 + Bx 3 , and f uu (x, y) = 20x 3 y 3 •<br />
55. w = Ju2 + v2<br />
(u<br />
uv ( 1) ( 2 2) - 3/ 2 ( ) . uv<br />
u + v2)3 2' .<br />
2 + v2)3/ 2 ' W uu = v -2 u +v 2u = ( 2 I<br />
1 · ~ - v · ~(u 2 + v 2 ) - 1 1 2 (2v) Juz + v1 - v 2 j .,Ju2 + v 2 u 2 + v 2 - v 2 u 2<br />
Wuv = ( Ju2 + v2)2 ' = u2 + v2 = (u2 + v2)3/2 = (u2 + v2)3/2.<br />
x + y<br />
57. z = arctan-- =?<br />
1 - xy<br />
1 (1)(1-xy)-(x+y)(- y)_ l+y 2 _ 1+y 2<br />
Zx = 1 + (..:.±1L..) 2 . (1 - x y)2 - (1 - x y)2 + (x +y)2 - 1 + x2 + y2 + x2y2<br />
1- xy<br />
1 + y 2 1<br />
= (1 + x2)(1 + y2) = 1 + x2 '<br />
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200 0 CHAPTER 14 PARTIAL DERIVATIVES<br />
1 (1)(1-xy)-(x+y)(-x) _ 1+x 2 1 +x 2 1<br />
zu = 1 + ( x+u )2 . (1- xy)2 (1- xy)2 +(x + y)2 (1 + x2)(1 + y2) = 1 + y2 ·<br />
1-xy<br />
2x 2 -2<br />
( 1<br />
+ x 2 )2, Zxy = 0, Zyx = 0, Zyy = -(1 + y ) . 2y =<br />
2y<br />
Thus U:cy = Uy:c.<br />
61. u = cos(x 2 y) => U:c = -sin(x 2 y) · 2xy = -2xysin(x 2 y),<br />
Uxy = -2xy · cos(x 2 y) · x 2 +sin(x 2 y) · (-2x) = -2x 3 ycos(x 2 y)- 2xsin(x 2 y) and<br />
uu = -sin(x 2 y) ·x 2 = - x 2 sin(x 2 y), 1t 11<br />
x = - x 2 ·cos(x 2 y) ·2xy+sin(x 2 y) · (- 2x) = -2x 3 ycos(x 2 y) -2xsin(x 2 y).<br />
Thus Uxy = Uux.<br />
63. f(x, y) = x 4 y 2 - x 3 y => fx = 4x 3 y 2 - 3x 2 y, fxx = 12x 2 y 2 - 6xy, fxxx = 24xy 2 - 6y and'<br />
f xv = 8x 3 y- 3x 2 , fxvx = 24x 2 y- 6x.<br />
67. u=er 8 sin(J => :~ . =er 8 cosfJ+sinfJ·er 8 (r)=e'. 0 (cosO+rsinO),<br />
;<br />
2<br />
;<br />
0 = er8 (sinfJ) +(cosO+ rsinO) erO (0) = ero (sinO+ OcosB + rOsinO),<br />
{)~:~O = ero (0 sin 0) +(sinO+ 0 cosO+ rOsinO) · e'' 8 (0) = Oero (2sin0 + Ocos 0 + rO sinO).<br />
X OW 8 2 w .<br />
69.w=y+2z=x(y+2z)- 1 => ax=(y+2z)- I, 8yax = -(y+2z)- 2 (1)=-(y+2z)- 2 ,<br />
8 3 w . . 4 ow<br />
{)z{)y{)x = -(-2)(y + 2z)- 3 (2) = 4(y + 2z)- 3 = (y + 2<br />
z)3 and {)y = x(-1)(y + 2z)- 2 (1) = -x(y + 2z)- 2 ,<br />
71. Assuming that the third partial derivatives off are continuous (easily verifi<strong>ed</strong>), we can write fxzy = fyxz· Then<br />
f(x, y, z) = :J;y 2 z 3 + arCSin ( X vz) =?<br />
fu = 2xyz 3 + 0, fvx = 2yz 3 , and fyxz = 6yz 2 = fxzy·<br />
73. By Definition 4, fx(3, 2) = lim !( 3 + h, 2 l-!( 3 , 2 ) which we can approximate by considering h = 0.5 and h = -0.5:<br />
h-0<br />
f ( 3 2<br />
) ~ !(3.5, 2) - f(3, 2) _ 22.4 - 17.5 _ 9 8<br />
f ( 3 2<br />
) ~ !(2.5, 2) - f(3, 2) _ 10.2- 17.5 _ 14 6<br />
A .<br />
"' ' ~ 0.5 - 0.5 - · ' "' ' ~ -0.5 - -0.5 - · · veragmg<br />
these values, we estimate f,.(3, 2) to be approximately 12.2. Similarly, f x(3, 2.2) = lim !( 3 + h , 2·2}-!( 3 , 2·2) which<br />
_ h"""""+O L<br />
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SECTION 14.3 PARTIAL DERIVATIVES 0 201<br />
· · · d I f ( 3 2 2<br />
) !(3.5, 2.2) - !(3, 2.2) 26.1- 15.9<br />
we can approximate by cons1denng h = 0.5 an 1 = - 0.5: "' , . ::::::<br />
= = 20.4,<br />
. . ' 0 . 5<br />
0.5<br />
f(2.5, 2.2) - f(3, 2.2) 9.3 - 15.9<br />
f x(3, 2 A . I I h f ( )'<br />
2.~) ~ _ _ = _ _ 0 5 0 5<br />
= 13. . veragmg t 1ese va ues, we ave "' 3, 2.2 ~ 16.8.<br />
To estimat~ f, 11 (3, 2), we first ne<strong>ed</strong> an estimate for .fx(3, 1.8):<br />
~ /(3.5, 1.8) - /(3, 1.8) - 20.0 - 18.1 - 3 8 f (3 1 8) ~ /(2:5, 1.8)- !(3, 1.8) - 12.5 - 18.1 -<br />
.fx( 3 • 1. 8 ) ~ 0.5 - 0.5 - . ' x ' . ~ -0.5 - - 0.5 - 1 1. 2 .<br />
. . a<br />
Averaging these values, we get f,(3, 1.8) ~ 7.5. Now fx 11 (x, y) = -;::;- [f.,(x, y)) and fx(x, y) is itself a function of two<br />
uy<br />
.. ()a[ ( )) li f,(x,y+h)-fx(x,y)<br />
variables, so Defimtton 4 says that fx 11 x, y = -;::;- fx x, Y = m I · =><br />
. uy h-o a<br />
( )<br />
f x(3, 2 +h)- fx(3, 2) W . I· · 1 · · k · 1 1<br />
h- o h<br />
f , 11<br />
3, 2 = lim . e can estimate t liS va ue usmg our prev1ous wor Wit 1 a = 0.2 and h = -0.2:<br />
) ~ f x(3, 1.8)- J,(3, 2) = 7.5 - 12.2 =<br />
f ( 3 2 ) ~ j,(3, 2.2)- f x(3, 2) = 16.8- 12.2 = 23 f ( 3 2<br />
xy • ~ 0.2 0.2 ' xy ' -0.2 -0.2<br />
Av~raging these values, we estimate f, 11 (3, 2) to be appr~ximately 23.25.<br />
23·5·<br />
U:r:x = -(x2 + y2 + z2)- 3/2 - x( -~) (x2 + y2 + z2) - 5f2(2x) = 2x2 - y2- z2 .<br />
. l . (x2 + y2 + z2)5/ 2<br />
2y2 - x2 - z·2 2z 2 - x2 - y2<br />
By symmetry, u uu = .(x2 + y2 + z2 )5/2 and U ::z = (x2 + y2 + z2)5/2 .<br />
2x2 - y2 - z2 + 2y2 - x2 - z2 + 2z2 - x2 - y2<br />
Thus Uxo:·+ u,,y + Uzz = ( 2 ? 2)"/2 = 0.<br />
. X +y- + Z "<br />
8[f(v) + g(w)] df(v) 8v dg(w) 8w ·<br />
79. Let v = x +at, w = x - at. Then Ut = f)t = ~ ot +~7ft = af'(v) - ag'(w) and<br />
uu = o[af'(v) a~ ag'(w)J = a[af"(v) + ag"(w)) = a 2 [f"(v) + y"(w)). Similarly, by using the Chain Rule we have<br />
u, = f'(v) + g'(w) and Uxx = f"(v) + g"(w). Thus Utt = a 2 Ux:r.·<br />
8z e"' 8z eY 8z 8z e" e·u e"' + eY<br />
81. z = in(e"' + e!l) => -'- = --- and - = --- , so - +- = - -- + --- = --- = 1.<br />
8x e"' + e!l 8y e"' + eY Dx 8y e"' + e11 e"' + eY e"' + eY<br />
D 2 z e"'(e"' + eY) .- e"'(e'") e"'+ 11 8 2 z 0 - e 11 (e"') ex+y<br />
8x2 = (e +eY) 2 = (e"' + e11 )2' 8x8y=(e"' + ev)2 = (e"' + eY)2' and<br />
8 2 z e 11 (e"' + e 11 )-e 11 (e 11 )<br />
8y2 (e"' + eY )2 ( ) 2 • Thus<br />
e" +eY<br />
e"'+Y )2 (e"'+u? (e"'+Y?<br />
(e"' + eu) 2 = (e"' + eY)1 - (ex + eu )1 = 0<br />
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202 0 CHAPTER 14 PARTIAL DERIVATIVES<br />
. 83. By the Chain Rule, taking the partial derivative of both sides with respect to R 1 gives<br />
8R- 1 8R 8[(1/Rl) + (1/R2) + (1/R3)] - 2 8R _2 8R R 2<br />
8R 8R1 = 8R1 or -R 8R 1<br />
= - R1 . Thus 8R 1<br />
= _Rf'<br />
85. If we fix K = Ko , P.(L,-Ko) is a function of a single variable L, and ~~ =a I is a separable differentia; equation. Then<br />
dP dL J dP J dL<br />
p = aL =>- p = a L =>- In IPI = a In 1-41 + C (Ko), where G_(Ko) can depend on K 0 • Then<br />
- .<br />
IPI = e lniLI-f;C(Ko), andsinceP > 0 andL >0, we have P = ec:dnLeC(Ko) = eC(T- f xv(x, y) = 4 and fv(x, y) = 3x- y =>- fvx(x, y) = 3. Since f xy and f vx are continuous<br />
everywhere but f:x:v(x, y) f- f 11<br />
.,(x, y), Clairaut's Theorem implies that such a function f(x, y) does not exist.<br />
95. By the geometry of partial derivatives, the slope of the tangent line is fx(l , 2). By implicit differentiation of<br />
4x 2 + 2y 2 + z 2 = 16, we get8x + 2z (8zj8x) = 0 => 8zf8x = - 4x/z, so when x = 1 and z = 2 we have ·<br />
8zj8x = -2. So the slope is fx(1, 2) = -2. Thus the tangent line is given by z - 2 = -2(x - 1), y = 2. Taking the<br />
parameter to bet = x - 1, we can write parametric equations for this line: x = 1 + t, y = 2, z = 2 - 2t.<br />
97. By Clairaut's Theorem, f xvv = (f:x:y) 11<br />
= (/ 11 x}y = fvxv = Uv).,'ll = Uv)yx = fvvx·<br />
99. Let g(x) = f(x, 0) = x(x 2 )- 3 1 2 e 0 = ·x ix J'- 3 • But we are using the point (1, 0), so near (1, 0), g(x) = x- 2 • Then<br />
g'(x) = - 2x- 3 and g'(1) = -2, so using (l) we have fx(l, 0) = g'(l) = - 2.<br />
101. (a) (b) For (x, y) f- (0, 0),<br />
(3x2y- y3)(x2 + y2) - (x3y- xy3)(2x)<br />
f x(x,y) =<br />
(x2+y2)2<br />
x4y + 4 x2y3 _ ys<br />
(x2 + y2)2<br />
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SECTION 14.4 TANGENT PLANES AND LINEAR APPROXIMATIONS D 203<br />
(c) f (0 0) = lim f(h, 0) - f(O, 0) = lim {O/ h2) - 0 = 0 and jy(O, 0) = lim j(O, h) -<br />
"' ' J•-O h h- o h h-o h<br />
f(O, 0) = '0.<br />
(d) B (3) f (0 0) = of, = lim f.,(O, h)- f , (O, O) = lim ( - h 5 - O) / h 4 = -1 while by (2),<br />
Y ' "'Y ' oy h- o h h- o h<br />
f ,(0, 0) = ofy = lim jy(h, 0) - jy(O, 0) = lim h5/ h4 = 1.<br />
!I ox h ..... o h ,, ..... o h<br />
(e) For (x,·y) i= (0, 0), we use a CAS to compute<br />
f xy(x, y)=<br />
x6 + gx4y2 _ gx2y4 _ y6 ·<br />
(x2+y2)3<br />
Now as (x, y) -> (0, 0) along the x-axis, fxv(x, y)-> 1 while~<br />
(x, y) -> (0, 0) along they-axis, fxy(X, y) -> - 1. Thus fx 11 isn't<br />
continuous at (0, 0) and Clairaut's Theorem doesn't apply, so there is<br />
no contradiction. The graphs of f, 11 and fyx are identical except at the<br />
origin, where we observe the discontinuity.<br />
14.4 Tangent Planes and Linear Approximations<br />
1. z = f(x, y) = 3y 2 - 2x 2 + x =? f .,(x, y) = -4x + 1, / 11 (x, y) = 6y, so f,(2, -1) = ~ 7, / y(2, -1) = -6.<br />
By Equation 2, an equation of the tangent plane is z - ( -3) = f, (2 , - 1)(x- 2) + fv(2, -1)[y- ( -:-1)] =:<br />
z+3= - 7(x - 2)-6(y + 1) or z= - 7x - 6y + 5.<br />
3. z = f(x, y) = .,fXY => f,(x, y) = ~(xy) - 112 · y = ~gx, fu(x, y) = ~(xy)- 1 1 2 · x = ~.../XlY, so f.,(1 , 1) = ~<br />
and fv(1, 1) = ~-Thus an equation of the tangent plane is z - 1 = f,(1, 1)(x- 1) + jy(1, 1)(y- 1) =?<br />
z- 1 = ~(x -1) + Hv - 1) or x + y- 2z = 0.<br />
5. z = f(x, y) = xsin(x + y) =? f , (x, y) = x · cos(x + y) + sin(x + y) · 1 = x cos(x + y) + sin(x + y),<br />
fv(x, y) = xcos(x + y), so f,( -1, 1) = ( -1) cosO+ sin 0 = - 1, / 11 ( -1, 1) = ( -1) cosO = - 1 and an equation of the<br />
tangent plane is z - 0 = ( -1) ( x + 1) + ( -1) (y - 1) or x + y + z = 0.<br />
7. z = f(x, y) = x 2 + xy + 3y 2 , so f,.(x, y) = 2x + y => f,(1, 1) = 3, fv(x, y) = x + 6y =? fv(1, 1) = 7 and an<br />
equation of the tangent plane is z - 5 = 3(x- 1) + 1(y- 1) or z = 3x + 7y - 5. After zooming in, the surface and the<br />
® 2012 Ccngogc Le;uning. All Rights Res
204 D CHAPTER 14 PARTIAL DERIVATIVES<br />
tangent plane become almost indistinguishable. (Here, the tangent plane is below the surface.) If we zoom in farther, the<br />
surface and the tangent plane will appear to coincide.<br />
_ xy sin {X - y)<br />
9 · !( x,y ) - 2 2<br />
1+x +y<br />
ACAS . f ( ' ) ysin(x - y) + xycos(x-y) 2x2 ysin{x- y) d<br />
g1ves "' x,y = - an<br />
1 + x2 + y2 (1 + x2 + y2)2<br />
fv(x, y) =<br />
x sin ( x - y) - xy cos ( x - y) 2xy 2 sin ( x - y) ·<br />
- ( 2 . We use the CAS to evaluate these at (1 , 1), and then<br />
+ x + y 1 + x2 + y2)<br />
1<br />
2 2<br />
substitute the results into Equation 2 to compute an equation oqhe tangent plane: z = tx - tY· The surface and tangent<br />
plane are shown in the first graph below. After zooming in, the surface and the tangent plane become almost indistinguishable,<br />
as shown in the second graph. (Here, the tangent plane is shown with fewer traces than the surface.) If we zoom in farther, the<br />
surface and the tangent plane will appear to coincide.<br />
z 0<br />
-1<br />
11. f(x, y) = 1 + x Jn(xy - 5): The partlal derivatives are f, (x, y) = x · -<br />
and fv(x, y) = x · -<br />
xy-5 xy- ,<br />
1 - (y) + Jn (xy - 5) · 1 = xy 5<br />
+ln(xy- 5)<br />
. xy - 5 xy -<br />
1 - (x) = __£_ 5<br />
, so f x(2 , 3) = 6 and / 11 (2, 3) = 4. Both f , and / 11 are continuous functions for<br />
xy > 5, so by Theorem 8, f is differentiable at (2, 3). By Equation 3, the linearization off at (2, 3) is given· by<br />
L(x, y) = /(2, 3) + f x (2, 3)(x- 2) + f v (2, 3)(y - 3) = 1 + 6(x - 2) + 4(y - 3) = 6x + 4y - 23.<br />
13. f(x, y) ~ _ x _ . The partial !;lerivatives are f x(x, y) = l(x t y) ) ~'t( 1 ) = y/ (x + y) 2 and<br />
x + y x +y<br />
fv(x, y) = x ( -1){x + y)- 2 • 1 = -x/ (x + y) 2 , so f x(2, 1) = ~and / 11<br />
(2; 1) = - ~.<br />
Both f, and j 11 are continuous<br />
functions for y =I - x, so f is differentiable at (2, 1) by Theorem 8. The linearization of f at (2, 1) is given by<br />
L(x, y) = !(2, 1) + f x(2 , 1)(x - 2) + / 11 (2;1)(y - 1) = 1 + ~ (x - 2) - ~(y - 1) = ~ X - ~ y + ~·<br />
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SECTION 14.4 TANGENT PLANES AND LINEAR APPROXIMATIONS 0 205<br />
15. f(x, y) = e-"' 11 cosy. The partial derivatives are f,.(x , y) = e-"' 11 ( -y) cosy .= -ye-"'!1 cosy and<br />
J 11 (x, y) = e- "'V(- siny) + (cosy)e-"' 11 ( - x) = -e-"' 11 (siny + xcosy), so f.,(1T, 0) = 0 and / 11(1T, 0) = -1r.<br />
Both f., and fv are continuous func;tions, so I is differentiable at (1r, 0), and the linearization of I at (1r, 0) is<br />
L(x, y) = j(1r, 0) + /:r(1r, O)(x- 1r) + j 11 (1r, O)(y - 0) = 1 + O(x -1r) - 1r(y - 0) = 1 -1ry.<br />
2x + 3 2 . ( ) ( )( )( )_2 - 8x - 12<br />
17.Letf(x,y) = --.Thenj.,(x,y)= - 4 1<br />
andj11 x,y = 2x+3 -1 4y+1 (4)=( 4 1<br />
)2 .Bothf.,andl 11<br />
4y + 1 y + y +<br />
are continuous functions for y =I= - ~. so by Theorem 8, f is differentiable at (0, 0) . We have f:,,(O, 0) = 2, f 11 (0, 0) = -12<br />
and the linear approximation off at (0, 0) is f(x, y) ~ f(O, 0) + f:,(O, O)(x - 0) + / 11 (0, O)(y - 0) = 3 + 2x- 12y.<br />
19. We can estimate /(2.2, 4.9) using a linear approximation off at (2, 5), given by.<br />
f(x, y) ~ /(2, 5) + f,.(2, 5)(x - 2) + / 11 (2, 5)(y - 5) = 6 + 1(x - 2) + ( -1)(y- 5) = x- y + 9. Thus<br />
!(2.2, 4.9) ~ 2.2- 4.9 + 9 = 6.3.<br />
21.j(x,y,z)"=Jx 2 +y 2 +z 2 =><br />
f.,(x,y,z)= J<br />
2 +x 2 + 2 ,f 11(x,y,z)= J 2<br />
y 2 2<br />
,and<br />
X y Z X +y + Z<br />
f:(x, y, z) = z , so f.,(3, 2, 6) = ~. fv(3, 2, 6) = ~; f,(3, 2, 6) = ~·Then the linear approximation off<br />
,jx2 +y2 +z2<br />
at (3, 2, 6) is given by<br />
l(x, y, z) ~ /(3, 2, 6) + j .,(3, 2, 6)(x - 3) + / 11 (3, 2, 6) (y- 2) + f z(3, 2, 6)(z' - 6)<br />
= 7 + ~(x- 3) + ~(y- 2) + ¥
206 0 CHAPTER 14 PARTIAL DERIVATIVES<br />
Thus when T = 95 and H = 78, !(95, 78) ~ 127 + 4(95- 94) + 1(78 - 80) = 129, so we estimate the heat index to be<br />
approximately 129°F.<br />
25. z = e- 2 "' cos 27rt =><br />
az f)z -2 ( ) . -2 . 2 2 .<br />
dz = -a dx + - 0<br />
dt = e "' -2 cos 21rt dx + e "' (- sm2?rt)(27r) dt = -2e- "'cos 27rt dx- 21re- x sin 21rt dt<br />
. X t<br />
"3 am om 43 "2<br />
ap oq<br />
27. m = p 0 q · => dm = - dp + - dq = 5p q dp + 3pa q dq<br />
29. R = o:(3 2 cos 1 =><br />
aR oR oR 2 2 .<br />
dR = oo: do: + 0<br />
(3 d(3 + O"f d"f = (3 cos '"Y do: + 2o:(3 cos 1 d(3 - o:(3 sm 1 d1<br />
• 2 2<br />
31. dx = ll.x = 0.05, dy = ll.y = 0.1, z = 5x + y , z., = lOx, Zy = 2y . . Thus when x. = 1 and y = 2,<br />
dz = zx(1, 2) dx + z 11 (1, 2) dy = (10)(0.05) + (4)(0.1) = 0.9 while<br />
ll.z = ! ,(1.05, 2.1)- /(1, 2) = 5(1.05? + (2.1) 2 - 5 -4 = 0.9225.<br />
8A 8A .<br />
33. dA = ox dx + oy dy = y dx + x dy and Jll.xJ :::; 0.1, Jll.yJ :::; ~.1. We use dx = 0.1, dy = 0.1 w1th x = 30, y = 24; then<br />
the maximum error in the area is about dA = 24(0.1) + 30(0.1) = 5.4 cm 2 •<br />
35. The <strong>vol</strong>ume of a can is V = 1rr 2 h and ll. V ~ dV is an estimate of the amount of tin. Here dV = 27rr h dr + 1rr~ dh, so put .<br />
dr = 0.04, dh = 0.08 (0.04 on top, 0.04 on bottom) and then ll. V ~ dV = 27r( 48)(0.04) + 7r(16)(0.08) ~ 16.08 cm 3 .<br />
Thus the amount of tin is about 16 cm 3 .<br />
37. T = ";'gR 2<br />
, so the differential ofT is<br />
2r +R<br />
dT = aT dR<br />
8T d = (2r 2 + R 2 )(mg) - mgR(2R) dR + (2r 2 +R 2 )(0) - mgR(4r) d<br />
oR + or . r (2r 2 + R 2 ) 2 (2r2 + R2)2 r<br />
mg(2r 2 - R 2 ) 4mgRr<br />
= (2r2 + R2)2 dR- (2r2 + R2)2 dr<br />
Here we have ll.R = 0.1 and ll.r = 0.1, so we take dR = 0.1, dr = 0.1 with R = 3, r = 0.7. Then the change in the<br />
tension T is approximately<br />
mg[2(0.7?- (3) 2 ]<br />
4mg(3)(0.7)<br />
dT = (2(0.7)2 + (3) 2]2 (O.l) - (2(0.7) 2 + (3)2)2 (0. 1 )<br />
0.802mg 0.84mg 1.642<br />
(9.98)2 - (9.98) 2 = - 99.6004 mg ~ - 0.0 165 mg<br />
Because the change is negative, tension· decreases.<br />
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SECTION 14.5 THE CHAIN RULE 0 207<br />
39. First we find :~ implicitly by taking partial derivatives of both sides with respect to R1 :<br />
oR . R 2<br />
oR<br />
1<br />
= Ri. Then by symmetry,<br />
oR R 2 2<br />
oR = R2 . When RI = 25, R2 = 40 and Ra = 50, -R 1 = 2 1 0 7 0 ¢::} R = 2 1°7° n. Since the possible error<br />
oR2 - ~· oR3 R 3<br />
for each Ri is 0.5%, the maximum error of R is attain<strong>ed</strong> by setting t:J.R, = 0.005R.. So<br />
oR oR oR 2 ( 1 1 1 ) 1<br />
!:l.R ~=:::: dR = oR !:l.R1<br />
1<br />
+ oR<br />
2<br />
!:l.R2 + oR<br />
3<br />
!:l.R3 = (0.005)R R<br />
1<br />
+ R 2<br />
+ Ra = (0.005)R = l7 ~=:::: 0.059 st.<br />
~ l!:l.wl l!:l.hl . . .<br />
41 . The errors in measurement are at ~ost 27o, so w $ 0 .. 02 and T $ 0.02. The relative error m the calculat<strong>ed</strong> surface<br />
area is<br />
!:l.S dS 0.1091(0.425w 0·425 - 1 )h 0·725 dw + 0.1091w 0 .4 25 (0.725h 0·725 - 1 } dh dw dh<br />
s 1':::: s= 0.1091w0.426h0.725 . = 0.425-:;:; + 0.725/i:<br />
To estimate the maximum relative error, we use : = I ~w I = 0.02 and ~~i = I ~hI = 0.02 =><br />
dS = 0.425 (0.02) + 0. 725 (0.02} = 0.023. Thus the maximum percentage error is approximately 2.3%.<br />
s<br />
43. !:l.z = f(a + !:l.x, b + !:l.y) - f(a, b) =(a+ !:l.x? + (b + !:l.y) 2 - (a 2 + b 2 )<br />
= a 2 + 2a!:l.x + (!:l.x) 2 + b 2 + 2b!:l.y + (!:l.y?- a 2 - b 2 = 2a !:l.x + (!:l.x? + 2b!:l.y + (!:l.y?<br />
But f,(a, b) = 2a and /y(a, b) = 2b and so !:l.z = f x(a, b) !:l.x + /y(a, b) !:l.y + !:l.x !:l.x + !:l.y !:l.y, which is Definition 7<br />
with e:1 = !:l.x and e: 2 = !:l.y. Hence f is differentiable.<br />
45. To show that f is continuous at (a, b) we ne<strong>ed</strong> to show that lim f(x, y) = f(a, b) or<br />
(z,y)~(a,b)<br />
equivalently lim f(a + !:l.x, b + !:l.y) = f(a, b). Since f is differentiable at (a, b),<br />
(6x,6y)-(O,O )<br />
f(a + !:l.x, b + !:l.y) - f(a, b)= !:l.z = f,(a, b) 6-x + /y(a, b) !:l.y + e:1 !:l.x + e:2 !:l.y, where €1 and €2 ---+ 0 as<br />
(!:l.x, !:l.y} --> (0, 0). Thus f(a + !:l.x, b + !:l.y) = f(a, b)+ f,(a, b) !:l.x + / 11 (a, b) !:l.y + e:1 !:l.x + e:2 !:l.y. Taking the limit of<br />
both sides as (.C:..x, .C:..y) ___.. (0, 0) gives lim f (a + .C:..x, b + .C:..y) = f (a , b). Thus f is continuous at (a, b).<br />
(6 x,61J)- (0 ,0) .<br />
14.5 The Chain Rule<br />
1. z = x? + y 2 + xy, x =sin t, y = et =><br />
~ &~ &~ t<br />
-d =,.--d +,.--d =(2x+ y}cost+(2y +x)e<br />
t uX t uy t<br />
3. z =· .J1 + x 2 + y 2 , x = In t, y = cost =><br />
dz oz dx 8z dy 1 2 2 -1/2 1 1 2 2 -1/ 2 . 1 (x . )<br />
- =-- + --= 2 (1+x + y) (2x)·- + 2 (1+x + y) (2y)(-smt) = - - ysmt<br />
dt ox dt 8y dt t · .J1 + x2 + y2 t<br />
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208 D CHAPTER 14 PARTIAL DERIVATIVES<br />
5. w=xevl =, x = t2, y = l-t, z=1+2t =><br />
dw = OW dx + ow dy + ow dz = ey/ z. 2t + xevl= (.!) .(- 1) + xeYI= (-J!...) . 2 = evl= (2t - ~ - 2xy)<br />
dt ox dt /Jy dt oz dt z z 2 z z2<br />
7. z = x 2 y 3 , x = scost, y = ssint =><br />
az oz ax oz 8y 3 2 2 •<br />
!i"'" = !i"'"' + -a -a = 2xy cost+ 3x y sm t<br />
us uxus y s .<br />
az . 8z Ox az 8y 3 . 2 2 s . 2<br />
ot = ox at+ oy at = (2xy )(- ssmt) + (3x y )(scost) = -2sxy smt + 3sx 2 y cost<br />
9. z = sin() cos, () = st 2 , ¢ = s 2 t =><br />
~; = ~; ~~ + ~~ ~! =(cos 8 cos q))(t~) + (- sinO sin q.,)(2st) = t 2 cosO cos¢ - 2stsin0 sin )(2st) + (-sinO sin )(s 2 ) = 2stcos0 cos¢- s 2 sinO sin
(p+r)2<br />
SECTION 14o5 THE CHAIN RULE D 209<br />
l<br />
19o<br />
w<br />
/ I~<br />
r s r<br />
/\ /\ 1\<br />
X J X y X J<br />
210 z = x 4 + x 2 y, x = s + 2t - u, y = stu 2 =><br />
w = f(r, s, t), r = r(x, y), s = s(x, y), t = t(x, y) =><br />
ow ow or ow as ow at aw aw ar aw as ow at<br />
fu = &ofu+asax+8tax·~=aroy+asay + Bt~<br />
oz = az ox+ az.ay = ( 4 xa + 2xy)( 1 ) + (x2)(tu2),<br />
OS OX OS ay as ' 0<br />
oz = oz ax+ az ay = (4x3 + 2xy)(2) + (x2)(su2)<br />
m &m ~m ·<br />
oz = az ax+ az ay = (4x3 + 2xy)( - 1} + (x2)(2stu).<br />
au ax au ay a·u<br />
- When s = 4, t = 2, and u = 1 we have x = 7 and y = 8,<br />
so az = (1484)(1} + {49)(2} = 1582, aaz = {1484) (2) + (49){4) = 3164, aaz = (1484)(-1) + {49)(16} = -700.<br />
as t u<br />
23o w = xy + yz + zx, x = r cos 0, y = r sin 8, z = r8 =><br />
aw = awax + aawaay + aawaaz ~(y +z}(cos8} + (x+z)(sin8}+(y+x) (O), .<br />
ar axar yr zr<br />
aw = aw ax+ aaw aay + aaw aaz8 = (y + z)(- rsinO) + (~ + z )(rcosO) + (y + x}(r·)o<br />
ao ox o8 . y 8 z<br />
aw<br />
When ro = 2 and()= ?r/2 we have x = 0, y = 2, and z = 1r, so ar = (2 + rr}(O) + (0 + rr){1) + (2 + 0)(1r / 2) = 27r and<br />
~~ = (2 + 1r)( - 2) + (0 + 1r)(O) + (2 + 0)(2) = -2rro<br />
..<br />
25o N = p + q, p = u + vw, q = v + uw, r = w + uv =9<br />
p+r<br />
aN = aN ap + aN aq + aN ar.<br />
au ap au aq au 8r au<br />
'<br />
= (p + r){1)- (p + q){1) ( 1<br />
) + (p + r)(1) - (p + q)(O) (w) + (p + r)(O) - (p + q)(1) (v)<br />
(p+r)2 (p + r)2 (p + r)2<br />
=<br />
(r- q) + (p+r)w - (p+q)v<br />
(p+r)2<br />
oN = aN ap + aN oq + aN or = r - q (w) + p + ro ( 1<br />
) + -(p + q) (u) = (r - q)w + (p + r} - (p + q)u'<br />
av op ov 8q 8v or ov 0 (p +r) 2 . (p+1')2 (p+ r)2<br />
aN _ aN ap + 8N aq + 8N !!!.._ = ro - q (v) + p+r (u)+ -(p +q) ( 1<br />
) = (r- q)v +(p+r)u- (p + q)<br />
ow - ap ow aq ow or OW (p + r )2 (p + r) 2 (p + r) 2 (p + r )2<br />
· aN -1 + (24)(4) - (25)(3) 20 5<br />
When u = 2, v = 3, and w = 4 we have p = 14, q = 11, and r = ~0, so ou = . ( 24<br />
)2 . = 576<br />
= 144<br />
,<br />
aN _ (-1)(4)+24-(25)(2) _ -30 __ ~ d 8N = · (- 1 )(3) + (24)(2)-25 =~ = ~<br />
av - (24)2 - 576 - 96' an aw (24)2 576 144 °<br />
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210 0 CHAPTER 14 PARTIAL DERIVATIVES<br />
27. y cosx = x 2 + y 2 , so let F(x, y) = ycosx - x 2 - y 2 = 0. Then by Equation 6<br />
dy<br />
dx<br />
-y sin x - 2x 2x + y sin x<br />
= --:-----'"----<br />
cos x - 2y cosx- 2y .<br />
and<br />
[2xy - (1 + y 2 ) (1 + x-iy 2 )]/(1 + x 4 y 2 ) _ (1 + y 2 )(1 + x 4 y 2 ) - 2xy<br />
(x2 - 2xy(1 + x4y 2 ))/(1 + x4y 2 ) - x 2 - 2xy(1+ x4y2)<br />
- 1 + x4y2 + y2 + x4y4 - 2xy<br />
- x 2 - 2xy - 2x 5 y3<br />
31. x 2 + 2y 2 + 3z 2 = 1, so let F(x, y, z) = x 2 + 2y 2 + 3z 2 - 1 = 0. Then by Equations 7<br />
f)z = .:_ Fx = _ 2x = -~ and oz = _ Fy = _ 4y = _ 2y<br />
ox Fz 6z 3z oy Fz QZ 3z .<br />
_ • oz F. . -yz yz<br />
33. e· = x yz, so let F(x, y, z) = e· - xyz = 0. Then~ = - -F = --.-- = --- and<br />
. ux 'z e· - xy ez - xy<br />
8z<br />
oy<br />
F 11 - xz xz<br />
- Fz = --e=---x-y = -ez ___ x_y<br />
35. Since x andy are eac~ functions oft, T(x, y) is a function oft, so by the Chain Rule, ~ = ~: + ~~~;.After<br />
dx 1<br />
3 seconds, x = .Jl+t = .,;r+3 = 2, y = 2 + ~ t = 2 + H 3) = 3, -d = .J1+t<br />
t 2 1 + t<br />
1 -.!. and dy -.!.<br />
2 v'1 + 3 - 4, dt - 3 .<br />
Then~~ = Tx(2, 3) ~~ + Ty(2, 3) ~; = 4(t) + 3(~ ) = 2. Thus the temperature is rising at a rate of2°C/ s.<br />
ac ac .<br />
37. C = 1449.2 + 4.6T- 0.055T 2 + 0.00029T 3 + 0.016D, so 8T = 4.6 - O.llT + 0.00087T 2 and oD = 0.016.<br />
According to the graph, the diver is experiencing a temperature of approximately 12.5°C at t = 20 minutes, so<br />
. c:;, = 4.6 - 0.11(12.5) + 0.00087(12.5) 2 ~ 3.36. By sketching tangent lines_ at t = 20 to the graphs given, we est~ate<br />
dD 1 dT 1 . dC 80 dT 80 dD 1 1 .<br />
dt ~ '2 and (jj ~ - 10<br />
. Then, by the Cham Rule, (jj = &T (jj + &Ddt ~ (3.36) (-Iii) + (0.016) ( 2) r::::: -0.33.<br />
Thus the spe<strong>ed</strong> of sound experienc<strong>ed</strong> by the diver is decreasing at a rate of approximately 0.33 m/s per minute.<br />
39. (a) V = l!wh, so by the Chain Rule,<br />
dV = 8V dl! + 8V dw + oV dh = wh dl! + l!h dw + l!w dh = 2 . 2 . 2 + 1 . 2 . 2 + 1 . 2 . ( _ 3) = 6 m3/s.<br />
dt 81! dt aw dt &h dt dt dt dt<br />
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SECTION 14.5 THE CHAIN RULE D 211<br />
(b) S = 2(£w + £h + wh), so by the Chain Rule,<br />
~~ = ~~: + ~! :~ + ~~ ~~ = 2(w +h):+ 2(£ + h):~ + 2(£+ w) ~~<br />
= 2(2 + 2)2 + 2(1 + 2)2 + 2(1 + 2)( - 3) = 10 m 2 fs<br />
(c) L 2 = £ 2 + w 2 + h 2 ::::? 2£ ~~ = 2f. ~ + 2w :~ + 2h ~: = 2(1)(2) + 2(2)(2) + 2(2)(-3) = 0 ::::?<br />
dL/ dt = 0 mjs.<br />
dP dT T dV 8.31 dT T dP<br />
41. dt = 0.05, dt = 0.15, V = 8.31? and dt = Pdt - 8.31 p 2 Yt· Thus when P = 20 andT = 320,<br />
dV = 8 31 [ 0.15 _ (0.05)(320)] ~ _ 0 . 27 Lf s.<br />
dt . 20 400<br />
43. Let x be the length of the first side of the triangle andy the lengtl1 of the second side. The area A of the triangle is given by<br />
A= ~xy sin B where 0 is the angle between the two sides. Thus A is a function of x, y, and (}, and x, y, and 0 are each in tum<br />
functions of timet. We are given that ~~ = 3, ~~ ·= -2, and because A is constant, :~ = 0. By the Chain Rule,<br />
dA 8A dx 8A dy 8A dO<br />
dt = ax dt + ay dt + ao dt<br />
dA 1 .<br />
0<br />
dx 1 .<br />
0<br />
dy<br />
1<br />
d8<br />
::::? dt = 2YSI.Il · dt + 2xsrn · dt + 2xycos(} · dt. When x = 20, y = 30,<br />
and 0 = 1r /6 we have<br />
0 = ~(30)(sin ~)(3) + ~(20)(sin ~)( - 2) + ~(20)(30)(cos ~) ~~ ·<br />
= 45 · l - 20 · l + 300 · vf3 · dO = ll + 150 '3 d(}<br />
2 2 2 dt 2 v,) dt<br />
S o I vmg<br />
.<br />
.or<br />
.,<br />
-d<br />
dO<br />
g1ves<br />
.<br />
-d<br />
dO<br />
= ---;;;<br />
- 25/2<br />
= -------:r.i·<br />
1<br />
sot<br />
h<br />
e ang<br />
I<br />
e<br />
be<br />
tween t<br />
h<br />
e s1<br />
·d<br />
es 1s<br />
· d<br />
ecreasmg<br />
·<br />
at a rate o<br />
f<br />
t t 150 v 3 . 12 v 3<br />
1/ (12 V3) ~ 0.048 radj s.<br />
(<br />
45. (a) By the Chain Rule, : = ~= cos(} + ~; sin B, ~; = ~= - r sin B) + ~~ r cos 0.<br />
( {Jz)2 ({)z ) 2 2 {)z {)z . ( {)z ) 2 • 2<br />
(b) or = ax cos 0 +.2 8<br />
x 8<br />
y cosO ~mO + By sm 0,<br />
( 8z ) 2<br />
2<br />
1 (az )<br />
2<br />
[(az )<br />
2<br />
(az) ]<br />
2<br />
(az )<br />
2<br />
(oz )<br />
2 2<br />
8r + r2 80 = 8x + 8y (cos 0 + sm<br />
.<br />
0) = 8x + 8y .<br />
8z dz 8u dz 8z dz 8z 8z<br />
47. Letu = x - y. Then - 8<br />
= d-- 8<br />
= -d and -a = -d (-1). Thus -a + - 8<br />
= 0.<br />
X UX U y U X y<br />
® 2012 Ccngogc lc:uning. All Rights Res
212 D CHAPTER 14 PARTIAL DERIVATIVES<br />
49. Let u =X + at, v =X - at. Then z = f(u) + g(v), so azjau ~ f'(u) and azjav ~ g 1 (v) .<br />
az f}z aU az av 1 1<br />
Thus at = au at + av at = af (u)- ag (v) and<br />
8 2 z = 2._ [/1( ) - I( )] = (df'(u) 8u - dg 1 (v ) au ) = 2/"( ) 2 " ( )<br />
[)t2 a at u g v a du 8t dv 8t a u + a g v .<br />
S .. , 8z fl( ) I( ) 8 2 z !"( ) "( )' 8 2 z 2 8 2 z<br />
tmt arly .,-- = u + g v and
SECTION 14.6 DIRECTIONAL DERIVATIVES AND THE GRADIENT VECTOR 0 213<br />
(b) Differentiating both sides of f(tx, ty) = t" f(x, y) with 'respect tot using the Chain Rule, we get<br />
8 a<br />
8 t f(tx, ty) = t [t" f(x,y)] ¢:><br />
8<br />
8 8(tx) 8 8(ty) 8 a -1<br />
8 (tx) f(tx, ty) · {j't + 8 (ty) f(tx, ty) · at = x a(tx ) f(tx, ty) + y 8<br />
(ty) f(tx, ty) = nt" f(x , y).<br />
8<br />
8<br />
Setting t = 1: x f(x , y) 8<br />
+ y f (x, y) 8<br />
= nf(x, y).<br />
X . y .<br />
57. Differentiating both sides of f(tx, ty) = tn f(x, y) with respect to x using the Chain Rule, we get<br />
8 8<br />
ax f(tx,ty) = 8x [t"'f(x, y)] ¢:><br />
8 8(tx) 8 8(ty) n 8<br />
a (tx) f.(tx , ty) . a;;- + 8 (ty) f(tx , ty). a;:- = t 8x f(x, y) ¢:><br />
tfx(tx, ty) = tn f,.(x, y).<br />
Thus f ,.(tx, ty) = t"·- 1 J,.(x, y).<br />
59. Given a function defin<strong>ed</strong> implicitly by F(x, y) = 0, where F is differentiable and F 11 =J: 0, we know that ~~ = - F,. . Let<br />
. ~ ~<br />
G(x, y) = - F,. so ddy = G(x, y). Differentiating both sides with respect to x and using the Chain Rule gives<br />
Fy X<br />
d 2 y = 8G dx + 8G dy where 8G = ,E_ ( - Fx) = _ F11Fxx - F,.F11., , 8G = ,E_ (-F.,) =<br />
dx2 8x dx ay dx 8x 8x Fy FJ ay 8y Fy<br />
T~us<br />
d 2 y = ( FuF:.,,. - F,.Fyx) (l)<br />
2<br />
+ (- FyFxll - FxF,m) (-Fa;)<br />
dx 2 F<br />
2<br />
11<br />
F 11<br />
F 11<br />
FxxF; - FyxFxFy - F,.uF11Fx + FyyF;<br />
FJ<br />
But F has continuous second derivatives, so by Clauraut's Theorem, F 11 ,<br />
= F., 11<br />
and we have<br />
d211 F.,.,F; - 2F., 11 F, Fu + FuuF; d . <strong>ed</strong><br />
dx 2 = - F. 3 as es1r .<br />
II<br />
14.6 Directional Derivatives and the Gradient Vector<br />
1. We can approximate the directional derivative of the pressure function at K in the direction of S by the average rate of change<br />
of pressure between the points where the r<strong>ed</strong> line intersects the contour lines closest to K (extend the r<strong>ed</strong> line slightly at the<br />
left). In the direction ofS, the pressure changes from 1000 millibars to 996 millibars and we estimate the distance between<br />
these two points to be approximately 50 km (using the fact that U1e distance from K to S is 300 km). Then the rate of change of<br />
pressure in the direction given is approximately 996 ; 0<br />
1000<br />
= -0.08 millibar /km.<br />
3. Du /{ -20, 30) = 'V j { -20, 30) · U = /T{ -20, 30) ( 7z) + /v{ - 20, 30) ( ?z).<br />
/T( - 20, 30) ~ lim !( - 20 + h, 30 )-!( - 20 • 30 ), so we can approximate /T( -20, 30) by considering h = ±5 and<br />
h -+0<br />
t<br />
© 2012 Ccngage Learning. All Rights Rc>en'Cd. Muy not be scnnncd. copi<strong>ed</strong>, or duplicot<strong>ed</strong>, or post<strong>ed</strong> to a publicly oe!Iiblc website, in whole or in part.
214 0 CHAPTER 14 PARTIAL DERIVATIVES<br />
using the values given in the table·: h (-20, 30) ~ f ( - 15 • 30 ) ~ · f (- 20 • 30 ) = - 26 -<br />
5<br />
( - 33) = 1.4,<br />
!( -25 30) - f( -20 30) -39 - ( - 33)<br />
fr( -20, 30) ~ ' _ 5<br />
' = _ 5<br />
= 1.2. Averaging these values gives fr( - 20, 30) ~ 1.3.<br />
Similarly, f,( -20, 30) = lim /( - 20 • 30 + h~- !( - 20 • JO), so we can approximate f,( --120, 30) with h = ±10:<br />
h-0<br />
f (-20 30) ~ !(- 20, 40) - J( - 20, 30) = -34 - (- 33) = - 0 1<br />
1J , 10 10 . '<br />
!( - 20 20) - J( -20 30) - 30- ( - 33)<br />
f,( - 20, 30) ~ ' _ 10<br />
' = _ 10<br />
= -0.3. Averaging these values gives f,( -20, 30) ~ -0.2.<br />
Then Duf(- 20,30) ~ 1.3(-}z) + (-0.2>(-}z) ~ 0.778.<br />
5. f(x, y) = ye- "' :::> fx (x, y) = -ye_., and fu(x, y) = e-"'. If u is a unit vector in the direction of 8 = 21r / 3, then<br />
from Equation 6, Du /(0, 4) = f x (0, 4) co~ en + fv(O, 4) sin en = -4 · (- ~) + 1 · {1 = 2 + 4.<br />
7. f(x, y) = sin(2x + 3y).<br />
• I<br />
(a) 'il f (x, y) = . 8<br />
8 1 i + ~! j = [cos(2x + 3y) · 2] i + [cos(2x + 3y) · 3]j = 2 cos {2x + 3y) i + 3 cos (2x + 3y)j<br />
X ~J •<br />
(b) 'il f( -6,4) = (2cos0) i + (3cosO)j = 2 i + 3j<br />
(c) By Equation 9, Du f( - 6, 4) = 'il f( -6, 4) · u = (2 i + 3j ) · H v'3 i - j ) = ~ (2v'3- 3) = v'3 - ~·<br />
9. f(x, y, z) = x 2 yz- xyz 3<br />
(a) 'il f(x, y, z) = (f,. (x, y, z), fv(x, y, z), f:(x, y, z)) = (2xyz - yz 3 , x 2 z- xz 3 , x 2 y - 3xyz 2 )<br />
(b) 'il/(2, -1, 1) = (-4 + 1,4- 2,-4 + 6) = (-3, 2, 2)<br />
(c) By Equation 14, Duf(2, - 1, 1) = 'il /(2, - 1, 1) · u = (-3, 2, 2) · ( 0, ~. - ~) = 0 + ~ - ~ = ~·<br />
11. f(x, y) = e"' si.ny => 'il f(x, y) = (e"' siny, e"' cosy}, '\1 f(O , Tr/3) = ( 4, ~ ) ,and a<br />
unit vector in the direction.ofv is u = J 2<br />
+8 2<br />
(- 6, 8) = 1~ (-6, 8) = ( -i, V. so<br />
- '
SECTION 14.6 DIRECTIONAL DERIVATIVES AND THE GRADIENT VECTOR 0 215<br />
17. h(r, s, t) = ln(37· + 6s + 9t) => 'ilh(r, s, t) = (3/(3r + 6s + 9t), 6/(3r + 6s + 9t), 9/(3r + 6s + 9t)),<br />
'ilh(1, 1, 1) = (k, ft, ~ ), and a unit vector in the direction ofv = 4 i + 12j + 6 k<br />
is u = .; 16<br />
+; 4 4+3& (4i + 12j + 6k) = ~ i .+ ~ j + ~ k, so<br />
( ) ' 'il f(x, y) = (Hxy)- 1 1 2 (y), Hxy)- 112 (x)) = / yt=' ~) ·so 'il!(2, 8) = (1, ~<br />
--<br />
).<br />
\2 yxy 2 yxy<br />
. .<br />
The unit vector in the direction of PQ = (5 - 2, 4 - 8) = (3, -4) is u = ( ~, - ~ ), so<br />
Duf(2,8)='il/(2,8)·<br />
.<br />
u=<br />
(<br />
1,4)·<br />
1 (35•-6")- -5.<br />
2<br />
21. f(~,y) = 4yy'X => 'ilf(x,y) = ( 4y · ~x - 1 1 2 ,4v'x) = (2yjy'x,4.,fi).'<br />
· 'il f(4, 1) = {1, 8) is.the direction of maximum rat~ ofch~ge, and the maximum rate is·/'il /(4, 1)/ = v'1 + 64 = v'65.<br />
23. J(x, y) = sin(xy) => 'il f(x, y) = (y cos(xy), x cos(xy)), 'il!(1, 0) = (0, 1). Thus the maximum rate of change is<br />
/'il f(1, 0)/ = 1 in the direction {0, 1).<br />
25. f(x,y,z) = Jx2 + y2 + .z2 =><br />
'il f(3, 6, -2) = ( -7,w. ~· ~) = ( ~. ¥, -¥ ). Th~ s the maximum rate ofchange is<br />
27. (a) As in the proof of Theorem 15, Du f = /'il!I cos 0. Since the minimum value of cos(} is - 1 occurring when(} = 11', the<br />
minimum value of Du f is - /'il f/ occurring when(} = 11', that is when u is in the opposite direction of 'il f.<br />
(assuming \7 f "f 0).<br />
(b) f(x, y) = x 4 y - x 2 y 3 => 'il f(x, y) = ( 4x 3 y - 2xy 3 , x 4 - 3x 2 y 2 ), so f decreases fastest at the point (2, -3) in the<br />
direction-'il /(2, - 3) = - {12, -92) = (-12, 92).<br />
29. The directionoffastestchange is 'ilf(x,y) = (2x- 2) i + (2y- 4)j , so we ne<strong>ed</strong> to find all points (x,y) where 'ilf(x,y) is<br />
parallel to i + j ¢'> (2x - 2) i + (2y - 4)j = k (i + j) ¢'> k = 2x- 2 and k = 2y - 4. Then 2x- 2 = 2y- 4 =><br />
y = x + 1, so the direction of fastest change is i + j at all points on the line y = x + 1.<br />
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216 0 CHAPTER 14 PARTIAL DERIVATIVES<br />
k<br />
Jx2 + y2 +z2 3<br />
31. T = and 120 = T(1, 2, 2) = - so k = 360.<br />
k<br />
( )<br />
- (1, - 1, 1)<br />
au- v'3 ,<br />
DuT(1, 2, 2) = 'VT(1, 2, 2) · u = [ .:._360( x 2 + y 2 + z 2 ) -S/ 2 (x, y, z)] . u = - .12 3 (1, 2, 2). -4.. (1, -1, 1) = -~<br />
. . {1,2,2) v3 3v3<br />
(b) From (a), 'VT = -360(x 2 + y 2 + z 2 ) - S/ 2 (x, y, z), and since (x, y, z) is the position vector of the point (x, y, z), the<br />
vector - (x, y, z), and thus 'VT, always points toward the origin.<br />
33. 'VV(x, Y •. z) = (lOx- 3y + yz, xz - 3x, xy), 'VV(3, 4, 5) = (38, 6, 12)<br />
{a) Du V(3,4,5) = {38, 6, 12} · ~(1, 1, -1} = '7a<br />
(b) 'VV(3, 4, 5) = (38, 6, 12), or equivalently, (19, 3, 6).<br />
(c) IV'V(3, 4, 5)1 = v'382 + 6 2 + 12 2 = v'l624 = 2 v'406<br />
-----t<br />
35. A unit vector in the direction o(AB is i and a unit vector in the direction of AC isj. Thus D_..... f(1, 3) = f.,(1, 3) ~ 3 and<br />
AB<br />
D- f(l, 3) = / y(1, 3) = 26. Therefore \7 /(1, 3) = (!.,(1, 3), /y(l, 3)) = (3, 26}, and by definition,<br />
AC<br />
-----t<br />
DAD f (1 , 3) = V' f · u where u is a unit vector in the direction of AD, which is (fa, H). Therefore,<br />
37 ()<br />
DAD f (1, 3) = (3, 26) . . ( 153, H) = 3. fa + 26. H = 312;.<br />
. a v au + v ~ ' a a ~ + {) , a ~- + ~ a a , ~ + ~ ' a<br />
'["7( b)=(o(au+bv) 8(au+bv)) =\ au bav au bav)= (au au; b\av au;<br />
.= a 'Vu + b 'Vv<br />
uX y ux X uy uy X uy ux y<br />
-----t<br />
(d)<br />
'["7 n ( o(un) 8(u")) ( n - 1 au n - 1 au) n - 1 '["7<br />
v u - , - nu , nu · - nu v u<br />
ax oy ox {)y<br />
39. f(x, y) = x 3 + 5x 2 y + y 3 . "*<br />
Duf(x,y) = 'Vf(x, y) · u = (3x 2 + lOxy, 5x 2 + 3y 2 ) · (~, ~) = ~x 2 + 6xy+ 4x 2 + ¥v 2 = ~x 2 + 6xy + ¥y 2 . Then<br />
D~.J(x ,y) = Du (Duf(x,y)] = \7 [Duf(x,y)] · u = (¥x + 6y, 6x + -\1Y) · (~, ~)<br />
= l;s4x + Jfy + ¥x + ~y = 2is4x + ~ar,oY<br />
and D~f(2 , 1) = 2iu4 (2) + 12866 (1) = 72754 .<br />
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SECTION 14.6 DIRECTIONAL DERIVATIVES AND THE GRADIENT VECTOR 0 217<br />
41. Let F(x, y, z) = 2(x - 2) 2 + (y - 1? + (z - 3) 2 . Then 2(x - 2) 2 + (y - 1f + (z - 3) 2 = 10 is a level surface of F.<br />
F:.(x, y,z) = 4(x - ·2) =? F,(3, 3, 5) = 4, F 11 (x,y,z) = 2(y- 1) =? F 11 (3,3,5) = 4, and<br />
Fz(x,y, z) =2{z - 3) =? Fz{3,3, 5) = 4.<br />
(a) Equation 19 gives an equation of the tangent plane at {3, 3, 5) as 4{x - 3) + 4{y - 3) + 4{z- 5) = 0 <br />
4x + 4y + 4z = 44 or equivalently x + y + z = 11.<br />
. I I' h . . x - 3 y -<br />
(b)<br />
3 z - 5 . I l<br />
By EquatiOn 20, the norma me as symmetnc equat10ns -<br />
- or eqUiva ent y<br />
4 - = - 4 - = - 4<br />
x - 3 = y - 3 = z - 5. Corresponding parametric equations are x = 3 + t, y =; 3 + t, z = 5 + t.<br />
43. Let F(x, y, z ) = xyz 2 • Then xyz 2· = 6 is a level surface ofF and 'V F(x, y, z ) = (yz 2 , xz 2 , 2xyz).<br />
(a) 'V F(3, 2, 1) ~ (2,3, 12) is a normal vector for the tangent plane at {3, 2, 1), so an equation of the tangent plane<br />
is 2(x - 3) + 3(y - 2) + 12(z - 1) = 0 or 2x + 3y + 12z = 24.<br />
(b) The normal line has direction (2, 3, 12), so parametric equations are x = 3 + 2t, y = 2 + 3t, z = 1 + 12t, and<br />
. . x-3 . y-2 z- 1<br />
symmetriC equations are -<br />
2 - = - 3 - = l2.<br />
45. Let F (x, y, z) = x + y + z - e"'Yz. Then x + y + z = e"' 11 :; is the level surface F(x, y , z) = 0,<br />
and 'V F(x, y, z) = (1 - yze"'Y:;, 1 - xze"' 11 "", 1 - x ye"' 11 z).<br />
(a) 'V F(O, 0, 1) = (1, 1, 1) is a normal vector for the tangent plane at {0, 0, 1), so an equation of the tangent plane<br />
is 1(x- 0) + 1{y - 0) + 1(z - 1) = 0 or x + y + z = 1.<br />
(b) The normal line has direction (1, 1, 1), so parametric equations are x = t, y = t, z = 1 + t, and symmetric equations are<br />
x = y =z- 1.<br />
47. F(x, y , z) = xy + yz + zx, 'V F (x, y, z) = (y + z, x + z, y + x), 'V F{1, 1, 1) = (2, 2, 2), so an equation of the tangent<br />
plane is 2:r; + 2y +2z = 6 or x + y + z = 3, and the normal line is given by x - 1 = y - 1 = z - 1 or x = y = z. To graph<br />
3 - xy<br />
the surface we solve for z : z = --- .<br />
- x+y<br />
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218 D CHAPTER 14 PARTIAL DERIVATIVES<br />
49. f(x, y) = xy => '\1 f(x, y) = (y, x}, '\1 !(3, 2) = (2, 3). '\1 !(3, 2)<br />
is perpendicular to the tangent line, so the tangent line has equation<br />
'\1](3, 2) · (x - 3, y- 2} = 0 => (2, 3) · (x- 3, x- 2) = 0 =><br />
2(x - 3) + 3(y - 2) = 0 or 2x + 3y = 12.<br />
I 2xo 2yo 2zo J . .<br />
51. '\1 F(xo, yo, zo) = \ (i2• 1)2• C2 . . Thus an equatiOn oflhe tangent plane at (x0 , y0 , zo) 1s<br />
2xo 2yo 2zo ( x5 y~ z~ ) ( ) . . . . .<br />
(i2 x + 1)2 Y + C2 z = 2 a 2 + b 2 + c 2 = 2 1 = 2 smce (xo, yo, zo) IS a pomt on the elhps01d. Hence<br />
xo · Yo zo<br />
a<br />
c<br />
2 x + b 2<br />
y + 2 z = 1s an equation o e tangent p ane.<br />
1 . . fth 1<br />
( ) I 2xo 2yo - 1 J ' . . 2xo 2yo 1 2x5 2y5 zo<br />
53. '\1 F xo, yo, zo = \ (i2• 1)2, c , so an equatiOn of the tangent plane 1s (i2 x -1: 1)2 y - c z = (i2 + 1)2 - c<br />
2xo 2yo z ( x 2 y 2 ) zo zo x 2 y 2 . .<br />
or - 2<br />
x + -b 2<br />
y =- + 2 -% + b~ - - . But - = -% + b~, so the equat1on can be written as<br />
a c a c c a<br />
2xo 2yo z + zo<br />
- x+ - y = --.<br />
a 2 b 2 c<br />
55. The hyperboloid x 2 - y 2 - z 2 = 1 is a level surface of F(x, y, z) = x 2 - y 2 - . z 2 and '\1 F (x, y , z) = (2x, -2y, -2z) is a<br />
nonnal vector to the surface and hence a nonnal vector for the tangent plane at (x, y, z). The tangent plane is parallel to the<br />
plane z = x + y or x + y- z = 0 if and only if the corresponding normal vectors are parallel, so we ne<strong>ed</strong> a point (xo, yo, zo)<br />
on the hyperboloid wh~re (2xo, -2yo, - 2zo) = c (1, 1·, ....,-1) or equivalently (xo, - yo, -zo) = k (1, 1, -1) for some k ¥: 0.<br />
Then we must have xo = k , yo = - k , zo = k and substituting into the equation of the hyperboloid gives<br />
k 2 - ( -k) 2 ~ k 2 = 1 {:} - k 2 = 1, an impossibility. Thus there is no such point on the hyperboloid.<br />
57. Let (xo, y 0 , z 0 ) be a point on the cone [other than {0, 0, 0)]. The cone is a level surface of F(x, y, z) = x 2 + y 2 - z 2 and<br />
'\1 F(x, y, z) = (2;t, 2y, - 2z), so '\1 F(xo, yo, zo) = (2xo, 2yo, -2zo) is a nomial vector to the cone at this point and an<br />
equation of the tangent plane there is 2xo (x - xo) + 2yo (y - y 0 ) - 2zo (z - zo) = 0 or xox + yoy - zoz = x~ + y5- z5.<br />
But x& + v5 = za so the tangent plane is given by xox + YOY - ZoZ = 0, a plane which always contains the origin.<br />
59. Let F(x, y , z) = x 2 + y 2 - z. Then the paraboloid is the level surface F (x, y, z) = 0 and '\1 F (x, y ; z) = (2x,2y, -1), so<br />
'\1 F(1, 1, 2) = '(2, 2,-1) is a nonnal vector to the surface. Thus the normal line at (1, 1, 2) is given by x = 1 + 2t,<br />
y = 1 + 2t, z = 2-t. Substitution into the equation of the paraboloid z = x 2 + y~ gives 2-t = {1 + 2t) 2 + {1 + 2t? {:}<br />
2 - t = 2 + 8t + Bf {:} 8t 2 + 9t = 0 {:} t(Bt + 9) = 0. Thus the line intersects the paraboloid when t = 0,<br />
corresponding to the given point (1, 1, 2), or when t = - ~,corres ponding to the point (-%, -~, ¥).<br />
I<br />
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SECTION 14.6 DIRECTIONAL DERIVATIVES AND THE GRADIENT VECTOR 0 219<br />
61. Let (xo, y 0 , zo) be a point on the surface. Then an equation of the tangent plane at the point is<br />
X y Z JXO+Jy0+ /Zo r;. . · ·<br />
--+ --+ r.;:: = . But JXo + ..;Yo + v'Zo = v c, so the equation IS<br />
2y'Xo 2..;Yo 2yZo 2<br />
-=-- + _]f_ + ~ = .;c. The x-, y-, and z-intercepts are foO, -ICYo and .jCZO respectively. (The X-intercept is found by<br />
y'Xo ..;Yo v zo<br />
setting y = z = 0 and solving the resulting equation for x, and they- and z-intercepts are found similarly.) So the sum of the<br />
intercepts is .jC ( JXo + ffo + v'Zo) = c, a constant.<br />
63. If f(x, y, z ) = z - x 2 - y 2 and g(x, y, z) = 4x 2 + y 2 + z 2 , then the tangent line is perpenoicular to both "il f and "ilg<br />
at ( - 1, 1, 2). The vector v = "il f x "ilg will therefore be parallel to the tangent line.<br />
We have "il f(x, y, z) = ( - 2x, -2y, 1) =? "il f ( - 1, 1, 2) = (2, - 2, 1), and "il g(x, y , z ) = (8x, 2y, 2z) =?<br />
"Vg(- 1, 1, 2)= (-8, 2,4).Hencev="Vf x "Vg =<br />
j k<br />
2 - 2 1 = - 10 i-16j-12k.<br />
- 8 2 4<br />
Parametric equations are: x = -1 - lOt, y = 1 - 16t, z = 2 - 12t.<br />
65. (a) The direction of the normal line ofF is given by "il F, and that of G by "VG. Assuming that<br />
"il F -1= 0 -1= "VG, the two normal lines are perpendicular at P if "il F · "VG = 0 at P -¢:?<br />
(8Fj8x, 8F/8y,8F/8z) · (8G j8x, 8G/ 8y, 8G/ 8z) = 0 at P # F_..,G, + FvGv + FzGz = 0 at P.<br />
(b) Here F = x 2 + y 2 - z 2 and G = x 2 + y 2 + z 2 - r 2 , so<br />
"il F · "VG = (2x, 2y, - 2z) · (2x, 2y, 2z) = 4x 2 + 4y 2 -<br />
4z 2 = 4F = 0, since the point (x, y, z ) lies on the graph of<br />
F = 0. To see that this is true without using calculus, note that G = 0 is the equation of a sphere center<strong>ed</strong> at the origin and<br />
F = 0 is the equation of a right circular cone with vertex at the origin (which is generat<strong>ed</strong> by lines through the origin). At<br />
any point of intersection, the sphere's normal line (which passes through the origin) lies on the cone, and thus is<br />
perpendicular to tl1e cone's normal line. So the surfaces with equations F = 0 and G = 0 are everywhere orthogonal.<br />
67. Let u = (a, b) and v = (c, d). Then we know that at the given point, Du f = "il f · u = afx + bfv and<br />
Dv I = "il I · v = clx + dfv· But these are just two linear equations in the two unknowns fx and fv, and since u and v are<br />
not parallel, we can solve the equations to find "V f = (f.,, f v) at the given point. In fact,<br />
"V f = I d Du f - b Dv f , a Dv f - c D u f ) .<br />
\ ad - be ad - be<br />
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220 D CHAPTER 14 PARTIAL DERIVATIVES<br />
14.7 Maximum and Minimum Values<br />
1. (a) First we compute D(1, 1) = f,x(1, 1) fvv(1, 1)- [f:cy(1, 1W = (4)(2) - (1? = 7. Since D(1, 1) > 0 and<br />
f o:x(1, 1) > 0, f has a local minimum at (1, 1) by the Second Derivatives Test.<br />
(b) D(1, 1) = f.xx (1, 1) fuy(1, 1)- [fxu(1, 1W = (4)(2)- (3) 2 = - 1. Since D(1, 1) < 0, f has a saddle point.at (1, 1) by<br />
'<br />
the Second Derivatives Test.<br />
3. In the figure, a point at approximately (1 , 1) is enclos<strong>ed</strong> by level curves wh'ich are oval in shape and indicate that as we move<br />
away from the point in any direc~ion the values off are increasing. Hence we would expect a local minimum at or near (1, 1).<br />
The level curves near (0, 0) resemble hyperbolas, and as we move away from the origin, the values off increase in some<br />
directions and decrease in others, so we would expect to find a saddle point there.<br />
To verify our pr<strong>ed</strong>ictions, we have f(x, y) = 4 + x 3 + y 3 - 3xy => f x (x, y) = 3x 2 - 3y, f v (x, y) = 3y 2 - 3x. We<br />
have critical points where these partial derivatives are equal to 0: 3x 2 - 3y = 0, 3y 2 - 3x = 0. Substituting y = x 2 from the<br />
first equation into the second equation gives 3(x 2 ) 2 - 3x = 0 => 3x(x 3 - 1) = 0 => x = 0 or x = 1. Then we have<br />
two critical points, (0, 0) and (1, 1). The second partial derivatives are f xx (x, y) = 6x, fxu(x, y) = -3, and fvu(x, y) = 6y,<br />
' ' 2 2<br />
so D(x, y) = fxx(x, y) fuv(x, y)- [fxu(x, y)] = (6x)(6y)- (-3) = 36xy- 9. Then D(O, 0) = 36(0)(0) - 9 = - 9,<br />
and D (1, 1) = 36(1)(1) - 9 = 27. Since D(O, 0) < 0, f has a saddle point at (0, 0) by the Second Derivatives Test. Since<br />
D (1, 1) > 0 and fxx(1, 1) > 0, f has a local minimum at (1 , 1).<br />
5. f(x, y) = x 2 + x y + y 2 + y => f x = 2x + y, fv = x + 2y + 1, fxx = 2, fx 11 = 1, fvv = 2. Then fx = 0 implies<br />
y=-2x,andsubstitutionintofy=x + 2y + 1=0 gives x+2(~2x) +1=0 => -3x=-1 => x = i·<br />
Then y =-~and the only critical point is (t, -~).<br />
D(x, y) = fxxfvv - (fru) 2 = (2)(2) - (1? = 3, and since_<br />
D(~, - D = 3 > 0 and fxx(i, -~) = 2 > 0, !(~ ,<br />
minimum by the Second Derivatives Test.<br />
-~) = -~ is a local<br />
7. f(x,y) = (x- y)(1- xy) = x -y- x 2 y+ xy 2 => f x = 1 - 2xy +y 2 , fv = - 1- x 2 '+ 2xy, fx:n = -2y,<br />
f xv = - 2x + 2y, fvv = 2x. Then fx = 0 implies 1 - 2xy + y 2 = 0 and / 11 = 0 implies -1 - x 2 + 2xy = 0. Adding the<br />
two equations gives 1 + y 2 - 1- x 2 = 0 => y 2 = x 2 => y = ±x, but ify = -x then f x = 0 implies<br />
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SECTION 14.7 MAXIMUM AND MINIMUM VALUES 0 221<br />
1 '+ 2x 2 + x 2 = 0 =} 3x 2 = -1 which has no real solution. lfy = x<br />
then substitution into f, = 0 gives 1 - 2x 2 + x 2 = 0 =} x 2 = 1 =}<br />
x = ±1, so the critical points are (1, 1) and (-1, -1). Now<br />
D (1, 1) = (- 2)(2) - 0 2 = - 4 < 0 and<br />
D( - 1, - 1) = (2)(-2)- 0 2 = -4 < 0, so (1, 1) and (-1, - 1) are<br />
saddle points.<br />
9. f(x, y) = y 3 + 3x 2 y - 6x 2 - 6y 2 + 2 =} f ,. = 6xy - 12x, ! 11 = 3y 2 + 3x 2 - 12y, f x:t = 6y- 12, f xu = 6x,<br />
fw = 6y - 12. Then f z = 0 implies 6x(y- 2) = 0, sox = 0 or y = 2. If x = 0 then substitution into j y = 0 gives<br />
3y 2 - 12y = 0 =} 3y(y- 4) = 0 =} y = 0 or y = 4, so we have critical points (0, 0) and (0, 4). If y = 2,<br />
substitution into jy = 0 gives 12 + 3x 2 - 24 = 0 =} x 2 = 4 =}<br />
x = ±2, so we have critical points (±2, 2).<br />
D(O, 0) = (- 12)( -12) - 0 2 = 144 > 0 and fu(O, 0) = - 12 < o •. so<br />
f(O, 0) = 2 is a local maximum. D{O, 4) = (12){12) - 0 2 = 144 > 0<br />
and f xx(O, 4) = 12 > 0, so f(O , 4) = -30 is a local minimum.<br />
D(±2, 2) = {0)(0)- {±12) 2 = -144 < 0, so (±2, 2) are saddle points.<br />
y<br />
11. f(x, y) = x 3 - 12xy + _8y 3 =} fx = 3x 2 - 12y, f u = - 12x + 24y 2 , fxx = 6x, fx 11 = - 12,/ 1111 = 48 y. Then fx = 0<br />
implies x 2 ·= 4y and / 11 = 0 implies x = 2y 2 • Substituting the second equation into the first gives {2y 2 } 2 = 4y =}<br />
4y 4 = 4y =} 4y(y 3 - 1) = 0 =} y = 0 or y = 1. lfy = 0 then<br />
x = 0 and if y = 1 then x = 2, so the critical points are (0, 0) and {2, 1).<br />
D(O, 0) = {0){0) - ( - 12) 2 = - 144 < 0, so (0, 0) is a saddle point.·<br />
D(2, 1) = (12)(48)- ( -12? = 432 > 0 and fxx(2, 1) = 12 > 0 so<br />
!(2, 1) = -8 is a local .minimum.<br />
- 4<br />
13. f(x,y)=e"'cosy =} f x=e"'cosy, / 11 =-e"'siny.<br />
Now f a: = 0 implies cosy = 0 or y = f + mr for nan integer.<br />
But sin ( f + mr) =1- 0, so there are no critical points.<br />
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222 D CHAPTER 14 PARTIAL DERIVATIVES<br />
2 2) 2 2 ( ) 2 2 2 2 . 2 2<br />
l" = ( x + y e'~~ - :r - 2x + 2xe'~~ -:r = 2x e'~~ - :r (1 - x - y ),<br />
fzx = 2xe'fl 2 - o: 2 (- 2x) + (1- x 2 - y 2 ) ( 2x ( -2xe'~~ 2 -:r 2) + 2e'fl 2 - :r 2) = 2e'~~ 2 -:r 2 ((1- x 2 - y 2 )(1 - 2x 2 ) - 2x 2 ),<br />
22 22 2 2 22<br />
f xv = 2xe'11 - o: (- 2y) + 2x(2y)eY -:r (1 - x - y ) = ~ 4xyeY - x (x 2 + y 2 ),<br />
f 11<br />
v = 2yeV<br />
2<br />
- o:<br />
2<br />
(2y) + (1 + x<br />
2<br />
+ y<br />
2<br />
)<br />
(<br />
2y<br />
(<br />
2yeV<br />
2 2)<br />
-:r + 2eY<br />
2 2 )<br />
- :r = 2e 71 2 2<br />
- :r ((1 + x 2 + y 2 )(1 + 2y 2 ) + 2y 2 ).<br />
fv = 0 implies y = 0, and substituting into fx = 0 gives<br />
2xe-"' 2 (1 - x 2 ) = 0 ~ x = o. or x = ± 1. Thus the critical points are<br />
(0, 0) and (±1, 0). Now D(O, 0) = (2)(2) - 0 > 0 and f :r:x (O, 0) = 2 > 0,<br />
so f(O, 0) = 0 is a local minimum. D(±l, 0) = ( - 4e- 1 )(4e- 1 )- 0 < 0<br />
so (±1, 0) are saddle points.<br />
17.f(x,y)=y 2 - 2ycosx ~<br />
j ; =2ysinx,fv=2y -2cosx,<br />
fxx = 2y cos x, f ., 11 = 2 sin x, f v 11 = 2. Then/:. = 0 implies y = 0 or<br />
sinx = 0 ~<br />
x = 0, 1r, or 211" for -1 :-::; x :-::; 7. Substituting y = 0 into<br />
f v = 0 gives cosx = 0 ~ x = ~or 3 ;, substituting x = 0 or x = 21r<br />
into f 71 = 0 gives y = 1, and substituting x = 1r into fv = 0 gives y = - 1.<br />
Thus the critical points are (0, 1), ( ~. 0), (1r, -1), ( 3 ;, 0), and (21r, 1).<br />
D (~ , 0) = D e 2<br />
"", 0) = -4 < 0 so (~, 0) and e;, 0) are saddle points. D(O, 1) = D(1r, -1) = D(21r, 1) = 4 > 0 and<br />
fa:x(O , 1) = fxx('Tr, - 1) = fxo:(27r, 1) = 2 > 0, so f(O, 1) = j(1r, -1) = f(27r, 1) = - 1 are local minima.<br />
19. f(x, y) = x 2 + 4y 2 - 4xy + 2 ~ fx = 2x- 4y, fv =By - 4x, f x:r: = 2, f,v = -4, / 1171 = 8. Then fx = 0<br />
and fv = 0 each implies y = ~x, so all points of the form (xo, ~xo) are critical points and for each ofth~se we have<br />
D(xo, ~xo) = (2)(8) - (-4) 2 = 0. The Second Derivatives Test gives no information, but<br />
f(x,y) = x 2 +4y 2 - 4xy + 2 = (x- 2y} 2 + 2 ;:: 2 with equality if and only ify = ~x. Thus f(x 0 , ~xo) = 2 are all local<br />
(and absolute) minima.<br />
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SECTION 14.7 MAXIMUM AND MINIMUM VALUES 0 223<br />
8 ·<br />
7<br />
8<br />
6<br />
4<br />
2<br />
6<br />
z 5 -2.5<br />
4<br />
3<br />
2 - 2.5<br />
- 2 - I 0 2<br />
y<br />
2.5<br />
From the graphs, there appear to be local minimaofabout/(1, ±1) = f(- 1,±1) ~ 3 (and no local maxima or saddle<br />
points). f, = 2x - 2x- 3 y- 2 , fv = 2y - 2x- 2 y- 3 , f xx = 2 + 6x- 4 y- 2 , fx11 = 4x- 3 y- 3 , / 1111 = 2 + 6x- 2 y- 4 • Then<br />
f, = 0 implies 2x 4 y 2 - 2 = 0 or x 4 y 2 = 1 or y 2 = x - 4 • Note that neither x nor y can be zero. Now f v = 0 implies<br />
2x 2 y 4 - 2 = 0, and with y 2 = x- 4 this implies 2x- 6 - 2 = 0 or x 6 = 1. Thus x = ±1 and if x = 1, y = ± 1; if x = - 1, ·<br />
y = ±1. So the critical points are (1, 1), (1, - 1),( - 1, 1) and (- 1, -1). Now D(1, ± 1) = D( -1, ±1) = 64 - 16 > 0 and<br />
fxx > 0 always, so f(1, ± 1) = f (-1, ± 1) = 3 are local minima.<br />
23. f(x,y) = sin x +siny+sin(x +y), 0 :S: x :S: 2rr, 0 :S: y :S: 2rr<br />
y<br />
From the graphs it appears that f has a local maximum at about (1, 1) with value approximately 2.6, a local minimum<br />
at about (5, 5) with value approximately -2.6, and a saddle point at about (3, 3).<br />
f :z: = cos x + cos(x + y), fv = co:;y + cos(x + y), f xx = - sinx- sin(x + y ), / 11 y = - siny- sin(x + y),<br />
f x y = - sin(x + 1J). Setting fx = 0 and / 11 = 0 and subtracting gives cos x - cos y = 0 or cos i = cos y. Thus x = y<br />
or x = 2rr - y. lfx = y, f, = 0 becomes cosx + cos 2x = 0 or 2 cos 2 x + cosx - 1 = 0, a quadratic in cosx. Thus<br />
cos x = - 1 or ~ and x = rr, i , or r; , giving the critical points ( rr, rr ), ( i, i) and ( 5 ;, 5 ;). Similarly if<br />
x = 2rr - y, f :z: = 0 becomes (cos x) + 1 = 0 and the resulting critical point is (rr, rr). Now<br />
D(x, y) = sin x sin y + sinx sin(x + y) +sin y sin(x + y). So D(1r, 1r) = 0 and the Second Derivatives Test doesn't apply.<br />
However, along the line y = x we have f(x, x) = 2sin x + sin2x = 2 sinx + 2sinx cos x = 2sinx (1 + cosx), and<br />
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224 0 CHAPTER 14 PARTIAL DERIVATIVES<br />
f(x, x) > 0 for 0 < x < 1r while f(x, x) < 0 for 1r < x < 21r. Thus every disk with center (1r, 1r) contains points where f is<br />
positive as well as points where f is negat~v e, so the graph crosses its tangent plane (z = 0) there and (1r, 1r) is a saddle point.<br />
D(~, ~) = * > 0 and fxx (~,·?f) < 0 so/( ~,~ ) = .tf! is a local maximum while D( 5 3 ", 6 ;) = ~ > 0 and<br />
f xx ( 5 ;, 5 ;) > 0, so f ( 6 3 " , 6 ;) = -¥ is a local minimum.<br />
4x(x 2 - 2y) = 0, sox= 0 or x 2 = 2y. lfx = 0 then substitution into f 11<br />
= 0 gives 4y 3 = -2 :::::}<br />
• 1<br />
y = - 1i2, so<br />
( 0, - ~) is a critical point. Substituting x 2 = 2y into fv = 0 gives 4y 3 - 8y + 2 = 0. Using a graph, solutions are<br />
approximately y = - 1.526, 0.259, and 1.267. (Alternatively, we could have us<strong>ed</strong> a calculator or a CAS to find these roots.)<br />
We have x 2 = 2y => x = ±.J2Y, soy = -1.526 gives no real-valu<strong>ed</strong> solution for x, but<br />
y = 0.259 => x ~ ± 0.720 andy = ,1.267 => x ~ ±1.592. Thus to three decimal places, the critical points are<br />
( 0, -~) ~ (0, - 0.794), (±0.720, 0.259), and (±1.592, 1.267). Now since f xx = 12x 2 - 8y, f ., 11 • = - 8x, f 1111 = 12y 2 ,<br />
and D = (12x 2 - 8y)(12y 2 )- 64x 2 , we have D(O, -0.794) > 0, fxx(O, - 0.794) > 0, D(±0.720, 0.259) < 0,<br />
D(±1.592, 1.267) > 0, and f xx (± 1.592, 1.267) > 0. Therefore /(0, _-0.794) ~ - 1.191 and /(±1.592, 1.267) ~ - 1.310<br />
are local minima, and (± 0. 720, 0.259) are saddle points. There is no highest point on the graph, but the lowest points are<br />
approximately (±1.592, 1.267, -1.310).<br />
20<br />
10<br />
-6<br />
0<br />
y<br />
27. f(x, y) = x 4 +y 3 - 3x 2 +y 2 +x - 2y + 1 => f x (x, y) = 4x 3 - 6x + 1 and f 11 (x, y) = 3y 2 + 2y- 2. From the<br />
graphs, we see thatto three decimal places, f ., = 0 when x ~ -1.301, 0.170, or 1.131, and / 11<br />
= 0 when y ~ -1.215 or<br />
0.549. (Alternatively, we could have us<strong>ed</strong> a calculator or a CAS to find these roots. We could also use the quadratic formula to<br />
find the solutions of / 11 = 0.) So, to three decimal places, f has critical points at (- 1.301, - 1.215), (- 1.301, 0.549),<br />
(0.170, -1.215), (0.170, 0.549), (1.131, - 1.215), and (1.131, 0.549). Now since /zz = 12x 2 - 6, fx 11 .= 0, f 11 y = 6y + 2,<br />
and D = (12x 2 - 6)(6y + 2), we have D( -1.301, -1.215) < 0, D( - 1.301, 0.549) > 0, f xx( -1.301, 0.549) > 0,<br />
D(0.170, - 1.215) > 0, f xx (0.170, - 1.215) < 0, D(0.170, 0.549) < 0, D(l.131, -1.215) < 0, D(l.131, 0.549) > 0, and<br />
/ xz (l.131, 0.549) > 0. Therefore, to three decimal places, f( -1.301, 0.549) ~ -3.145 and /(1.131, 0.549) ~ - 0.701 are<br />
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SECTION 14.7 MAXIMUM AND MINIMUM VALUES 0 225<br />
local minima, !(0.170, - 1.215) ::::: 3.197 is a local maximum, and ( -1.301, -1.215), (0.170, 0.549), and (1.131, -1.215)<br />
are saddle points. There is no highest or lowest point on the graph.<br />
3<br />
z<br />
29. Since f is a polynomial it is contint,Jous on D, so an absolute maximum and minimum exist. Here f x = 2x- 2, jy = 2y, and<br />
setting fx = jy = 0 gives (1, 0) as the only critical point (which is inside D), where f (1, 0) = - 1. Along £ 1 : x = 0 and<br />
f(O, y) = y 2 for -2 ~ y ::=:; 2, a quadratic function which attains its minimum at y = b, where f(O, 0) = 0, and its maximum<br />
·aty = ±2, wheref(0,±2) = 4. Along £2: y = x- 2 forO ::=:; x ~ 2, and f(x,x- 2) = 2x 2 - 6x + 4 = 2 (x _ ~) 2 _ ~ .<br />
a quadratic which attains its minimum at x = £, where f ( ~, - ~) = - ~, and its maximum at x = 0, where f ( 0, - 2) = 4.<br />
Along £3: y = 2 - x for 0 ~ x ::::; 2, and<br />
f(x, 2- x) = 2x 2 - 6x + 4 = 2(x- ~) 2 - ~.a quadratic which attains<br />
its minimum at x =~~where f(~, ~) =-~,and its maximum at x = 0,<br />
where f (O, 2) = 4. Thus the absolute maximum off on D is f(O, ±2) = 4<br />
and the absolute minimum is !(1, 0) = -1.<br />
y<br />
(0,2)<br />
L,<br />
(0, - 2)<br />
(2,0)<br />
X<br />
31. f x(x,y) = 2x +2xy, jy(x,y) = 2y +x 2 , and setting f.,= f" = 0<br />
gives (0, 0) as the only critical point in D , with .f(O, 0) = 4.<br />
On £ 1: y = -1, j(x, - 1) = 5, a constant.<br />
On £2: x = 1, /(1, y) = y 2 + 11 + 5, a quadratic in y which attains its<br />
maximum at (1, 1), /(1, 1) = 7 and its minimum at (1, - ~) ,<br />
f(1, - t) =.!f.<br />
On £ 3: f(x, 1) = 2x 2 + 5 which attains its maximum at (-1, 1) and (1, 1)<br />
with f(±l , 1) = 7 and its minimum at (0, 1), f(O, 1) = 5.<br />
(-1, 1)<br />
L•<br />
(-1,-1)<br />
y<br />
L, (I, I)<br />
L2<br />
0<br />
L, (1,-1)<br />
X<br />
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226 D CHAPTER 14 PARTIAL DERIVATIVES<br />
On .i4 : f( -1, y) = y 2 + y + 5 with maximum at ( -1, 1 ), f( - 1, 1) = 7 and minimum at ( -1, -D. f( - 1, -~) = Jf.<br />
Thus the absolute maximum is attain<strong>ed</strong> at both (±1, 1) with /(±1, 1) = 7 and the absolute minimum on Dis attain<strong>ed</strong> at<br />
(0, 0) with f(O, 0) = 4.<br />
33. f(x, y) = x 4 +y 4 - 4xy + 2 is a polynomial and hence continuous on D, so Y<br />
it has an absolute maximum and minimum on D. fx(x, y) = 4x 3 -<br />
4y and<br />
(0, 2)+---L--'3'-------1(3, 2) .<br />
fv(x, y) = ·4y 3 -<br />
4x; then f x = 0 implies y = x 3 , and substitution into<br />
fv = 0 => x = y 3 gives x 9 - x = 0 => x(x 8 - 1) = 0 => x = 0<br />
or x = ±1. Thus the critical points are -(0, 0), (1, 1), and ( - 1, - 1), but only<br />
(0,0) (3,0) X<br />
(1, 1) with f(1, 1) = 0 is inside D . On £1: y = 0, f(x, 0) = x 4 + 2,<br />
0 ~ x ~ 3, a polynomial in x which attains its maximum at x = 3, !(3, 0) = 83, and its minimum at x = 0, f (O, 0) = 2.<br />
On £ 2 : x = 3, /(3, y) = y 4 - 12y + 83, 0 ~ y ~ 2, a polynomial in y which attains its minimum at y = {/3,<br />
/(3, {13) = 83- 9 .{/3 ~ 70.0, and its maximum at y = 0, ! (3, 0) = 83.<br />
On L 3 : y = 2, f(x, 2) = x 4 -<br />
Bx + 18, 0 ~ x ~ 3, a polynomial in x which attains its minimum at x = q'2,<br />
f( q'2, 2) = is- 6 ij2 ~ 10.4, and its maximum at x = 3, /(3, 2) = 75. On £ 4: x = 0, f(O, y) = y 4 + 2, 0 ~ y ~ 2, a<br />
polynomial in y which attains its maximum at y = 2, f(O , 2) = 18, and its minimum at y = 0, f(O, 0) = 2. Thus the absolute<br />
maximum off on Dis /(3, 0) = 83 and the absolute minimum is /(1, 1) = 0.<br />
35. f,(x, y) = 6x 2 and fv(x, y) = 4y 3 • And so f x = 0 and fv = 0 only occur when x = y = 0. Hence, the only critical point<br />
.inside the disk is at x = y :::o 0 where /(0, 0) = 0. Now on the circle x 2 + y 2 = 1, y 2 = 1 - x 2 so let<br />
g(x) = f(x, y) =·2x 3 + (1- x 2 ) 2 = .x 4 + 2x 3 - 2x 2 + 1,-1 ~ x ~ 1. Then g'(x) = 4x 3 + 6x 2 - 4~ = 0 => x = 0,<br />
-2, or ~ - /(0,±1) = g(~) = 1, !(~, ±1 ) = g(~) = ~· and ,(-2,-3) is not in D. Checking the endpoints, we get<br />
.f( -1, 0) = g( -1) = -2 and f(1 , 0) = g(1) = 2. Thus the absolute maximum and minimum off on Dare /(1, 0) = 2 and<br />
f(- 1,0) = - 2.<br />
Another method: On the boundary x 2 + y 2 = 1 we can write x = cos (;I, y = sin 8, so f (cos 8, sin 8) = 2 cos 3 8 + sin 4 (;I,<br />
0 ~ () ~ 27T.<br />
37. f(x, y) = - (x 2 - 1) 2 - (x 2 y- x - 1) 2 => f,(x, y) = - 2(x 2 - 1)(2x) - 2(x 2 y- x - 1)(2xy- 1) and<br />
jy(X, y) = -2(x 2 y- x - 1)x 2 . Setting j y(x, y) = 0 gives either x = 0 or x 2 y - x- 1 = 0.<br />
There are no critical points for x = O,.since f:.,(O, y) = .:_2, so we set x 2 y - x- 1 = 0 {:} y = x ~ 1 [x =/= 0],<br />
X<br />
so f, ( x, x~ 1 ) = ·-2(x 2 -<br />
1){2x)- 2(x 2 x ~ 1 - x - 1) (2x x ~ 1 - 1) = -4x(x 2 - 1). Therefore<br />
f,(x, y) = fv(x, y)- = 0 at the points (1, 2) and ( -1, 0). To classifY these critical points, we calculate<br />
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SECTION 14.7 MAXIMUM AND MINIMUM VAL4ES 0 227<br />
f :u:(x, y) = - 12x 2 -<br />
12x 2 y 2 + 12xy + 4y + 2, fvv(x, y) = - 2x",<br />
and f zv(x, y) = -8x 3 y + 6x 2 + 4x. In order to use the Second Derivatives<br />
Test we calculate<br />
- I<br />
Of-----1~<br />
D(-1,0) = fxx(-1, 0) / 7171 (-1,0) - [f:c 71 (- 1 , 0)] ~ = 16 > 0, -2<br />
/ :r:c ( - 1, 0) = -10 < 0, D (1, 2) = 16 > 0, and f:r:c{1, 2) = -26 < 0, so<br />
both ( -1, 0) and (1, 2) give local maxima.<br />
X<br />
39. Let d be the distance fr~m {2, 0, -3) to any point (x, y, z ) on the plane x + y + z = 1, sod= J(x - 2) 2 + y 2 + (z + 3)2<br />
where z = 1-x- y, and we minimize d 2 ~ f(x, y) = (x - 2) 2 + y 2 + {4 -<br />
x- y) 2 • Then<br />
j;,(x,y) = 2(x- 2) + 2{4- x - y)( -1) = 4x + 2y -12, f 71 (x,y) = 2y + 2(4 - x- y){-1) = 2x + 4y - 8. Solving<br />
4x + 2y- 12 = 0 and 2x + 4y- 8 = 0 simultaneously gives x = ~. y = ~.so the only critical point is(~,~). An absolute<br />
minimum exists (since there is a minimum di.stance from the point to the plane) and it must occur at a critical point, so the<br />
shortest distance occurs for x = ~. y = ~ for which d = J (~ - 2) 2 + (j) 2 + (4 - ~ - ~) 2 = ji = ~-<br />
41. Let d be the distance from the point {4, 2, 0) to any point {x, y, z) on the cone, sod = ,j(x - 4)2 + (y- 2)2 + z2 where<br />
z 2 = x 2 + y 2 , and we minimize d 2 = (x - 4) 2 + (y - 2) 2 + x 2 + y 2 = f(x, y). Then<br />
fz(x, y) = 2 (x - 4) + 2x = 4x - 8, f 71 (x, y) = 2 (y - 2) + 2y = 4y- 4, and the critical points occur when<br />
f., = 0 => x = 2, fu = 0 => y = 1. Thus the only critical point is (2, 1). An absolute minimum exists (since there is a<br />
minimum distance from the cone to the point) which must occur at a critical point, so the points on the cone closest<br />
to ( 4, 2, 0) are (2, 1, ±v'5).<br />
43. x + y + z = 100, so maximize f(x, y) = xy(l OO - x- y). f x = 100y- 2xy- y 2 , f 71 = 100x - x 2 - 2xy,<br />
f,:r. = - 2y, f vu = -2x, fxv = 100-2x- 2y. Then J,r; = 0 implies y = 0 or y = 100- 2x. Substituting y = 0 into<br />
fv = 0 gives x = 0 or x = 100 and substi~ting y = 100 - 2x into fv = 0 gives 3x 2 - 100x = 0 sox = 0 or 1 ~ .<br />
Thus the critical points are (0, 0), (100, 0), (0, 100) and ( 1 ~ 0 ,<br />
D(O, 0) = D{100, 0) = D {O, 100) = -10,000 while D(1~ 0 ,<br />
1 ~ 0 ) = 10·~ 00 and f xx (1~ 0 ,<br />
1 ~ 0 ) = - 2 ~ < 0. Thus (0, 0),<br />
(100, 0) and {0, 100) are saddle points whereas f( 1 ~ 0 ,<br />
1 ~ 0 ) is a local maximum. Thus the numbers are x = y = z = 1 ~ 0 .<br />
45. Center the sphere at the origin so that its equation is x 2 + y 2 + z 2 = 1·2 , and orient the inscrib<strong>ed</strong> rectangular box so that its<br />
<strong>ed</strong>ges are parallel to the coordinate axes. Any vertex of the box satisfies x 2 + y 2 + z 2 = r 2 , so take (x, y, z) to be the vertex<br />
1<br />
~ 0 ) .<br />
in the fi rst octant. Then the box has length 2x, width 2y, and height 2z = 2 ,jr 2 - x 2 -<br />
y 2 with <strong>vol</strong>ume given by<br />
V(x, y) = (2x)(2y)(2 Jr 2 - x 2 - y 2 ) = 8xy Jr 2 - x 2 - y 2 forO < x < r, 0 < y < r. Then<br />
8y(r 2 - 2x 2 - y 2 ) 8x(r 2 - x 2 - 2y 2 )<br />
,j r2 _ x2 _ y2<br />
. J r2 _ x2 _ y2<br />
V., = (Bxy) . ~(r2- x2- y2)-1f2( -2x) + ,jr2 - x2 - y2. By= . ·and Vv = .<br />
Setting Vx = 0 gives y = 0 or 2x 2 + y 2 = r 2 , but y > 0 so only the latter solution applies. Similarly, V 71 = 0 with x > 0<br />
© 201 2 Ccngoge lcruniny. All Rights Rcscn"Cd. May nol be scwliiCd. copi<strong>ed</strong>. or duplical<strong>ed</strong>. or post<strong>ed</strong> loa publicly accessible website, in \\bole or in pn.
228 0 CHAPTER 14 PARTIAL DERIVATIVES<br />
implies x 2 + 2y 2 = r 2 . Substituting, we have 2x 2 + y 2 = x 2 + 2y 2 ::::> x 2 = y 2 ::::> y = x. Then x 2 + 2y 2 = r 2 '*<br />
3x 2 = r 2 ::::> x = .jr"i]3 = r I vf3 = y. Thus the oni,Y critical point is (r I vf3, r I ,/3). There must be a maximum<br />
<strong>vol</strong>ume and here it must occur at a critical point, so the maximum <strong>vol</strong>ume occurs when x = y = r 1 V3 and the maximum<br />
.47. Maximize f(x, y) = x; (6-x- 2y), then the maximum <strong>vol</strong>ume is V = xyz.<br />
f., = i(6y ~ 2xy- y 2 ) = h(6- 2x- 2y) and fv = ~x (6 - x - 4y). Setting fx =:= 0 and j 11 = 0 gives the critical point<br />
(2, 1} which geometrically must give a maximum. Thus the <strong>vol</strong>ume of the largest such box is V = (2)(1) ( ~) = t·<br />
49. Let the dimensions be x, y, and z; then 4x + 4y + 4z = c and the <strong>vol</strong>ume is<br />
V = xyz = xv(ic- x - y) = ic:cy- x 2 y - xy 2 , x > 0, y > 0. Then V, = ~cy- 2xy- y 2 and Vy = ~ex- x 2 -<br />
2xy,<br />
so Vx = 0 = Vy when 2x + y = ic and x + 2y =~c. Solving, we get x =f-ie, y = 1<br />
12<br />
c and z = ic- x- y = f-ie. From<br />
the geometrical nature of the ·problem, this critical point must give an absolute maximum. Thus the box is a cube with <strong>ed</strong>ge<br />
length -f2c.<br />
51. Let the dimensions be x, y and z, then minimize xy + 2(xz + yz) if xyz = 32,000 cm 3 • Then<br />
f(x,y) = xy + [64,000(x + y)fxy] = xy + 64,000(x- 1 + y- 1 ), fx = y- 64,000x- 2 , fv = x- 64,000y- 2 .<br />
And fx = 0 implies y = 64,000/x 2 ; substituting into /y = 0 implies x 3 = 64,000 or x = 40 and then y = 40. Now<br />
D(x, y) = [(2)(64,000Wx - 3 y- 3 -<br />
dimensions of the box are x = y = 40 em, z = 20 em.<br />
1 > 0 for ( 40, 40) and 'f.,, ( 40, 40) > 0 so this is inde<strong>ed</strong> a minimum. Thus the<br />
53. Let x, y, z be the dimensions of the rectangular box. Then the <strong>vol</strong>ume of the box is xyz and'<br />
L = J xz + y2 + z2 ::::> L 2 = x 2 + y 2 + z 2 ::::> z = J L 2 - x 2 - y 2 .<br />
Substituting, we have <strong>vol</strong>ume V(x,y) = xy JL2- x 2 - y 2 (x,y > 0).<br />
2<br />
Vy = xJL2 -x 2 -y 2 - xy . V., =Oimpliesy(L 2 - x 2 -y 2 )=x 2 y<br />
JP - x2 - y2<br />
::::> y(L 2 -2x 2 -y 2 ) =0 ::::><br />
2x 2 + y 2 = L 2 (since y > 0), and v; 1<br />
= o ·implies x(L 2 - x 2 - y 2 ) = xy 2 ::::> x(L 2 - x 2 - 2y 2 ) = 0 ::::><br />
x 2 + 2y 2 = L 2 (since x > 0). Substituting y 2 = L 2 - 2x 2 into x 2 + 2y 2 = L 2 gives x 2 + 2L 2 - 4x 2 = L 2 ::::><br />
3x 2 = L 2 ::::> x = L/vf3 (since x > 0) f!nd then y = J L2- 2(L/vf3) 2 = L/vf3.<br />
So the only critical point is (Lf,/3, Llvf3 ) which, from th~ geometrical nature of the problem, must give an absolute<br />
maximum. Thus th~ maximum <strong>vol</strong>ume is V(LI,/3, Ll,/3) = (LI,/3) 2 j L2- (Livf3) 2 - (LI ,/3) 2 = L 3 I (3 ,/3)<br />
cubic units.<br />
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. n<br />
SECTION 14.8 LAGRANGE MULTIPLIERS 0 229<br />
55. Note that here the variables are m and b, and f{m, b) = :L: [y,- (mxi +b)] . Then /m. = :L: - 2xi [Vi - (mxi + b)J = 0<br />
. i =l i=l<br />
n. · n n n · n<br />
implies :L: (x•Y•- mx; - bxi) = 0 or .:L: XiYi = m .:L: x~ + b .:L: Xi and fb = :L: -2[yi- (mxi +b)] = 0 implies<br />
i= l t. = l \=1 •=1 1.=1<br />
n n n (n )<br />
.~ Yi = m i~l Xi + i~l b = m i~t Xi + nb. Thus we have the two desir<strong>ed</strong> equations.<br />
1<br />
n n n .<br />
Now fmm = :L: 2x~, fbb = :L: 2 = 2n and fmb = :L: 2Xi. And fmm{m, b) > 0 always and<br />
i =l i = l i= l<br />
2<br />
n<br />
equations do inde<strong>ed</strong> minimize f= d;.<br />
i = 1<br />
14.8 Lagrange Multipliers<br />
1. At the extreme val';les off, the level curves off just touch the curve g(x, y) = 8 with a common tangent line. (See Figure I<br />
and the accompanying discussion.) We can observe several such occurrenc~ on the contour map, but the level curve<br />
f(x, y) = c with the largest value of c which still intersects the curve g(x, y) = 8 is approximately c =59, and the smallest<br />
value of c corresponding to a level curve which intersects g(x, y) = 8 appears to be c = 30. Thus we estimate the maximum<br />
value off subject to the constraint g(xj y) = 8 to be about 59 and the minimum to be 30.<br />
3. f(x, y) = x 2 + y 2 , g(x, y) = xy = 1, and \1 f = >.. \lg => (2x, 2y) = (>..y, >..x), so 2x = >..y, 2y = >..x, and xy = 1.<br />
From the last equation, x =I= 0 andy =I= 0, so 2x = >.y => >.. = 2xjy. Substituting, we have 2y = (2xjy) x =><br />
y 2 = x 2 => y = ±x. But xy = 1, so X= y = ± 1 and the possible points for the extreme values off are {1, 1) and<br />
( -1, - 1). Here there is no maximum value, since the constraint xy = 1 allows x or y to become arbil:(arily large, and hence<br />
f(x, y) = x 2 + y 2 can be made arbitrarily large. The mi~imum value is !{1, 1) = f( - 1, -1) = 2.<br />
5. f(x, y) = y 2 - x 2 , g(x, y) = tx +<br />
2 y 2 = 1, and \1 f = >.\lg =><br />
{-2x, 2y) = (t>..x, 2>..y), so -2x = t>.x, 2y = 2>..y,<br />
and ix 2 + y 2 = 1. From the first equation we have x{ 4 + >..) = 0 => x = 0 or>. = - 4. If x = 0 then the third equation<br />
gives y = ±1. If), = - 4 then the second equation gives 2y = -8y => y = 0, and substituting into the third equation,<br />
we have x = ±2. Thus the possible extreme values off occur at the points (0, ± 1) and (± 2, 0). Evaluating fat these points,<br />
we see that the maximum value is f{O, ± 1) = 1 and the minimum is !{±2, 0) = -4.<br />
7. f(x, y, z) = 2x + 2y + z, g(x, y, z) = x 2 + y 2 + z 2 = 9, and \1 f = >.. \1 g => (2, 2, 1) = (2>..x, 2>..y, 2>.z), so 2>.x = 2,<br />
2>.y = 2, 2)..z = 1, and x 2 + y 2 + z 2 = 9. The first three equations imply x = l• y = ·l· and z = 2<br />
\ . ~ut substitution into<br />
2 2<br />
the fourth equation gives (l)2<br />
+ ( l) + ( \ ) = 9 2<br />
=><br />
9<br />
-<br />
4)..<br />
2 = 9 => ).. = ±t, so f has possible extreme values at<br />
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230 0 CHAPTER 14 PARTIAL DERIVATIVES<br />
the points (2, 2, 1) ~nd ( - 2, - 2, - 1). The maximum value off on x 2 + y 2 + z 2 = 9 is f(2, 2, 1) = 9, and the minimum is<br />
J( -2, -2, - 1) = - 9.<br />
9. f(x, y, z) = xyz, g(x, y, z) = x 2 + 2y 2 '+ 3z 2 = 6. V f = >. Vg =:. (yz, xz, x y) = .>. {2x, 4y, Gz). If any ofx, y, or z is<br />
zero then x = y = z = 0 which contradicts x 2 + 2y 2 + 3z 2 = 6. Then.>.= (yz) /(,2x) = '(xz)/ (4y) = (xy)/(6z) or<br />
x 2 = 2y 2 and z 2 = ~y 2 .<br />
Thus x 2 + 2y 2 + 3z 2 = 6 implies 6y 2 = 6 or y = ±1. Then the possible points are<br />
( .,/2, ±1, /j), (.,/2, ±1, -/j), (-.,/2, ± 1, /j), (- .,/2,±1,-/j). The maximum valu~ off on the ellipsoid is<br />
-J:i, occurring when all coordinates are positive or exactly two are negative and the minimum is -7a occurring when 1 or 3 of<br />
the coordinates are negative.<br />
11. f(x,y,z) = x 2 +y 2 +z 2 , g(x,y, z) = x~ +y 4 + z 4 = 1 =? Vf = (2x,2y,2z)~ >.Vg = (4>.x 3 , 4>.y 3 ,4>.z 3 ).<br />
Case 1: lf x f. 0, y f. 0 and z f. 0, then V f = >. Vg implies.>. = l /(2x 2 ) = 1/(2y 2 ) = l /(2z 2 ) or x 2 = y 2 = z 2 and<br />
giving the points (±-L -L -L) (±-L _ _!_ -L) (±-L -L _ _!_) (±-L _ _!_ _ _!_)<br />
3x 4 = 1 or x = ± - 1 -<br />
V3 VJ' VS' V3 ' · VJ' VS' V3 ' VS' VJ ' V3 ' VJ' W' V3<br />
all with an !-value of v'3.<br />
Case 2: If one of the variables equals zero and the other two are not zero, then the squares of the two nonzero coordinates are<br />
equal with common value ~ and corresponding f value of .,/2.<br />
Case 3: If exactly two of the variables are zero, then the third variable has value ± 1 with the corresponding f value ofl. Thus<br />
on x 4 + y' 1 + z 4 = 1, the maximum value off is y'3 and the minimum value is 1.<br />
13. f(x, y, z, t) = X -1- y -1- Z -1- t, g(x, y, z, t) = x 2 -1- y 2 -1- z 2 -1- e = 1 => (1, 1, 1, 1) = (2>.x, 2).y, 2).z, 2>.t), SO<br />
). = 1/(2x) = 1/(2y) = 1/(2z) = 1/(2t) and x = y ~ z = t. But x 2 -1- y 2 -1- z 2 -1- t2 = 1, so the possible points are<br />
(±~ , ±~, ±~, ±~). Thus the maximum value off is f(~, ~~ ~~ t) = 2 and the minimum value is<br />
15. f(x,y, z) = x + 2y, g(x,y,z) = x +y + z = 1, h(x,y, z ) = y 2 + z 2 = 4 =:. Vf = (1,2,0), .>.Vg = (>.,>., >.)<br />
and p.Vh = (0, 2p.y, 2p.z). Then 1 = .>., 2 = .>. + 2p.y and 0 = >. + 2~tz so p.y = ~ = - p.z or y = 1/ (2p.), z = -1/ (2p.).<br />
Thus x + y + z = 1 implies x = 1 and y 2 + z 2 = 4 implies p. =±~.Then the possible points are (1, ±.J2, =FV2)<br />
and the'mfJ.Ximum value is f (1, .J2, -.J2) = 1 + 2 .J2 and the minimum value is f (1, - .,/2, .J2) = 1 - 2 .J2.<br />
17. f(x, y, z ) = yz + xy, g(x, y , z ) = xy = 1, h(x, y, z) = y 2 + z 2 = 1 =:. V f = (y, x + z, y), >.Vg = (.>.y, >.x, 0),<br />
p.Vh = (0, 2p.y, 2p.z). Then y = >.y implies>. = 1 [y f. 0 since g(x, y, z) = 1], x + z = .>.x + 2p.y andy= 2p.z. Thus<br />
p. = z/(2y) = y/(2y) or y 2 = z 2 , and so y 2 + z 2 = 1 implies y = ±72, z = ±72. Then xy = 1 implies x = ±.J2 and<br />
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SECTION 14.8 LAGRANGE MULTIPLIERS 0 231<br />
the possible points are ( ±-/2, ± 7:i, 72), ( ±V-2, ±~, - );) . Hence the maximum off subject to the constraints is<br />
f(±J2,±~,±jz ) = ~ andtheminimumisf(±J2,±~, 'f72) = ~·<br />
Note: Since xy = 1 is one of the constraints we could have solv<strong>ed</strong> the problem by solving f (y, z) = yz + 1 subject to<br />
y2 + z2 = 1.<br />
19. f(x, y) = x 2 + y 2 + 4x - 4y. For the interior of the region, we find the critical points: fx = 2x + 4, f u = 2y - 4, so the<br />
only-critical point is ( - 2, 2) (which is inside the region) and f ( - 2, 2) = -8. For the boundary, we use Lagrange multipliers.<br />
g(x, y) = x 2 + y 2 = 9, so "iJ f = >. "ilg =;. (2x + 4, 2y - 4) = (2>.x, 2>.y). Thus 2x + 4 = 2>.x and 2y - 4 = 2>.y.<br />
Adding the two equations gives 2x + 2y = 2>.x + 2>.y =:. x + y = >.(x + y) =:. (x + y)(>.- 1) = 0, so<br />
x + y = 0 =;. y = -x or>. - 1 = 0 =;. >. = 1. But >. = 1 leads to a contradition in 2x + 4 = 2>.x, soy = - x and<br />
x 2 + y 2 = 9 implies 2y 2 = 9 => y = ±12. We have f ( ~· -72) = 9 + 12-/2 ~ 25.97 and<br />
r( - 12, ~ ) = 9- 12-/2 ~ - 7.97, so the maximum value off on the disk x 2 ~ y 2 :s: 9 is f ( 72· -12) = 9 + 12-/2<br />
and the minimum is f( - 2, 2) = -8.<br />
21. f(x, y) = e- xy. For the interior ofthe region, we find the critical points: f x = - ye-xu, / 11<br />
= -xe-"'Y, so the only<br />
critical point is (0, 0), and f(O, 0) = 1. For the boundary, we use Lagrange multipliers. g(x, y) = x 2 + 4y 2 = 1 =;.<br />
>. "iJ g = (2>.x, 8>.y), so setting "iJ f = >. "iJ g we get -ye-"' 11 = 2>.x and -xe-"' 11 = 8>.y. The first of these gives<br />
e- "' 11 = -2>.xjy, and then .the second gives - x( - 2>.xjy) = 8>.y => x 2 = 4y 2 • Solving this last equation with.the<br />
constraint x 2 + 4y 2 = 1 gives X= ±72 and y=±~ . Now t( ±~, 'f 2~) = e 1 1 4 ~ 1.284 and<br />
f ( ±~ , ±;~ ) = e- 1 / 4 ~ 0.779. The former arc the maxima on the region and the latter are the min.ima.<br />
23. (a) f(x, y) = x , g(x,y) = y 2 + x 4 - x 3 == 0 =;. "ilf = (1, 0) = >.Vg = >.(4x 3 - 3x 2 ,2y). Then<br />
1 = >.( 4x 3 - 3x 2 ) (1) and 0 = 2>.y (2). We have >. i= 0 from (1), so (2) gives y = 0. Then, from the constraint equation,<br />
x 4 - x 3 =. 0 =;. x 3 (3; - 1) = 0 =;. x = 0 or x = 1. But x = 0 contradicts (1), so the only possible extreme value<br />
subject to the constraint is /(1, 0) = 1. (The question remains whether this is inde<strong>ed</strong> the minimum of f.)<br />
(b) The constraint is y 2 + x 4 - x 3 = 0 ¢} y 2 = x 3 - x 4 . The left side is non-negative, so we must have x 3 - x 4 :2: 0<br />
which is true only for 0 :S: x :S: 1. Therefore the minimum possible value for f(x, y) =x is 0 which occurs for x = y = 0.<br />
However,>. "iJ g(O, 0) = >. (0- 0, 0) = (0, 0) and "iJ f(O, 0) = (1, 0), so "iJ f(O, 0) # >. "iJ g(O, 0) for all values of>..<br />
(c) Here "iJ g(O, 0) = 0 but the method of Lagrange multipliers requires that "iJ g # 0 everywhere on the constraint curve.<br />
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232 D CHAPTER 14 PARTIAL DERIVATIVES<br />
25. P(L,K)=bLo.Kl-o., g(L,K)=mL+nK=p '* 'VP=(abLo:- 1 K 1 - o.,(1-a)bL'""JC'""), >..'Vg=(>..m, >..n).<br />
Then ab(K/ L) 1 -o. = >..m and (1- a)b(L/ K)"' =>..nand mL + nK = p, so ab(K/ L?-"'/m = (1 - a)b(L/ K) 0 /n or<br />
naj[m(1-a)]= (L/ K)"'(L/ K) 1 -"'. or L = Kno/[m{1- a)]. Substituting into mL + nK = p gives K = (1- a)pjn<br />
and L = apjm for the maximum production.<br />
27. Let the sides of the rectangle be x andy. Then f(x, y) = xy, g(x, y) = 2x + 2y = p '* 'V f(x, y) = (y, x)·,<br />
>.. 'Vg = (2>.., 2>..). Then A= ~y = ~x i~pli es x = y and the rectangle with maximum area is a square with side length iP·<br />
29. The distance from {2,0, - 3) to a point (x,y,z) on the plane is d = .j(x- 2)2 + y 2 + (z + 3) 2 , so we seek to minimize<br />
d 2 = f(x, y, z) = (x- 2) 2 + y 2 + (z + 3) 2 subject to the constraint that (x, y, z) lies on the plane x + y + z = 1, that is,<br />
that g(x, y, z) = x + y + z = 1. Then 'V f =A 'Vg =* (2(x- 2), 2y, 2(z + 3)) = (A, .A, .A), sox= (A+ 4)/2,<br />
y = >./2, z = (>.. - 6)/2. Substituting into the constraint equation gives .A~ 4 + i + A; 6 = 1<br />
>.. = ~ . sox = ~. y = ~.and z = -i· This must correspond to a minimum, so the shortest distance is<br />
d = J ( ~ - 2) ~ · + ( ~ )2 + (- i + 3) 2 = ..fi = ~ -<br />
'* 3A- 2 = 2 '*<br />
31. Let f(x, y , z) = d 2 = (x- 4) 2 + (y- 2) 2 + z 2 • Then we want to minimize f subject to the constraint<br />
g (x,y,z) = x 2 +y 2 - z 2 = 0. 'Vf = )..'\Jg '* (2 (x --; 4) ,2(y- 2) ,2z) = (2>..x,2>..y, -2-Xz), sox- 4 = J..x,<br />
y - 2 = .Ay, and z = -J..z. From the last equation we have z + >..z = 0 '* z (1 +.A) = 0, so either z = 0 or>..= -1.<br />
But from the constraint equation we have z = 0 '* x 2 + y 2 = 0 '* x = y = 0 which is not possible from the first<br />
two equations. So ).. = - 1 and x - 4 = >..x '* x = 2, y - 2 = >..y '* y = 1, and x·2 + y 2 - z 2 = 0 '*<br />
4 + 1 - z 2 = 0 '* z = ±v'S. This must correspond to a minimum, so the points on the cone closest to ( 4, 2, 0)<br />
are (2, 1, ±v'S).'<br />
33. f(x,y,z) = xyz,g(x,y,z) = x +y + z = 100 =* 'Vf = (yz,xz, xy) = )..'\Jg =(>.., >..,>..). Then>.. = yz = xz = xy<br />
implies x = y = z = 1 ~ 0 .<br />
35. If the dimensions are 2x, 2y, and 2z, then maximize f(x, y, z) = (2x)(2y)(2z) = 8xyz subject to<br />
g(x, y, z) := x 2 + y 2 + z 2 = r 2 (x > 0, y > 0, z > 0). Then 'V f = .A 'V g '* (8yz, Bxz, 8xy) = .A (2x, 2y, 2z) =*<br />
41JZ 4xz 4xy . . 2 2 2 2<br />
8yz = 2-Xx, Bxz = 2>.y, and 8xy = 2.Xz, so>.= - ·- = - = - .This gives x z = y z '* x = y (since z t= D)<br />
X Y Z<br />
'<br />
and xy 2 = xz 2 '* z 2 = y 2 , so x 2 = y 2 = z 2 '* x = y = z, and substituting into the constraint<br />
equation gives 3x 2 = r 2<br />
'* x = r/../3 = y = z. Thus the largest <strong>vol</strong>ume of such a box is<br />
. f ( ~, ~, ~ ) = 8 ( ~) ( 7J) ( ~) = 3<br />
~ r 3 .<br />
'<br />
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37. f(x,y,z) = xyz, g(x,y,z) = x +2y + 3z = 6 => \lf = {yz,xz,xy) = >..Vg = (>.., 2>.,3>.).<br />
SECTION 14.8 LAGRANGE MULTIPLIERS D 233<br />
Then >. = yz = ~xz = ~xy implies x = 2y, z = %Y· But 2y + 2y + 2y = 6 soy = 1, x = 2, z = ~ and the <strong>vol</strong>ume<br />
is V = ~-<br />
39. f(x, y, z) = xyz, g(x, y ,.z) = 4(x + y + z) = c => \7 f = (yz, xz, x y), >.Vg = (4>., 4>., 4>.). Thus<br />
4>. = yz = xz = xy or x = y = z = fie are the dimensions giving the maximum <strong>vol</strong>ume.<br />
41 . If the dimensions of the box are given by x, y, and z, then we ne<strong>ed</strong> to find the maximum value of f(x, y, z) = xyz<br />
[x, y, z > 0) subject to the constraint L = ..jx 2 + y 2 + z 2 or g(x, y, z) = x 2 + y 2 + z 2 = £ 2 . \7 f = ).. \lg =><br />
yz<br />
xz<br />
(yz , xz, xy ) = >.(2x, 2y, 2z), so yz = 2>.x => >. = - , xz = 2>.y => >. = - , and xy = 2.Xz =><br />
2 X 2 Y ·<br />
Thus). = yz = xz .=> x 2 = y 2 [since z =f. 0) => x = y and >.= y 2<br />
z = x 2<br />
y. => x = z [since y =f. 0].<br />
~ ~ X Z<br />
Substituting into the constraint equation gives x 2 + x 2 + x 2 = L 2 => x 2 = £ 2 / 3 · => x = L / V3 = y = z and the<br />
maximum <strong>vol</strong>ume is (L/V3) 3 = £ 3 / (3 V3).<br />
43. We ne<strong>ed</strong> to find the extreme values of f(x, y, z) = x 2 + y 2 + z 2 subject to the two constraints g(x, y , z) = x + y + 2z = 2<br />
and h(x, y, z) = x 2 + y~ - z = 0. \lf = ( ~x , 2y, 2z}, >. \lg = (>., >., 2>.) and ~\l h = (2J.Lx_, 2~y , - ~ ). Thus we ne<strong>ed</strong><br />
2x = ). + 2J.LX (1), 2y = >. + 2~y (2), 2z = 2>.- f.t (3), x + y + 2z = 2 (4), and x 2 + y 2 - z = 0 (5).<br />
From (l) and (2), 2(x - y) = 2 ~ (x - y), so if x =f. y, ~ = 1. Putting this in (3) give,s 2z = 2>.- 1 or >. = z +~. bu t p ~1ttin g<br />
~ = 1 into (l) says>. = 0. Hence z + ~ = 0 or z = -~. Then (4) and (5) become x + y- 3 = 0 and x 2 + y 2 + ~ ~ 0. The ·<br />
last equation cannot be true, so this case gives no solution. So we must have x = y. Then (4) and (5) become 2x + 2z = 2 and<br />
2x 2 -<br />
z = 0 which imply z = 1 - x and z = 2x 2 . Thus 2x 2 = 1 - x or 2x 2 + x - 1 = (2x - 1)(x + 1) = 0 s o x= ~ or<br />
x = - 1. The two points to check are (t, ~ .<br />
~)and ( - 1, - 1, 2): /(~. ~. ~) = ~and f(-1, -1, 2) = 6. Thus(~,~. ~) is<br />
the point on the ellipse nearest the origin and ( - 1, - 1, 2) is the one farthest from the origin.<br />
45. f(x, y , z) = yex-::, g(x, y, z) = 9x 2 + 4y 2 + 36z 2 = 36, h(x, y , z) = xy + yz = 1. \7 f = >.Vg + J-L\lh =><br />
(yex-::, ex - z, - yex- z) = >.(18x, 8y, 72z) + i'(V, x + z, y), so yex-z = 18>.x + J.l.V, ex- z = 8>.y + ~(x + z),<br />
- yex- :: = 72>.z + ~y . 9x 2 + 4y 2 + 36z 2 .= 36, x y + yz = 1. Using a CAS to solve these 5 equations simultaneously for x,<br />
y, z , >. , and ~ (in Maple, use the all values command), we get 4 real-yalu<strong>ed</strong> solutions:<br />
X~ 0.222444, y ~ · -2.157012, z ~ - 0.686049, >. ~ - 0.200401, ~ ~ 2.108584<br />
X~ - 1.951921, y ~ - 0.545867, z ~ 0.119973, ). ~ 0.003141, ~ ~ - 0.076238<br />
X ~ 0.155142, y ~ 0.904622, z ~ 0.950293, >. ~ - 0.012447, f.' ~ 0.489938<br />
X~ 1.138731, y ~ 1.768057, . z ~ -0.573138, >. ~ 0.317141, f.' ~ 1.862675<br />
Substituting these values into f gives /(0.222444, -2.157012, - 0.686049) ~ -5.3506,<br />
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234 0 CHAPTER 14 PARTIAL DERIVATIVES<br />
f( -1.951921, - 0.545867, 0.119973) ~ -0.0688, !(0.155142, 0.904622, 0.950293) ~ 0.4084,<br />
!(1.138731, 1.768057, -0.573138) ~ 9.7938. Thus the maximum is approximately 9.7938, and the minimum is<br />
approximately -5.3506.<br />
47. (a) We wish to maximize j(x1, x2, ... , xn) = ytx1x2 · · · Xn subject to<br />
g(x1,x2, .. . , Xn) = X1 +x2 + · · · +xn = c and X.;> 0.<br />
\1 f = ( ~ (X!X2 · .. Xn).; -l(X2 .. · Xn) , ~(X1X2 .. · Xn)'~-l (X!X3 · .. Xn), ... , ~(X t X2 · .. Xn) -!,-l (xl · .. Xn-1) ~<br />
and >. V g = (>., >., ... , >.),so we ne<strong>ed</strong> to solve the system of equations<br />
1/n 1/n 1/n ,<br />
x 1 x 2 · · · Xn = nAx1<br />
1 ( ).l.-1(<br />
;;: XtX2 · • • Xn " XtX3 • · • Xn) = A<br />
1/n 1/n 1/n \<br />
X 1 X 2 · · · Xn = nAX2<br />
This implies n>.x1 = n>.x2 = · · · = n>.x ... Note >. =I 0, otherwise we can't have all x.; > 0. Thus X1 = x2 -= · · · = Xn.<br />
But X ! + X2 + · · · + Xn = c => nx1 = c => X1 = ~ = X2 = X3 = · · · = Xn - Then the only point where f can<br />
n<br />
have an extreme value is (~ , ~, ... , ~). Since we can choose yalues for ( Xt, x2, ... , Xn) that make f as close to<br />
n n n<br />
zero (but not equal) as we like, f has no minimum value. Thus the maximum value is<br />
t(;,;, .... ;) = \};-; ..... ;=;.<br />
(b) From part (a), ~ i s the maximum value of f. Thus j(x1; x2, . .. , x n ) = ytx1x2 · · · Xn :::; ~-<br />
n<br />
,-,-,---,------~· -- X1 + X2 + ··• + Xn . .<br />
x 1 + x2 + ··· + Xn = c, so ytx1X2 · · · Xn :::;<br />
n<br />
n<br />
But<br />
. These two means are equal when f attams 1ts<br />
maximum value ~' but this can occur only at the point(;,*' . .. , * ) we found in part (a). So the means are equal only<br />
- c<br />
when Xt = X2 = X3 = · · · = Xn = -.<br />
- n<br />
14 Review<br />
CONCEPT CHECK<br />
1. (a) A function f of two variables is a rule that assigns to each order<strong>ed</strong> pair (x, y) of real numbers in its domain a unique real<br />
number denot<strong>ed</strong> by f(x, y).<br />
(b) One way to visualize a function of two variables is by graphing it, resulting in the surface z = f(x, y). Another method for<br />
visualizing a function of two variables is a contour map. The contour map consists of level curves ofthe function which are<br />
horizontal traces of the graph of the function project<strong>ed</strong> onto the xy-plane. Also, we can use an arrow diagram such as<br />
Figure I in Section 14.1.<br />
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CHAPTER 14 REVIEW 0 235<br />
2. A function f of three variables is a rule that assigns to each order<strong>ed</strong> triple (x, y, z) in its domain a unique real number<br />
f(x, y , z). We can visualize a function of three variables by examining its level surfaces f(x, y, z) = k, where k is a constant.<br />
3. lim f(x, y) = L means the values of f(x, y) approach the nwnber Las the point (x, y) approaches the point (a, b)<br />
(:z:, 11 )-+{a ,b)<br />
along any path that is within the domain of f. We can show that a limit at a point does not exist by finding two different paths<br />
approaching the point along which f(x, y) has different limits.<br />
4. (a) See Definition 14.2.4.<br />
(b) Iff is continuous on JR 2 , its graph will appear as a surface without holes or breaks.<br />
5. (a) See (2) and (3) in Section 14.3.<br />
(b) See "Interpretations of Partial Derivatives" on page 927 [ET 903].<br />
(c) To find f ., regard 1J as a constant and diffe~entiate<br />
differentiate f(x, y) with respect toy.<br />
6. See the statement of Clairaut's Theorem on page 931 [ET 907].<br />
7. (a) See (2) in Section 14.4.<br />
(b) See (19) and the prec<strong>ed</strong>ing discussion in Section 14.6.<br />
f(x, y) with respect to x. To find / y, regard x as a constant and<br />
8. See (3) and (4) and the accompanying discussion in Section 14.4. We can interpret the linearization off at (a, b) geometrically<br />
as the linear function whose graph is the tangent plane tothe graph off at (a, b) . Thus it is the linear function which best<br />
approximates f near (a, b).<br />
9. (a) See Definition 14.4.7.<br />
(b) Use Theorem 14.4.8.<br />
10. See (10) and the associat<strong>ed</strong> discussion in Section 14.4.<br />
11. See (2) and (3) in Section 14.5.<br />
12. See (7) and the prec<strong>ed</strong>ing discussion in Section 14.5.<br />
13. (a) See Definition 14.6 .2. We can interpret it as the rate of change off at ( Xo, yo) in the direction of u . Geometrically, if P is<br />
the point (xo, yo , f(xo, Yo)) on the graph off and C is the curve of intersection of the graph off with the vertical plane<br />
that passes through P in the direction u , the directional derivative off at (xo, Yo) in the direction ofu is the·slope of the<br />
tangent line to Cat P. (See Figure 5 in Section 14.6.)<br />
(b) See Theorem 14.6.3.<br />
14. (a) See (8) and (13) in Section 14.6.<br />
(b) Du f(x, y) = "\7 f (x, y) · u or Du f(x, y, z) = "\7 f(x, y, z) · u<br />
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236 0 CHAPTER 14 PARTIAL DERIVATIVES<br />
(c) The gradient vector of a function points in the direction of maximum rate of increase of the function. On a graph of the<br />
function, the gradient points in the direction of steepest ascent.<br />
15. (a) f has a local maximum at {a, b) if f(x, y) :5 f(a, b) when {x, y) is near {a, b).<br />
(b) f has an absolute maximum at {a, b) if f(x, y) :5 f(a, b) tor all points (x, y) in the domain of f .<br />
(c) f has a local minimum at (a, b) if f(x, y) ~ f (a, b) when (x, y) is near (a, b).<br />
(d) f has an absolute minimum at (a, b) if f(x, y) ~ f(a, b) for aU points (x, y) in the domain of f.<br />
(e) f has a saddle point at (a, b) if f(a, b) is a local maximum in one direction but a local minimum in another.<br />
16. (a) By Theorem 14.7.2, iff has a local maximum at (a, b) and the first-order partial derivatives of j exist there, then<br />
/ :r(a, b) = 0 and / 11 (a, b)= 0.<br />
(b) A critical point off is a point (a, b) such that [.(a, b) = 0 and j 11 (a, b) = 0 or one of these partial derivatives does<br />
not exist.<br />
17. See (3) in Section 14.7.<br />
18. (a) See Figure II and the accompanying discussion in Section 14.7.<br />
(b) See Theorem 14.7.8.<br />
(c) See the proc<strong>ed</strong>ure outlin<strong>ed</strong> in (9) in Section 14.7.<br />
19. See the discussion beginning on page 981 [ET 957]; see "Two Constraints" on page 985 [ET 961].<br />
TRUE-FALSE QUIZ<br />
1. True. fv(a, b) = lim f(a, b +h) - f(a, b) from Equation 14.3.3. Let h = ·y- b. Ash~ 0, y --+ b. Then by substituting,<br />
h~o h ,<br />
f(a,y)-f(a,b)<br />
we get f 11 ( a, b) = li m b .<br />
u~ b y-<br />
5. False. See Example 14.2.3.<br />
7. Tru~. Iff has a local minimum and f is differentiable at (a, b) then by Theorem 14.7.2, f x(a, b) = 0 and fu (a, b) = 0, so<br />
V f(a,b) = (f,(a,b),fv(a,b)) = (0,0} = 0.<br />
9. False. V f(x, y) = (0, 1/ y).<br />
11. True. V f = (cosx, cosy), so IV / I = v'cos 2 x + cos 2 y . But lcosBI :5 1, so IV !I :5 .J?.. Now<br />
Du f(x, y) = V f · u = IV f l l.u l cos{:}, but u is a unit vector, so IDu f (x, y)l :5 .J2 · 1 · 1 = ./2.<br />
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l<br />
CHAPTER 14 REVIEW 0 237<br />
EXERCISES<br />
1. ln(x + y + 1) is defin<strong>ed</strong> only when x + y + 1 > 0 ~ y > - x - 1,<br />
so the domain off is {(x, y) I y > - x - 1}, all those points above the<br />
line y = -x - 1.<br />
',<br />
' ' ' ' ' ' '<br />
y<br />
3. z = f(x, y) = 1 - y 2 , a parabolic cylinder 5. The level curves are J 4x2 + y2 = k or 4x 2 + y 2 = k 2 ,<br />
k ::2::: 0, a family of e ll ipses.<br />
y<br />
X<br />
7. y 9. f is a rational function, so it is continuous on its domain.<br />
2<br />
I<br />
(©J (©J<br />
(©J (©J<br />
0 I 2 X<br />
Since f is defin<strong>ed</strong> at (1, 1), we use direct substitution to<br />
1 th I" . lim 2x y 2(1)(1) 2<br />
eva uate e 1m1t:<br />
2<br />
2 2 = 12<br />
+ ( ) 2 1 2 = - .<br />
(z,v)-(1 ,1) X + y<br />
3<br />
11 . (a) T.,(6 , 4) = lim T(B + h , 4 ~ - T (B, 4 ), so we can approximateT:c (6, 4) by considering h = ± 2 and<br />
h- o<br />
using the values given in the table: Tx(6, 4) ~ T( 8 , 4 ) ; T ( 6 , 4 ) = 86 ;<br />
80<br />
= 3,<br />
. T.,(6, 4) ~ T( 4 , 4 ) ~ 2 T(B, 4 ) = 72 ~ 2 80 = 4. Averaging these values, we estimate Tx(6, 4) to ~e approximately<br />
3.'5°C/ m . Similarly, Tv (6, 4} = lim T(B, 4 + h~ - T ( 6 , 4 ) , which we can approximate with h = ±2:<br />
h - 0<br />
,.., (B ) _ T(6, 6} - T(6, 4) _ 75 80 _ _ ,.,., ( ) _ T(6, 2) - T(6, 4) _ 87- 80 _<br />
4 2 5 6 4 3 5 A . th<br />
.Lv ' - 2 - - - 2<br />
- - · ' .L v , - _ 2<br />
- ~- - .. veragmg ese<br />
values, we estimate Tv(6, 4) to be approximately - 3.0°C/ m.<br />
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238 0 CHAPTER 14 PARTIAL DERIVATIVES<br />
(b) Here u = ( 12• )?.). so by Equation 14.6.9, Du T(6,4) = VT(6,4) · u = T, (6,4) ~ + Tv(6,4) ~·Using our<br />
estimates from part (a), we have Du T(6, 4) l':;j (3.5) ~ + ( -3.0) ~ = ~ l':;j 0.35. This means that as we move<br />
through the point (6, 4) in the direction ofu , the temperature increases at a rate of approximately 0.35°C/m.<br />
. r(6+h~,4+h~)-T(6,4 )<br />
Alternatively, we can use Definition 14.6.2: Du T( 6, 4) = lim ,<br />
h-o<br />
h<br />
which ·we can estimate with h = ± 2 ,;2. Then Du T(6, 4) ~ T( 8 , 6 ) ~T( 6 ,<br />
4 ) =<br />
80 - ~O = 0,<br />
2 2 2v2<br />
D T(6 4)<br />
T(4, 2) - T(6,4) 74-80 3 . ( ) 3 0 I<br />
- 2v2 -2v2 v2<br />
u , ~ tn = ~ = tn' Averagmg these values, we have D u T 6, 4 ~ 27z ~ 1.1 C m.<br />
() T. ( ) _ !_ [T. ( )] - lim T,('x, y +h) - T:r.(x,y) T. ( 6 4 ) _lim T., (6,4 +h) - T.,(6, 4) 1 . h<br />
C :V!/ X 1 y - a x X 1 y - h , SO xy 1 - h W liC We Can<br />
. y h-o , h-o<br />
estimate with h = ±2. We have Tx (6, 4) ~ 3.5 from part (a), but we will also ne<strong>ed</strong> values for T.,(6, 6) and T,(6, 2). If we<br />
use h = ±2 and the values given in the table, we have<br />
T,( 6<br />
, 6<br />
) l':;j T(8, 6) - T(6, 6) = 80 - 75 = 2<br />
. 5<br />
, T, ( 6 , 6<br />
) ~ T(4, 6)- T (6, 6) = 68 - 75 = 3<br />
. 5<br />
.<br />
2 2 -2 -2<br />
Averaging these values, we estimate T.,(6, 6) ~ 3.0. Similarly,<br />
Tx( 6 , 2<br />
) l':;j T(8, 2) ~ Tx(6, 2) = 90 ; 87 = l.S, T.,( 6 , 2 ) ~ T(4, 2) ~<br />
2 T(6 , 2) = 74 ~ 87 = 6 . 5 .<br />
Averaging these values, we estimate T, (6, 2) ~ 4.0. Finally, we esti~ate T., 11 (6, 4):<br />
T. (6 4) ~ T.,(6, 6)- T.,(6, 4) _ 3.0 - 3.5 _ _ 0 25 T. ( 6 4 ) ~ T.,(6, 2)- T.,(6, 4) _ 4.0 - 3.5 __ 0 25<br />
:vy ! ~ 2 - 2 - • > X)J 1 ~ -2 - -2 - • •<br />
Averaging these values, we have Txy (6, 4) ~ -0.25.<br />
13. f(x, y) = (5y 3 + 2x 2 y) 8 => f.,= 8(5y 3 + 2x 2·yf(4xy) = 32xy(5y 3 + 2x 2 y) 7 ,<br />
fv = 8(5y 3 + 2x 2 yf (15y 2 + 2x 2 ) = (16x 2 + 120y 2 )(5y 3 + 2x 2 y) 7<br />
17. S(u,v,w) = uarctan(vyfw)<br />
S w = u · 1<br />
+ (:JW) 2 (v · ~ w - 1 1 2 )<br />
1 ufo<br />
S,. = arctan(vy'W), Su = u · 1<br />
+ (vfo)2 (fo) = 1 +v2w'<br />
= 2Vw (~: v2w)<br />
19. f(x,y) = 4x 3 - x y 2 => f x = 12x 2 - y 2 , f v = -2xy, f.,., = 24x, fv 11 = - 2_:~: , fxv = fvx = -2y<br />
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CHAPTER 14 REVIEW 0 239<br />
21. f(x, y , z) = xky 1 z'" => f x = kxk-lylzm., ! 11 = lxky 1 - 1 z m, fz = mxky 1 z""'- 1 , f:"x = k(k- 1)xk- 2 y 1 z"",<br />
fvv = l(l- 1)xky 1 - 2 zm, f zz = m(m -1)xky 1 z"'- 2 , f xv = fvcx: = klx k- 1 y 1 - 1 z"', ! ex::= f ::;r; = kmx"- 1 y 1 zm-t,<br />
fvz = f ::v = lmxkyl-1 zm-1<br />
23. z = x y + xe 11 1"' =><br />
25. (a) zx = 6x + 2 => zx (l ,-2) = 8 and z 11 = - 2y => zv(1, -2) = 4, so an equation of the tangent plane is<br />
z - 1 = 8(x - 1) + 4(y + 2) or z = 8x + 4y + 1.<br />
(b) A normal vector to the tangent plane (imd the surface) at (1, - 2, 1) is (8, 4, - 1). Then parametric equations for the normal<br />
·<br />
· d . . x-1 y + 2 z- 1<br />
I me there are x = 1 + 8t, y = - 2 + 4 t, z = 1 - t, an symmetnc equations are - - = - - = 8 4<br />
--=t·<br />
27. (a) Let F(x , y ,z) = x 2 + 2y 2 - 3z 2 • Then F.,= 2x, Fy = 4y, Fz = - 6z, so F,.,(2, - 1, 1) = 4, F 11 (2, - 1, 1) = - 4,<br />
F: (2, - 1, 1) = - 6. From Equation 14.6.1 9, an equation ofthetangent planeis 4(x - 2)- 4(y + 1) - 6(z - 1) = 0<br />
or, equivalently, 2x - 2y - 3z = 3.<br />
· 14620 · · fi h II' X- 2 Y + 1 z- 1<br />
(b) From Equatwns . . , symmetnc equatwns or t e norma 1 ~e are - 4<br />
- = --=4 = --=fl·<br />
29. (a) Let F (x, y , z ) = x + 2y + 3z- sin(xyz ) .. Then F,., = 1-yz cos(xyz ), Fv = 2 - xz cos(xyz), F: = 3 - x y cos(xyz),<br />
so F :z: (2, -1, 0) = 1, Fy{2, - 1, 0) = 2,·F: (2, - 1, 0) = 5. From Equation 14.6. 19, an equation of the tangent plane is<br />
1(x - 2) + 2(y + 1) + 5(z - 0) = 0 or x + 2y + 5z = 0.<br />
' . 14 6 20 . .
240 0 CHAPTER 14 PARTIAL DERIVATIVES<br />
linear approximation off at (2, 3, 4) is<br />
f(x;y,z) ~ /(2,3,4) + J.,(2,3,4)(x ~ 2) + fv(2,3,4)(y - 3) + /z(2,3, 4)(z - 4)<br />
= 40 +60(x- 2) + ¥CY - 3) + 3 2 5<br />
(z- 4) = 60x+ 2 4<br />
y+ 3 2<br />
z - 120<br />
5 5<br />
Then (1.98) 3 J(3.01) 2 + (3.97) 2 = /(1.98,3.01, 3.97) ~ 60(1.98) + ¥(3.01) + ¥(3.97) -120 = 38.656.<br />
du au dx au dy au dz 3 2 2 3<br />
35. -d = ,---d +-;;--d +-;;--d =2xy (1+6p) + 3x y (per>+er>)+4z (pcosp+sinp)<br />
p uX p uy p u z p<br />
. az az ax az ay<br />
37. By the Cham Rule, as = ax as + ay as. When s = 1 and t = 2, X = g(l, 2) =;:: 3 andy = h(1, 2) = 6, so<br />
& . . & &~ &~<br />
as = f,(3, 6)gs(1, 2) + fv (3, 6) h. (1, 2) = (7)( - 1) + (8)( -5) = - 47. Stmiiarly, Bt = ax 8t + ay at, so<br />
az<br />
at = f,(3, 6)gt(1, 2) + fv (3, 6) ht(1, 2) = (7)(4) + (8)(10) = 108.<br />
az !'( 2 2) az !'( 2 2) [ h !' df ]<br />
39. ax = 2x X - y ' ay = 1 - 2y X - y w ere = d( x2 - y2) . Then<br />
az az 2 2 I 2 2<br />
Y Bx + x {)y = 2xyf'(x - y ) + x - 2xyf (x - y ) = x.<br />
az az az - y<br />
41. -a = -a v+ -a 2 and<br />
X U V X<br />
a 2 z a (az). 2y az - y a (az) 2y az (a 2 z 8 2 z - y) - y (a 2 z - y a 2 z )<br />
ax2 = Yax au +xa8v + ~ax av =xaav+Y au2 ~ + 8v8u~ +~ 8v2 ~+auavY<br />
2y az 2 8 2 z 2y 2 a 2 z y 2 8 2 z<br />
= x3 8v + Y &u2 - X2 &uav + x 4 av2<br />
Thus<br />
. uv 2<br />
smcey = xv = - ory = uv.<br />
y<br />
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CHAPTER 14 REVIEW D 241<br />
45. j(x, y) = x 2 e- 11 => 'V f = (2xe- 11 , - x 2 e- 11 ), 'V f( -2, 0) = (-4, -4). The direction is given by (4, - 3), so<br />
u =<br />
1<br />
yf42+(-3)2 ... ...<br />
(4,-3} = { (4,-3) and Duf(-2,0} = 'Vf{-2,0} · u = (-4, -4) · t (4, -3) = t(-16 + 12} = -~.<br />
47. 'V f = (2xy,x 2 + 1/(2JY)), I'Vf(2, 1)1 = 1(4, ~) I · Thus the maximum rate of change off at (2, 1} is o/ in the<br />
direction ( 4·, !).<br />
49. First we draw a line passing through Homestea'd and the eye of the hurricane. We can approximate the directional derivative at<br />
Homestead in the direction of the eye of the hurricane by the average rate of change of wind spe<strong>ed</strong> between the points where<br />
this line intersects the contour lines closest to Homestead. In the direction of the eye of the hurricane, the wind spe<strong>ed</strong> changes<br />
from 45 to 50 knots. We estimate the distance between these two points to be approximately 8 miles, so the rate of change of<br />
wind spe<strong>ed</strong> in the direction given is approximately 50 8 45 = i = 0.625 knot/ mi.<br />
51 . f(x,y)=x 2 - xy+y 2 + 9x -6y+10 => f, = ·2x-y+9,<br />
j 11 = - x + 2y- 6, f xx = 2 = / 1111 ,<br />
f xv = - 1. Then f x = 0 and ! 11 = 0 imply<br />
y = 1, x = -4. Thus the only critical point is ( - 4, 1} and fx:r:( -4, 1) > 0,<br />
D(- 4, 1) = 3 > 0, so f( - 4, 1) = -11 is a local minimum.<br />
53. f(x, y) = 3xy - x 2 y- xy 2 => fx = 3y - 2xy - y 2 , / 11 = 3x- x 2 - 2xy,<br />
f,, = - 2y, fv·u = - 2x, f xv = 3-2x- 2y. Then f, = 0 implies<br />
y(3- 2x- y) = 0 soy= 0 or y = 3 - 2x. Substituting into / 11 = 0 implies<br />
x(3 - x) = 0 or 3x( - 1 + x )' = 0. Hence the critical points are (0, 0), (3, 0},<br />
(0, 3) and (1, 1). D(O, 0} = D(3, 0} = D(O, 3) = ·-9 < 0 so (0, 0), (3, 0), and<br />
(0, 3) are saddle points. D(1, 1) = 3 > 0 and f xx (1, 1} = - 2 < 0, so<br />
!(1, 1) = 1 is a local maximum.<br />
55. First solve inside D. Here f ., = 4y 2 - 2xy 2 - y 3 , f 11 = 8xy - 2x 2 y - 3xy 2 • y<br />
Then f, =·o implies y = 0 or y = 4- 2x, but y = 0 isn't inside D. Substituting (O, 6 )<br />
y = 4- 2x into / 11 = 0 implies x = 0, x = 2 or x = 1, but x = 0 isn't inside -D,<br />
and when x = 2, y = 0 but (2, 0} isn't inside D. Thus the only critical point inside<br />
D is (1, 2) and f (l , 2) = 4. Secondly we consider the boundary of D.<br />
On L1: f(x, 0) = 0 and so f = 0 on L 1. On L2: x = -y + 6 and<br />
(6,0)<br />
.t<br />
f( - y + 6, y) = y 2 (6 - y)( - 2) = -2{6y 2 - y 3 ) which has critical points<br />
at y = 0 and y = 4. Then !{6, 0) = 0 while !(2, 4) = -64. On La: f(O , y) = 0, so f = 0 on £ 3. Thus on D the absolute<br />
maximum off is !{1, 2} = 4 while the absolute minimum is /{2, 4) = - 64.<br />
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242 0 CHAPTER 14 PARTIAL DERIVATIVES<br />
57. f(x, y) = x 3 - 3x + y 4 - 2y 2 2<br />
z 0 l\ "'' 0 and f xx( -1, 0) = - 6 0 and / u(1, ±1} = 6 > 0 indicating local minima / (1, ± 1) = - 3; and D ( -1, ± 1) = - 48 and<br />
D(1, 0) = - 24, indicating saddle points.<br />
59. f(x,y) = x 2 y, g(x,y) = x 2 +y 2 = 1 ~ 'ilf = (2xy,x 2 ) = )..'iJg = (2>.x,2>.y). Then 2xy = 2)..x impliesx = Oor<br />
y = >.. Ifx = 0 then x +<br />
2 y 2 = 1 gives y = ±1 and we have possible points (0, ±1) where f (0, ± 1) = 0. lfy =).. then<br />
x 2 = 2)..y implies x 2 = 2y 2 and substitution into x 2 + y 2 = 1 gives 3y 2 = 1 ~ y = ± 7:J and x = ±jf. The<br />
corresponding possible points are ( ±jf, ±7:J). The absolute maximum is f ( ± jf, -Ja) = ~ while the absolute<br />
61. f(x, y, z) = xyz, g(x,y,z) = ~ 2 + y 2 + z 2 = 3. 'ilf = )..'iJg ~ (yz,xz,xy) = >.(2x, 2y,2z). lfany ofx, y, or z is<br />
zero, then x = y = z = 0 which contradicts x 2 + y 2 + z 2 = 3. Then ).. = YZ ·=x 2 2 z = xy ~ 2y2 z = 2x 2 z ~<br />
x y 2z<br />
. .<br />
y 2 = x 2 , and similarly 2yz 2 = 2x 2 y ~ z 2 = x 2 • Substituting into the constraint equation gives x 2 + x 2 + x 2 = 3 ~<br />
I<br />
x 2 = 1 = y 2 = z 2 . Thus the possible points are (1, 1, ± 1), (1, -1, ± 1), ( - 1, 1, ±1), ( - 1, -1, ±1). The absolute maximum<br />
is / (1, 1, 1) = /(1, -1, - 1) = f( -1, 1, -1) = f( - 1, - 1, 1) = 1 and the absolute<br />
minimum is /(1, 1, -1) = /(1, ....:.1, 1) = f( -1,1, 1) = f( - 1, - 1, - 1) = - 1.<br />
63. f(x,y,z) = x 2 +y 2 +z 2 , .g(x,y,z) = xy 2 z 3 = 2 ~ 'ilf = (2x, 2y,2z) = X'ilg = (>.y 2 z 3 , 2>.xyz 3 ,3)..xy 2 z 2 ).<br />
Since xy 2 z 3 = 2, x =I 0, y =I 0 and z =I 0, so 2x = )..y 2 z 3 (I), 1 = )..xz 3 (2), 2 = 3>.xy 2 z (3~ . Then (2) and (3) imply<br />
~ = - 3<br />
2<br />
2<br />
or y 2 = f z 2 soy = ±z q. Similarly (1) and (3) imply ~x 3<br />
= - 3<br />
2<br />
? or 3x 2 = z 2 sox = ± -jsz. But<br />
xz xy z V 3 y z xy-z<br />
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CHAPTER 14 REVIEW D 243<br />
xy 2 z 3 = 2 so x and z must have the same sign, that is, x = ~ z. Thus g(x, y, z) = 2 implies ~z G z 2 )z 3 = 2 or<br />
z = ±3 1 / 4 and the possible points are (±3- 1 / 4 , 3- 1 / 4 y'2, ±3 1 1 4 ), (±3- 1 / 4 , --,-3...:. 1 ; 4 ._/2, ±3 1 1 4 ) . However at each of these<br />
points f takes on the same value, 2 J3. But (2, 1, 1) also satisfies g(x, y , z) = 2 and / (2, 1, 1) = 6 > 2 v'3. Thus. f has an<br />
absolute minimum value of 2 J3 and no absolute maximum subject to the constraint xy 2 z 3 = 2. ·<br />
2 3 2 . I' ., 2 . . . f( ) 2 2 2 .I<br />
xz<br />
xz<br />
Alternate solution: g ( x , y, z ) = xy z = tmp tes y- = -<br />
3 , so mmtmtze x, z = x + - 3 + z . T 1en<br />
2<br />
fx = 2x -<br />
2<br />
3 , f~ = - --;. + 2z, fxx = 2 + 3<br />
4<br />
3 , f ~~ = 24<br />
5 + 2 and fr~ = 2<br />
X Z X Z X Z' X Z X Z<br />
6<br />
4<br />
• Now f x = 0 implies<br />
2x 3 z 3 -<br />
2 = 0 or z = 1/x. Substituting into / 11 = 0 implies -6x 3 + 2x- 1 = 0 or x .= ~·so the two criti 0, so each point<br />
is a minimum. Finally, y 2 = _2_ 3<br />
, so the four points closest to the origin are (± 4~ , .:fl, ± :y'3), (±+ , - ~ , ± V'3).<br />
= · v3" V3 V3<br />
65.<br />
The area of the triangle is ~ ca sinO and the area of the rectangle is be. Thus,<br />
c<br />
b<br />
the area of the whole object is f(a, b, c)= ~ca sinO+ be. The perimeter of<br />
the object is g( a, b, c) = 2a + 2b + e = P. To simplifY sin 0 in terms of a, b,<br />
· and e notice that a 2 sin 2 o + ( ~ e) 2 = a 2 => sino= ..!. v4a 2 - c2.<br />
c<br />
Thus f(a, b, c)= 4 J4a 2 -<br />
- 2a<br />
. '<br />
c 2 +be. (Instead of using 0, we could just have<br />
us<strong>ed</strong> the Py1hagorean Theorem.) As a result, by Lagrange's method, we must find a, b, c, and A by solving "il f = A "il g which<br />
gives the following equations: ca(4a 2 - c 2 ) - 1 1 2 = 2A (1), c = 2A (2), *(<br />
4a 2 - c 2 ) 1 1 2 - i c 2 (4a 2 - c 2 ) - 1 1 2 + b = A<br />
(3),and2a+2b +c = P (4). From(2),A= t candso(l)producesca(4a 2 - c 2 ) ....: 1 1 2 =c => (4a 2 - c 2 ) 1 1 2 = a =><br />
2<br />
4a 2 - e 2 = a 2 => · c = v'3 a (5). Similarly, since {4a 2 - c 2 ) 112 = a and A= tc, (3) gives ~ - !:__ + b = ~ . so from<br />
4 4a 2<br />
(5), ~ - 3 a + b = .;;a => -~ - v;a = -b => b =% (1 + vf3) (6). Substituting (5) and (6) into (4) we get:<br />
4<br />
2a + a(l + v'3) + v'3a=P => 3a + 2v'3a=P => a = p = 2 J3 - 3 P and thus<br />
3+ 2 J3 3<br />
b = ·(2 J3- 3~ (1 + J3) p = 3 - 6 J3 p and c = (2- V3)P.<br />
© 2012 Ccngagc learni ng. Atl Ril:l>ts Rcset'·cd. May nol be SC3M
D PROBLEMS PLUS<br />
1. The areas of the smaller rectangles are A1 = xy, A2 = (L - x)y,<br />
A 3 = (L- x)(W -y), A 4 = x(W - y). ForO :::;: x :::;: L,O :::;: y:::;: W, let<br />
f(x, y) = Ai +A~+ A~ +A~<br />
= x 2 y 2 + (L - x?y 2 + (L- x) 2 (W - v? + x 2 (W- y) 2<br />
y<br />
W-y<br />
X<br />
L<br />
L-x<br />
T<br />
w<br />
1<br />
= [x2 + (L- x?JIY2 + (W - y)2]<br />
Then we ne<strong>ed</strong> to find the maximum and minimum values of f(x , y). Here<br />
f,(x, y) = [2x - 2(L- x)Jiy 2 + (W - y) 2 ] = 0 => 4x '-- 2L = 0 or x = ~L, and<br />
· j 11<br />
(x, y) = [x 2 + (L - x) 2 JI2y- 2(W- y)] = 0 => 4y- 2W = 0 or y = W / 2. Also<br />
f :tx = 4[y 2 + (W - u?J, f v 11 = 4(x 2 + (L- x?J, and fxv = (4x- 2L)(4y- 2W). Then.<br />
D = 16(y 2 + (W - y) 2 J!x 2 + (L- x) 2 )- (4x- 2L) 2 (4y- 2W) 2 . Thus when x =~Landy= ~W, D > 0 and<br />
f x:r: = 2W 2 > 0. Thus a minimum off occurs at (~L , %W) and this ·minimum value is f(~L, ~W) = ~L 2 W 2 .<br />
There are no other critical points, so the maximum must occur on the boundary. Now along the width ofthe rectangle let<br />
g(y) = j(O, y) = f(L, y) = L 2 [y 2 + (W- Y?]. 0 :::;: y:::;: W . Then g'(y) = L 2 [2y - 2(W- y)) = 0 ~ y = ~W .<br />
And g(%) = ~L 2 W 2 .<br />
Checking the endpoints, we get g(O) = g(W) = L 2 W 2 • Along the length of the rectangle let<br />
h (x) = f(x, 0) = f(x, W) = W 2 [x 2 + (L- x?]. 0 :::;: x :::;: L. By symmetry h'(x) = 0 ~ x = ~Land<br />
h( ~L) = %L 2 W 2 • At the endpoints we have.h(O) = h(L) = L 2 W 2 • Therefore L 2 W 2 is the maximum value of f .<br />
This maximum value off occurs when the "cutting" lines correspond to sides of the rectangle.<br />
3. (a) The area of a trapezoid is ~ h(b1 + b2), where his the height (the distance between the two parallel sides) and b 1 , ~ are<br />
the Ieng!hs of the bases (the parallel sides). From the figure in the text, we see that h = x sinO, b 1 = w - 2x, and<br />
b 2 = w - 2x + 2x cos 8. Therefore the cross-sectional area of the rain gutter is<br />
~(x, 8) = ~xsin 0 [(w - 2x) + (w- 2x + 2xcos 8)) = (x sinO)(w - 2x + xcos8)<br />
= wx sine - 2x 2 sin B + x 2 sin B cos e. 0 < X ::; ~w. 0 < B :::;: i<br />
We look for the critical points of A : oAf ax = w sin B -<br />
4x sin B + 2x sin B cos B and<br />
EJAfEJO = wxcos B- 2x 2 cosB + x 2 (cos 2 8 - sin 2 8), so {)Ajax= 0 ~ sinO (w - 4x + 2x cosO) = 0 ~<br />
cos B = 4 x - w = 2 -<br />
2x<br />
w (0 < 8 :::;: ~ => sin 8 > 0). If, in addition, oAf EJB = 0, then<br />
2x<br />
0 = wxcosB - 2x 2 cosO+ x 2 (2cos 2 8 - 1)<br />
= wx ( 2.- ~) - 2x 2 ( 2 - ~) + x 2 [ 2 ( 2 - ~) 2 - 1]<br />
= 2wx - ~w 2 - 4x 2 + wx + x 2 [ 8 - 4 : + ~: - 1] = -wx + 3x 2 = x(3x - w)<br />
® 2012 Ccngage Learning. All RighiS Rescn·<strong>ed</strong>. Mny not be scann<strong>ed</strong>. copi<strong>ed</strong>. or duplicat<strong>ed</strong>, or postL-d lo u publicly accessible websilt.:, in whole or in p3t1. 245
246 0 CHAPTER 14 PROBLEMS PLUS<br />
Since x > 0, we must have x = ~w, in which case cosO=~. soB = f, sinO=~ .<br />
k = '{!w, b1 = ~w,ln = ~w.<br />
and A= {!:w 2 • As in Example 14.7.6, we can argue from the physical nature of this problem that we have found a local<br />
maximum of A. Now checking the boundary of A, let<br />
g( B) = A( w /2, B) = ~ w 2 sin B - ~w 2 sin B + iw 2 sin B cos 6 = i w 2 sin 26, 0 < B ::; ~. Clearly g is maximiz<strong>ed</strong> when<br />
sin2B = 1 in which case A= iw 2 . Also along the line B = ~. let h(x) = A(x, ~) = wx- 2x 2 , 0 < x < ~w . =><br />
h'(x) = w- 4x = 0<br />
*? x = i w, and h(~w) = w(~w) - 2(tw) 2 = iw 2 . Since -kw 2 < "If w 2 , we conclude that<br />
the local maximum found earlier was an absolute maximum.<br />
2 2<br />
(b) If the metal were bent into a semi-circular gutter of radius r, we would have w = 1rr and A= ~11'r 2 = ~1r(~) = ~7r.<br />
S . ../3 w2 w 2 mce - > ---, 1t • wou ld b e b etter to be n d th e meta l · mto a gutter WI 'th a sem1c1rcu · · I ar cross-section. .<br />
211' 1 2<br />
5. Letg(x,y) = xf(~) · Theng,.(:,y) = !(~) +xf'(~) ( - : 2 ) = !(~ ) - ~ f'(~;) and<br />
gy ( x, y) = xf' ( ~) (;) = f' ( ~ ). Thus the tange?t plane at ( x0 , y 0 , zo) on the surface has equation<br />
[t ( ;~ ) - y0 x0 1 !' ( ~~ )] x + [r ( ~~ )] y - z = 0. But any plane whose equation is of the form ax+ by+ cz = 0<br />
passes through the origin. Thus the origin is the ~ommon point of intersection.<br />
7. Since we are minimizing the area of the ellipse, and the circle lies above the x -axis,<br />
y<br />
the ellipse will intersect the circle for only one value ofy. This y-value must<br />
satisfY both the equation of the circle and the equation of the ellipse. Now<br />
2 2 2<br />
x 2<br />
+ Y 2<br />
= 1 => x 2 = a 2<br />
(b 2 -y 2 ). Substitutingintotheequationofthe<br />
a b b<br />
2<br />
circle gives ~2 (b 2 - y 2 ) + y 2 - 2y = 0 =><br />
In order for there to be orlly one solution to this quadratic equation, the discrirninant must be 0, so 4- 4a 2 2 2<br />
b ~ a<br />
b 2 - a 2 b 2 + a 4 = 0. The area of the ellipse is A(a, b) = 1rab, and we minimize this function subject to the constraint ·<br />
g(a, b) = b 2 - a 2 b 2 + a 4 = 0.<br />
= 0 =><br />
-rrb<br />
2a(2a 2 -<br />
2) (1),<br />
b<br />
.>. = 2 b(l7r~ a 2 ) (2), b2 - a 2 b 2 + a 4 = 0 (3). Comparing (1) and (2) gives 2<br />
a( 2<br />
; b _ b2).<br />
21rb 2 = 47ra 4 -:=:> a 2 = ~ b. Substitute this into (3) to get b = ~ => a = jf.<br />
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15 D MULTIPLE INTEGRALS<br />
15.1 Double Integrals over Rectangles<br />
1. (a) The subrectangles are shown in the figure.<br />
The surface is the graph of f(x, y) = xy and 6.A = 4, so we estimate<br />
3 2<br />
v~ I: I: f(x;,y 1 )6.A<br />
i =l j=l<br />
= j(2, 2) 6.A + !(2, 4) 6.A + f(4, 2) 6.A + /(4,4) 6.A + !(6, 2) LU + !(6, 4) 6.A<br />
= 4(4) + 8(4) + 8(4) + 16(4) + 12(4) + 24(4) = 288<br />
y<br />
4<br />
2<br />
0 2 4 6 X<br />
3 2<br />
(b) V ~ ,L: ,L: j (x.:, v 1 ) 6.A = /(1, 1) 6.A + f(1, 3) 6.A + !(3, 1) 6.A + !(3, 3) 6.A + !(5, 1) LU + !(5, 3) 6.A<br />
i=lj=l<br />
= 1(4) + 3(4) + 3(4) + 9(4) + 5(4) + 15(4) = 144<br />
3. (a) The subrectangles are shown in the figure. Since 6.A = 1 · ~ = ~ .we estimate<br />
2 2<br />
J fn. xe-"' 11 dA ~ I: I: f ( x;1, Yi;) 6.A<br />
i=l j = l<br />
= /(1, ~) 6.A + /(1, 1) 6.A + /(2, ~) 6.A + /(2, 1) 6.A<br />
= e - 1 1 2 ( ~ ) +e - 1 (~) +2e - 1 (~) + 2e - 2 ( ~) ~ 0 .990<br />
. 2 2<br />
(b) ffn xe-"' 11 dA ~ I: I: f(x;, v 1 ) 6.A<br />
. i=l j =l<br />
= !(~, t) 6.A + !(~ , ~) 6.A + f(~, t) 6.A + ! (~. ~) 6.A<br />
= ~e- l /8a) + ~e - 318(~) + ~e-3 18(~) + ~e - 9/ sa) ~ u 51<br />
y<br />
I<br />
I<br />
2<br />
0 1 2 X<br />
y<br />
1<br />
. .<br />
I<br />
2<br />
. .<br />
0 1 2 X<br />
5. (a) Each subrectangle and its midpoint are shown in the figure.<br />
The area of each subrectangle is 6.A = 2, so we evaluate f<br />
at each midpoint and estimate<br />
2 2<br />
.ffn.f(x,y)dA ~ I: I: f(x.:,y 1<br />
) 6.A<br />
i = lj = l<br />
= /(1, 2.5) 6.A + /(1, 3.5) AA<br />
+ /(3, 2.5) AA + /(3, 3.5) AA<br />
= - 2(2) + ( -1)(2) + 2(2) + 3(2) = 4<br />
y<br />
4<br />
. 3<br />
. .<br />
2<br />
0 2 4 X<br />
® 20 12 Ccngage learning. All Ril:l•ts ReserY<strong>ed</strong>. May rn>l be scann<strong>ed</strong>, copiC
248 0 CHAPTER 15 MULTIPLE INTEGRALS<br />
(b) The subrectangles are shown in the figure.<br />
In each subrectangle, the sample point closest to the origin<br />
is the lower left comer, and the area of each subrectangle is .6.A = ~.<br />
Thus we estimate<br />
4 4<br />
ffnf(x,y) dA ~ L:: L:: f(xii,Yii) .6.A<br />
i= l j = l .<br />
= f (O, 2) .6.A + f(O , 2.5) .6.A + f(O , 3) . .6.A + f(O, 3.5) .6.A<br />
+ /(1, 2) .6.A + f(1, 2.5) .6.A + /(1, 3) .6.A + /(1, 3.5) .6.A<br />
+ / (2, 2) .6.A + /(2, 2.5) .6.A + /(2, 3) .6.A + /(2, 3.5) .6.A<br />
y<br />
4<br />
3<br />
2<br />
0 I 2<br />
3 4 X<br />
+ / (3, 2) .6.A + /(3, 2.5) .6.A + /(3, 3) .6.A + !(3, 3.5) .6.A<br />
= - 3(~) + < -5) G) + < - 6) (~) + < -4)(~) + < - 1) (~) + < - 2){ ~) + < -3)(~) + < -1)G)<br />
= - 8<br />
+ 1G) + o(~) + ( -1 ){~) + 1 (~) + 2(~) + 2G) ·;~- 1 (~) + 3G)<br />
7. The val~es off ( x, y) = .j52 - x 2 - y 2 get smaller as we move farther from the origin, so on any of the subrectangles in the<br />
problem, the function will have its largest value at the lower left comer of the subrectangle and its smallest value at the upper<br />
right comer, and any otl).er value will lie between these two. So using these subrectangles we have U < V < L . (Note that this<br />
is true no matter how R is divid<strong>ed</strong> into subrectangles.)<br />
9. (a) With m = n = 2, we h~ve .6.A = 4. Using the contolir map to estimate the value off at the center of each subrectangle,<br />
we have<br />
2 2<br />
ffn f(x, y) dA ~ L:; L:; J(xi, Yi) .6.A = .6.A[f(1, 1) + /(1, 3) -t /(3, 1) + /(3, 3)] i'::j 4(27 + 4 + 14 + 17) = 248<br />
i = lj=l<br />
(b) /ave= Aln) ffn f(x, y) dA i'::j ft(248) = 15.5<br />
11. z = 3 > 0; so we can interpret the integral as the <strong>vol</strong>ume of the solid S that lies below the plane z = 3 and above the<br />
rectangle [- 2, 2] x [1, 6]. Sis a rectangular solid, thus ffn 3 dA = 4 · 5 · 3 = 60.<br />
13. z = f(x, y) = 4- 2y :2: 0 for 0 :::; y :5 1. Thus the integral represents the <strong>vol</strong>ume of that<br />
part ofthe rectangular.solid [0, 1] x [0, 1] x (0, 4] which lies below the plane z = 4 - 2y.<br />
So<br />
ffn(4 - 2y) dA = (1)(1)(2) + H 1)(1)(2) = 3<br />
X<br />
© 2012 C.ngnge L1cd too publicly accessible website, in whole or in port.
SECTION 15.2 ITERATED INTEGRALS D 249<br />
15. To calculate the estimates using a programmable calculator; we can use an algorithm<br />
similar to that of Exercise 4.1.9 [ET 5.1.9). In Maple, we cari define the function<br />
f(x, y) = v'1 + xe 11 (calling it f), load the student package, and then use the<br />
command<br />
middlesum(middlesum(f, x=0 . . 1,m),<br />
y=O .. 1 , m);<br />
to get the estimate with n = m 2 squares of equal size. Mathematica has no special<br />
Riemann sum command, but we can define f and then use nest<strong>ed</strong> Sum commands to<br />
calculate the estimates.<br />
,.<br />
n estimate<br />
' 1 1.141606<br />
4 1.143191<br />
16 1.143535<br />
64 1.143617<br />
256 1.143637<br />
1024 1.143642<br />
17. If we divideR into mn subrectangles, ffn k dA:::::: .f _t f (xi; . Yi;) ~A for any choice of sample points (xi;, Yi;).<br />
250 0 CHAPTER 15 MULTIPLE INTEGRALS<br />
13. J~ f 0<br />
1f r sin 2 8 d8 dr = f~ r dr J; sin 2 8 d8 [as in Example 5] = J 0<br />
2<br />
r dr J 0<br />
1f t (1 - cos 28) d8<br />
= [ tr 2 )~· ~ [0- ~ sin28)~ = (2 - 0) · ~ [(1r- ~ sin 27r) - (0- ~sinO)]<br />
= 2. ~[(7r- 0)- (0 - 0)] = 7r<br />
. 15. ffn sin(x - y) dA = Io1f 12 Iorr~ 2 sin(x- y) dyd:r: = I; 12 [cos(x - y)]~~~l 2 dx = Iorrl 2 [cos(x - ~)- cosx] dx<br />
17.ffn X:~ 1 dA = 11 /_33 X::<br />
19. Io1riG j~1rl 3 x sin(x + y) dy dx<br />
1f 12 .<br />
= [ sin(x-~)-si?x ] 0<br />
= sinO-sin~-,- (sin(-~) - sinO]<br />
=0-1-(-1-0) = 0<br />
1 dydx = 11 :r;2:<br />
= ~ (In 2 -In 1) . t(27 + 27) = 91n 2<br />
1 dx ;_: y2 dy = [t ln(x2 + 1)]: [h3] ~3<br />
= IorriG [-xcos(x + y)]~=~ 13 dx = J; 16 [x cosx- xcos(x +f)] dx<br />
= x (sin x - sin ( x + f) )~ 16 - J 0<br />
1f<br />
16 [sin x - sin ( x + f)] dx [by integrating by parts separately for each term]<br />
= i [~- 1)- (-cosx + cos(x + f)]~ 16 = - -&- [ -~ + 0 - (-1 + ~)] = -q- 1 - f2<br />
21. IInye-:z:il dA =I: I: ye-"' 11 dxdy = J 0<br />
3<br />
(-e -"' 11 J ::~ dy = I:(-e- 2 il + 1) dy = [~e- 2 Y + y]~<br />
= ~ e- 6 + 3 - ( ~ + 0) = t~-6 + ~<br />
23. z = f(x, y) = 4- x - 2y ~ 0 for 0 ~ x ~ 1 and 0 ~ y ~ 1. So the solid<br />
is the region in the first octant which lies below the plane z = 4 - x -<br />
and above [0, 1] x {0, 1].<br />
2y<br />
X<br />
25. The solid lies under the plane 4x + 6y - 2z + 15 = 0 or z = 2x + 3y + 1/ so<br />
V = fin(2x + 3y + 1 ;)'dA = I~ 1 I~ 1 (2x + 3y + Jf) dxdy = j~ 1 [x 2 + 3xy + ¥xJ::~ 1 dy<br />
= j~ 1 ((19 + 6y) - ( - 1 23 - 3y)) dy = I~ 1 (¥' + 9y) dy = [Yf y + ¥Y 2 )~ 1 = 30- ( -21) = 51<br />
27. V = I~ 2 f~ 1 (1 - tx 2 - h 2 ) dx dy = 4 J; I 0<br />
1<br />
(1 - ix 2 - ~y 2 ) dx dy<br />
4 f2 [ 1 3 1 2 ] :z; = 1 d 4 f2 ( 11 1 2) d [ 11 l 3] 2 83 166<br />
= Jo X- fiX - iiY X :z:=O y = Jo i2- iiY y = 4 i2Y - vY 0 = 4. 54= 27<br />
29. Here we ne<strong>ed</strong> the <strong>vol</strong>ume of the solid lying under the surface z = x sec 2 y and above the rectangle R = (0, 2] ~ [0, 1r / 4] in<br />
the xy-plane.<br />
V = I: Iorr 14 x sec 2 y dy dx = I: x dx Io" 14 sec 2 y dy = [ ~ x 2 ] ~ [tan y] ~ 14<br />
= (2- O)(tan f - tan 0) = 2(1 - 0) = 2<br />
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l<br />
SECTION 15.3 DOUBLE INTEGRALS OVER GENERAL REGIONS 0 251<br />
31. The solid lies below the surface z = 2 + x 2 + (y- 2? and above the plane z = 1 for -1 ~ x ~ 1, 0 ~ y ~ 4. The <strong>vol</strong>ume<br />
of the solid is the difference in <strong>vol</strong>umes between the solid that lies under z = 2 +x 2 + (y- 2) 2 over the rectangle<br />
R = (-1, 1] x (0, 4) and the solid that lies under z = 1 over R.<br />
V = J; J~ 1 (2 + x 2 + (y :._ 2?J dxdy- f 0<br />
4<br />
f~ 1 (1) dxdy = f 0<br />
4<br />
[2x + 4x<br />
3<br />
+ x(y - 2?J:: ~ 1<br />
dy- t 1<br />
dx J;dy<br />
= J; [(2 + ~ + (y - 2) 2 )- ( -2- ~- (y- 2) 2 )] dy- [x]~ 1 (y]~<br />
4<br />
= f 0<br />
(lf + 2(y- 2) 2 ) dy- (1 - (-1)][4- OJ = ( 1 4<br />
3 y + t(Y- 2?)~- (2)(4)<br />
= [e; + ¥)- (o - nJ 1 -8 = ¥-8 = ¥<br />
33. In Maple, we can calculate the integral by defining the integrand as f<br />
and then using the command int ( int ( f, x=O .. 1) , y=O . . 1) ; .<br />
In Mathematics, we can use the command<br />
Int egrate[f,{x,0,1}, {y,0,1}]<br />
We find that J n x 5 y 3 e
252 D CHAPTER 15 ' MULTIPLE INTEGRALS<br />
7. IIv y2 dA = I~t I~~ -2 Y2 dx dy = I~t (xy2J ::~v - 2 dy = I~l y2 [y - ( - y- 2)] dy<br />
= J 1 (2y3 + 2y2)dy = [ly4 + 1y3J 1 = 1 + 1 - l + 1 = :!.<br />
- 1 2 3 - 1 2 3 2 3 3<br />
9. ffv x dA = f " ro•ln"' x dy dx = ~ " [xy]Y:'"oin "' dx = f" x sin x dx [ . integrate by parts ]<br />
· .lo J t Jo Y- Jo wtth v.=x, rlv = sm x dx<br />
= [ - x cos x + sin x J ~ = -1r cos 1r + sin 1r + 0 - sin 0 = 1r<br />
11. (a) At the right we sketch an example of a region D that can be describ<strong>ed</strong> as lying<br />
y<br />
betw~en the graphs of two continuous functions of X (a type I region) but not as<br />
lying between graphs of two continuous functions ofy (a type IJ region). The<br />
regions shown in Figures 6 and 8 in the text are additional examples.<br />
(b) Now we sketch an example of a region D that can be describ<strong>ed</strong> as lying between<br />
the graphs of two continuous functions ofy but not as lying between graphs of two<br />
continuous functions of x. The first region shown in Figure 7 is another example.<br />
0<br />
y<br />
X<br />
0<br />
X<br />
13.<br />
y<br />
(1, 1)<br />
As a type I region, D lies between the lower boundary y = 0 and the upper<br />
boundary y = x for 0 ::; x ::; 1, soD= {(x, y) I 0 $ x $ 1, 0 $ y $ x}. lfwe<br />
x =l<br />
describeD as a type ll region, D lies between the left boundary x = y and the<br />
right boundary x = 1 for 0 ::; y ::; 1, soD = {(x, y) I 0 ::; y::; 1, y ::; x::; 1}.<br />
X<br />
1 1 2<br />
Thus IIv xdA =I; IC: xdydx = I 0 [xy] ~ =~ dx = I 0 x dx = ! x 3 ]~ = !(1 - 0) = ~or<br />
1 1<br />
.rf'v x dA = f 0<br />
I: xdxdy = I 0<br />
(tx 2 J : : ~ dy = t J;(1 - y 2 ) dy = t[y - h 3 ] ~ = t((1-!) - 0] = ~ -<br />
15.<br />
y<br />
The curves y = x - 2 or x = y + 2 and x = y 2 intersect when y + 2 = y 2 ~<br />
y 2 - y- 2 =0 ~ (y-2)(y+ 1)=0 ~ y = -1, y =2,so thepoints.of<br />
intersection arc (1, - 1) and (4, 2). Tf we de~c ribe D as a type I region, the upper<br />
boundary curve is y = ..fii but the lower boundary curve consists of two parts,<br />
y = -vx for 0 ::; x ::; 1 andy = x - 2 for 1 ::; x ::; 4.<br />
Thus D = {(x, y) I 0::; X::; 1, - vx::; y::; vx} u {(x, y) 11 ::; X::; 4, X- 2::; y::; vx} and<br />
I I 0<br />
y dA = Io 1 I!5z y dy dx + f 1<br />
4<br />
f.,~ y dy dx. If we describe D as a type II region, D is enclos<strong>ed</strong> by the left boundary<br />
x = y 2 and the right boundary x = y + 2 for -1 ::; y ::; 2, so D = {( x, y) I - 1 ::; y ::; 2, y 2 ::; x ::; y + 2} and<br />
. . .<br />
SECTION 15.3 DOUBLE INTEGRALS OVER GENERAL REGIONS 0 253<br />
J J 0<br />
y dA = f~ 1<br />
J:/ 2 y dx dy. In either case, the resulting iterat<strong>ed</strong> integrals are not difficult to evaluate but the region D is<br />
more simply describ<strong>ed</strong> as a type II region, giving one iterat<strong>ed</strong> integral rather than a sum of two, so we evaluate the latter<br />
integral:<br />
ffo ydA = f~tf:/2 ydxdy = f~t (xyJ ::~~2 dy = J~1 (y + 2- y2)ydy = f~ t (y2 + 2y - ya) dy<br />
= [jy3+y2- h4]~1 = (! +4-4) - (-~ +1 - i) = ~<br />
19.<br />
0 X<br />
= [§.y 3 - y 4 ] 2 = .§1 - 16- ~ + 1- ll<br />
3 1 3 3 - 3<br />
21.<br />
-2<br />
y<br />
! 2 !~ (2x - y) dydx<br />
- 2 -~<br />
2<br />
- /<br />
- 2 y=-y4- x 2<br />
- 2xy - 2Y r.--;; dx<br />
[ 12] !1=~<br />
= }~ 2 [2x v'4 - x 2 :- H4 - x 2 ) + 2x v'4- x2 + H4 - x 2 )] dx<br />
=J~24xv'4-x2dx= -H4-x2)3/2]2 = 0<br />
-2<br />
[Or, note that 4x v'4 - x 2 is an odd function, so J~<br />
2 4x v'4- x2 dx = 0.]<br />
23.<br />
y<br />
f1 J1-a: 2 ( ) f1 [ 2] y-l-x 2<br />
V= 10 1 _., 1- x+2y dydx= 10 y-xy+ y !1;: 1<br />
_., dx<br />
=<br />
1<br />
[ ((1- x 2 ) - x(1- x 2 ) + (1 - x 2 ?)<br />
- ((1- x)- x(1- x) + (1- x) 2 )] dx<br />
X<br />
= J 0<br />
1<br />
[(x 4 + x 3 - 3x 2 - x + 2)- (2x 2 - 4x + 2)] dx<br />
25.<br />
0 y<br />
J2J7- 3y J2 [ 1 2 ] "' 7- 3y<br />
V =<br />
1 1<br />
xy dx dy = 1 2 x y ., = dy<br />
1<br />
0<br />
X<br />
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254 0 CHAPTER 15 MULTIPLE INTEGRALS<br />
27.<br />
y<br />
(0,3)<br />
V = f 0<br />
2<br />
f~ - ~ "'<br />
(6- 3x - 2y)dydx<br />
r2( 2] v= 3- 1x<br />
= Jo 6y - 3xy - y v = o 2 dx<br />
= J; (6(3 - ~x)- 3x(3 - ~x) - (3 - ~ x) 2 ] dx<br />
= J; (~x 2 - 9x + 9) dx = [~x 3 - ~ x 2 + 9x] ~ = 6 -0 = 6<br />
29.<br />
31.<br />
y<br />
1~ 11 [ V = 2 ] Y=~<br />
y dy dx = 1L . dx<br />
1 1<br />
0 0 0 2 y= O<br />
{ 1 1 - X 2 1 1 3 1 1<br />
= Jo -2- dx = 2 [x - 3·-c Jo = 3<br />
X<br />
33.<br />
From the graph, it appears that the two curves intersect at x = 0 and<br />
at x ~ 1.213. Thus the desir<strong>ed</strong> integral is<br />
Jf dA rl.213J3x- x 2 d .J. fl.213 [ ] y = 3 2<br />
"'- "'<br />
D X ~ Jo x•l X y u.'C - Jo xy d<br />
X<br />
y = x4.<br />
r<br />
= Jo L213( 3 X -X - X X= X - 4X - aX O<br />
~ 0.713<br />
2 3 6) d [ 3 1 4 1 G] 1.213<br />
35. The two bounding curves y = 1 - x 2 and y = x 2 - 1 intersect at (±1, 0) with 1 - x 2 ~ x 2 - 1 on (-1, 1]. Within this<br />
region, the p la n~ z = 2x·+ 2y + 10 is above the plane z = 2 - x - y, so<br />
V = J_ •r1-x2 ( ) rl 1 Ji-~ 2 ( )<br />
1<br />
J,. 2 _ 1<br />
2x+2y + 10 dydx -, _ 1<br />
.,2 _ 1<br />
2-x-y dydx<br />
J •l J1-x2 ( ' ( )) •<br />
= _ .,2_ 1 1<br />
2x + 2y + 10- 2- x - y dy dx<br />
= j _ 1<br />
, 2 _ 1<br />
(3x + 3y + 8) dy dx = f_ 1<br />
3xy + 2Y + 8y y=z _ 2<br />
dx<br />
1<br />
•l fl-:c2 1 [ 3 2 J y = l - :c2<br />
= J~ 1 [3x(l- x 2 ) + ~(1 - x 2 ) 2 + 8(1- x 2 ) - 3x(x 2 - 1) - ~(x 2 - 1) 2 - 8(x 2 - 1)] dx<br />
1 "3 1<br />
= J (-6x - 16x 2 + 6x + 16) dx = [-.:!x 4 - .l&x 3 + 3x 2 + 16x]<br />
- 1 2 3 -1<br />
= -~ - ~ 6 + 3 + 16 + ~ - ¥ - 3 + 16 = 6 3 4<br />
@) 2012 C
SECTION 15.3 DOUBLE INTEGRALS OVER GENERAL REGIONS D 255<br />
37. The solid lies below the plane z = 1 - x - y<br />
or x + y + z = 1 and above the region<br />
y<br />
D={(x,y)IO~x~l,O~y~ 1 -x}<br />
in the xy-plane. The solid is a tetrah<strong>ed</strong>ron.<br />
X ,<br />
X<br />
39. The two bounding curves y = x 3 - x andy = x 2 + x intersect at the origin and at x = 2, with x 2 + x > x 3 - x on (0, 2).<br />
· Using a CAS, we find that the <strong>vol</strong>ume is<br />
V = Z y X= X y . XY y -X= "<br />
1o x3- x 0 xa - :z: 14,549,35<br />
21x2 +x d d '121:r:2 +:r:( 3 4 ,+ 2) d d 13,984,7:'!5,616<br />
41 . The two surfaces intersect in the circle x 2 + y2 = 1, z = 0 and the region of integration is the disk D : x 2 + y<br />
2<br />
~ 1.<br />
43.<br />
Using a CAS, the <strong>vol</strong>ume is (1- x<br />
!In 2 1 11~<br />
- y 2 ) dA = . (1 - x 2 - y 2 ) dydx = ~.<br />
' D - 1 -Vl-x2 2<br />
y<br />
Because the region of integration is<br />
D = {(x,y) I 0 ~ x ~ y, 0 ~ y ~ 1} = {(x, y) I x ~ y ~ 1, 0 ~ x ~ 1}<br />
we have f 0<br />
1<br />
J~ f(x, y) dx dy = ffv f(x, y) dA = f 0<br />
1<br />
J: f(x, y) dy dx.<br />
45.<br />
Because the region of integration is<br />
D = {(x,y) I 0 ~ y ~ cosx,O ~ x ~ 1rj2}<br />
= {(x,y) I 0 ~ x ~ cos- 1 y, O ~ y ~ 1}<br />
we have<br />
foTr/2 focou f(x,y)dydx = ffv f(x, y)dA = fol focos-lv f(x,y)dx dy.<br />
47.<br />
49.<br />
In 2<br />
y<br />
Because tlie reg_ion of integration is -<br />
D = {(x, y) I 0 ~ y ~' Inx, 1 ~ x ~ 2} = {(x,y) J e 11 ~ x ~ 2, 0 ~ y ~ ln2}<br />
we have<br />
/,<br />
211" x ~~ 11n 2[2<br />
f(x,y).dydx = f(x,y)dA = f(x,y) dxdy<br />
l 0 D 0 cV<br />
113<br />
2<br />
2 131:r./3 2 13 [ ]u=x/3<br />
ex dxdy = ex dydx = ex y dx<br />
o 3u o o o u=O<br />
(3, 1)<br />
X<br />
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256 0 CHAPTER 15 MULTIPLE INTEGRALS<br />
51.<br />
y<br />
53.<br />
y<br />
y = sinx or<br />
x= arcsiny<br />
\<br />
1 111
61 . Since m :S j(x, y) :S M, ffD mdA ~ ffD j(x, y) dA ~ ffD M dA by (8) =><br />
SECTION 15.3 DOUBLE INTEGRALS OVER GENERAL REGIONS 0 257<br />
mjf 0<br />
ldA ~ fj~ j(x,y)dA ~ lVI .ffD ldA by(?) => mA(D) ~ ffDJ(x,y)dA ~ MA(D) by (10).<br />
63.<br />
y<br />
- 3 X<br />
First we' can write JJ 0 (x + 2) dA = JJD xdA + JJ 0<br />
2dA. But f(x, y) =xis<br />
an odd function with respect to x [that is, f( -x, y) = - f(x, y)] and Dis<br />
symmetric with respect to x. Consequently, the <strong>vol</strong>ume above D and below the<br />
graph off is the same as the voiume below D and above the graph off, so<br />
JJD x dA = 0. Also, .f.fD 2 dA = 2 ·A( D) = 2 · ~7r(3) 2<br />
= 97r5ince Dis a half<br />
disk of radius 3. Thus JJD(x + 2) dA = 0 + 97r = 97r.<br />
65. We can write J J 0<br />
(2x + 3y) dA = J J 0<br />
2x dA + J J D 3y dA. J J 0 2x dA represents the <strong>vol</strong>ume of the solid lying under the<br />
plane z = 2x and above the rectangle D . This solid region is a triangular cylinder with length band whose cross-section is a<br />
triangle with width a and height 2a. (See the first figure.)<br />
(a,0,2a) (O,b, 3b)<br />
z= 3y<br />
y<br />
X<br />
Thus its <strong>vol</strong>ume is~· a· 2a · b = a 2 b. Similarly, JJ 0<br />
3ydA represents the <strong>vol</strong>ume of a triangular cylinder with length a,<br />
triangular cross-section with width band height 3b, and <strong>vol</strong>ume t · b · 3b · a ~ ~ab 2 • (Sec the second figure.) T~us<br />
JJ 0<br />
(2x + 3y) dA = a 2 b + ~ab 2<br />
67. ff 0<br />
(ax 3 + In/+ va 2 - x 2 ) dA = ff 0<br />
ax 3 dA + JJ 0<br />
by 3 dA + JJ'a Ja 2 - x 2 dA. Now ax 3 is odd with respect<br />
to x and by 3 is odd witb respect to y , and the region of integration is symmetric with respect to both x and y,<br />
so ffD ax 3 dA = ffD'by 3 dA = 0.<br />
JJ'v J a 2 -<br />
x2 dA represents the <strong>vol</strong>ume of the solid region under the<br />
graph of z = Ja 2 -<br />
x 2 and above the rectangle p, namely a half circular<br />
cylinder with radius a and length 2b (see the figure) whose <strong>vol</strong>ume is<br />
t · 1rr 2 h = ~7ra 2 (2b) = 1ra 2 b. Th~s<br />
JJD (ax 3 + by 3 + Ja 2 - x 2 ) dA = 0 + 0 + 1ra 2 b = 1ra 2 b.<br />
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258 D CHAPTER 15 MULTIPLE INTEGRALS<br />
15.4 Double Integrals in Polar Coordinates<br />
1. The'region R is more easily describ<strong>ed</strong> by polar coordinates: R = { (r, B) I 0 $ r::; 4, 0::; B::; 3 ; }.<br />
Thus JJR f~x, y) dA = J:'~~' 12 4<br />
f 0<br />
f(r cos fJ, r sinO) r dr dB.<br />
3. The region R is more easily describ<strong>ed</strong> by rectangular coordinates: R = { ( x, y) I -1 :=:; x :=:; 1, 0 :=:; y :=:; ~ x + ~}.<br />
Thus JJRf(x,y) dA = J~ 1 J 0<br />
("'+ 1 )/ 2 f(x,y) dydx.<br />
5. The integral J:;t J 1<br />
2<br />
r dr dB represents the area of the region .<br />
R = {(r,B) 11::; r::; 2, 7r/4 $ '0::; 37r/4}, the top quarter portion of a<br />
ring (annulus):<br />
0=31T<br />
4<br />
y<br />
J:; t j 1<br />
2 r dr dO = ( J:; t d()) ( f 1<br />
2 r dr)<br />
= [B]3'11'/4 [!r2]2 = {3.,. _ .!!.) . 1 (4 _ 1) = 1!.. i! = 3'11'<br />
'II' /4 2 - 1 4 4 2 2 2 4<br />
X<br />
7. The half disk D can be describ<strong>ed</strong> in polar coordinates as D = { ( r, B) I 0 ::; r ::; 5, 0 ::; B ::; 1r}. Then<br />
ffv· x 2 y dA = J 0<br />
" J; (r cos B) 2 (r sin B) r dr dB= (J 0<br />
.,. cos 2 Osin B dB) ( J~ r 4 dr)<br />
= [-% cos 3 OJ~ [ir 5 ]~ = -%(-1-1) · 625 = 12 i 0<br />
9. JJR sin(x 2 + y 2 ) dA = JJ:/ 2 J{ sin(r 2 ) r dr dB= (! 0<br />
.,. 12 dO) (J 1<br />
3<br />
r sin(r 2 ) dr)<br />
= [0]~ 12 [-~ cos(r 2 )]~<br />
= (:~) [-~(cos9- cos 1)] = -;f(cos 1 - cos9)<br />
_ ,2_ u2<br />
11 dA- J'll'/ 2 f2 -r2 d dB- J"'/2 dB r2 -r2 d<br />
. f·r<br />
Jn e - -'11'/2 Jo e r r - -'11'/2 Jo re r<br />
, = [B]"'/ 2 [-le-r 2 )<br />
2<br />
= 1r{-l)(e-<br />
4 - e 0 ) = 1!.(1- e- 4 )<br />
-'11'/2 2 0 2 2<br />
13. R is the region shown in the figure, and can be describ<strong>ed</strong><br />
by R = {(r,fJ) I 0::; B ::; rr/4, 1$ r::; 2}. Thus<br />
JJR ~ctan(y/x) dA = f0 '~~' 14 f 1<br />
2 arctan( tan B) r dr dO since yjx =tan 0.<br />
Also, arctan( tan B) = B for 0 ::; B ::; 1r /4, so the integral becomes<br />
. f":r/4 f20 d -'"- f'11'/ 4"d(} f2 d - [!02)"'/4 [l 2]2 - .,-2. ~- _1_ 2<br />
.lo 1 r r uu - Jo u 1 r r - 2 o 2 r 1 - 32 2 - a.t7r ·<br />
15. One loop is given by the region<br />
D = {(r,O) 1-rr/6 S 0 $ rr/6, 0 S r $ cos30}, so the area is<br />
11 !<br />
7r /61cos SO . ~.,. /6 [ 1 2<br />
] r=coo 30 _<br />
dA = rdrdO =<br />
2 r dO<br />
D -'11'/6 0 -'11'/6 r=O<br />
= {"' 16 _! cos 2 30 dfJ = 2 f" 16 _! ( 1 +cos 60 ) dO<br />
}_71: 16 2 } 0 2 2<br />
y<br />
y=x<br />
1[ 1 ]"'/ 6<br />
sin 60<br />
0<br />
= 2<br />
0 + G<br />
= ;;<br />
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SECTION 15.4 DOUBLE INTEGRALS IN POLAR COORDINATES 0 259<br />
17. In polar coordinates the circle (x- 1} 2 + y 2 = 1 ¢} x 2 + y 2 = 2x is r 2 = 2r cos 0 ~ 1' = 2 cos 0,<br />
and the circle x 2 + y 2 = 1 is r = 1. The curves intersect in the first quadrant when<br />
2 cos 0 = 1 ~ cos 0 = ~ · .~ 0 = 7r / 3, so the portion of the region in the first quadrant is given by<br />
D = {(r, 0} 11 ~ r ~ 2 cos 0, 0 ~ 0 ~ 7r /2}. By symmetry, the total area<br />
is twice the area of D:<br />
2A(D} = 2ffvdA = 2Ja"/3 j l2coao rdrdO = 2fo"/3 ar2)~~~coao dO<br />
O= Tr/3<br />
,/ r=2oos(J<br />
= J;/ 3 (4cos 2 0- 1) d(} = J;/ 3 [4 · ~( 1 +cos 20}- 1] dO<br />
= J 0<br />
" 13 (1 + 2 cos 20} dO= [0 + s~n20] ~;a = f + ~<br />
21. The hyperboloid of two sheets - x 2 - y 2 + z 2 = 1 intersects the plane z = 2 when -x 2 - y 2 + 4 = 1 or x 2 + y 2 = 3. So tlie<br />
solid region lies above the surface z = )1 + x 2 + y 2 and below the plane z = 2 for x 2 + y 2 :::; 3, and its <strong>vol</strong>ume is<br />
23. By symmetry,<br />
V = I I ( 2 - ) 1 + x 2 + y 2 ) dA = la 2 " 1av"J ( 2 - ) 1 + r2) r dr dO<br />
:r:2 +y2 ~3<br />
= f~" dO f ov"J (2r- rv'f+'Ti) dr = [ 0] ~.,.. [r 2 - t {1 + r 2 ?1 2 ] :<br />
= 27r (3-i- 0 + 1) = ~7r<br />
2<br />
V = 2 // Ja 2 - x 2 - y 2 dA = 21a<br />
"1aa )a 2 - r 2 rdrdO = 21az... dO loa<br />
~+~~~ .<br />
r Ja2 - r2 dr<br />
26. The cone z = Jx 2 + y 2 intersects the sphere x 2 + y 2 + z 2 = 1 when x 2 + y 2 2<br />
+ ( Jx 2 + y 2 ) = 1 or x<br />
2<br />
+ y 2 = ~· ~0<br />
V = /1 ()1- x2- y2- )x2 +y2) dA = la 2 7r lal/-/2 () 1- r2 - r)rdrdO<br />
.,2 + y2 ~ 1/ 2<br />
= f~1r dO f 0<br />
11 V2 (r Jl-T2- r 2 ) dr = [ 0]~ 7r [ - ~( 1 - r 2 ) 3 1 2 - ~r 3 ) :/-.12 = 271'( -~) ( ?z - 1) = f(2 - V2)<br />
27. The given solid is the region inside the cylinder x 2 + y 2 = 4 between the surfaces z = J64 - 4x2 - 4y2<br />
and z = - )64- 4x 2 - 4y 2 • So<br />
J 2 J 64 - 4x2 - 4y2 dA<br />
x2 + 11 2 s 4 +2 + 11 2J:<br />
2 ~ 4<br />
= 4J~ .,.. J~ ~ rdrdO = 4J:.,.. dO J: r v'16 - r 2 dr = 4 [ OJ~". [-1(16- r 2 ) 3 1<br />
V = I I [ J 64 - 4x 2 - 4y 2 - ( --: J 64 - 4x 2 - 4y 2 ) ] dA = J<br />
= 8Tr( - ~)( 12 3 1 2 -16 2 '<br />
3 ) = 8<br />
; .(64 - 24 v'3)<br />
© 2012 Cengage Learning. All Rights Rc"S
260 D CHAPTER 15 MULTIPLE INTEGRALS<br />
29.<br />
/<br />
3<br />
{~ sin(x 2 + y 2 )dy dx = {" r sin e) r dr dB<br />
-3lo · Jo Jo<br />
= fo" dB J; r sin (r 2 ) dr = [B) ~ [-~cos (r 2 )J~<br />
-3 0 3 X<br />
= 1r (-~) (cos9 -1) = ~ (1 - cos9)<br />
31.<br />
y<br />
f 0<br />
" 14 fov'2 (r cos B + r sin fJ) r dr dB == f 0<br />
" 14 (cos B + sin B) dB f 0<br />
v'2 1·2 dr<br />
== [sin(}- cosB ) ~/ 4 [ i r 3 ]~<br />
= ( ¥ - ¥-O+ 1] . i(2v'2- o) = ¥<br />
33. D = {( r-, B) [ 0::; r ::; 1, 0 ::; 8::; 271' }, so<br />
ff D e
SECTION 15.5 APPLICATIONS OF DOUBLE INTEGRALS 0 261<br />
·Y<br />
?r/4 2<br />
= r r 1rr/4 [<br />
r 3 4 ] r=2<br />
cos 8 sinfJdrd8= .!:.... cos8sin8 d()<br />
Jo J 1 o 4 r = 1<br />
151rr/ 4 . 15 [sin 2 ()] .,.; 4 15<br />
= - sm8cos9d(J = - -- = -<br />
4 0 4 2 0 16<br />
• . 2 2<br />
41. (a) We integrate by parts with u = x and dv =·xe-x dx. Then du = dx and v = - ~e-x , so<br />
f "" x 2 e- "' 2 dx = lim rt x 2 e-"' 2 dx = lim ( - l xe- "' )t 2 + rt l e- "' 2 dx)<br />
Jo t-+oo Jo t-+oo 2 0 Jo 2<br />
= lim (_.!.te-t 2) + .!. roo e-x 2 dx = 0 + .!. roo e- "' 2 dx<br />
t-+oo 2 2 Jo 2 Jo [by !'Hospital's Rule]<br />
1 f "" _.,2 d<br />
= 4 -oo e X<br />
[by Exercise 40(c)]<br />
2<br />
[since e-"' is an even function]<br />
(b) Let u = ..JX. Then u 2 = x .::? dx = 2udu .::?<br />
00<br />
f. 0<br />
.../Xe- "'dx = lim I; .../Xe-"'dx = lim I 0<br />
,/f.ue-" 2 00<br />
2u du = 2I 0<br />
u 2 e- u 2 du = 2 U.J7T)<br />
· t-oo # t-oo<br />
[by part(a)] = ~ft.<br />
15.5 Applications of Double Integrals<br />
1. Q = IIv O"(x, y) dA = I~ I: (2x + 4y) dydx =I~ [2xy + 2y 2 ] ~=~ dx<br />
= I~ (lOx + 50- 4x- 8) dx = J~ (6x + 42) dx = [3x 2 + 42x]~ = 75 + 210 = 285 C<br />
3. m = IIv p(x, y) dA = I 1<br />
3<br />
I 1<br />
4<br />
ky<br />
2<br />
dydx = k I 1<br />
3<br />
dx .{ 1<br />
4<br />
y<br />
2<br />
dy = . k [xJ ~ [h 3 ]~ = k(2)(21) = 42k,<br />
x = ~ IIv xp(x,y) dA = 4 ~k I : I 1<br />
4<br />
kxy<br />
2<br />
dydx = 4<br />
1<br />
2 I 1<br />
3<br />
xdx I 1<br />
4 y<br />
2<br />
dy = f2 [ tx 2 ]~ [k y 3 ]~ = 4\(4)(21) = 2,<br />
- 1 ff ( )dA - 1 f a f 4k a d dx- 1 f sdx r4 ad _ 1 [ ]3 [1 4] 4 - 1 ( 2 )( 2 11:;) 85<br />
Y = m D yp x , Y - 42k 1 . 1 Y y - 42 1 Jt y y - 42 X 1 4Y 1 - 42 -4- = 28<br />
Hence m = 42k, (x, Y) = (2, ~) .<br />
r2f3-x( ) r2 [ 1 2]v=3-:z: dx r2 [ ( a ) 1 ( 3 )2 1 2]<br />
5. m= Jo .,12 x+y dydx = Jo xy+2Y v=x/ 2 =J~ x 3- 2x + 2 - x - 8x dx<br />
= J 0<br />
2<br />
(-~x 2 + ~) dx = [- ~(tx 3 ) + ~x ] ~ = 5,<br />
. r2J·3-x( 2 )d dx r2 ( 2 1 2]11=8- :z: dx r2 (9 9 3) d 9<br />
lvfv = Jo :r:/ 2 x + xy y = Jo x y + 2xy 1l=x/2 = Jo 2x - gX x = 2•<br />
lvfx = Jo<br />
f2J3-v(<br />
.,<br />
2)d d r2.[1 2 1 3)11=3-x d r 2 (g 9 )dx 12 xy + y y x = Jo 2xy + 3Y<br />
9<br />
11 =, 12 x = Jo - 2x = .<br />
--) (lvfy Mx) (3 3)<br />
Hencem = 6, ( x_, y = m '--;:; = 4' 2 .<br />
7. m = J~ 1 J~ -"' 2 kydydx = k f~ 1 [ ~y 2 ]~=~-"' 2 dx = ~k f~ 1 (1 - x 2 ? dx = ~ k f~ 1 (1 - 2x2 + x 4 ) dx<br />
= ~k [x- ix 3 + %x 5 ] ~ 1 = ~k (1 - i + -k + 1- % + %) = 1<br />
85 k,<br />
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262 D CHAPTER 15 MULTIPLE INTEGRALS<br />
Mv ·= .f~ 1 ,{0 1 -:r 2 kxydydx = kf~ 1 axy 2 ]~=~- :r 2 dx = ~kf~ 1 x(1 - x 2 ) 2 dx = ~ kf~ 1 (x- 2x 3 + x 5 )dx<br />
- lk(lx 2 - lx 4 + lx 6 ] 1 - l k (1- l + l - l + l _ l) - 0<br />
- 2 2 2 6 -1 - 2 2 . 2 6 2 2 6 - '<br />
M x = J 1 r 1 -"'~ ky 2 dydx = kf 1 (1y 3 ] v=l-x2<br />
dx = lkf 1 (1 - x 2 ) 3 dx = lk { 1 (1'- 3x 2 + 3x 4 - x 6 )dx<br />
- 1 Jo -1 3 v=o s -1 3 . - 1<br />
H _ 8 k (- -) _ (o 32k/105) _ (o 4)<br />
encem - 15 , x , y - ,· Bkf l ~ - •7 ·<br />
9. Note thatsin(7Tx/ L) ~ 0 for 0 ::::; x::::; L.<br />
m = J:' J;in(rrxfL) ydydx = g t sin 2 (7Tx/L) dx =tax- 4 ~ s in (27rx /L) j~ = ~ L ,<br />
M. _ j 'Lj·sin(7rx/L) . d dx _ .!<br />
y - o o<br />
rL · 2( jL) dx<br />
X Y Y - 2 Jo X SID 11'X<br />
[<br />
integrate by parts with ]<br />
u = x , dv = sin 2 (7rx/ L) dx<br />
= t · x(tx - 4r;, sin (27Tx/L)) ] ~ - ~ J 0<br />
L [tx - 4 ~ sin(27Tx/ L)] dx<br />
= l£2- l [.!x2 + L2 cos(211'xj L)]L = .!£2 - l (.!£2 + L2 - L2) = l£2<br />
4 2 4 41f2' 0 4 2 4 41f2' 41f2' 8 '<br />
M:z: = J 0<br />
L J;in("'xf L ) y · y dy dx = f 0<br />
L ~ sin 3 (7Tx/ L ) dx = ~ f 0<br />
L [1 - .cos 2 (7Tx/ L )] sin(7Tx/ L ) dx<br />
[substituteu=cos(7Tx/L)) ::} du= -fsin(7Tx/L))<br />
= t(- ~)[cos(7Tx / L) - 1 cos 3 (7Tx/L)]~ = - 3~ (- 1 + ~ -1 + i) = 9~L .<br />
L _ _ (L 2 /8 4£/ (911')) (L 16)<br />
Hencem = 4' (x,y) = L/ 4 ' L /4 = 2 ' 971' .<br />
11 . p(x, y) = ky = kr sin O, m = )~,. 12 J; kr 2 sinO dr dB= ik )~.,. 12 sinO dtl = ik[- cosB]~ 12 = ftk,<br />
Mv = J 0<br />
,. 12 J; kr 3 sinBcos BdrdB = i k f 0<br />
" 12 sinB cosB dB= ~k [- cos 28] ~ 12 = lk,<br />
M x = J 0<br />
,. 12 J 0<br />
1<br />
kr 3 sin 2 B dr dB = ik f 0<br />
"' 12 sin 2 B dB = lk[B + sin 28] ~ 1 2 = fak.<br />
Hence (x, y) = (i, ~~) .<br />
13.<br />
y<br />
p(x, y) = k jx2 +y 2 = kr,<br />
m = ffo p(x, y)dA = f 0<br />
"' f 1<br />
2<br />
kr · r dr dB<br />
= k f 0 "' dB f 1<br />
2<br />
r 2 dr = k(11') [ ir 3 ]~ = ~11'k,<br />
= k [sinO]"' [.!r 4]2 = k(O) (.!&) = O . [this istobe expect<strong>ed</strong>astheregionanddensity<br />
0<br />
4<br />
1<br />
4<br />
function are symmetric about they-axis]<br />
M:r = ffo yp(x, y)dA = fo,. t<br />
SECTION 15.5 APPLICATIONS OF DOUBLE INTEGRALS D 263<br />
15. Placing the vertex opposite the hypotenuse at (0, 0), p(x, y) = k(x 2 + y 2 ). Then<br />
m = J; foa-"' k (x 2 + y 2 ) dy dx = k foa [ax 2 - x 3 + 4 (a - x) 3 ] dx = k[~ax 3 - tx 4 -<br />
By symmetry,<br />
1<br />
12 (a - x) 4 ]~ = ~ ka 4 •<br />
Hence (x, y) = (~a, ~a).<br />
17. I :r: = JJ 0<br />
y 2 p(x, y)dA = .{~ 1 J; -x 2 y 2 · ky dy dx = k j~ 1 [h 4 J~:~ -x 2 dx = tk f~ 1 (1 - :t2 ) 4 dx ·<br />
- l kf 1 (x 8 -4x 6 +6x 4 - 4x 2 + l)dx = l k [lx 9 - ix 7 + !!x 5 - :!x 3 + x] 1 = .M..k<br />
- 4 -1 4 9 7 5 3. - 1 315 '<br />
Iy = fio x 2 p(x, y) dA = }~ 1 j~ -"' 2 kx~ydydx = k I~ 1 [ ~x 2 y 2 J ~:~-"' 2 dx = ~k t 1<br />
x 2 (1- x 2 ? dx<br />
1 k It < 2<br />
2<br />
4 6) d 1 k [ 1 3 2 s 1 1] 1 _ 8 k<br />
= 2 • - 1 X - X +X X= 2 aX - s X + "'fX - 1 - 105 '<br />
and Io = I, + Iv = 36~k + t~sk = 38t8sk.<br />
19. As in Exercise 15, we place the vertex opposite the hypotenuse at (0, 0) and the equal sides along the positive axes.<br />
Ix = I~' I; -xy2k(x2 +y2)dydx = k.fo" .fo"- "'(x2y2 +y4)dydx = k.fo"[~x2y3 + h5J~:~-x<br />
dx<br />
= k f'a [lx 2 (a- x) 3 + l(a - x) 5 ] dx = k [l (la 3 x 3 - 1a 2 x 4 + !l.ax 5 - lx 6 ) - l (a- x) 6 ] a = - 1 -ka 6<br />
. 0 3 5 3 3 4 5 6 30 0 180 '<br />
Iv = I; I;-x x 2 k(x 2 +y 2 ) dydx = k J 0<br />
a.f;-"'(x 4 +x 2 y 2 )dydx = k .f 0<br />
a[x 4 y + ~x 2 y 3 J::~ - x<br />
dx<br />
- k j ·a [x 4 (a - x) + l x 2 (a - x) 3 ) dx = k [lax 5 - lx 0 + l (l a 3 x 3 - !l.a 2 x 4 + !!ax 5 - lx 0 ) ) a - - 1 - ka 6<br />
- 0 3 5 6. 3 3 4 5 6 0 - 180 '<br />
21. I,. = IIo y 2 p(x, y)dA = Io" I~ py 2 dx dy =pI: dx I~' y 2 dy = p[ x ]~ [ ~y 3 ]~ = pb(th 3 ) = tpbh 3 ,<br />
Iv = IIo x 2 p(x, y)dA = Io" I~ px 2 dxdy =pI: x 2 dx I~' dy = P[ tx 3 ]~ (y) ~ = ipb 3 h,<br />
l l b 3 h b 2<br />
and m = p (area of rectangle) = pbh since the lamina is homogeneous. Hence ¥ 2 = ...JL = L_ = -<br />
m pbh 3<br />
= 2 I :r: tpbh 3 h 2<br />
andy = - = --= -<br />
m pbh 3<br />
=><br />
23. In polar coordinates, the region is D = { (r, 8) I 0 ~ r ~ a, 0 ~ 8 ~ ~},so<br />
! 11 = .{[ 0<br />
x 2 pdA = f 0<br />
"" 12 J; p(r cos 8) 2 r dr d/J = p I 0<br />
"" 12 cos 2 dB .f 0<br />
o. r 3 dr<br />
= p(t8 + t sin 28)~ 12 (tr 4 )~ =~Of) (ta 4 ) = ftPa 4 rr,<br />
.l...pa4rr<br />
and m = p · A(D) = p · trra 2 since the lamina is homogeneous. Hence ¥ 2 = y 2 = ~ = - => x = y = ~<br />
2 .<br />
4pa 1r 4<br />
a2<br />
® 2012 Ccngogc Lc:uning. All Rights Re.~rwd . May not be scann<strong>ed</strong>. copie-d. or duplicat<strong>ed</strong>. or post<strong>ed</strong> to a publicly accessible website, in whole or in port.
264 D CHAPTER 15 MULTIPLE INTEGRALS<br />
25. The right loop of the curve is given by D = {(r, 0) I 0 ~ r :::; cos 20, - r./4 ~ () ~ 71'/ 4}. Using a CAS, we<br />
find m = f fo p(x,y) dA = ff 0 (x 2 + y 2 ) dA = r:~;4 f oco" 20 r 2 rdrd9 = ::. Then<br />
1 i'!r 64 j "'/4 [ os20 64 ~ Tr/4 1cos20 16384J2<br />
·x= - xp(x,y)dA= - (rcos9)r 2 rdrd8= - r 4 cos9dr d8= and<br />
m o 3 7r -Tr/4 o 371' - Tr/4 0 1039571'<br />
1 /lr 64 ~ rr/4 1 cos28 64 j"'/4 cos20<br />
y =- yp(x, y) dA = - 3<br />
. (rsin 9) r 2 rdrd(J = - 3<br />
r 4 sin 9drd0 = 0, so<br />
m D 7r -rr/4 o 7r -Tr/4 1o<br />
(- _) ( 16384v'2 o)<br />
x, y = 1039571' ' .<br />
The moments of inertia are<br />
I _ [J 2 ( )dA -'J "/4 rcos20( . (})2 2 d dB - J"/4 j ·cos20 s . 2(J d d(J- 571' 4<br />
:r -. D y p x, y - -Tr/ 4 Jo r sm r r r - - 7r/4 o r sm r - 384 - 105,<br />
ly = JJ 0 x 2 p(x, y) dA = J~~;4 J; 0 "<br />
571'<br />
Io = I, + Iy = 192<br />
28 (Hos0)<br />
2 r<br />
2 r dr dO = J~~~ 4 J 0 c"" 20 r 5 cos 2 0 drdO = :~ +<br />
1 ~ 5 , and<br />
27. (a) f(x, y) is a joint density function, so we know JfR2 j(x, y) dA = 1. Since f(x, Y).= 0 outside the<br />
rectangle [0, 1) x [0, 2], we can say<br />
Then 2C = 1 => C = t.<br />
J JR2 f(x,y)dA = f~oo f~oo f(x,y)dy 'dx.= f 0<br />
1<br />
J; Cx(1 + y ) dy dx<br />
= C f 1 x [y + ly 2 ]v= 2 dx = c' f 1 4xdx ~ C (2x 2 ] 1 = 2C<br />
Jo 2 v=;O Jo o<br />
1<br />
(b) P (X :::; 1, Y ~ 1) = f~oo f~oo f(x, y) dy dx = J 0<br />
J; tx(1 + y) dy dx<br />
r l 1 [ 1 2] Y = 1 d r1 1 ( 3) dx 3 [ 1 2 ] 1 3<br />
0 375·<br />
= Jo 2x Y + 2Y v=O x = Jo 2x 2 = 4 2x o = 8 or ·<br />
(c) P (X+ Y :::; 1) = P ((X , Y) ED) where Dis the triangular region shown in<br />
the figure. Thus<br />
· If 1 12: 1<br />
P(X+Y :s; 1)=<br />
0 f(x, y)dA = f 0 f 0 - 2x(1+y)dydx<br />
Y<br />
= t 1 lx[y + ly 2 ] v=l -:~: dx = f 1 lx(lx 2 - 2x + ~) dx<br />
Jo 2 2 11 = o Jo 2 2 2 D<br />
3 2 [ •I 3 2] 1<br />
= l 1 (x - 4x + 3x) dx = l L - 4L + 3L<br />
40 44 3 2()<br />
= 4~ ~ 0. 1042<br />
0<br />
1 X<br />
. 29. (a) f(x, y) ;;::: 0, so f is a joint density function if Jf'it2 f(x, y) dA = 1. Here, f(x, y) = 0 outside the fi rst quadrant, so<br />
JJR2 f(x,y)dA= f ooo fo":'0.1e-(o.ox+?·211) dydx = 0.1 J~oc f ooo e-0.5:re-0.2y dy dx = O.l fooo e-o.sx dx f ooo e-0.2y dy<br />
= 0. 1 lim r t e- 0·5"' dx lim rt e- 0·2 Y dy = O.l. lim (- 2e- 0·5"'] t lim [- se- 0·211 ] t<br />
L--+oo Jo t-oo Jo t.--+eo 0 t --+oo 0<br />
= 0.1lim [- 2(e- 0 ·5t<br />
- 1)] lim [- 5(e- 0·2 t - 1)] = (0.1). (-2)(0 - 1) · (-5)(0 - 1) = 1<br />
t --+oo<br />
t --+oo<br />
Thus f(x, y) is a joint density function.<br />
. '<br />
© 2012 Ceng;~gc Le:uning. All Rights Rcsc:n·cd. lvfuy not be ocanncd, copi<strong>ed</strong>, or duplicat<strong>ed</strong>. or JXh1cd 10 a publicly acccssiblc website, in whole or in pan.
SECTION 1505 APPLICATIONS OF DOUBLE INTEGRALS 0 265<br />
(b) (i) No restriction is plac<strong>ed</strong> on X, so<br />
P(Y ~ 1) = f~oo I 1<br />
00<br />
f (x,y) dy dx = I oo Jt 0°1e-(OoSx+0. 2 y) dydx<br />
= o ."1 roo e- 0 ·5"'<br />
dx f 00 e- 0·211 dy = oo1 tim rt e - o.s., dx lim It e- 0· 211 dv<br />
Jo 1 t-+oo Jo t-oo 1<br />
= 001 lim [-2e- 0 ·5"']t lim [-5e- 0·211 ]t = 001 lim [- 2(e- 0·5 ' - 1)] lim [- 5(e- 0 ·2 ' - e- 0 ·2 )]<br />
t-oo 0 t-oo 1 t-+oo t-oo<br />
(0.1) 0 ( -2)(0- 1) 0 ( - 5)(0 - e- 0·2 ) = e- 0 ·2 ~ 008187<br />
(ii) P(X $2, Y $ 4) = I~ oo I~ oo<br />
(c) The expect<strong>ed</strong> value of X is given by<br />
f (x,y) dyd.x = I 02<br />
J 0<br />
4 Oo1e-<<br />
0·5"'+ 0·2Y) dydx<br />
= 001 1~ 2 e- o.sx dx 1~ 4 e- 0 ·2 ll dy = 0.1 [ -2e- 0 ·5"']<br />
~ [.- 5e - 0 · 211 ]~<br />
= (0.1) · (-2)(e- 1 -1) · (- 5)(e- 0·8 -1)<br />
= (e- 1 - 1)(e- 0 ·8 - 1) = 1 + e-J.B - e- o.s - e- 1 ~ 0.3481<br />
Mt = IIR2 x f(x, y) dA = 1~ 00 1~ 00 :z; [oo1e-(o.sx,+o. 2 v)J dydx<br />
= 0.1 ]; 00 xe-O.ux dx ]; 00 e- 002 !1 dy = 001 lim ];t xe- 005 "' dx lim rt e- 0 ·2 Y dy<br />
0 0 t~oo o t-oo ./o<br />
To evaluate the first inteirat, we integrate by parts with u. = x and dv = e- 0·5"' dx (or we can use Formula 96<br />
in the Table of integrals): I xe- 0 ·5"'<br />
dx = - 2xe- 0·5"'- J - 2e- 0·5"' do'l = -2xe- 0·5"' -<br />
4e- 0 ·5 "' = - 2(x + 2)e- 0 ·5"'0<br />
Thus<br />
JJ. 1<br />
= 0°1 lim [-2(x + 2)e- 0·5 "'] '· lim [- 5e- 0·2ll] t<br />
t-+oo 0 t -+oo 0<br />
= 001 lim ( -2) [ (t + 2)e- 0 ·5t<br />
- 2] lim ( - 5) [e- 0·2 t - 1]<br />
t -...oo<br />
t-+oo<br />
The expect<strong>ed</strong> value ofY is given by<br />
= 0.1(-2)(lim t +. 2 - 2)(- 5)(- 1) = 2 [by !'Hospital's Rule]<br />
t -+oo eO.at<br />
J1.2 = Ifa2 y f(x , y) dA = Io 00 fa"" v [oo1e-(0.5+0. 2 y)] dy dx<br />
= 001 ]; 00 e- 0 ·5"'<br />
dx {; 00 ye- 0 ·211 dy = 001 lim {;t e-o.sx dx lim rt ye- 0 · 2Y dy<br />
0 . 0 t-oo. o t-oo _Jo<br />
To evaluate the second integral, we integrate by parts with u = y and dv = e - 0·2 !1 dy (or again we can use Formula 96 in<br />
the Table oflntegrals) which gives I ye- 0·211 dy = - 5ye- 0·2ll + .r se- 0·2 11 dy = -5(y + 5)e- 0 ·2U<br />
0 Then<br />
JJ. 2 = 0.1 lim [- 2e- 0 · 0·"] t lim [- 5(y + 5)e- 0 ·2 Y] '<br />
t-oo 0 t-+oo 0<br />
= Oollim [- 2(e-o.st _ 1)] lim (- 5[(t + 5)e- 0 ·2t- 5])<br />
t-+oo<br />
t -oo<br />
0<br />
=Oo1(- 2)(- 1)o (-5) lim -<br />
t+5 )<br />
( 0<br />
.,t - 5 = 5<br />
t- oo e ....<br />
[Py )'Hospital's Rule]<br />
310 (a) The random variables X and Y are normally distribut<strong>ed</strong> with JJ. 1 = 45, JJ. 2 = 20, 0'1 = 005, and a 2 = Ool.<br />
The individual density functions for X and Y, then, are !J (x) =<br />
h (y) =<br />
~ e-
266 D CHAPTER 15 MULTIPLE INTEGRALS<br />
f(x, y) = /I(x)/2(y) = 1 e-(x-45) 2 /0.s 1 e-(y-20) 2 /0.02 = lQe - 2 (x-45) 2 -so(v -2~)2.<br />
0.5 ..,l2ir 0.1 ..,f2ir " .<br />
Then P(40
-I<br />
SECTION 15.6 SURFACE AREA 0 267<br />
15.6 Surface Area<br />
1. Here z = f(x, y) = 2 + 3x + 4y and Dis the rectangle [0, 5) x [I, 4), so by Formula 2 the area of the surface is<br />
A(S) = ffD J[f,(x,y)] 2 + [! 11 (x,y)]2 + I dA= ffo -../3 2 +4 2 +IdA= ../26ffDdA<br />
= ../26 A( D) = ../26 (5)(3) = I5 ../26<br />
3. z = f(x, y) = 6- 3x - 2y which intersects the xy-plane in the line 3x + 2y = 6, 'soD is the triangular region given by<br />
{(x,y) I 0:5 x :5 2,0:5 y :53- ~x}. Thus<br />
A(S) = JJD J( -3) 2 + ( -2) 2 +IdA= .JI4 ffo dA = JI4 A( D ) = JI4 (~ · 2 · 3) = 3 .JI4<br />
5. y 2 + z 2 = 9 =* z = ~ · fx = 0, fv = -y(9 - y 2 ) - 112 =*<br />
A(S) = h412 Jo2 + [- y(9- y2)-1/2)2 + 1 dy dx = h4 h 2 J 9 ~2y2 + I dy dx<br />
= t { 2 3<br />
lo lo J9 - y 2 lo · 3 v=O<br />
7. z = f(x, y) = y 2 - x 2 with I ::; x 2 + y 2 ::; 4. Then<br />
dydx = 3 t [sin- 1 IL] 11 = 2 dx = 3 [(sin- 1 (~))x]~ = I 2sin- 1 (~)<br />
A( S) = J f D J 1 + 4x 2 + 4y 2 2 2<br />
dA = .{ 0<br />
2" f 1 Vf+"4r2 r dr dO = J:.,. dO J 1<br />
r -../I + 4r2 dr<br />
= (0]~" u2(I + 4r 2 ) 3 1 2 J: = {f(17vT7 - 5v'5)<br />
9. z = f(x, y) = xy with x 2 + y 2 :5 1, so / ex = y, f v =X =*<br />
A(S) = JJ Jy 2 + x 2 + 1dA = t" r 1 -../r 2 +I r dr dO= J; 2 " [l(1·2 + 1) 3 1 2 ] r=l dO<br />
. D 0 J o 0 3 r=O<br />
= I:'lr H2-../2 - I) dO= 2 I(2-../2 - 1)<br />
A(S) = !L<br />
x2 +y2<br />
2 2 2+ IdA<br />
a - x - y<br />
"/2 1 acos6 / r2<br />
= /_ --+ 1rdrd0<br />
- 1r/2 o a2 - r2<br />
a coso ar<br />
= drdO<br />
/_<br />
- .r/2 o · Va 2 - r 2<br />
'lr f21<br />
= /_"/2 [-a Ja2...:. r2 ] r=acose dO<br />
- 1rj2 r=O<br />
1r/2 1 -rr/2<br />
= -a( J a2 - a 2 cos 2 0-a) dO = 2a 2 (I - Vl - cos 2 0) dO<br />
/_<br />
-1r/2 0<br />
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268 0 CHAPTER 15 MULTIPLE INTEGRALS<br />
2 2 2 2 2 2<br />
13. z = f(x, y) =e-x - v , f.,= -2xe-x - v , fv = -2ye-x - v . Then<br />
A(S)= II v(- 2xe- xLv 2 )2+(- 2ye- x 2 -v 2 )2+ 1dA =<br />
x2+y2_54<br />
Converting to polar coordinates we have<br />
If .j4(x2-t-y2)e- 2(x 2 +v 2 l-t- ldA.<br />
x2+1125-1<br />
A(S) = J:" .{ 0<br />
2<br />
J 4r 2 e-' 2 r 2 + 1 r dr d(J = ;;11: dl1 .{ 0<br />
2<br />
r V 4T 2 e- 2 '' 2 + 1 dr<br />
= 27!' .r; r V 4r 2 e- 2 r 2 + 1 dr ~ 13.9783 using a calculator.<br />
15. (a) The midpoints of the four squares are (i 1 i ), (i 1 %) , (%, ~)~and ( i 1 ~). Here f(x 1 y) = x 2 + y 2 , so the Midpoint Rule<br />
gives<br />
A(S) = JJD .j[fx(X 1 y)F + [fv(x, y)J2 + 1 dA = jjD .j(2x) 2 + (2y) 2 + 1 dA<br />
~ ~ ( J[2(~)J 2 + (2(iW + 1 + V[2(~) J 2 + [2(t~] 2 + 1<br />
+ V[2(~)] 2 + [2(i)] 2 + 1 + V[2(%W + [2(~)] 2 + 1)<br />
= ~ ( .ft + 2 JJ + /¥) ~ 1.8279<br />
(b) A CAS estimates the integral to be A(S) = J[D .)1 + (2x ) 2 + (2y )2 dA = .1; J 0<br />
1<br />
.)1 + 4x 2 + 4y 2 dy dx ~ 1.8616.<br />
This agrees with the Midpoint estimate only in the first decimal place.<br />
17. z = 1 + 2x + 3y -t--4y 2 , so<br />
4<br />
·A(S) = 11 1 + ( ~~r + (~~r dA = 1.f Using a CAS, we have J 1<br />
4<br />
45 'P4 15 ln 11 J5 + 3 v'70<br />
or- V J. '% + -<br />
8 16 3 J5 +. v'70 .<br />
4<br />
v 1 +4+(3 + 8y)2dydx = 1<br />
\<br />
1<br />
V 14+48y+64y2dydx.<br />
I~ .)14 + 48y + 64y 2 dy dx = ¥ .Jl4 + ~ ln(ll J5 + 3 .Jl4J5) - t~ ln(3 J5 + v'i4 J5)<br />
19: f(x, y) = 1 + x 2 y 2 => f, = 2xy 2 , fv = 2x 2 y. We use a CAS (with precision r<strong>ed</strong>uc<strong>ed</strong> to five significant digits, to spe<strong>ed</strong><br />
up the calculation) to estimate the integral<br />
A(S) =<br />
1<br />
V<br />
!1~ 1!1~ '<br />
fi + F~ + 1 dy dx = ..j 4x 2 y 4 + 4x 4 y 2 + 1 dy dx, and find that A(S) ~ 3.3213.<br />
- 1 - vh-x.2 . - 1 -~ ·<br />
21. Here z = f(x , y) = ax+ fnJ + c, f.,(x 1 y) =a, / 11 (x, y) = b, so<br />
I<br />
A(S) = JJD ,Ja 2 + bz + 1 dA = ,ja2 + b 2 + 1JJD dA = ,Ja 2 + b 2 + 1 A( D).<br />
23. If we project the surface onto the xz-plane, then the surface lies "above" the,' disk x 2 + z 2 ~ 25 in the x z-plane.<br />
We have y = f(x, z) = x 2 + z 2 and, adapting Formula 2, the area of the surface is<br />
A(S) = J{ .j(f,.(x,z)]2+[fz(x,z)]2+1dA = JJ ,J4x 2 +4z 2 +1dA<br />
,;2 +z2 .$25 ,2 +=2 .$25<br />
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SECTION 15.7 TRIPLE INTEGRALS 0 269<br />
Converting to polar coordinates x = r cos fJ; z = r sin fJ we have<br />
15.7 Triple Integrals<br />
If<br />
, r 2 ri r3 J2 2 d d dx ri r3 [ 1 2 2] v=2 d d · ri r3 3 2<br />
1. JB<br />
d d<br />
xy~ dV = Jo Jo _ 1 xyz y z = Jo Jo 2xy z v=- 1 z x = Jo Jo 2 x z z x<br />
= r1 (lxz3]==3 dx = fl TI xdx = 27 x2] 1 = TI<br />
Jo 2 ==O Jo 2 4 o 4<br />
2 " 2 ru- · . r2 r· 2 [ 2 ] x=y- z r2 r= 2 [( 2 ]<br />
3. J 0 J; Jo "(2 x - y)cJ:J: dydz=Jo J; x - xy x=O dydz =Jo Jo y- z) - (y-z)y dydz<br />
2<br />
2<br />
= J; J 0 " (z - yz) dydz = J~ 2 [yz 2 - h 2 z ] ~ =~ 2 2<br />
dz = J 0<br />
(z 4 - %z 5 ) dz<br />
5 ! 2 r2= r tn x - y d dxd = !2 r2= [- -v] v=ln:r; d d = !2 r2z (- - ln x + 0) dxd<br />
. 1 Jo Jo x e y z 1 Jo x e v=O x z 1 Jo xe xe z<br />
= Jt J:= (-i + x) dxdz = g [-x + %x 2 J::~= dz<br />
· = f 1<br />
2 (- 2z + 2z<br />
2 ) dz = [- z 2 + ~ z 3 ] ~ = - 4 + 1 36<br />
+ 1 - ~ = ~<br />
7. J 0<br />
.,. 12 J~ J 0 "' cos(x+y+z)dzdxdy = j~rr/ 2 g [sin(x+y +z)J::~ dxdy<br />
= f 0<br />
.,. 12 J~ (sin(2x + y) - sin(x + y)J dx dy<br />
= j~,./ 2 [- % cos(2x + y) + cos(x + y)J::~ dy<br />
= f 0<br />
,. 12 [- ~ cos 3y + cos 2y + ~ cos y - cos y] dy<br />
1. . 3 1 . 2 1 . ] .,./2 1 1 1<br />
= [-6 SID y + 2 S Ln y - 2 sm y 0 = G. - 2 = -S<br />
9. JJJE ydV = J; fa"' J;::: ydzdydx = J; J; [yz]~== ~~ dydx = f 0<br />
3<br />
f0"' 2y 2 dydx<br />
- r3 (1y3] y = x dx - r3 1x3 dx - lx4] 3 - 81 = ll<br />
- Jo 3 11=o - Jo 3 - G o - 6 2<br />
[z · .! tan -l ~] x = z dz dy<br />
} } E X + Z 1 71 0 X + Z 1 71 Z Z x =O<br />
11. / " ( ( - 2<br />
- z-<br />
2 dV = 14141= ~ dx dz dy = 1414<br />
4 4<br />
= f 1 f 11<br />
(tan- 1 (1)-tan- 1 4 4<br />
(o)] dzdy = f 1 I:(~- o) dz' dy = ~ f 1<br />
[zJ::: dy<br />
= ~ f 1<br />
4<br />
(4-y)dv= ~ [4v - h<br />
2<br />
]~ · = ~ (16-8 - 4+~) = 9 ;<br />
13. Here E = {(x,y,z) I 0 :::; x:::; 1, 0 :::; y:::; -fo, O :::; z :::; 1 +x +y}, so<br />
JJJE 6xydV = f 0<br />
1<br />
fov'X J;+.,+u 6xydzdydx = J~ fov'X [6xy zJ : : ~+"'+ 11 dydx<br />
= f~ fov'X 6xy(1 + x + y) dy dx = / 0<br />
1<br />
(3xy 2 + 3x 2 y 2 + 2x~ 3 ) ~=~ dx<br />
= t (3x2 + 3x3 + 2x5f2) ·dx = [x3 + ~x·l + :!x7f 2] 1 = 65<br />
0 4 7 0 28<br />
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270 D CHAPTER 15 MULTIPLE INTEGRALS<br />
15.<br />
HereT = {(x, y,z) I 0 $ x $ 1, 0 $ y $ 1 - x, O $ z $ 1- x - y}, so<br />
Jffr x2 dV ::::: fot J;-a: fot -a:-y x2 dz dydx = fot fot-x x2(1-x- y) dy dx<br />
= J; fot -a:(x2- ~3 - x2y) dy dx = fot [x2y.- xay - t X2y2J::~- a: dx<br />
= J; [x 2 (1 - x)- x 3 (1 - x) - t x 2 (1- x?] dx<br />
= j' 1 (lx 4 - x 3 + lx 2 ) dx = [...!. x 5 - l x 4 + l x 3 ) 1<br />
0 2 2 10 4 . 6 0<br />
1 1 1 1<br />
- w - 4+ s = oo<br />
17.<br />
The projection of E on the yz-plane is the disk y 2 + z 2 $ 1. Using polar<br />
coordinates y = r cos 8 and z = r sin 8, we get<br />
f.ffe x dV = ffv [J4~2 +4:2 x d.x] dA = ~ ffv [4 2 -<br />
(4y 2 + 4z 2 ) 2 ) dA<br />
= 8 J;"" J:(1- r 4 ) 1·drd8 = 8 J~ 2 " dB J 0 \ r - r 5 ) dr<br />
X<br />
_ 8(211") [lr2 _ lr6] I _<br />
16rr<br />
- 2 6 0- 3<br />
19. The plane 2x + y + z = 4 intersects' the xy-plane when<br />
2x + y + 0 = 4 => y = 4 - 2x, so<br />
4<br />
E = {(x, y, z) I 0 $ x $ 2, 0 $ y $ 4- 2x, 0 $ z $ 4 - 2x - y} and<br />
V = fo2 J~l -2 a: fo4-2x-v dz dy dx = fo2 fo4- 2:r. ( 4 - 2x - y) dy dx<br />
r2 [4 2 1 2] y =4.- 2x dx<br />
= Jo y - xy - 2 Y v= O<br />
= J; [4(4 - 2x)- 2x(4 - 2x) - t(4 - 2x?] dx<br />
= J; (2x 2 - 8x + 8) dx = [ ~x 3 - 4x 2 + 8x]~ ='¥<br />
X<br />
y<br />
21. The plane y + z = 1 intersects the xy-plane in the line y = 1, so<br />
E = { (x, y, z) I. - 1 $ x $ 1, x 2 $ y $ 1, 0 $ z $ 1 - y} and<br />
V = JJJ ~ dV = .C 1<br />
J:2 f 0<br />
1 - 11 dz dy dx = f~ 1 f:.2 (1 - y) dy dx<br />
1 [ 1 2] v=1<br />
!<br />
d }'1 (1 '.!<br />
=<br />
1 ") d<br />
-1 y- 2Y y=x2 X = - 1 2 - X + 2X X<br />
_ [ 1 1 3 + 1 5 ] 1 1 1 1 l 1 1 _ R<br />
- 2X - .3 X lOX· - 1 = 2 - 3 + 10 + 2 - J + lO - 15<br />
(- 1, 1, 0)<br />
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SECTION 15.7 TRIPLE INTEGRALS D 271<br />
23. (a) The w<strong>ed</strong>ge can be describ<strong>ed</strong> as the region<br />
D = {(x,y,z) I y 2 + z 2 ::; 1, 0 ::; x::; 1, 0 ::; y ::; x}<br />
= { (x, y , z) I 0 ::; x ::; 1, 0 ::; y ::; x, 0 ::; z ::; } 1 - .Y 2 }<br />
So the integral expressing the <strong>vol</strong>ume of the w<strong>ed</strong>ge is<br />
JffodV=fo1 fo"'J/1 -v2 dz dydx.<br />
(b) C S . r l f"' n /l.-vz d d d " l<br />
A A gives Jo Jo Jo z y x = 4 - 3·<br />
(Or use Formulas 30 and 87 from the Table of Integrals.)<br />
25. Here f(x, y, z) = cos(xyz) and~ V = ~ · t · ~ = ~,so the Midpoint Rule gives<br />
1 [!(1 1 1) J(l 1 3) !(1 3 l) f(l 3 3)<br />
= s 4 • 4•4 + 4•4• 4 + 4•4• 4 + 4 • 4 •4<br />
+ !(~ , t. t) + !(~ , t. ~) + !(~, ~, t) + ! (~, ~. ~ ) ]<br />
= ~ [cos -l4 +cos ~ +cos 6~ +cos f4: +cos 6~ +cos i 4 +cos f4: + cos~~ ) ~ 0.985<br />
27. E = {(x, y, z) I 0 ::; x::; 1, 0::; z::; 1- x, 0 ::; y::; 2- 2z },<br />
the solid bounde~ by the three coordinate planes and the planes<br />
z = 1 - x, y = 2 - 2z.<br />
X<br />
29.<br />
y<br />
4<br />
X<br />
I<br />
---~"--x' + 4z 1 = 4<br />
.t<br />
- I<br />
If D 1 , D 2 , D 3 are th~ projections of Eon the xy-, yz-, and xz-planes, then<br />
D 1 = {(x, y) I - 2::; X::; 2, 0 ::; y::; 4 - x 2 } = {(x, y) I 0 ::; y::; 4, - v4 - y::; X::; .J4=Y }<br />
I .<br />
D2 = {(y,z) 1 o ::; v ::; 4, - ~v4- y::; z::; ~v4- v} = {(y,z) 1-1 ::; z::; 1, o::; v ::; 4 - 4z 2 }<br />
DJ = {(x,z) I x 2 +4z 2 ::; 4}<br />
[continu<strong>ed</strong>]<br />
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272 0 CHAPTER 15 MULTIPLE INTEGRALS<br />
Therefore<br />
E = { (x y z) 1 -2 < x < 2 0 < y < 4 ·_ x 2 - 1 yl 4 - :z;2 - y < z < 1 yl 4 - :z;Z - y }<br />
, , - -'-- , 2 - - 2<br />
= { (x, y,z) I 0:::; y:::; 4, - y'4""='1j:::; x:::; y'4""='1j, - h/4 - x 2 - y:::; z ~ ~yf4- x 2 - y}<br />
= { (x,y,z) I -1 $ z 5: 1, 0 5: y 5:4 - 4z 2 , - yf4 - y- 4z 2 :::; x:::; yf4 - y - 4z 2 }<br />
= { (x, y, z) I 0 5: y 5: 4, - ty'4""='1j $ z $ ty'4""='1j, - j4-y- 4z 2 :::; x:::; yf4 - y- 4z 2 }<br />
Then<br />
= { ( x, y, z) I - 2 5: x 5: 2, - ~ J 4 - x 2 :::; z :::; t J 4 - x~, 0 :::; y :::; 4 - :z: 2 - 4z 2 }<br />
= { (x, y, z) I -1 5: z 5: 1, - .J4 - 4z 2 :::; x:::; .J4- 4z 2 , 0:::; y:::; 4- x 2 - 4z 2 }<br />
Iff sf ( x,y,z ) dV = !_ 2 ; -~/ 0 2 / x,y,z dzdydx=:Jo --14-ii -~ 1 2<br />
2 r4-x2 JJ4-x2-yf2 ( ) r4 f-14-ii JJ4-:z:2-yj 2 ( )<br />
! x,y,z dzdxdy<br />
I I f4- 4z2 JV4-y-4z2 ( ) , f4 f-/4-!i/2 J V4-y-4z 2 (<br />
= - tJo -.J _ _ z 4 11 4 2 f x, y, z dx dy dz = Jo --14-ii/ 2 - J _ _ : 4 11 4 2 f x, y, z) dx dz dy<br />
f 2 J~/2 r4-x2 -4:2' . ) fl J J4- 4:;2 r4-o: 2 -4: 2 ( )<br />
= _ 2<br />
r;--:; Jo f(x,y,z dydzdx= _ 1 ~Jo f x,y,z dydxdz<br />
-y4- :z:2/2<br />
-y4-4z2<br />
31 .<br />
y<br />
(- 2,4,0;<br />
JC<br />
(2,4,0)<br />
X<br />
4 y - 2 2 JC<br />
If D1, Dz, and Da are the projections of Eon the xy-, yz-, and xz-planes, then<br />
D1 = { (x, y) I -2 $ x 5: 2, :z: 2 ~ y 5: 4} = { (x, y) I 0 :::; y :::; 4, -y'Y $ x:::; vY },<br />
D2 = { (y,z) I 0 :::; y $ ,4, 0 :::; z 5: 2- ~y } = { (y,'z) I 0:::; z 5: 2,0 ~ y:::; 4 -<br />
2z}, and<br />
Ds = { (x, z) I - 2 $ x :::; 2, 0 5: z ~ 2 - ~~ 2 } = { (x, z) I 0 :::; i ~ 2, - .J4- 2z :::;. x $ v'4 - 2z}<br />
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Therefore<br />
E = { (x, y, z) I -2 ~ X~ 2, x 2 ~ y ~ 4, o' ~ z ~ 2- h}<br />
= { (x,y,z) I 0 ~ y ~ 4, -Jv ~ x ~ V"Y. 0 ~ z ~ 2 - tv}<br />
= { (x,y,z)I O ~y~4,0 ~z~ 2-ty , -vv~ x~ JY}<br />
= { (x, y, z) I 0 ~ z ~ 2, 0 ~ y ~ 4- 2z, - VY ~ x ~ vY}<br />
SECTION 15.7 TRIPLE INTEGRALS 0 273<br />
= { (x,y,z) l-2 ~ x ~ 2, 0 ~ z ~ 2 - tx 2 , x 2 ~ y ~ 4 - 2z}<br />
= { (x, y, z) I 0 ~ z ~ 2, - v'4 - 2z ~ x ~ y'4- 2z, x 2 ~ y ~ 4 - 2z }<br />
Then 1JJ~ f(x, y, z ) dV = I~ 2 I:2 I~ - 1112 f(x,y,z) dzdydx = I 0<br />
4<br />
I~ I 0<br />
2<br />
-v<br />
12<br />
f (x, y, z) dz dxdy<br />
33. y<br />
= 1~ 4 J~ - v/ 2 1~f(x , y ,z) dxdz dy =I~ Io 4 - 2 " J~f (x,y,z)dxdydz<br />
2 f 2 - %2 / 2 !4- 2::<br />
J<br />
J( ) d d d r2 Jv'4=2z f4- 2z ( ) .<br />
= -2 Jo z2 x , y,z y z ·X= Jo - "'4 - 2z z2 I x ,y ,z dydxdz<br />
The diagrams show the projections<br />
of Eon the xy-, yz-, and xz-planes.<br />
-~------+--+<br />
0 I X<br />
Therefore<br />
Io 1 1<br />
I{z 1~l - v f (x, y, z ) dz dydx =I: I/ I:-rJ f(x, y, z) dzdx dy = f 0<br />
Iol-,z It f(x, y, z) dx dy dz .<br />
rt rt- u1'tJ 2 J( ) d d rt rt-../XJ1- z (<br />
= JoJo 0 x,y,z x zdy= Jo Jo ..;x f x,y,z)dydzdx<br />
fl /·(l - z) 2 Jl-z ( )<br />
= J o. 0 ..;x f x,y,z dydxdz<br />
35.<br />
y<br />
z<br />
X<br />
I: I:Icf f(x,y, ~)dzdxdy = IIIE f(x, y ,z)dV where E = {(x,y,z) I 0 ~ z ~ y, y ~ x ~ 1, 0 ~ y ~ 1}.<br />
If D 1, D2, and D3 are the projections of Eon the xy-, yz- and xz-planes then<br />
Dt = [(x, y) I 0 ~ y ~ 1,, y ~ x ~ 1} = {(x, y) I 0 ~ x ~ 1, 0 ~ y ~ x},<br />
D2 = [(y,z) I 0 ~ y ~ 1, 0 ~ z ~ y} = {(y,z) I 0 ~ z ~ 1, z ~ y ~ 1}, an~<br />
D3 = [(x,z) I 0 ~ x ~ 1, 0 ~ z ~ x} = { (x, z ) I 0 ~ z ~ 1, z ~ x ~ 1}.<br />
[continu<strong>ed</strong>]<br />
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274 0 CHAPTER 15 MULTIPLE INTEGRALS<br />
Thus we also have<br />
E = {(x, y, z) I 0 ::::; x::::; 1, 0 ::::; y ::::; X, 0 ::::; z::::; y} = {(x, y, z) I 0 ::::; y::::; 1, 0::::; z.::::; y, y ::::; x ::::; 1}<br />
Then<br />
= {(x,y,z) I 0 ::::; z::::; 1, z::::; y::::; 1, y::::; x ::::; 1} = {(x,y,z) I 0 ::::; x $ 1,0::::; z ::::; x, z::::; y ::::; x}<br />
= {(x,y, z) I 0::::; z::::; 1, z::::; x::::; 1, z::::; y::::; x}.<br />
J; fv 1 f~ f(x,y, z) dzdx_dy = f 0<br />
1<br />
J; f~ f(x,y,z) dz dydz = f 0<br />
1<br />
g 1: f(x,y, z ) dxdz dy<br />
{=]; J; I: f(x,y,z) dxdyd~ = J; J; f:'" f(x,y, z) dydzdx<br />
"~--<br />
= f 0<br />
1<br />
f~ 1 [;"' f(x, y, z) dy dx dz<br />
37. The region Cis the solid bound<strong>ed</strong> by a circular cylinder of radius 2 with axis the z -axis for -2 ::::; z $ 2. We can write<br />
fff 0<br />
(4 + 5x 2 yz 2 ) dV = fffc 4dV + fffc 5x 2 yz 2 dV, but f(x,y,z) = 5x 2 yz 2 is an odd function with<br />
. respect toy. Since Cis symmetrical about the xz-plane, we have fffc 5x 2 yz 2 dV = 0. Thus<br />
39. m = JJJE p(x, y,z) dV = f 0<br />
1<br />
fofi fol+x+v 2dz dydx = J: J 0<br />
fi 2(1 + x + y) dydx.<br />
= rl (2y + 2xy + y2] y=fi dx = rl (2 v'x + 2x3/ 2 + x) dx = [ix3/ 2 + ix5/2 + lx2] 1 = 79<br />
Jo y=D Jo 3 s 2<br />
'<br />
My: = fffE xp(x, y, z ) dV = f 0<br />
1<br />
fofi J~ +o:+v 2x dz dy dx = J; fofi 2x(1 + x + y) dydx<br />
Mx~<br />
0<br />
so<br />
, = r1 [2xy + 2 x2y + xy2]v=fi dx = r1(2xs12 + 2xsf2 + x2) dx = [ixo/2 + ix7/2 + J.x3] 1 = 179<br />
J o v=D J o . s 1 3<br />
= JJJE yp(x,y,z)dV = f 0<br />
1<br />
fofi J;+x+v 2ydzdydx = J 0<br />
1<br />
f 0<br />
fi2y(1 +x+y)dydx<br />
= r1 [y2 + xy2 + 1y3] v=fi dx = r1 (x + x2 + 1x3/2) dx = [lx2 + lx3 + ~x:> /2) 1 = 11<br />
Jo 3 v=O Jo 3 2 3 15<br />
. fff ( ) dV r l rfi r1+:r+v 2 d d .d J·l J·./i. [ 2] ~=l+ :r+v d. d r1 r./i. (1 )2 d d<br />
lvlxv = E zp x, y, z = Jo Jo Jo z z y x = 0 0 z ==D y x = Jo Jo + x + y y x<br />
= J 0<br />
1<br />
J/"(1 + 2x + 2y + 2xy + x 2 + y 2 ) dydx = J; [y + 2xy + y 2 + xy 2 + x 2 y.+ ty 3 J::f dx<br />
_ r1 ( 'x + 1x3/2 + x + x2 + x5/2) dx _ ['lx3/2 + llx5/ 2 + l x2 + lx3 + 'lx7/2) 1 _<br />
- Jo<br />
571<br />
v"' 3 - 3 1s 2 · 3 7<br />
0 - 2 10<br />
. 79 . (- __ ) (My= M:r= Mxv ) (358 33 571)<br />
Thus the mass IS 30 and the Center Of mass IS X, Y, Z = --:;;:;:--, --:;;:;:--, m = 553 ' 79 > 553 ·<br />
0<br />
0<br />
10<br />
105<br />
Mv: = J; f 0<br />
a f 0<br />
a [x 3 + x(y 2 + z 2 )] dxdydz = J; J; [ta 4 + ~a 2 (y 2 + z 2 ) ] dydz<br />
_ ru (.1 s 1 s 1 a 2) d 1 6 1 6 1 6 M M b fE d ( )<br />
- Jo 4a + 0 a + 2a Z Z = 4a + 3a = na = :z:z = xy y symmetry 0 an p X, y, Z<br />
H ence (---) x,y,z = (1 7 ' 7)<br />
12<br />
a, 12 a, 12<br />
a.<br />
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43. I,. = foL foL foL k(y2 + z2)dz.dydx = k foL foL (Ly2 + ~L3) dydx = k foL ~L4 dx = ~kLs.<br />
By symmetry, l x = 1 11 =I.= ~k£ 5 •<br />
SECTION 15.7 TRIPLE INTEGRALS 0 275<br />
45. 1::: = JJJE(x 2 + y 2 ) p(x, y, z) dV = JJ [J;• k(x 2 + y 2 ) dz] dA = JJ k(x 2 + y 2 )hdA<br />
z2+y2:5a2<br />
:z:2+y2:5a.2<br />
(b) (x,y, z) wherex = "* f~ 1 !:2 f0 1 - 11<br />
~ = ~ t 1<br />
J:2 J;- v z Jx 2 + y 2 dzdydx.<br />
x Jx 2 + y 2 dzdyd$, y = "* f 1 J:z f 0<br />
1 -v y Jx 2 + y 2 dz dy dx, and<br />
) f 1 r~ f"(1 ) d d d<br />
3 ""<br />
11<br />
49. (a m = Jo Jo Jo + x + y + z z y x = 32 + 24<br />
(b) {X, y, z) = ( m- 1 J; fo~ f~ x(1 + x + y + z) dzdydx,<br />
m- 1 f 0<br />
1<br />
f0~ J~ y(1 + x + y + z) dzdydx,<br />
m - 1 f 0<br />
1<br />
fo~ j~ z(1 + x + y +z)dzdydx)<br />
. 28 307r + 128 457r + 208 )<br />
= ( 9n + 44 ' 45n + 220 ' 135n + 660<br />
t {~ {" . 68 + 1571'<br />
(c) 1::: = fo f o fo (x2 + y2)(1 + x + y + z) dz dy dx = 240<br />
51. (a) f(x, y, z) is a joint density function, so we know JJJR 3 f(x, y, z) dV = 1. Here we have<br />
JJJR3 f(x,y,z) dV = f~oo f~oo }:" 00<br />
f(x,y, z) dzdydx = f 0<br />
2<br />
J: J: Cxyzdzdydx<br />
Then we must have 8G = 1 => C - 1 - 8'<br />
= C J: xdx f 0<br />
2<br />
ydy f 0<br />
2<br />
zdz = C [~x 2 )~ [h 2 )~ [~z 2 )~ = 8C<br />
'<br />
1 rt d r1 d rl d 1 [ 1 2]1 [ 1 2] 1 [ 1 2] 1 1 ( 1 )3 1<br />
= ii Jo x X Jo 11 Y Jo z z = ii 2x o 2Y o 2z o = 8 2 = 1M<br />
(c) P(X + Y + Z::; 1) = P((X, Y, Z) E E) where E is the solid regio·n ·in the first octant bound<strong>ed</strong> by the coordinate planes<br />
and the plane x + y + z = 1. The plane x + y + z = 1 meets the xy-plane in the line x + y = 1, so we have<br />
P(X.+ Y + Z::; 1) = fff Ef(x, y, z) dV = j~ 1 j~ -x j 0<br />
1 -:z:-y kxyz dz dy dx<br />
= k J; fot - :z: xy[~z2 J ;:~ - :z:-y dydx = fs fo1 fol -z xy(l - x - y)2 dydx<br />
= 1~ f 0<br />
1<br />
f 0<br />
1<br />
- "'[(x<br />
3 - 2x 2 + x)y + (2x 2 - 2x)y 2 + xy 3 ] dydx<br />
= 110 fo1 [(x3 - 2x2 + xHy2 + (2x2- 2x)h3 + x(fy'')J::- :r: dx<br />
- 1 f1( 4 2 6 3 . 4 4 + 5) d - 1 ( 1 ) - 1<br />
- 192 Jo X- X + X . - X X X - 192 30 - 57iii)<br />
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276 0 CHAPTER 15 MULTIPLE INTEGRALS<br />
1 L2 L2 L2 La<br />
£ 3·222 = 8<br />
55. (a) The triple integral will attain its maximum when the integrand 1 - x 2 - 2y 2 - 3z 2 is positive in the region E and negative<br />
everywhere else. For if E contains some region F where the integrand is negative, the integral could be increas<strong>ed</strong> by<br />
excluding F from E, and if E fails to contain some part G of the region where the integrand is positive, the integral could<br />
be increas<strong>ed</strong> by including Gin E. So we require that x 2 + 2y 2 + 3z 2 $ 1. This describes the region bound<strong>ed</strong> by the<br />
ellipsoid x 2 + 2y 2 + 3z 2 = 1.<br />
(b) The maximwn value of JjJE (1 - x 2 - 2y 2 -<br />
3z 2 ) dV occurs when E is the solid region bound<strong>ed</strong> by the ellipsoid<br />
x 2 + 2y 2 + 3z 2 = 1. The projection of E on the xy-plane is the plaf1ar region bound<strong>ed</strong> by the ellipse x 2 + 2y 2 = i, so<br />
and<br />
using a CAS.<br />
15.8 Triple Integrals in Cylindrical Coordinates<br />
.1. (a) From Equations I, x = r cos 9 = 4 cos i = 4 · ~ = 2,<br />
y '= rsinfJ = 4sin ~ = 4 · ~ = 2.J3, z = ~ 2,<br />
3<br />
so the point is<br />
X<br />
I<br />
- 2 •<br />
I<br />
l (4, 3. -2)<br />
Y · (2, 2.J3, -2) in rectangular coordinates ..<br />
(b)<br />
(2.-¥. 1)<br />
x = 2cos(-~) = 0, y = 2sin( - ~l) = - 2,<br />
and z == 1, so the point is (0, - 2, 1) in rectangular coordinates.<br />
X<br />
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SECTION 15.8 TRIPLE INTEGRALS IN CYLINDRICAL COORDINATES 0 277<br />
3. (a) From Equations 2 we have r 2 = ( - 1) 2 + 1 2 = 2 so r = v'2; tan{;/= _: 1 = -1 and the point ( -1, 1) is in the second<br />
quadrant of the xy-plane, so 8 = s; + 2mr; z = 1. Thus, one set of cylindrical coordinates is ( v'2, 3 4 '~~' , 1).<br />
(b) r 2 = ( -2) 2 + (2J3) 2 = i6 so r = 4; tanB = ~ = - v'3 and the point ( - 2, 2v'3) is in the second quadrant of the<br />
xy-plane, so (} = 2 ; + 2mr; z = 3. Thus, one set of cylindrical coordinates is ( 4, 2 ; , 3).<br />
5. Since (;I = f but r and z may vary, the surface is a vertical half-plane i~clud ing the z -axis and intersecting the xy-plane in the<br />
half-line y = x, x ~ 0.<br />
1. z = 4- r 2 = 4 - (x 2 + y 2 ) or 4 - x 2 - y 2 , SQ the surface is a circular paraboloid with vertex (0, 0, 4), axis the z-axis, and<br />
opening downward.<br />
9. (a) Substituting x 2 + y 2 = r 2 and x = r cos 8, the equation x 2 - x + y 2 + z 2 = 1 becomes r 2 - r cos 8 + z 2 = 1 or<br />
z 2 = 1 + r cos 8 - r 2 .<br />
(b) Substituting x = r cosO andy = r sinO, the equation z = x 2 -<br />
y 2 becomes<br />
z = (rcos8) 2 - (rsinB? = r 2 (cos 2 8- sin 2 B) or z = r 2 cos 28. 1<br />
11.<br />
0 ~ r ~ 2 and 0 ~ :z ~ 1 describe a solid circular cylinder with<br />
radius 2, axis the z -axis, and height 1, but - 11' /2 ~ 8 ~ 11' /2 restricts<br />
the solid to the first and fourth quadrants of the xy-plane, so we have<br />
a half-cylinder.<br />
13. We can position the cylindrical shell vertically so that its axis coincides with the z-axis and its base lies in the xy-plane. If we<br />
use centimeters as the unit of measurement, then cylindrical coordinates conveniently describe the shell as 6 ~ r ~ 7,<br />
o ~ o ~ 211', o .~ z ~ 20.<br />
15.<br />
The region of integration is given in cylindrical coordinates by<br />
E = {(r,B,z) 1- 11'/2 ~ 8 ~ 11'/2, 0 ~ r ~ 2, 0 ~ z ~ r 2 } . This<br />
represents the solid region above quadrants I and IV of the xy-plane enclos<strong>ed</strong><br />
by the circular cylinder r = 2, bound<strong>ed</strong> above by the circular paraboloid<br />
z = r 2 (z = x 2 + y 2 ), and bound<strong>ed</strong> below by the xy-plane (z = 0).<br />
I<br />
T<<br />
-71'/ 2 0 Jo rdz r = - -rr/2 Jo rz z=O r = -71'/ 2 Jo r drd<br />
/ 2 j '2 rr<br />
2<br />
d d(} I" / 2 f 2 [ ] z = r<br />
2 d dB I 'll' / 2 f 2 3 (}<br />
= I 1f 12 dB f 2 r 3 dr = toJr. 12 (lr 4 ] 2<br />
c-'lf/2 Jo - 71'/2 4 o<br />
= 11'(4- 0) = 411'<br />
© 2012 Ccngage luming. All RighiS Rescrvctl. May nol be scannctl, copiod, orduplicalcd. or poslc'
278 D CHAPTER 15 MULTIPLE INTEGRALS<br />
17. In cylindrical coordinates, E is given by {(r, B, z} I 0 ~ B ~ 21r, 0 ~ r ~ 4, - 5 ~ z ~ 4}. So<br />
19. The paraboloid z = 4 - x 2 - y 2 = 4 - r 2 intersects the xy-plane in the circle x 2 + y 2 = 4 or r 2 = 4 =? r = 2, so in<br />
cylindrical co~rd inate s, E is given by { (1·, B, z} I 0 ~ (} ~ 1r / 2, 0 ~ r ~ 2, 0 ~ z ~ 4- r 2 }. Thus ·<br />
JJJE (x + y + z) dV = Io" 12 I; I 0<br />
4<br />
-r<br />
2<br />
(rcos{J + r sin(}+ z) r dzdr d(} = I 0 " 12 I; [r 2 (cos 0 + sin O) z + ~7·z 2 ]~=~-r 2 dr d{J<br />
= Io"/ 2 J; [(4r 2 - r 4 )(cos 0 + sin 9) + ~r(4- r· 2?] dr d{J<br />
= f '1r / 2 [(1r 3 - l r 5 ) (cos(} +sin 0) - .1.. (4 - r 2 ) 3 ] r = Z d9<br />
.Jo 3 5 12 r=O<br />
= I 0 " 12 [~~(cosO + sin 0) + Jf] dO= [ ~(sin 0 - cos B)+ JfB] ; 12<br />
= lfi(1 - 0) + ¥. ~ - ~(0 - 1)- 0 = ~7r + \ 2 5 8<br />
21 . In cylindrical coordinates, E is bound<strong>ed</strong> by the cylinder,. = 1, the plane z = 0, and the cone z = 2r. So<br />
E = {(r, 0, z) I 0 ~ 8 ~ 27r, 0 ~ r ~ 1, 0 ~ z ~ 2r} and<br />
I I IE x 2 dV = J;'T( f 0<br />
1<br />
J;r r 2 cos 2 9 r dz dr dO = I 0<br />
21f J 0<br />
1<br />
[r 3 cos 2 0 z ] ::~·· dr d9 = I;" Io 1 2r 4 cos 2 8 dr dO<br />
= I 02<br />
'T( [%r 5 cos 2 B]~=~ dO= % I 0<br />
2 " cos<br />
2<br />
0 d(} = ~I;"~ (1 +cos 28) dO,;, t [B + t sin 28] ~1f = 2 ;<br />
23. In cylindrical coordinates, E is bound<strong>ed</strong> below by the cone z = r and above by the sphere r 2 + z 2 = 2 or z = v'2 - r 2 . The<br />
con~ and the sphere·intersect when 2r 2 = 2 =? r = 1, so E = { (r, 8, z) I 0 ~ 0 ~ 21r, 0 ~ r ~ 1, r ~ z ~ v'2 - r 2 }<br />
and the <strong>vol</strong>ume is<br />
IIIE dV = J;" I0 1 .fr~ rdzdrdO =I;" I; [rzJ~::P drdO = I 0<br />
2<br />
" J 0<br />
1<br />
(r~- r 2 ) drd(}<br />
= J 0<br />
21f dO Io 1 (rv'2 - r 2 - r 2 ) dr = 21r [-~(2- r 2 ) 3 1 2 - ~r 3 J:<br />
= 27r (- ~) (1 + 1-2 3 1 2 ) = - j7r (2- 2V2) = ~7r (V2 - 1)<br />
25. (a) The paraboloids intersect when x 2 + y 2 = 36 - 3x 2 - 3y 2 =? x 2 + y 2 = 9, so the region of integration<br />
is D = {(x,y) I x 2 + y 2 ::; 9}. Then, in cylindrical coordinates,<br />
. E = {( r, B, z) I r 2 ~ z ~ 36 - 3r 2 , 0 ~ r ~ 3, 0 ~ (} ~ 27r} and<br />
V = g" I; fr 3 26<br />
-<br />
3<br />
r 2 r dz dr d(} = J;" I; (36r - 4r 3 ) dr dB = J;" [18r 2 - r 4 ] ::~dB= J 0<br />
2<br />
" 81 dO= 16271".<br />
(b) For constant density K , m = J(V = 1627rJ( from part (a). Since the region is homogeneous and symmetric,<br />
Nfyz = Mxz = 0 and<br />
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M xv = I~1f I: I:2 6 - 3 r 2 (zK) r dz dr d() = J( I~ 1f<br />
SECTION 15.8 TRIPLE INTEGRALS IN CYLINDRICAL COORDINATES 0 279<br />
I: r [ ~ z 2 ) ; :~~- 3 ,. 2 dr d()<br />
=if I 0<br />
2 " I: r((36- 3r 2 ) 2 - r 4 ) dr d() = if I~-:r d8 I 0<br />
3<br />
(8r<br />
5 - 216r 3 + 1296r) dr<br />
- K (27r) [.!!.r6 - 216 r4 + 1296 r2] 3 = 11" !((2430) = 243011" J(<br />
-2 6 4 ~ 0 .<br />
(<br />
- __) ( M 11z Mxz M:xv ) (O O 2430rr<br />
Thus x, y, z = --,--,--. = , I
280 0 CHAPTER 15 MULTIPLE INTEGRALS<br />
15.9 Triple Integrals in Spherical Coordinates<br />
1. (a)<br />
X<br />
(6.¥. f)<br />
If<br />
I I<br />
1 I<br />
I I<br />
'1r I I<br />
7i /6 :<br />
I<br />
I<br />
I<br />
I<br />
I<br />
y<br />
From Equat!ons 1, x = psin¢cos9 = 6 sin 2J cos i = 6 · ~ · ~ = !.<br />
y = p sin rP sin 9 = 6 sin ~ sin ~ == 6 · ~ · :1/- = ~, and<br />
z = pcos¢' = 6cos 2J = 6 · :1/- = 3VJ, so the point is(~ . ~ ,3../3) in<br />
rectangular coordinates.<br />
(b)<br />
x = 3 sin 3 _; cos ~ = 3 · ~ · 0 = 0,<br />
y = 3 sin 3 ;<br />
sin ~ = 3 · ~ · 1 = ¥, and<br />
X<br />
z= 3cos 3 ; = 3 (- ~) = ·~¥. so thepointis (o.¥,-¥) in<br />
·rectangular coordinates.<br />
. . z 0 71'<br />
3. (a) From Equations 1 and 2, p = .jx2 + y2 + z2 = ../0 2 + (-2)2 + 0 2 = 2, cos¢ = p = '2 = 0 => ¢' = 2'' and<br />
cos 9 = p s: rP = 2<br />
sin~ 71' / 2<br />
) = 0 => 9 = 3 ; [since y < 0). Thus spherical coordinates are ( 2, 3 ; , -i).<br />
z - J2 371'<br />
(b) p =. v'1 + 1 + 2 = 2, cos¢ = P = - 2<br />
- ::::} =<br />
4 , and<br />
X - 1<br />
cos 9 = -- = ::-:--:-::---:-:7<br />
• psin¢ 2sin(311'/4)<br />
371' 37!')<br />
are (<br />
2, 4 , 4<br />
.<br />
- 1 1 371'<br />
-,..-::::--:- - -- => 9 = - 4<br />
[since y > 0). Thus spherical coordinates<br />
2 (J2/2) - J2<br />
5. Since ¢ = ~, the surface is the top half of the right circular cone with vertex at the origin and axis the positive z-axis.<br />
x 2 + (y - ~) 2 + z 2 = ~ . Therefore, the surface is a sphere of radius ~ center<strong>ed</strong> at ( 0, ~, 0).<br />
9. (a) x = psin¢>cos8, y = psin ¢>sinO, and z = pcos ¢,so the equation z 2 = x 2 + y 2 becomes<br />
(p cos¢ ) 2 = (p sin cos 8) 2 + (p sin ¢> sin 8) 2 or p 2 cos 2 ¢J = p 2 sin 2 ¢'. If p ;f 0, this becomes co~ 2 = sin 2 rp. (p = 0<br />
corresponds to the origin which is includ<strong>ed</strong> in the surface.) There are many equivalent equations in spherical coordinates,<br />
such as tan 2 ¢ = 1, 2 cos 2 ¢' = 1, cos 2¢ = 0, or even rP = ~ . ¢' = :~ 4 ,..<br />
(b) x 2 + z 2 = 9<br />
SECTION 15.9 TRIPLE INTEGRALS IN SPHERICAL COORDINATES 0 281<br />
11. 2 :S p :S 4 represents the solid region between and including the spheres of<br />
radii 2 and 4, center<strong>ed</strong> at the origin.<br />
0 :S ¢> :S f restricts the solid to that<br />
portion on or above the cone ¢> = f, and 0 :S 8 :S 1r further restricts the<br />
solid to that portion on or to the right of the xz-plane.<br />
13. p :::; 1 represents the solid sphere of radius I center<strong>ed</strong> at the origin.<br />
3 ; :S ¢> :S 1r restricts the solid to that portion on or below the cone ¢> = ~;.<br />
15. z 2': Jx2 + y 2 because the solid lies above the cone. Squaring both sides ofthis inequality gives z 2 2': x 2 + y 2 · :::?<br />
2z 2 2:_ x 2 + y 2 + z 2 = p 2 :::? z 2 = p 2 cos 2 ¢> 2': ~p 2 :::? cos 2 ¢> 2': ~· The cone opens upward so that the inequality is<br />
cos¢> 2': ~, or equivalently 0 :S ¢> :S *. In spherical coordinates the sphere z = x 2 +. y 2 + z 2 is p cos lp = p 2 :::?<br />
p = cos¢. 0 :S p :S cos¢> because the solid lies below the sphere. The solid can therefore be describ<strong>ed</strong> as the region in<br />
spherical coordinates satisfYing 0 :S p :S cos ¢>, 0 :S ¢> :S *.<br />
17.<br />
The region of integration is given in spherical coordinates by<br />
E = {(p, 8, ¢>) I 0 :S p :S 3, 0 :S 8 :S 7r / 2, 0 :S ¢> :S 1r / 6}. This represents the solid<br />
region in the first octant bound<strong>ed</strong> above by the sphere p = 3 and below by the cone<br />
¢> = 1T/6.<br />
J 0 " 16 J 0<br />
" 12 .1; p 2 sin ¢> dp dB d4> = ./~" 16 sin 4> d¢> f 0 " 12 dB J: p 2 dp<br />
X<br />
= [-cos¢>] ~ 16 ( 8]~ 12 [~Pa]~<br />
= (1- ~) G)(9) = 9 ; (2- v'3)<br />
19. The solid E is most conveniently describ<strong>ed</strong> if we use cylindrical coordinates:<br />
E = {(r,B,z) I 0 :s e :s i.O :s r :s 3,0 :s z :s 2}. Then<br />
JJJE f(x, y, z) dV = f 0 " 12 .1; ./~ f(r cos f)_, rsinB, z) 1' dz d1·d8.<br />
@ 20 12 Ccngag~ Lc:uning. All Rights Resc~rvcd. May not be: scnnnc:d. copi<strong>ed</strong>. or duplictuctl, or poS1ec.l 10 u publicly ucccssible wcbsilc~ in whole or in p3rl.
282 0 CHAPTER 15 MULTIPLE INTEGRALS<br />
21. In spherical coordinates, B is represent<strong>ed</strong> by {(p, 0, 4>) I 0 ~ p ~ 5, 0 ~ 0 ~ 27r, 0 ~ cf> ~ 1r}. Thus<br />
JJJ 8<br />
(x 2 + y 2 + z 2 ) 2 dV = J; f 0<br />
2<br />
"' J:(p 2 ) 2 p 2 sin cf>dpdO dl/> = Io"' sin 1/>dl/> J~"' dB I~ p 6 dp<br />
= [- coscf>]~ [0]~" [ tp 7 ]~ = (2)(27r)(1s.~211)<br />
= 312 tJ07r ~ 140,249.7<br />
23. In spherical coordinates, E is represent<strong>ed</strong> by {(p, 0, ¢) 12 ~ p ~ 3, 0 ~ 0 ~ 27r, 0 ~ cf> ~ 1f} and<br />
IIIs(:c 2 + y 2 ) dV = ]~"'<br />
g •r I 2 3 (p 2 sin 2 4>) p 2 sin cf>dpdO dcf> = Io" sin 3 cf>dcf> I~"' dO I: p 4 dp<br />
= I 0<br />
"'(1- cos 2 1/>) sinl/>dl/> [ 0]~,. [ip 5 )~ = [-coscf>+ ~ cos 3 4>]~ (27r) : i(243 - 32)<br />
= (1 - ~ + 1 - U (27r) Cil) = 16~:"<br />
25. In spherical coordinates, E is represent<strong>ed</strong> by {(p, 0, 4>) I 0 ~ p ~ 1, 0 ~ 8 ~ "i· 0 ~ 4> ~ ~}.Thus<br />
JJJE xe"' 2 + 112 +z 2 dV = J 0<br />
"/ 2 J 0<br />
Tr 12 I:(psin cf>cosO)eP 2 p 2 sin = Io"' 12 sin 2 1/>d¢ J; 12 cos(} d(} j 0<br />
1<br />
p 3 eP 2 dp<br />
.= fo" 12 H 1 - cos 2¢>) d¢ IoTr 12 cos(} d(} ( ~ p 2 eP2<br />
J:<br />
-I: peP 2 dp)<br />
[integrate by parts with u = p 2 , dv = peP 2 dp]<br />
= a¢- i sin 2¢]~ /Z [sin(})~/ 2 [ ~p 2 eP 2 - ~eP 2 J: = (i- 0) (1- 0) (0 + ~) = i<br />
27. The solid region is given byE= { (p, 0, cf>) I 0 ~ p ~ a, 0 ~ 0 ~ 21r, ~ ~ cf> ~ ~}and its <strong>vol</strong>ume is<br />
V = ffis dV = I:/ 6<br />
3<br />
J~ "' I; p 2 sin cf>dpd(}drjJ = I:/: sincf>dl/> I~ 11' dO I; p 2 dp<br />
. = [-coscf>J;~: [0) ~11' [~p 3 )~ = ( -~ + 4) (27r) (ta 3 ) = '\- 1 7ra 3<br />
29. (a) Since p = 4 cos 4> implies p 2 = 4p cos 1/>, the equation is that of a sphere of radius 2 with center at (0, 0, 2). Thus<br />
(b) By the symmetry of the problem Mv= = M x= = 0. Then<br />
Hence (x,y,z) = (0,0, 2.1).<br />
M, 11<br />
= I:" I; 13 I 0<br />
4 co• 4> p 3 cos cf> sin cf> dp d¢ d(} = J~ " I; 13 cos cf> sin r/> ( 64 cos 4 rjJ) dcf> dO<br />
- f 2 "' 64 [-1 () -•] =71' 13 dO - f 2 71' .ll dO - 21<br />
- Jo 6 cos '~' ¢=0 - Jo 2 - 7r<br />
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SECTION 15.9 TRIPLE INTEGRALS IN SPHERICAL COORDINATES 0 283<br />
31. (a) By the symmetry of the region, My: = 0 and Mx: = 0. Assuming constant density K,<br />
m = fffE K dV = K fffE dV = iJ< (from Example 4). Then ·<br />
M .,y = Jjj~ z K dV = K J; ,. }~,. 14 j~o• ¢>(pcos ¢) p 2 sin r/J dp drpd8 = K J;" J0,. 14 sin ¢cos¢ [tP 4 ] ==~os¢ d¢ d8<br />
=if< J:w f 0<br />
"' 14 sin¢cos¢ (cos 4 ifJ) .d(j)d(} = iK J:" d(} J; 14 cos 5 {j}sin (j)d¢<br />
= iK (8]~ " [ -i cos 6 t/>] ~ 14 = iK(27r)( -i) [ c~r - 1] = -fiK ( -~) = ;~ K<br />
.. (- __) (My: !vfu M.,u) ( 77rK/96) ( 7)<br />
Thusthecentro1d1s x,y,z = m'm'm = 0, 0, 1rf(jB = 0,0, 12.<br />
(b)' As in Exercise 23, x 2 + y 2 = p 2 sin 2 ¢ and<br />
I: ~ JJJE (x2 + y2) J( dV = K J;" fo"/4 focos¢ (p2 sin2 ifJ) p2 sin¢ dpd¢d8 = K J:.,.. fo"'/4 sin3 t/> [kPsJ::~os ¢ d¢d8<br />
= iK J:" J; 14 sin 3 ¢cos"' ¢d¢d8 =if< .r:w d(} j~w/ 4 cos 5 ¢ (1 - cos 2 ¢) sin¢d¢<br />
= tK [8]~"' [- t cos 6 ¢ + ~- cos 8 ¢]~ 14<br />
= 1J
284 0 CHAPTER 15 MULTIPLE INTEGRALS<br />
39. The region E of integration is the region above the cone z = .J x 2 + y 2 and below the sphere x 2 + y 2 + z 2 = 2 in the first<br />
octant. Because E is in the first octant we have 0 ~ B ~ %. The cone has equation 1/> = "i (as in Example 4), so 0 ~ 1/> ~ .;f,<br />
and 0 ~ p ~ ../2. So Ute integral becomes<br />
f rr 14 0<br />
.fa"'' 12 fov"i (p sin¢ cos B) (p sin c/> sin t1) p 2 sin¢ dp dB d¢ ·<br />
= J~-r / 4 sin 3 1/>d¢ J 0<br />
rr/ 2 sinB cost1dB .J 0<br />
v"i p 4 dp = (Jorr/ 4. (1 - cos 2 1/>) sin 1/> dl/>) [~ sin 2 B]~ 12 [ip 5 ]:<br />
- [1 cos 3 A .. - cos A..] rr/4.. 1. l ( '2) 5 - [~ - ~ - ( l - 1) ] . M - 4<br />
fl- 5<br />
- 3 '~' '~' o 2 5 V"' - 12 2 a 5 ~ 15<br />
41 . The.region of integration is the solid sphere x 2 + y 2 + (z - 2? ~ 4 or equivalently<br />
p 2 sin 2 c/> + (p cos lj> - 2) 2 = p 2 - 4pcos + 4 ~ 4 =><br />
p ~ 4cos¢, so 0 ~ 8 ~ 211', 0 ~ c/> ~ %• and<br />
0:::; p ~ 4cos ¢.Also (x 2 + y 2 + z 2 ) 3 1 2 = (p 2 ) 3 1 2 = p 3 , so the integral becomes<br />
J" 12 0<br />
J:" J4. cos 0<br />
z = rcot r/> 0<br />
y<br />
= 2:r [- (a2 - a2 sin2 c/Jo)3/2 - a3 sin3 ¢o cot¢o + a3]<br />
;, ~7ra 3 [1 - ( cos 3 c/> 0 + sin 2 ¢ 0 cos 1/> 0 )] = lrra 3 (1 - cos ¢ 0 )<br />
® 2012 Ccngagc Learning. AU Rjg.hts Rcscn·<strong>ed</strong>. Muy not be scann<strong>ed</strong>, copi<strong>ed</strong>, ur dupllcntcd, or rostcd to n publicly acccsslblc website, in whole or in pttrt.
(b) The w<strong>ed</strong>ge in question is the shad<strong>ed</strong> area rotat<strong>ed</strong> from B = 81 to B = 82.<br />
Letting<br />
V.i = <strong>vol</strong>ume of the region bound<strong>ed</strong> by the sphere of radius P;<br />
and the cone with angle ¢; (B = 81 to 82)<br />
SECTION 15.1 0 CHANGE OF VARIABLES IN MULTIPLE INTEGRALS 0 285<br />
and letting V be the <strong>vol</strong>ume of the w<strong>ed</strong>ge, we have<br />
y<br />
V = (1122- 1l21)- (V12 - Vn)<br />
· = 1(82 - B1)[pg(1- cos¢ 2 ) - p~(l - cos¢ 1 )- p~(1- cos¢ 2 ) + p~(1- cos¢ 1<br />
) ]<br />
= 1(82- 81) [(P~- p~)(1- cos ¢ 2 ) - (P~- pi) (1- cos ¢1)] = i(B2 - (Jt)[(p~- pi) (cos ¢ 1 - cos¢ 2 )]<br />
Or: Show that V =<br />
1<br />
021 P2 s in 21 r cot ¢ 1<br />
01 p 1 s in 4> 1 r cot 4> 2<br />
r dz dr dB.<br />
(c) By the Mean Value Theorem with f(p) = p 3 there exists some p with p 1 ~ p ~ p 2 such that<br />
f(p 2 ) - f(p 1 ) = !'(p)(p 2 - p 1 ) or p~ - p~ = 3Ji D.p. Similarly there exists¢ with ¢1 ~ ~ ~ ¢ 2<br />
such that cos ¢ 2 - cos ¢ 1 ·= (-sin¢) D.¢. Substituting into the result from (b) gives<br />
D. V = (p 2 D.p)(B2- B1)(sin¢) D.¢ = p 2 sin¢> D.pD.¢ D.B.<br />
15.10 Change of Variables in Multiple Integrals<br />
1. x = 5u- v, y = u + 3v.<br />
TheJacobianis<br />
a(x<br />
~( ,y)<br />
1)<br />
=<br />
l8xf8u 8xf8v I<br />
= =5(3) -(-1)(1) =16.<br />
v u,v 8yf8u oyfav 1 3<br />
1<br />
3. x = e-r sinB, y :;= er cos B.<br />
5 -ll<br />
8(x,y) 18xf8r 8xj8fJ I ~ -e - ''sinB e-rcosBI · ·<br />
-- - - r . = e-•·er sin 2 B- e-re'· cos 2 B = sin 2 fJ- cos 2 ()or- cos 2()<br />
8(r, B) - ayj8r ayjaB - er cos() -e smfJ . .<br />
5. x = ufv, y = vfw, z = wfu.<br />
a(x,y,z) -<br />
a(u,v,w)-<br />
oxfou 8xfav 8x/8w 1/ v - u/v 2 0<br />
ayfau ayf8v 8yf8w 0 1/w -vjw 2<br />
azjou 8z/8v 8z/8w - wfu2 0 1/u<br />
= .!_11/w - vjw 2 ! ( u) I 0 -vjw 2 ! I 0 1/w I<br />
v 0 1/u - - V 2 .:..wfu 2 1/ u + 0 -wfu 2 0<br />
=.!. (...!_ - o) + ~ (o- ~) + o = - 1 - - - 1 - = o<br />
v uw v 2 u 2 w uvw uvw<br />
1. The transformation maps the boundary of S to the boundary of the imageR, so we first look at side 8 1 in the uv-plane. S 1 is<br />
describ<strong>ed</strong> by v = 0, 0.::; u.::; 3, sox = 2u + 3v = 2u andy = u- v = u. Eliminating u, we have.x = 2y, 0 _::; x ~ 6. 8 2 is<br />
@ 20 12 Ccng:~ge Learning. All Right5 Resen-cd. MBy not be sccum<strong>ed</strong>. copi<strong>ed</strong>. or duplicat<strong>ed</strong>. or post<strong>ed</strong> 10 a publicly acccssiblo wcb~ itc, in whole or in prut.
286 0 CHAPTER 15 MULTIPLE INTEGRALS<br />
the line segment u = 3, 0:::; v :::; 2, sox= 6 + 3v andy = 3 - v. Then v = 3- y =? x = 6 + 3(3- y) = 15- 3y,<br />
6 :::; x :::; 12. S 3 is the line segment v = 2, 0 :::; u :::; 3, sox = 2u + 6 andy = u- 2, giving u = y + 2 =? x = 2y + 10,<br />
6 $ X $ 12. Finally, S4 is the segment U = 0, 0 $ V $ 2, SO X = 3v and y = -V :=} X = -3y, 0 $ X $ 6. The image of<br />
set S is the region R shown in the xy-plane, a parallelogram bound<strong>ed</strong> by these four segments.<br />
IJ<br />
sl<br />
(0, 2) (3,2)<br />
-<br />
(6, 3)<br />
T<br />
s. s sl<br />
(12,1)<br />
0 s, (3, 0) II 0 X<br />
y<br />
(6,- 2)<br />
9. sl is the line segment u = v, 0$ u ~ 1, soy= v = 'U and X = u 2 = y 2 • Since 0 ~ u ~ 1, the image is the portion of the<br />
parabola X = y 2 ' 0 :::; y :::; 1. s2 is the segment v = 1, 0 :::; u ~ 1, thus y = v = 1 and X = u 2 ' so 0 :::; X :::; 1. The image is<br />
the l.ine segment y = 1, 0 $ X $ 1. 83 is the segment U = 0, 0 $ V $ 1, SO X = u 2 = 0 and y = V :=} 0 $ y $ 1. The<br />
image is the segment x = 0, 0 :::; y :::; 1. Thus, the image of S is the region R in the first quadrant bound<strong>ed</strong> by the parabola<br />
x = y 2 , they-axis, and the line y = 1.<br />
-T<br />
y<br />
u 0 X<br />
11. R is a parallelogram enclos<strong>ed</strong> by the parallel lines y = 2x- 1, y = 2x + 1 and the parallel lines y = 1- x, y = 3 - x. The<br />
first pair of equations can be written as y - 2x = -1, y - 2x = 1. If we let u = y - 2x then these lines are mapp<strong>ed</strong> to the<br />
'<br />
vertical lines u = - 1, u = 1 in the uv-plane. Similarly, the second pair of equations can be written as x + y = 1, x + y = 3,<br />
and setting v = x + y maps these lines to the horizontal lines v = 1, v = 3 in the uv-plane. Boundary curves are mapp<strong>ed</strong> to<br />
boundary curves under a !ransformation, so here the equations u = y- 2x, v = x + y define a transformation r - 1 that<br />
maps R in the xy-plane to the square S enclos<strong>ed</strong> by the lines u = -1, u = 1, v = 1, v = 3 in the uv-plane. To find the<br />
transformation T that maps S to R we solve u = y - 2x, v = x + y for x, y: Subtracting the fi_rst equation from the second<br />
gives v - u = 3x =? x = ~ ( v - u) and adding twice the second equation to the first gives u + 2v = 3y =?<br />
y = Hu + 2v). Thus one possible transformation T (there are many) is given by x = Hv- u),y = ~('u + 2v).<br />
3 v=3<br />
u '=-1 s u = 1<br />
1 v - 1<br />
T<br />
--+-<br />
1 y=2t-1<br />
- I 0 u '',f'<br />
: •<br />
X<br />
SECTION 15.10 CHANGE OF VARIABLES IN MULTIPLE INTEGRALS 0 287<br />
13. R is a portion of an annular region (see the figure) that is easily describ<strong>ed</strong> in polar coordinates as<br />
R = { ( r , 8) 11 ::; r ::; ../2, 0 ::; (} ::; 7T / 2}. If we convert<strong>ed</strong> a double integral over R to polar coordinates the resulting region<br />
of integration is a rectangle (in the rO-plane), so we can create a transformation T here by letting u play the role of rand v the<br />
role of 0. Thus T is defin<strong>ed</strong> by x = u cos v , y = u sin v and T maps the rectangle S = { ( u, v) I 1 ::; u ::; ../2, 0 ::; v. ::; 7T / 2}<br />
in the uv-plane to R in the xy-plane.<br />
v<br />
y<br />
E.<br />
2<br />
s<br />
-T<br />
0<br />
II<br />
X<br />
2 1<br />
15 B(x, y) = 1 1 = 3 and x - 3y = (2u + v) - 3(u + 2v) = -u- 5v. To find the regionS in the uv-plane that<br />
· 8(u, v) 1 2 ·<br />
corresponds toR we first find the corresponding boundary under the given transformation. The line through {0, 0) and (2, 1) is<br />
y = ~x which is the image of u + 2v = ~(2u + v) => v = 0; the line through (2, 1) and (1, 2) is x + y = 3 wh ich is the<br />
image of (2u + v) + (u + 2v) = 3 => u + v = 1; the line through (0, 0) and (1, 2) is y = 2x which is the image of<br />
u + 2v = 2(2u + v) =><br />
u = 0. Thus S is the triangle 0 ::; v ::; 1 - u, 0 ~ u ::; 1 in the uv-plane and<br />
jj~ (x - 3y) dA = j 0<br />
1<br />
j 0<br />
1 -u ( -u - 5v) 131 dv du = -3 f 0<br />
1<br />
[uv + ~v 2 J::~ - u du .<br />
= - 3 j; (u - u 2 + ~{ 1 - u?) du = - 3[~u 2 - iu 3 - ~( 1- u?]~ = -3(~ - ~ + ~) = - 3<br />
17 B(x, y) = 12 0<br />
I = 6, x 2 = 4u 2 and the planar ellipse 9x 2 + 4y 2 ::; 36 is the image ~fthe disk u 2 + v 2 < 1. Thus<br />
. 8( u, v) 0 3 . -<br />
ffn x 2 dA =<br />
JJ (4u 2 )(6) dudv = f 0<br />
2<br />
"' f 0<br />
1<br />
(24r<br />
2<br />
cos 2 8)rdrd8 = 24J;.,.. cos 2 8d8 J 0<br />
1<br />
r 3 dr<br />
u2 +v2~ 1<br />
= 24[~ x +~sin 2x]~"' ar 4 ] ~ = 24(7r)(i) = 67!'<br />
19. 8(x, y) = , 1 2<br />
/v - ufv ' = .!., xy = u, y =xis the image of the parabola v 2 = u, y == 3x is the image of the parabola<br />
8(u, v) o 1 v<br />
j L x y dA = 1<br />
31: u ( ; ) dv du = 13<br />
v 2 = 3u, and the hyperbolas xy = 1, xy =a are the images of the lines u = 1 and u = 3 respectively. Thus<br />
21 . (a) :ix,<br />
a 0 0<br />
y, z~ = 0 b 0 = abc and since u = ~. v<br />
u,v,w a c<br />
0 0 c<br />
u (In v3u - ln vfu) du = J: u In J3 du = 4ln J3 = 2 ln 3.<br />
= J!.b' w = ~the solid enclos<strong>ed</strong> by the ellipsoid is the image of the<br />
JJJE dV = JJJ abcdudvdw = (abc)(<strong>vol</strong>umeof tlie ball) = ~1rabc<br />
u2+v2+w2 :S 1<br />
© 201 2 Ccngage Learning. All Rights Reserv<strong>ed</strong>. May not be sc.um~ copi<strong>ed</strong>. or dupliea.tcd. or post<strong>ed</strong> too publicly accessible website, in whole or in part.
288 0 CHAPTER 15 MULTIPLE INTEGRALS<br />
2 2 2<br />
(b) If we approximate the surface. of the earth by the ellipsoid 6<br />
;., 82<br />
+ 6<br />
i 782<br />
+ 6<br />
; 562<br />
_ = 1, then we can estimate<br />
the <strong>vol</strong>ume of the earth by finding the <strong>vol</strong>ume of the solid E enclos<strong>ed</strong> by the ellipsoid. From part (a), this is<br />
fffE dV = ~11"(6378)(6378)(6356) ~ 1.083 X 10 12 km 3 •<br />
(c) The moment ofintertia about the z-axis is I% = JJJE (x 2 + y 2 ) p(x, y, z) dV, where E is the solid enclos<strong>ed</strong> by<br />
xz yz zz 18(x y z) I<br />
2 + b 2<br />
+ 2 = 1. As in part (a), we use the transformation x = au, y = bv, z = cw, so 8<br />
( ' ' ) = abc and<br />
a . c u,v,w<br />
I , = JJJE (x 2 + y 2 ) kdV = f.(.[ k(a 2 u 2 + b 2·u 2 )(abc) dudv dw<br />
u2+ v2+w2 ~ 1<br />
= abck f 0<br />
" f 0<br />
2 " J 0<br />
1 ( a<br />
2 p<br />
2<br />
sin 2 ¢ cos 2 () + b 2 p 2 sin 2 ¢ sin 2 8) p 2 sin ¢ dp d8 drjJ<br />
= abck [a 2 f 0 ,.. f~,.. f 0<br />
1 (p 2 sin<br />
2<br />
rP cos 2 B) p 2 sin 1/> dp d() dlj> + b 2 f 0 " f~,.. f 0<br />
1 (p<br />
2<br />
sin 2 rjJ sin 2 B) p 2 sin rP dp dB dr!>]<br />
=. a 3 bck J 0<br />
" sin 3 rjJ drjJ f 0<br />
2<br />
" cos 2 B dB J; p 4 dp + ab 3 ck fo.,.. sin 3 rjJdrjJ f~" sin 2 B dB J; p 4 dp<br />
= a 3 bck [ ~ cos 3 ¢-cos I/>] ~ [ ~B + i sin2 B]~" [iP 5 ] ~ + ab 3 ck [t cos 3 rjJ- cosr/J]~ [~B- i sin 28] ~" [tP 5 ]~<br />
= a 3 bck (t) (1r) ( t) + ab 3 ck (1) (1r) (i) = rt7r (a 2 + b 2 ) abck<br />
23. Letting u = x - 2y and v = 3x - y, we have x = f.(2v- u) and y = f(v- 3u). Then 8 8 ((x, y)) = ~ - 1 / 5 215 1 = .!.<br />
0 0<br />
u,v -3/5 1/5 5<br />
and R is the image of the rectangle enclos<strong>ed</strong> by the lines u = 0, u = 4, v = 1, and v = 8. Thus<br />
8(xy) 1.:...1/2 1/ 21 1<br />
25. Letting u = y- x, v = y + x, we nave y = !(u + v), x = Hv - u). Then (u: v) = i = 8 1 2 112<br />
- and R is the<br />
2<br />
image of the trapezoidal region with vertices ( - 1, 1), ( -2, 2), {2, 2), and {1, 1). Thus<br />
y -x<br />
!h 121" u I 11 11 2 [ u]u=v 112<br />
R Y +X '1 -v V 1 V u= -v 1<br />
cos--dA-= cos - - - 2<br />
du dv = - 2<br />
v sin- dv = - 2<br />
2v sin{1) dv = ~sin 1<br />
.<br />
27. Letu=x+yandv=-x+y.Thenu+v = 2y => v=Hu+v) andu - v=2x => X=~(u-v).<br />
8(x y) 11/2 -1/ 2! 1<br />
- 8<br />
( ') = =- 2<br />
.Now[u[= [x+y[< [x[+[y[ - l
CHAPTER 15 REVIEW 0 289<br />
15 Review<br />
CONCEPT CHECK<br />
m<br />
n<br />
1. (a) A double Riemann sum off is ,L ,L f (xi;, Yi;) b.A, where b. A is the area of each subrectangle and (xi;, Yi;) is a<br />
i=lj = l<br />
S3J!lple point in each subrectangle. If j(x, y) ;::: 0, this sum represents an approximation to the <strong>vol</strong>ume of the solid that lies<br />
above the. rectangle R and below the graph off.<br />
(b) ffn. f(x, y) dA = lim<br />
m,n-+oo i = 1 j = 1<br />
f f= f(xi;. Yi;) b.A<br />
(c) If f(x, y) ;::: 0, ffn f(x, y) dA represents the <strong>vol</strong>ume of the solid that lies above the rectangle Rand below the surface<br />
z = f(x, y). Iff takes on both positive and negative values, ffn j(x, y) dA is the difference of the <strong>vol</strong>ume above R but<br />
below the surface z = f(x, y) and the <strong>vol</strong>ume below R but above the surface z = f(x, y).<br />
(d) We usually evaluate ffn. j(x, y) dA as an iterat<strong>ed</strong> integral according to Fubini's Theorem (see Theorem 15.2.4).<br />
(e) The Midpoint Rule for Double Integrals says that we approximate the double integral JJR f(x, y) dA by the double<br />
Riemann sum f f: f(xi, fi;) b.A where the sample points (:x,, fi;) are the centers of the subrectangles.<br />
i=lj= l<br />
(f) fnvc = A ~R) I In f(x, y) dA where A (R) is the area of R.<br />
2. (a) See ( I) and (2) and the accompanying discussion in Section 15.3.<br />
(b) See (3) and the accompanying discussion in Section 15.3.<br />
(c) See (5) and the prec<strong>ed</strong>ing discussion in Section 15.3.<br />
(d) See (6)- (1 I) in Section 15.3 .<br />
3. We may want to change from re~tangular to polar coord inates in a double integral if the region R of integration is more easily<br />
describ<strong>ed</strong> in polar coordinates. To accomplish this, we use ffn f(x, y) dA = J: J: f (rcosB, rsin 9) r dr d(} where R is<br />
given by 0 ::::; a ::::; r ::::; b, a ::::; B ::::; {3.<br />
4. (a) m = JJD p(x, y) dA<br />
(b) M , = ffv yp(x, y) dA, M 11 = ffv xp(x, y) dA<br />
(c) The center of mass is (x, y) where x == M" and y= lvfx .<br />
m m<br />
(d) I, = ffv y 2 p(x, y) dA I v = ffv x 2 p(x, y) dA, Io = ffv (x 2 + y 2 )p(x, y) dA<br />
5. (a) P(a ::::; X::::; b, c:=; Y::::; d)= 1: J: j(x, y)dydx<br />
(b) f(x , y) ;::: 0 and JJ~~ f(x, y) dA = 1.<br />
· (c) The expect<strong>ed</strong> value of X is JJ. 1 = ffR2 xf(x, y) dA; the expect<strong>ed</strong> value ofY is p..z = JJT:/.2 yf(x, y) dA.<br />
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290 0 CHAPTER 15 MULTIPLE INTEGRALS<br />
6. A(S) = JJ 0 J [f.,(x, y))2 + [f11(x, y))2 + 1 dA<br />
(b) We usually evaluate f[JB f(x, y , z) dV as an iterat<strong>ed</strong> integral according to Fubini's Theorem for Triple Integrals<br />
(see Theorem 15.7.4).<br />
(c) See the paragraph following Example 15.7.1.<br />
(d) See (5) and (6) and the accompanying discussion in Section 15.7.<br />
(e) See (10) and the accompanying discussion in Section 15.7.<br />
(f) See (11) and the prec<strong>ed</strong>ing discussion in Section 15.7.<br />
8. (a) m = JJJE p(x, y, z) dV<br />
. (b) M 71z = JJJE xp(x, y, z) dV, Mxz = JJJE yp(x, y, z) dV, Mxy = JJJE zp(x, y, z) dV.<br />
· (- - - ) J - Myz _ M xz d _<br />
( c ) Th e cen<br />
lvfx<br />
t er o f mass ts x, y, z w 1ere x = --, y = --,an z = --. 11<br />
m m m<br />
9. (a) See Formula 15.8.4 and the accompanying discussion.<br />
(b) See Formula 15.9.3 and the accompanying discussion.<br />
(c) We may want to change from rectangular to cylindrical or spherical coordinates in a triple integral if the region E of<br />
integration is more easily describ<strong>ed</strong> in cylindrical or spherical coordinates or if the triple integral is easier to evaluate using<br />
cylindrical or spherical coordinates.<br />
a (x, y) I ax;au ax;av I ax ay ax By<br />
10· (a) a (u, v) = ay;au ayjav = au av - av au<br />
(b) See (9) and the accompanying discussion in Section 15.10.<br />
(c) See (13) and the accompanying discussion in Section 15.10.<br />
TRUE-FALSE QUIZ<br />
1': This is true by Fubini's Theorem.<br />
3. True by Equation 15.2.5.<br />
5. True. ~y Equation 15.2.5 we can write J~ f0 1 f(x) f(y) dy dx = I~ f (x) dx I~ f(y) dy. But J~ f(y) dy = J~ f(x) dx so<br />
this becomes I; f(x) dx .{ 0<br />
1<br />
f(x) dx = [J; f(x) dxf.<br />
7. True: J 0<br />
J 4 - x 2 - y 2 dA = the <strong>vol</strong>ume under the surface x 2 + y 2 + z 2 = 4 and above the xy-plane<br />
= ~ (the <strong>vol</strong>ume ofthe sphere x 2 + y 2 + z 2 = 4) = ~ · t1r(2) 3 = 1 36<br />
1r<br />
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CHAPTER 15 REVIEW 0 291<br />
l<br />
9. The <strong>vol</strong>ume enclos<strong>ed</strong> by the cone z = ...jx2 + y2 and the plane z = 2 is, in cylindrical coordinates,<br />
V = J~" J~ J: r dz dr dO i= f 0<br />
2<br />
" J~ J: dz dr dO, so the assertion is false.<br />
EXERCISES<br />
1. As shown in the_contour map, we divideR into 9 equally siz<strong>ed</strong> subsquares, each with area b.A = 1. Then we approximate<br />
ffn f(x, y) dA by a Riemann sum with m = n = 3 and the sample points the upper right comers of each square, so<br />
3 3<br />
ffnf(x,y)dA>:::J L: L: f(xi, yj)b.A<br />
i =lj=l<br />
= b.A [!(1, 1) + /(1, 2) + /(1, 3) + !(2, 1) + !(2, 2) + f(2, 3) + !(3, 1) + !(3, 2) + !(3, 3)]<br />
Using the contour lines to estimate the function values, we have<br />
ffnf(x,y) dA >:::! 1(2.7 + 4.7 + 8.0 + 4.7 + 6.7 + 10.0 + 6.7 + 8.6 + 11 .9] ~ 64.0<br />
3. f 1<br />
2<br />
J~ (y + 2xeY) dx dy = J? [xy + x 2 e 11 ] ::~ dy = f 1<br />
2 (2y + 4eY) dy = [y 2 + 4eY] ~<br />
= 4 + 4e 2 - 1 - 4e = 4e 2 - 4e + 3<br />
7. J; J; J0~ ysinx.dzdy dx = f 0<br />
" J; [(ysinx)zJ::F dydx·= f 0<br />
" f 0<br />
1<br />
y ...jl- y 2 sinxdydx<br />
= f 0<br />
" [- ~ (1- y 2 ) 3 1 2 sinx)·u:l dx = f 0<br />
" ~ sin x dx = -~ cosx]~ = ~<br />
. . y~<br />
9. The regi?n R is more easily d~sc rib<strong>ed</strong> by polar coordinates: R = { ( r, 0) I 2 s; r s; 4, 0 s; 8. s; 71"}. Thus<br />
ffn f(x, y) dA = fo" f 2<br />
4<br />
f (rcosO, r sinO) rdrdO.<br />
11.<br />
Th . h . . b fTr / 2 r•in 20 d dO .<br />
e reg1on w ose area IS g.tven y Jo Jo r r IS<br />
· { ( r, 0) I 0 s; 0 s; %, 0 s; r s; sill 20}, which is th~ region contain<strong>ed</strong> in the<br />
loop in the first quadrant of the four-leav<strong>ed</strong> rose r =sin 28 .<br />
13.<br />
./~ J: 1 cos(y 2 ) dy dx = J; g cos(y 2 ) dx dy<br />
= f 0<br />
1<br />
cos(y 2 ) [ x] ::~ dy = J 0<br />
1<br />
y cos(y 2 ) dy<br />
X<br />
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292 0 CHAPTER 15 MULTIPLE INTEGRALS<br />
17. y<br />
r..rx , . 11 1 2 y-..rx<br />
1 D 1 + x2 dA = o 1 o 1 : x2 dy dx = o 1 + x2 [ h ) y:o dx<br />
! .<br />
r y 11<br />
1<br />
X 1<br />
= l --dx = [! In(1 + x 2 )) = ! In 2<br />
2 0 1 + x2 ·I o 4<br />
0 X<br />
19.<br />
fj ,<br />
f 2J,B-y~<br />
0 ydA = J o 112 ydxdy<br />
= fo2 y[x] ::~~v2 dy = J; y(8 ~ y2- y2) dy<br />
= f 0<br />
2<br />
(8y - 2y<br />
3) dy = [4y 2 - tY 4 ]~ = 8<br />
21 .<br />
y<br />
/L (xz + y2)3/ 2 dA = 17r/313 (r2)3/2r drd0<br />
= 11r/3 dO 13 r4dr = [0)~/3 [ir:;)~<br />
1f 3 5<br />
811f<br />
=35=5<br />
23. fffe xy dV = .1: j~x J;+v xy dz dy dx = .{ 0<br />
3<br />
.fox xy [ z J::~+v dy dx = J; ./~"' xy(x + y) dy dx<br />
= ]~ 3 J;(x 2 y + xy 2 ) dydx = .r; [tx 2 y 2 + ~x1lJ;;:~ dx = J; (tx 4 + tx 4 ) dx<br />
6 / '3 4 d [ 1 5] 3 81<br />
=<br />
40 5<br />
6, 0 X X = GX 0 = 2 = . '<br />
25. Iff v 2 z 2 d\l=f 1 J~ ]' 1 -v 2 - z<br />
2<br />
y2 z 2 dxdzdv=t r v'~-~~ 2 y 2 z 2 (1-v 2 - z 2 ) dz dv<br />
E - 1 -~ 0 - 1 · -Vl -y2<br />
= J;1r J; (r 2 cos 2 O)(r 2 sin 2 O)(l - 1·2)rdrd()<br />
= J 0<br />
2 Tr Jd ;t sin 2 2(J(r 5 - r 7 ) drdO<br />
r2rr 1 (1 40) [ 1 6 1 8 ] r = 1 d() 1 [0 1 . 49] 2,- 2Tr "<br />
= Jo S - COS 6r - ii 1 ' r=O = TIJ2 - 4 SID 0 = 192 = 96<br />
= ll f " sin 3 () dO = lll [- cos 0 + i- cos 3 OJ " = !11<br />
s Jo 5 3 . o 15<br />
29 V _ r 2 r4 ( 2 2) d d _ 1 .2 [ 2 4 3J u=4 d . _ r2 ( 3 2 8 ) d . _<br />
. - J o JJ X + 4y y X - 0 X y + 3 y u= l , X - Jo X + 4 :r. - 176<br />
31:<br />
V = f 2 j' 11 r< 2 - ul/ 2 dzdxdy = rz rv (1-ly) dxdy<br />
Jo o .lo .Jo .lo 2<br />
y<br />
X<br />
CHAPTER 15 REVIEW D 293<br />
33. Using the w<strong>ed</strong>ge above the plane z = 0 and below the plane z = mx and noting that we have the same <strong>vol</strong>ume for .m < 0 as<br />
for m > 0 (so use m > 0), we have<br />
V = 2 foa/3 J/a2-9y2 mx.dxdy = 2 foal~ tm(a2 - 9y2) dy = m[a2y - 3y3)~/3 = m(ia3 - ia3) = ~ma3 .<br />
35. (a) m = fo1 fo1 - v2 ydxdy = J~1 (y - y3) dy = ~- ~ = ~<br />
rl rl- v 2 d d ri 1 ( 2)2 d 1 ( 1 2)3) 1 1<br />
(b) Mv = Jo Jo xy x Y = .Jo 2Y 1 - Y Y = -u - 'y o = 12'<br />
M, = Jo r l Jo r1 - v 2 y 2 d x d y = r1 c 2 4) d. 2 H c- -) ( 1 s )<br />
10 y - y y = 15 . ence x, y = _ 3<br />
, 15<br />
.<br />
r1 r1-v 2 3 J·1 ( 3 s) 1<br />
(c)lx=JoJo y dxdy= 0 y - y dy=u,<br />
r1 rl- v 2 2 d d _ r1 1 ( 2)3 d _ 1 ( 1 2)4] 1 _ 1<br />
ly = Jo Jo yx x Y - Jo 3Y 1 - Y Y- -24 - Y o - 24'<br />
.l _ I I _ 1 =2 _ .!.L_g _ 1 = _ 1 d = 2 _ 1/24 _ 1 = 1<br />
o - "'+ v- 8• Y - 1/ 4 - 3 =? Y - 73• an x - 1/ 4 - 6 =? x = ?s·<br />
37. (a) The equation of the cone with the suggest<strong>ed</strong> orientation is ( h - z) = ~ .j x 2 + y 2 , 0 $ z $ h. Then V = ~ 1ra 2 h is the<br />
• !! 11•-(11/a),jx2+y2 12,.1a1(hfa)(a-r) 1a. h2 ;<br />
M;r;y = z dz dA = rzdzdrd/J = 1r r 2 (a- r) dr<br />
o ooo o· a<br />
;r;2+y2 ~ 112 ·<br />
<strong>vol</strong>ume of one frustum of a cone; by symmetry M 11 :: = !vfx:: = 0; and<br />
1rh 2 1a 2 2 3 1rh 2 (a 4 2a 4 a 4 ) 1rh 2 a 2<br />
= - (a r- 2ar + r ) dr = - - - - + - = --<br />
a2 0 a 2 2 3 4 12<br />
Hence the centroid is (x,y,z) = (0,0, ~h).<br />
2"1"1(/•/.a)(a-r) 3 1" h 3 4 27rh (a5 a5) 7ra4 h<br />
10 o o 0 a a 4 5 10<br />
(b) I , = r. dz dr d() = 27r - (ar - r ) dr = - - - - = - -<br />
39. Let D represent the given triangle; then D can be describ<strong>ed</strong> as the area enclos<strong>ed</strong> by the x- and y-axes and the line y = 2 - 2x,<br />
or equivalently D = { (x, y) I 0 $ x $ 1; 0 $ y $ 2 - 2x }. We wantto find the surface area of the part of the graph of<br />
z = x 2 + y that lies over D, so using Equation 15.6.3 we have<br />
A(S)= Jfv 1+ (~;y + (~~Y dA = Jfv v 1+(2x)2+(1)2dA=<br />
112<br />
-<br />
2<br />
;r; )2 + 4x2dydx<br />
= .r; .../2 + 4x 2 [y J~:~- 2 ;r; dx = J;(2 - 2x) .../2 + 4x 2 dx = · J 0<br />
1<br />
2 J2 + 4x2 dx- JC: 2x J2 + 4x2 dx<br />
Using Formula 21 in the Table oflntegrals with a = J2. u. = 2x, and du = 2 dx, we have<br />
J 2 J2 + 4x2 dx = x J2 + 4x 2 + ln(2x + J2 + 4x 2 ). lfwe substitute u = 2 + 4x 2 in the second integral, then<br />
du = Bx dx and J 2x ,J2 + 4x 2 dx = ~ J .JU du = i · tu 3 1 2 = i (2 + 4x 2 ? 1 2 . Thus<br />
. 1<br />
A(S) = [xJ2 + 4x 2 + 1n(2x + J2 + 4x2) - t{2 + 4x 2 ) 3 1 2 Jo<br />
= J6 + ln (2 + JB) - ~(6) 3 1 2 - lnJ2 + 4- = ln 2 t# + {1<br />
= ln( J2 + v'3) + 4- ~ 1.6176<br />
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294 0 CHAPTER 15 MULTIPLE INTEGRALS<br />
41.<br />
f 3<br />
rJ 9 2<br />
- x<br />
(x 3 + xy 2 )dydx = { 3 {~ x(x 2 +y 2 )dydx<br />
1o 1 -J9-x2 1o 1 -~<br />
= J::~~2 f~ (r cos O)(r2 ) r dr dO<br />
43. From the graph, it appears that 1 - x 2 = e"' at x ~ - 0.71 and at<br />
x = 0, with 1 - x 2 > e"' on ( -0.71, 0). So the desir<strong>ed</strong> integral is<br />
= J::~~2 cos 0 dO J; r 4 dr<br />
= (sino]:::/2 (ir5] ~ = 2 ·. i (243) = 4 : 6 = 97.2<br />
JJ"n y2dA ~ f~o.n J.1.,-x2 y2 dydx<br />
= l Jo [(1 - x2)3 - e3"'] dx<br />
3 -0.71<br />
=-o.25<br />
45. {a) f(x, y) is a joint density function, so we know that JJR2 f(x, y) dA = 1. Since f(x, y) = 0 outside the rectangle<br />
(0, B) x (0, 2], we can say<br />
Then 15C = 1 :::?<br />
JJR2 f (x,y) dA = f~oo f~oo f(x,y) dydx = J: J: C(x + y) dydx<br />
C = ft.<br />
= C .r; [xy + h 2 J ~:~ dx = C J0 3 (2x + 2) dx = C[x 2 + 2x)~ = 15C<br />
(c) P(X + Y ~ 1) = P ((X, Y) E D) where Dis the triangular regiol} shown in<br />
the figure. Thus<br />
y<br />
P(X +Y ~ 1) = JJ0 f(x,y) dA = f 01 J;-x -ft (x+y)dydx<br />
=· ..1.. rl [ + 1 2]u= 1 -"' dx<br />
15 Jo xy 2Y u=O<br />
= fs f; [x(1 - x) + ~ ( 1 - x?] dx<br />
- 1 f1(1 .2)d - 1 [ 1 3]1 1<br />
- 30 Jo - X X - 30 X- 3X 0 = 45<br />
0 I X<br />
47.<br />
. z<br />
J_ l 1<br />
j'l .,2 j '1- IJ ( ) J.l r.1-=Jv"ii ! ( ) d d d<br />
0<br />
fx,y ,zdzdydx= 0<br />
. 0<br />
- v"ii x,y,z xyz<br />
X<br />
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49. Since u = x- y and v = x + y, x = t(u + v) andy= ~(v- u).<br />
dv<br />
8(u,v) -1/2 ·1/2 2 Rx+y 2 _ 2 v 2 , 2 v ·<br />
8(x y) I 1/2 1/21 1 fl x- y 14<br />
/ .o u (1) £4<br />
Thus --'- = = - and --dA = - - du dv = - - = -In 2<br />
CHAPTER 15 REVIEW 0 295<br />
51 . Let U = y - X and V = y +X SO X = y :... U = ( V - X) - U =:> X = ~ ( V - U) and y = V - ~ ( V - U) = ~ ( V + u).<br />
I~~::~~ I = I ~:~ -. ~~~~I = 1-~ ( ~) - ~(~)I = J -'~I = ~. · R is the image under this transformation .of the square<br />
with vertices (u, v) = (0, 0), ( -2, 0), (0, 2), and ( -2, 2). So<br />
This result could have been anticipat<strong>ed</strong> by symmetry, since the integrand is an odd function of y and R is symmetric about<br />
the x-axis.<br />
53. For each r such that Dr lies within the domain, A(Dr) = 71'1' 2 , and by the Mean Value Theorem for Double Integrals there<br />
exists (xr, Yr) in D,. such that f (xr, Yr) = ~ j' { f(x, y) dA. But lim (xr, Yr ) = (a, b),<br />
7TT lv,. r-o+<br />
so lim ~ j' { f (x, y) dA = lim f(xr, y,.) = f(a, b) by the continuity of f.<br />
r-o+ 1fT } Dr r--+O+<br />
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D PROBLEMS PLUS<br />
1.<br />
y<br />
5<br />
4<br />
3<br />
2<br />
x + y = 5<br />
Let R, = u; = 1 Ri, where<br />
R , = { (x, y) I x+ y ~ i + 2, x +y < i+3, 1 ~ x ~ 3, 2 ~ y ~ 5}.<br />
5 5<br />
ffn[x+y) dA = I: f fn .[x+y)dA= L: [x+ylffn . dA, since<br />
i =1 1. i=l '<br />
[x + y] = cons tant = ~+ 2 for (x, y) E R;. Therefore<br />
0<br />
2 3 X<br />
I rb I t [ t ?<br />
]<br />
3. f,, .• = b _ala f (x)dx = I- 0<br />
Jo J. cos(t-)dt dx<br />
Jfn[x + y) dA = E?= 1 (i + 2) [A(R;)]<br />
= 3A{RI) + 4A(R2) + 5A(R3) + 6A(&) + 7 A(Rs)<br />
= 3(4) + 4(~) + 5{2) + 6(~) + 7 ( ~) = 30<br />
= J; J: cos(t 2 ) dt dx = ,{ 0<br />
1 J; cos(t 2 ) dx dt [changing the ordeqlf intcsrntion]<br />
x = t<br />
0<br />
5. Since lxvl < 1, except at (1, 1), the formula for the sum of a geometric series gives -I- 1 - = f (xy)", so<br />
. - ~ n~<br />
00 00<br />
= "'<br />
L.., n+1<br />
1<br />
' n+1<br />
1 = "'<br />
L.., (n+<br />
1<br />
1)2 = 1!! 1 + 21<br />
1<br />
+ 32"<br />
1<br />
+ · · · =<br />
"'""<br />
L..,n = 1 ~ 1<br />
n=O<br />
n= O<br />
•.<br />
7. (a) Since ixyz i < 1 except at (I , I , I), the formula for the sum of a geometric se;ies gives - - 1 - = E (xyz )n, so<br />
1- X 1JZ n=O<br />
1<br />
11 11 1 11lll100 00 111·1<br />
- _--d:r;dydz= I: (xyztdxdy dz = I: (xyztdxdydz<br />
o o o 1 xyz o . o o n =O n = O 0 o 0<br />
00<br />
00<br />
= I:<br />
[.f1 ] [ 1 ] [ f 1 ] 1 1 1<br />
0<br />
x" dx ,[ 0<br />
y"' dy Jo z" dz = I: - - · - - · - - 1 1 1<br />
n = O . n =O n + n + n +<br />
00<br />
1 1 1 1 .<br />
00<br />
1<br />
= ... ~o (n + 1)3 = 13 + 23 + 33 + ... = n~1 n 3<br />
(b) Since 1-xyzl < 1, except at (1, 1, I ), the formula for the sum of a geometric series gives<br />
1 +xyz n = O<br />
1<br />
111 1 11·1·1 00 00 11.1111<br />
1<br />
1<br />
= f= (- x yz)" , so<br />
dx dy dz = I: ( -xyz)" dx dy dz = 2: ( -xyz)" dx dy dz ·<br />
o o o + xyz o o o n=O n=O o o o<br />
= f (-1)". (1~ 1 x" dx] [J; y"dy] [J; z" dz] = f (-1)"- 1 -- -<br />
1 - . -<br />
1 -<br />
n=o n=O n + 1 n + 1 n + 1<br />
00<br />
(-1)" 1 1 1<br />
00<br />
(-1)" - 1<br />
= n~o (n + 1)3 = 13 - 23 + :P - . . . = n~o n 3<br />
[continu<strong>ed</strong>]<br />
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298 D CHAPTER 15 PROBLEMS PLUS<br />
To evaluate this sum, we first write out a few tenns: s = 1 - ; 3<br />
+ ; 3<br />
- ; 3<br />
+ ;~ - ; 3<br />
~ 0.8998. Notice that<br />
a1 = \ < 0.003. By the Alternating Series Estimation Theorem from Section 11.5, we have Is- s61 ~ a1 < 0.003.<br />
. 7 .<br />
This error of0.003 :-viii not affectlhe second decimal place, so we have s .~ 0.90.<br />
. au. 8u ox au ay au az au au. .<br />
9. (a)x=rcosO, y =rsiDO, z=z. Then or= axar + ayar + ozor =ox cosO+ oysmOand<br />
S . . 1<br />
I ou ou . (} 8u (} d<br />
lffil ary {)(} =-ax TSID + oy rcos an<br />
8 2 u a 2 u ' 2 . 2<br />
0<br />
8 2 u. 2 2 8 2 u 2 . au 8u .<br />
ao2 = 8x2 r SID + 8y2 r cos (} - 2 oy ax r SID 0 cos (} - OX r cos (} - {}y r SID 0. So<br />
8 2 u lau 1 8 2 u _ 8 2 u a 2 u 2 8 2 u . 2 a 2 v. . au cosO 8u sinO<br />
!:i:2 + - -a + 2 o2 + !i'2 = !i"2 cos 0 + !i'2 sm 0 + 2 .
CHAPTER 15 PROBLEMS PLUS D 299<br />
d<br />
au au . A. • f) au . A. () h'l<br />
An afJ = - ax psm.I'Slll + ay pSID'I'COS , W I e<br />
Therefore<br />
a 2 u a 2 u · a 2 u ·<br />
-2 = -2 - a p 2 sin 2 if> cos() sin 0 + -a 2 p 2 sin 2 ¢ sin 2 ()<br />
afJ y X X<br />
a 2 u 2 au cot¢> au 1 8 2 u 1 a 2 u<br />
- + --+ ---+ --+ ~--;:oap2<br />
pap p2 a¢ p2 aq? p2 sin2 if> aB2<br />
+ a2 u 2 . 2 A. 2 0 au . A. 0 au . A. • 0<br />
a 2 p sm 'l'cos --a psm'l' cos - -a p sm 'l' sm<br />
y X y<br />
= ::~ [ (sin 2 ¢> cos 2 B) + ( cos 2 ¢> cos 2 B) + sin 2 B)<br />
+ ~~ [ (sin 2 ¢> sin 2 0) + ( cos 2 ¢> sin 2 0) + cos 2 OJ + ~~ [cos 2 ¢> .+ sin 2 ¢)<br />
+ au [ 2 sin 2 if> cos()+ cos 2 ¢> c~ 0 - sin 2 ¢> cos 0 - cos 0]<br />
ax<br />
psm¢><br />
+ au [ 2sin 2 ¢> sinB + cos 2 ¢> s~n()- sin 2 ¢> sin()- sinO]<br />
ay<br />
psm¢><br />
But 2 sin 2 ¢>cos 0 + cos 2 if> cos 0 - sin 2 ¢>cos 0 - cos 0 = (sin 2 if>+ cos 2 if> - 1) cos 0 = 0 and similarly the coefficient of<br />
aujay is 0. Also sin 2 ¢co?> 2 0 + cos 2 if>cos 2 0 + sin 2 e = cos 2 fJ (sin 2 ¢ + cos 2 ¢) + sin 2 () = 1, and similarly the<br />
coefficient of a 2 uj 8y 2 is 1. So Laplace's Equation in spherical coordinates is as stat<strong>ed</strong>.<br />
11. Io"' I~ Io" f(t)dtdz dy = IIIE f(t)dV, where. y<br />
· E = {(t, z, y) I 0 ~ t ~ z, 0 ~ z ~ y, 0 ~ y ~ x}. t =z<br />
If we let. D be the projection of Eon the yt-plane then<br />
D = {(y, t) I 0 S t S x, t S y S x}. And we see from the diagram<br />
that E = { (t, z, y) I t ~ z ~ y, t ~ y ~ x, 0 ~ t ~ x }. So<br />
} ~"' I~ I; f(t) dt dz dy = Io"' I/' .ftv f (t ) dz dy dt = Io"' [ft"'(y- t) f(t) dy] dt<br />
= J-;' [(h 2 - ty)f(t) J~:; dt = Io"' [~x 2 - tx- ~t 2 + t 2 ]f(t) dt<br />
= Io"' [~x 2 - tx + tt 2 ] f(t) dt = I o"' ax 2 - 2tx + t 2 )f(t) dt<br />
= ~ I;(x - t) 2 f(t) dt<br />
© 2012 Cengoge Looming. All Righos Reserv<strong>ed</strong>. Moy not be: scann<strong>ed</strong>. copi<strong>ed</strong>. or duplicat<strong>ed</strong>, or post<strong>ed</strong> to a publicly oe
300 0 CHAPTER 15 PROBLEMS PLUS<br />
13. The <strong>vol</strong>ume is V = Jff R dV where R is the solid region given. From Exercise 15.10.2 l(a), the transformation x =au,<br />
y = bv, z = cw maps the unit ball u 2 + v 2 + w 2 :::; 1 to the solid ellipsoid<br />
x2 y2 z2 i o(x y z)<br />
2 + -b<br />
2<br />
+ 2 :::; with o( ' ' ) = abc. The same transformation maps the<br />
a c u,v,w<br />
X y z . T. h tl . R.<br />
plane u + v + w = 1 to - + - + - = 1. us 1e regton 111 xyz-space<br />
a b c<br />
corresponds to the regionS in uvw-space consisting of the smaller piece of the<br />
. unit ball cut offby the plane u + v + w = 1, a "cap of a sphere" (see the figure).<br />
We will ne<strong>ed</strong> to compute the <strong>vol</strong>ume of S, but first consider the general case<br />
where a horizontal plane slices the upper portion of a sphere of radius r to produce<br />
a cap of height h. We use spherical coordinates. From tile figure, a line through tile<br />
origin at angle ¢ from tile z-axis intersects the pl_ane when cos¢ = ( r - h)/ a =:><br />
a = (r· - h)/ cos¢, and the line passes through the outer rim of the cap when<br />
a = r =:> cos¢ = (r.- h)/r =:> ¢ = cos- 1 ((r- h)/r). Thus the cap<br />
is describ<strong>ed</strong> by { (p, 8, ¢) I (r- h)/ cos¢:::; p:::; r, 0:::; (}:::; 21r, 0:::; ¢:::; cos- 1 ((r- h)/ r)} and its <strong>vol</strong>ume is<br />
v = f27r rcos- 1 ((r-h)/r) rr<br />
2 sinA.d dd.d(}<br />
Jo .J o J(r-h)/ cos 4> P "~' P 'I'<br />
= f27r f COS- l ((r-h)/r) [l 3 sin¢] P=•· d¢ d(}<br />
Jo Jo 3P p = (•·-h)/cos¢<br />
112" 1cos-<br />
1<br />
((r-h)/r) [ (r _ h? ]<br />
= - r 3 sin¢ - ¢ sin rjJ d¢ dO<br />
3 3 0 0 · cos<br />
_ 1 f27r [- 3 A._ 1 ( _ h)3 - 2 ,;.] ¢=cos- 1 ((r- h)/•·) d()<br />
- 3 .I 0 r cos 'I' 2 r cos ~- ¢ =O .<br />
1 r7f' [ 3 (r-h) 1 3 (r- h)- 2 3 1 ( 3]<br />
= 3 ./o -r - r- - 2(r-h) - r- +r + 2<br />
r -h) d()<br />
(This <strong>vol</strong>ume can also be comput<strong>ed</strong> by treating the cap as a solid of re<strong>vol</strong>ution and using the single variable disk method;<br />
see Exercise 5.2.49 [ET 6.2.49].)<br />
To determine the height h ofthe cap cut from tl1e unit ball by the plane<br />
u + v + w = 1, note that the line u = ·u = w passes through the origin with<br />
direction vector (1, 1, 1) which is perpendicular to the plane. Therefore this line<br />
coincides with a radius of tile sphere that passes through the center of the cap and<br />
h is measur<strong>ed</strong> along this line. The line intersects the plane at ( t, t, t) and the<br />
plane u + u + w = I<br />
sphere at ( '7a, '7a, '7a). (See the figure.)<br />
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CHAPTER 15 PROBLEMS PLUS 0 301<br />
The distance between these points is h = V 3 ( -jg - t Y = v'3 (-jg - t) = 1 - 73. Thus the <strong>vol</strong>ume of R is<br />
V= IlL dV= Ills 1:(~:~:~~ ~ dV = abc Ills dV = abcV(S)<br />
= abc· 1rh 2 (r - th) = abc· 1r ( 1 - 7-J Y [ 1 - i _( 1-73)]<br />
= abC7r ( ~ - "7a) ( j + aTa) = abC7r ( ~ - ~ ) ~ 0.482abc<br />
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16 0 VECTOR CALCULUS<br />
16.1 Vector Fields<br />
1. F (x, y) = 0.3 i - 0.4j<br />
All vectors in this field are identical, with length 0.5 and<br />
parallel to (3, -:-4) .<br />
3. F (x,y) = - ~ i + (y - x) j<br />
y<br />
The length of the vector - ~ i + (y - x) j is<br />
vt. + (y - x) 2 . Vectors along the line y =X are<br />
horizontal with length ~.<br />
5 F ( ) - y i + x j<br />
. x,y - Jx2 +y2<br />
The length of the vector ~ is 1.<br />
x2 + y2<br />
7. F (x, y , z) = k<br />
All vectors in this field are parallel to the z-axis and have<br />
length 1.<br />
9. F(x,y,z) = x k<br />
At each point (x, y, z), F (x, y, z) is a vector of length lxl.<br />
For x > 0, all point in the direction of the positive z-axis,<br />
while for x < 0, all are in the direction of the negative<br />
z-axis. In each plane x = k, all the vectors are identical.<br />
X<br />
,,<br />
]j<br />
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304 0 CHAPTER 16 VECTOR CALCULUS<br />
11. F(x, y) ·= {x, -y) corresponds to graph IV. In the first quadrant all the vectors have positive x-components and negative<br />
y-components, in the second quadrant all vectors have negative x- and y-components, in the third quadrant all vectors have<br />
negative x -components and positive y-components, and in the fourth quadrant all vectors have ~os itive x - and y-compdnents.<br />
In addition, the vectors get shorter as we approach the origin.<br />
13. F (x, y) = (y, y + 2) corresponds to graph I. As in Exercise 12, aJI vectors in quadrants I and II have positive x~components<br />
while all vectors in quadrants III and IV have negative x-components.Vectors along the line y = -2 are horizontal, and the<br />
vectors are independent of x (vectors along horizontal lines are identical).<br />
15. F(x, y, z) = i + 2j + 3 k corresponds to graph IV, since all vectors have identical length and direction.'<br />
17. F(x, y, z)' =·xi+ y j + 3 k corresponds to graph III; the projection of each vector onto the xy-plane is x i + y j , which points<br />
away from the origin, and the vectors point generally upward because their z-components are all 3.<br />
19. 4.5'<br />
1~ \ '<br />
'<br />
~ I I<br />
I I<br />
- 4.5 1 I I . l<br />
I I<br />
' '<br />
I .<br />
I<br />
-4.5<br />
'<br />
.<br />
'<br />
'<br />
'<br />
I<br />
. I<br />
I<br />
I I<br />
' \<br />
\ \<br />
I<br />
I<br />
I<br />
I<br />
I<br />
4.5<br />
I<br />
I<br />
~<br />
t<br />
The vector field seems to have very short vectors near the line y = 2x.<br />
For F(x, y) = (0, o'} we must have y 2 - 2xy = 0 and 3xy- 6x 2 = 0.<br />
The first equation holds if y = 0 or y = 2x, and the second holds if<br />
x = 0 or y = 2x. So both equations hold [and thus F ( x, y) = 0] along<br />
the line y = 2x.<br />
21. f(x, y) = xe"'Y =><br />
'i7 f(x, y) = f:,(x, y) i + jy (x, y) j = (xe'"Y · y + e'"Y) i + (xe"'Y. x) j = (xy + l )e"'Y i + x 2 e'"!l j<br />
23. 'ilf(x,y, z) =f:z:(x,y,z)i+fy(x,y,z)j +f::(x,y, z) k =..) 2<br />
x i +.J<br />
2 2<br />
y j +..)<br />
2 2 2 2<br />
z 2<br />
~ k<br />
x +y +z x +y +z x +y +z<br />
25. f(x,y) = x 2 - y => 'ilf(x,y) = 2xi - j .<br />
The length ofV' f(x, y) is ..)4x 2 + 1. When x =I= 0, the vectors point away<br />
from they-axis in a slightly downward direction with length that increases<br />
as the distance from the y-axis increases.<br />
2x<br />
4y<br />
27.Wegraph\lf(x, y)=<br />
2 22 i + 1<br />
2<br />
l +x+ y +x+ jalongwith 22 y<br />
a contour map off.<br />
The graph shows that the gradient vectors arc perpendicular to the<br />
level curves. Also, the gradient vectors point in the direction in<br />
which f is increasing and are longer where the level curves are closer<br />
together.<br />
© 2012 Ccngogo Learning. All RigltlS Rc:scrvcd. May nol be scann<strong>ed</strong>, copi<strong>ed</strong>, or duplicat<strong>ed</strong>. or post<strong>ed</strong> 10 o publicly occc:ssible website, in whole or in pan.<br />
I
SECTION 16.2 LINE INTEGRALS 0 305<br />
29. f ( x , y) = x 2 + y 2 => V f( x, y) = 2x i + 2y j . Thus, each vector V f ( x, y) lias the same direction and twice the length of<br />
the position vector of the point (x, y), so the vectors all point directly away from the origin and their lengths increase as we<br />
move away from the origin. Hence, V f is graph III.<br />
31 . f (x, y) = (x + y) 2 => V f (x , y) = 2(x + y) i + 2(x + y) j. The x- andy-components of each vector are equal, so all<br />
vectors are parallel to the line y = x . The vectors are 0 along the line y = --:x and their l~ngth increases as the distance from<br />
this line increases. Thus, V f is graph II.<br />
33. At t = 3 the particle is at (2, 1) so its velocity is V (2, 1) = (4, 3). After 0.01 units of time, the particle's change in<br />
location should be approximately 0.01 V(2, 1) = 0.01 (4, 3) = (0.04, 0.0~) . so the particle should be approximately at the<br />
point (2.04, 1.03) .<br />
35. (a) We sketch the vector field F(x; y ) = x i - y j along with<br />
several approximate flow lines. The flow lines appear to<br />
be hyperbolas with shape similar to the graph of<br />
y = ±11x, so we might guess that the flow lines have<br />
equations y = C / x.<br />
(b) If x = x(t) andy = y(t) are parametric equations of a Aow line, then the velocity vector of the flow line at the<br />
point (x, y) is x' (t) i + y' (t) j . Since the velocity vectors coincide with the vectors in the vector field, we have<br />
x' ( t) i + y' ( t) j = x i - y j => dx Nt = x, dy I dt = - y. To solve these differential equations, we kno~<br />
dxldt = x => dx/ x = dt => In lxl = t +. C => x = ± ~ t + c = Aet for some cons~nt A, and<br />
dy I dt = - y => dy / y = -dt => In I vi = - t + I< => y = ±e- t + K = B e- t for some constant B. Therefore<br />
xy = A et B~- t = AB = constant. If the flow line passes through (1, 1) then (1) (1) = constant= 1 => xy = 1 =><br />
y = 1l x, x > 0.<br />
16.2 Line Integrals<br />
1. x = t 3 andy = t, 0 :::; t :::; 2, so by Formula 3<br />
.Ia y 3 ds ~ 1\3<br />
( ~~ Y + ( ~~ Y<br />
2<br />
dt = 1<br />
t 3 y'(3t 2 )2 + (1)2 dt = 1\a yf9t 4 + 1 dt<br />
= ..l. . 1 (9t 4 + 1) 3 / 2 ) 2 = ..l. (145 3 1 2 - 1) or ..l. (145 v'I45- 1)<br />
:ill 3 O G4 54<br />
3. Parametric equations for C are x = 4 cos t, y = 4 sin t, - ~ :::; t :::; ~ . Then<br />
fc xy 4 ds = J:~~2 (4 cos t)(4sin t) 4 y'( -4sin t )2 + (4cos t)2 dt = J:~~ 2 45 cost sin 4 t yf16(sin 2 t + cos2 t ) dt<br />
= 4 5 Jrr/ 2 (sin 4 t cos t)(4) dt = (4) 6 [l sinG t) Tr/ 2 = 2 · 40 = 1638.4<br />
- -rr/ 2 5 - Tr/ 2 5<br />
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308 0 CHAPTER 16 VECTOR CALCULUS<br />
5. If we choose x as the parameter, parametric equations for Care x = x, y = v'xfor 1:::; x:::; 4 and<br />
fc ( x2y3- v'x) dy = J;' [xz. (Vx)3 - v'x] 2 ~ dx = 4 J14 (x3- 1) dx<br />
= 1 [1 x ·1 - x] 4 - ! (64 - 4 - 1 + 1) - ill<br />
2 4 1- 2 4 - 8<br />
7.<br />
y<br />
(2, 1)<br />
On cl: X = x, y = ~X => dy = ~ dx, 0 :::; X :::; 2.<br />
On Cz: x = x, y = 3 - :v => dy = -dx, 2 :::; x:::; 3.<br />
Then<br />
f 0<br />
(x + 2y) dx + x 2 dy = fc<br />
1<br />
(x + 2y} dx + x 2 dy +.J 02<br />
(x + 2y} dx + x 2 dy<br />
=J: [x+2(~x)+x 2<br />
(~)]dx+J: [x+2(3-x}+x 2 (-1}]dx<br />
= f 0<br />
2<br />
(2x + ~x 2 ) dx + J 2<br />
3<br />
(6 - x- x 2 ) dx<br />
9. x = 2sint, y = t , z = -2cost, 0:::; t:::; 7T. Then by Formula 9,<br />
fc xyz ds = J 0<br />
7T (2 sin t)(t)( - 2 cos th/ (!!fd + { !fltf + C¥tf dt<br />
= J; -4t sin t cost .J(2 cos t)2 + 11)2 + (2 sin t) 2 dt = J 0<br />
7T - 2tsin 2t .J 4(cos 2 t + sin 2 t) + 1 dt<br />
= -2 y'5 j~'ff t sin2tdt = -2 y'5 [-~ t cos 2t +'i sin2t] ~<br />
= -2¥'5(-~ - 0) = J57T<br />
integrate by parts with ]<br />
[<br />
u = t , dv = s in 2t dt<br />
11 . Parametric equations for Care x = t, y = 2t, z = 3t, 0 :::; t :::; l. Then<br />
15. Parametric equations for Care x = 1 + 3t, y = t, z = 2t, 0 :::; t :5. 1. Then<br />
fc z 2 dx + x 2 dy + y 2 dz = J;(2t? · 3 dt + (1 + 3t}~ dt + t 2 · 2 dt = J; (23t 2 + 6t + 1) dt<br />
= [¥tl + 3t 2 + t]~ = ¥ + 3 + 1 = ¥<br />
17. (a) Along the line x = - 3, the vectors ofF have positive y-components, so since the path goes upward, the integrand F ·Tis<br />
always positive. Therefore f.e F · dr = f.e F · T ds is positive.<br />
l 1 .<br />
© 2012 Cengoge L
SECTION 16.2 LINE INTEGRALS 0 307<br />
(b) All of the (nonzero) field vectors along the circle with radius 3 are point<strong>ed</strong> in the clockwise direction, that is, opposite the<br />
direction to the path. So F · Tis negative, and therefore fc 2<br />
F · dr = fc 2<br />
F · T ds is negative.<br />
19. r(t) = llt 4 i + t 3 j ,, so F (r (t)) = (llt 4 )(t 3 ) i + 3(t 3 ) 2 j = llt 7 i + 3t 6 j and r'(t) =· 44t 3 i + 3t 2 j . Then<br />
fc F · dr = J; F(r(t)) · r'(t)dt = J;(ne · 44e + 3t 6 · 3t 2 )dt = J;(484t 10 + 9t 8 )dt = [44t 11 +t 9 )~ = 45.<br />
21 . fc F · dr = J; (sin t\ cos( -t 2 ), t 4 ) · (3t 2 , - 2t, 1) dt<br />
= J;(3t2 sint 3 - 2tcose + t 4 ) dt = [- cost 3 - sint 2 + tt 5 )~ =~-cos 1-sinl<br />
fc ~ ·<br />
d~ = i<br />
2<br />
= i<br />
2<br />
2<br />
F (r (t )) · r'(t)dt = i [et-t 2 et :rsm(e-t 2 )<br />
[ e 2 t- t 2 2 2<br />
- 2te - t sin ( e - t ) J dt ::::J 1.9633<br />
. ( - '2te- t 2 ) ] dt<br />
25. x = t 2 , y = t 3 , z = t 4 so by Formula 9,<br />
fc xsin(y + z) ds = J;(e) sin(t 3 + t 4 )vf(2t)2 + (3t2)2 + {4t 3 )2 dt<br />
= f 0<br />
5 t<br />
2<br />
sin(t 3 + t 4 ) J4t 2 + 9t 4 + 16t6 dt ::::: 15.0074<br />
27. We graph F ( x, y) = ( x - y) i + x y j and the curve C. We see that most of\he vectors starting on C poi~t in roughly the same<br />
direction as C, so for these portions of C the tangential compon~nt F · T is positive. Although some vectors in the third<br />
quadrant which start on C point in roughly the opposite direction, and hence give negative tangential components, it seems<br />
reasonable that the effect of these portions of C is outweigh<strong>ed</strong> by the positive tangential components. Thus, we would expect<br />
fc F · dr = fc F · T ds to be positive.<br />
To verifY, we evaluate fc F · dr: The curve C can be represent<strong>ed</strong> by r (t) = 2 cost i + 2 sin t j , 0 $ t $ 3 ;,<br />
so F (r(t)) = (2 cost- 2 sin t) i + 4 cos tsin t j and r ' (t) = - 2 sin t i + 2 cos t j . Then<br />
fc F · dr = ./;" 12 F (r (t)) · r ' (t) dt<br />
= J;" 12 [-2 sin t(2 cost - 2 sin t) + 2 cos t(4cos t sin t )) dt<br />
3<br />
= 4J ,.<br />
12<br />
0<br />
(sin<br />
2<br />
t- sintcost + 2sintcos 2 t) dt<br />
[using a CAS)<br />
@) 2012 Ccngogc Lcruning. All Rights Reserv<strong>ed</strong>. May nol be 5Cl!Mcd. copi<strong>ed</strong>, or duplicnl<strong>ed</strong>, or poslcd lo • publicly accessible wcbsilc, in whole or in part.
308 0 CHAPTER 16 VECTOR CALCULUS<br />
(b) r (O) = 0, F(r(O)) = (e- 1 , 0); 2.1<br />
1 ) _ (l 1 ) F( ( 1 )) _ / -1/2 1 )·<br />
r ( 72 - 2' 272' , .r 72 - \ e ' 4,72 '<br />
F(r(ll)/<br />
r(1) = (1, 1), F (r(1)) = (1, 1).<br />
In order to generate the graph with Maple, we usc the 1 ine command in<br />
the plot t ools package to define each of the vectors. For example,<br />
vl :=line ( [0, 0),. [exp(-1), 0 ] ):<br />
0 ~--....:.....:....c!..._ _ __ +i 2. 1<br />
-0.2<br />
generates the vector from the vector field at the point (0, 0) (but without an arrowhead) and gives it the name vl. To show<br />
everything on the same screen, we use the display command. In Mathematica, we use List Plot (with the<br />
Plot Join<strong>ed</strong> - > True option) to generate the vectors, and then Show to show everything on the same screen.<br />
31. x = e-t cos4t, y = e-t sin4t, z = e-t, 0 ~ t :::; 2rr .<br />
Then:= e- 1 (-sin4t}(4) - e-tcos4t = - e- 1 (4sin4t+cos4t),<br />
dy = e- 1 (cos4t)(4)- e-t sin4t = -e- t( - 4cos4t + sin4t), and dd.z = - e- L, so<br />
dt<br />
t<br />
(<br />
~~ y + ( ~; y + ( ~: y<br />
= v(-e- 1 )2((4sin 4t + cos4t)2 + ( - 4cos4t + sin4t)2 + 1]<br />
= e-t J16(sin 2 4t + cos 2 4t) + sin 2 4_t + cos 2 4t + 1 = 3 ..f2 e-t<br />
Therefore<br />
J~ x 3 1/ z ds = J 0<br />
211" (e-t cos4t) 3 (e-t sin 4t) 2 (e- 1 ) (3 ..f2 e-t) dt<br />
- r 2 -.r3.,J2e- 7tcos 3 4tsin 2 4tdt- 172·704 ..f2(1 - e- 14 -.r)<br />
- Jo - 5,632,706<br />
33. We use the parametrization x = 2 cost, y = 2 sin t, - ~ :::; t :::; 'i. Then<br />
ds = (',~n 2 + (~~) 2 dt = v(-2sint) 2 + (2cost) 2 dt = 2dt, som = J~ kds = 2kJ::~~ 2 dt ~ 2k(7r),<br />
Hence (x,·il) = · (~, 0).<br />
35. (a) x = ..!._ { xp(x, y , z) ds, y = ..!._ ; · yp(x, y, z) ds, z = ..!._ { zp(x, y, z) ds where m = ./~ p(x , y, z) ds.<br />
m}c m c m }c<br />
(b) m = fc kds = k J~ -.r V4sin 2 t + 4cos 2 t + 9 dt = k vTIJ;"" dt = 2rrkJI3,<br />
1 1 211" i 1 2-.r<br />
x = JI3 2kVlJ sintdt = 0, y = M 2k v'i3 costdt = 0,<br />
27rk 13 0 27rk 13 0<br />
z =<br />
1 M { 2 -.r (k Vi3) (3t) dt = 2<br />
3 (2rr<br />
2 ) = 3rr. Hence (x, y, z) = (0, 0, 37r).<br />
27rk 13 /o 7r<br />
® 2012 CcngJgc LcJ.ming. All Rights Rcsr.:rvcd. Mny not l>c scumH:~I. copic(l, oi· duplicntcd. or post<strong>ed</strong> to 11 publicly nccl-sslblc wcbsilc. in whole or in r:1rt.
SECTION 16.2 LINE INTEGRALS D 309<br />
37. From Example 3, p(x,y) = k(1- y), x = cost, y = sint, and ds = dt, 0 :S t :S 1r ::::}<br />
1:. = Icy 2 p(x,y) ds = IoT< sin 2 t [k(1- sint)]dt = ki 0<br />
" (sin 2 t- sin 3 t) dt<br />
= ~k I 0<br />
.,.(1- cos2t) dt- k Io"(1- cos 2 t)sintdt<br />
Let u = cost, du = - sin t dt ]<br />
[ in the second integral<br />
= k[~ + I1- 1 (1-u 2 )du] =k (~ - ~)<br />
lv = Icx 2 p(x, y)ds = ki 0<br />
.,. cos 2 t(1 - sint)dt = ~ I 0<br />
.,.(1 +cos2t)dt- kio" cos 2 tsintdt<br />
= k(~- i), using t?e same substitution as above. ·<br />
39. W = Ic F · dr = I;.,. (t-sint,3 - cost)· (1 - cost,sint) dt<br />
= I; .,.(<br />
t - t cost - sin t + sin t cos t + 3 sin t - sin t cos t) dt<br />
=I;,. (t- tcost + 2sin t) dt = [~t 2 - (tsin t + cost) - 2 cos t)~ ,..<br />
integrate by .parts ]<br />
[ in the second term<br />
41. r (t) = (2t, t, 1 - t), 0 :::; t :::; 1.<br />
W = Ic F · dr = Io 1 (2t- t 2 , t- (1 - t?, 1 - t- (2t) 2 ) · (2,1, - 1) dt<br />
= Io 1 (4t- 2t2 + t- 1 + 2t - t 2 - 1 + t + 4t 2 ) dt =I; (t 2 + 8t - 2) dt = [lt 3 + 4t2 - 2t)~ = ~<br />
43. (a) r (t) = at2 i + bt 3 j ::::} v (t) = r' (t) = 2at i + 3bt 2 j ::::} a(t) = v' (t) = 2a i + 6btj, and force is mass times<br />
acceleration: F(t) = ma(t) = 2mai + 6mbtj.<br />
(b) W = fc F · dr = J 0 1 (2mai + 6mbtj) · (2at i + 3bt 2 j) dt = I 0<br />
1<br />
(4ma 2 t + 18mb 2 t 3 ) dt<br />
= [2ma 2 t 2 + ~mb 2 t 4 )~ = 2ma 2 + ~mb 2<br />
45. Let F = 185 k. To parametrize the staircase, let x = 20 cost, y = 20 sin t, z = ~~ t = 1!-t, 0 :::; t :::; 61r ::::}<br />
W = fc F · dr =I:.,. (0, 0, 185) · ( -20 sin t, 20 cost, ~) dt = (185) ~ J:.,.. dt = (185)(90) :::::: 1.67 x 10 4 ft-lb<br />
47. (a) r (t) = (cost, sin t}, 0 :S t:::; 271", and let F =(a, b). Then<br />
W = Ic F · dr = I;" (a, b) · (- sin t,cost) dt = I~'lf(-asint + bcost) dt = [a cost + bsin t)~ .,.<br />
(b) Yes. F (x, y) = k x = (kx, ky) and<br />
=a+O - a+ O=O<br />
W = Ic F · d r =I;,. (k cost, k s~ t} · (-sin t , cost} dt = J;" ( - k sin t cost+ k sin t cost) dt = I~ .,.. 0 dt = 0.<br />
49. Let r(t) = (x(t), y(t), z(t)) and v = (v1, v2, va). Then<br />
Ic v · dr = I: (v1, v2, va) · (x' (t), y' (t), z' (t)) dt = I: [v1 x' (t} + v2 y' (t) + va z' (t)] dt<br />
= [v1 x(t) + v2 y(t) + va z(t)): = [v1 x (?) + V2 y(b) + va z(b)J - [v1 x(a) + V2 y(a) + vs z(a)]<br />
= v 1 [x(b)- x(a)J + v2 [y(b)- y(a)] + va [z(b) - z(a)]<br />
=: (v1, v2, va) · (x(b)- x(a), y(b) - y(a), z(b)- z(a))<br />
= (v1, v2, va) · [(x(b), y(b), z(b)}- (x(a), y(a), z(a))] = v · (r(b)- r (a)]<br />
@ 2012 C
310 0 CHAPTER 16 VECTOR CALCULUS<br />
51. The work done in moving the object is fc F · dr = fc F · T ds. We can approximate this integral by dividing C. into<br />
7 segments of equal length t:.s = 2 and approximating F · T , that is, the tangential component of force, at a point (xi , yi) on<br />
each segment. Since C is compos<strong>ed</strong> of straight line segments, F · T is the scalar projection of each force vector onto C.<br />
If we choose (xi, yi) to be the point on the segment closest to the origin, then the work done is .<br />
7<br />
fc F · T d.s ~ E (F(xi, yi) · T(xi, yi)] t:.s = (2 + 2 + 2 + 2+ 1 + 1 + 1)(2) = 22. Thus, we estimate the work done to<br />
i=l .<br />
be approximately 22 J.<br />
16.3 The Fundamental Theorem for Line Integrals<br />
1. C appears to be a smooth curve, and since \J f is continuous, we know f is differentiable. Then Theorem 2 says that the value<br />
of fc \J f · dr is simply the difference of the values off at the terminal and initial points of C. From the graph, this is<br />
50- 10 = 40.<br />
3. 8(2x- 3y)j8y = - 3 = 8( -3x + 4y- 8)j8x and the domain ofF is IR 2 which is open and simply-connect<strong>ed</strong>, so by<br />
Theorem 6 F is conservative. Thus, there exists a function f such that \J f = F , that is, f:r:(x, y) = 2x -<br />
3y and<br />
f 11<br />
(x, y) = -3x + 4y - 8. But f,.(x, y) = 2x - 3y implies f(x, y) = x 2 -<br />
3xy + g(y) and differentiating both sides of this<br />
equation with respect to y gives f 11<br />
( x, y) = - 3x + g' (y). Thus - 3x + 4y - 8 = -3:~.: + g' (y) so g' (y ) = 4y - 8 and<br />
g(y) = 2y 2 - By+ [(where I< is a constant. Hence f(x, y) = x 2 - 3xy + 2y 2 - 8y +I< is a potential function for F .<br />
5. 8(e"' cosy)j8y := - e"' sin y, 8(e"' siny)f8x = e" sin y. Since these are not equal, F is not conservative.<br />
7. o(ye"' + sin y)j8y =ex+ cosy= o(e"' + X cosy)f8x and the domain ofF is IR 2 • Hence F is conservative so there<br />
exists a function f such that \J f =F. Then fx (x, y) = ye" +sin y implies f(x, y) = ye" + x sin y + g(y) and<br />
fv(x, y) = e"' + xcos y + g'(y). But f 11 (x, y) = e"' + x cosy so g(y) = K and f(x, y) = ye" + xsin y + K is a potential<br />
function for F.<br />
9. 8(lny + 2xy 3 )j8y = 1/y + 6xy 2 = 8(3x 2 y 2 + xfy)j8x and the domain ofF is {(x, y) I y> 0} which is open and simply<br />
connect<strong>ed</strong>. Hence F is conservative so there exists a function f such that \J f = F . Then f:z:(x, y) =In y + 2xy 3 implies<br />
f(x,y) = xlny + x 2 y 3 + g(y) and ! 11<br />
(x,y) = xfy + 3x 2 y 2 + g'(y). But f 11 (x,y) = 3x 2 y 2 + xjy so g'(y) = 0 =}<br />
g(y) = [(and f(x, y) = xlny + x 2 y 3 + [(is a potential function for F .<br />
11. (a) F has continuous first-order partial derivatives and 8<br />
8 2xy = 2x = 8<br />
8 (x<br />
2 ) on JR 2 , which is open and simply-connect<strong>ed</strong>.<br />
y X .<br />
Thus, F is conservative by Theorem 6. Then we know that the line integral ofF is independent of path; in particular, the<br />
value of fc F · dr depends only on the endpoints of C. Sin.ce all three curves have the same initial and terminal points,<br />
J~ F · dr will have the same value for each curve.<br />
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SECTION 16.3 THE FUNDAMENTAL THEOREM FOR LINE INTEGRALS 0 311<br />
(b) We first find a potential function f, so that \l f = F. We know ! :r:(x, y) = 2xy and jy(x, y) = x 2 • Integrating<br />
fx(x, y) with respect to x, we have f(x, y) = x 2 y + g(y). Differentiating both sides with respect toy gives<br />
· J 11<br />
(x, y) = x 2 + g'(y), so we must have x 2 + g'(y) = x 2 =? g'(y) = 0 => g(y) = K, a constant.<br />
Thus f(x, y) = x 2 y + K. All three curves start at (1, 2) and end at (3, 2), so by Theorem 2,<br />
fe F · dr = !(3, 2) - !(1, 2) = 18 - 2 = 16 for each curve.<br />
13. (a) f r(x, y) = xy 2 implies J(x, y) = ~x 2 y 2 + g(y) and fy (x, y) = x 2 y + g'(y). But /y(x, y) = x 2 y so g'(y) = 0 =><br />
g(y) = K, a constant. We can take K = 0, so f(x, y) = ~x 2 y 2 .<br />
(b) The initial point of C is r (O) = (0, 1) and the terminal point is r(1) = (2, _1), so<br />
fe F · dr = /(2, 1) - f(O, 1) = 2 - 0 = 2.<br />
15. (a) f x(x, y, z ) = yz implies f(x, y, z) = xyz + g(y, z) and so jy(x, y , z) = xz + g 11 (y, z). But jy(x, y, z) = xz so<br />
gy(y, z ) = 0 => g(y, z) = h(z). Thus f(x, y, z) = xyz + h(z) and f:(x, y, z) = X'!f + h'(z). But<br />
f:(x, y, z) = xy + 2z, so h'(z) = 2z => h(z) = z 2 + K.. Hence f(x, y, z) = xyz + z 2 (taking J( = 0).<br />
(b) fe F · dr = !(4,6,3)-!(1, 0, - 2) = 81 - 4 = 77.<br />
17. (a) f:r:(x, y, z) = yze"'= implies f(x, y, z) = ye"'= + g(y, z) and so fv(x, y, z) = e"'= + gy(y, z). But jy(x, y, z ) = e"'= so.<br />
gy{y,z) = 0 =? g(y,z) = h(z). Thus f(x,y,z)·= ye"'= + h(z) and fz(x, y,z) = xyex= + h'(z). But<br />
f:.:(x,y,z) = xyex=, so h'(z) = 0 =? h(z) = K. Hence f(x,y,z) = ye"'= (taking!(= 0).<br />
(b) r (O) = (1, -1, 0), r(2) = (5, 3, 0) so fe F · dr = /(5, 3, 0) - /(1, - 1, 0) = 3e 0 + e 0 = 4.<br />
19. The functions 2xe-Y and 2y - x 2 e- v have continuous first-order derivatives on JR 2 and<br />
~ (2xe-v) = -2xe- Y = ~ (2y- x 2 e:-v), so F(x, y) = 2xe-Y i + (2y- x 2 e-Y) j is a conservative vector field by<br />
Ulj vx . .<br />
Theor~m 6 and hence the line integral is independent of path. Thus a potential function f exists, and fx(x, y) = 2xe-v<br />
implies f(x, y) = x 2 e- 11 + g(y) and fv(x, y) = - x 2 e-Y + g'(y). But fv(x, y) = 2y- x 2 e-v so<br />
g'(y) = 2y =? g(y) = y 2 + K. We can take K = 0, so f(x,y) = x 2 e- 11 + y 2 • Then<br />
.fe 2xe-v dx + (2y - x 2 e- v) dy = /(2, 1) - f(l, 0) = 4e- 1 + 1 - 1 = 4/e.<br />
21. IfF is conservative, then fe F · dr is independent of path. This means that the work done along all piecewise-smooth curves<br />
that have the describ<strong>ed</strong> initial and terminal points is the same. Your reply: It doesn't matter which curve is chosen.<br />
23. F (x, y) = 2y 3 1 2 i + 3x Jyj, W = .fe F · dr. Since 8(2y 3 1 2 )/8y = 3 y'Y = 8(3x Jy )/ ax, there exists a function f<br />
suchthat\lf=F.Infact,fx(x, y)='2y 3 1 2 => f(x,y)=2xy 312 +g(y) => fv(x,y)=3xy 1 1 2 +g'(y). But<br />
/y (x, y) = 3x y'Y so g'(y) = 0 or g(y) = K. We can take I< = 0 =? f(x, y) = 2xy 3 1 2 • Thus<br />
W = fe F · dr = j(2, 4) :.... j(I,·l) = 2(2)(8)- 2(1) .= 30.<br />
© 2012 Ccngogc Leoming. All Rights Resen'Od. May nol b< scann<strong>ed</strong>, copi<strong>ed</strong>. orduplicalcd. or posl
312 0 CHAPTER 16 VECTOR CALCULUS<br />
25. We know that if the vector field (call it F) is conservative, then around any clos<strong>ed</strong> path C, fc F · dr = 0. But take G to be a<br />
circle center<strong>ed</strong> at the origin, orient<strong>ed</strong> counterclockwise. All of the field vectors that start on G are roughly in the direction of<br />
motion along C, so the integral around G will be positive. Therefore the field is not conservative.<br />
27. r1 I I '<br />
I<br />
'<br />
\<br />
1 I<br />
t ' I \<br />
f r r I<br />
' ' ' '<br />
I I I<br />
-2'11'<br />
l I I<br />
'<br />
'<br />
. I<br />
\<br />
'<br />
' ..<br />
t 1 I I I<br />
r I I , '<br />
U I I<br />
'<br />
'<br />
\ : ! l<br />
.<br />
' I I<br />
I I I<br />
' ' r 1 r t<br />
' , 1 i<br />
' ' I I<br />
'<br />
I<br />
'<br />
I : ~ ~<br />
' I<br />
I<br />
From the graph, it appears that F is conservative, since around all clos<strong>ed</strong><br />
paths, the number and size of the field vectors pointing in directions similar<br />
to that of the path seem to be roughly the same as the number and size of the<br />
vectors pointing in the opposite direction. To check, we calculate<br />
. .<br />
~ + x cos y). Thus F is conservative, by<br />
y<br />
X<br />
a o (sin y) = cos y = 0° (<br />
Theorem 6.<br />
29. Since F is conservative, there exists a function f such that F = \1 f, that is, P = /z, Q = ! 11 , and R = f= · Since P ,<br />
Q, and R have continuous first order partial derivatives, Clairaut's Theorem says that 8P / EJy = fx 11 = fvx = EJQf EJx,<br />
EJP/8z = fxz = /zx = 8R /.8x, and 8Q/8z = fyz = f zv = 8Rf8y.<br />
31. D = {(x, y) I 0 < y < 3} consists of those points between, but not<br />
on, the horizontal lines y = 0 andy = 3.<br />
(a) Since D does not include any of its boundary points, it is open. More<br />
formally, at any point in D there is a disk center<strong>ed</strong> at that point that<br />
lies entirely in D.<br />
'(b) Any two points chosen in D can always be join<strong>ed</strong> by a path that lies<br />
y<br />
3<br />
----------- -----------<br />
---------·--.:<br />
0 X<br />
entirely in D , so D is com1ect<strong>ed</strong>. (D consists of just one "piece.")<br />
(c) Dis connect<strong>ed</strong> and it has no holes, so it's simply-connect<strong>ed</strong>. (Every simple clos<strong>ed</strong> curve in D encloses only points that are<br />
in D.)<br />
33. D = { (x, y) II ::; x 2 + y 2 ::; 4, y 2: 0} is the semiannular region<br />
in the upper half-plane between circles center<strong>ed</strong> at the origin of radii<br />
l and 2 (including all boundary points).<br />
(a) D includes boundary points, so it is not open. [Note that at any<br />
boundary point, (1, 0) for instance, any disk center<strong>ed</strong> there cannot lie<br />
entirely in D .]<br />
(b) .The region consists of one piece, so it's connect<strong>ed</strong>.<br />
(c) D is connect<strong>ed</strong> and has no holes, so it's simply-connect<strong>ed</strong>.<br />
y 8P y 2 - x 2 x 8Q y 2 - x 2 8P 8Q<br />
35. (a)P= - x 2 +y 2 , .A..=<br />
2 andQ=--- -- = 2 .Thus - VIJ (x2 + y2) x2 +y2' OX (x2 +y2) 8 y UX<br />
(b) G 1: X = cost, y = sin t, 0 ::; t ::; ;rr' c 2: X = cost, y = sin t, t = 21!' to t = 1l'. Then<br />
y<br />
=,.-.<br />
{ F ·dr = ("'(- sin t)(-s~n t)+.(~ost)(cost)dt= ("' dt = 7rand { F· dr = ("' dt= - 1!'<br />
Jc, } 0 cos t + sm t ./ 0 lc 2 J 21r<br />
© 2012 Ccngogc Lcoming. All RlgbLS Restl'\'<strong>ed</strong>. Moy no1 be scann<strong>ed</strong>, copi<strong>ed</strong>. or duplicalcd, or poSicd lo o publicly occessiblc wcbsilc, in whole or in p.1n.
SECTION 16.4 GREEN'S THEOREM D 313<br />
Since these aren't equal, the line integral ofF isn't independent of path. (Or notice that J~<br />
3<br />
F · dr = f 0<br />
2 1r dt = 21r where.<br />
Cs is the circle x 2 + y 2 = 1, and apply the contrapositive of Theorem 3.) This doesn't contradict Theorem 6, since the<br />
domain ofF , which is IR 2 except the origin, isn't simply-connect<strong>ed</strong>.<br />
16.4 Green's Theorem<br />
1. (a) Parametric equations for C are x = 2 cos t, y = 2 sin t, 0 :::; t :::; 27r. Then<br />
fc(x - y) dx + (x + y) dy = J;" [(2 cost- 2sin t )(-2 sin t) + (2 cos t+ 2 sin t)(2 cost)] dt<br />
= J;"(4sin 2 t+ 4 cos 2 t)dt = J~" 4dt = 4 t]~"' = 81r<br />
(b) Note that Cas given in part (a) is a positively orient<strong>ed</strong>, smooth, simple clos<strong>ed</strong> curve. Then by Green's Theorem,<br />
fc(x - y)dx + (x + y) dy= ffv [: ., (x+y) - : 11<br />
(x - y)] dA = ffv [1- (-1)] dA = 2ffvdA<br />
= 2A(D) = 27r(2) 2 = 81r<br />
3. (a) y<br />
(1,2)<br />
Ct: x = t => dx = dt, y = D => dy = Ddt, D :::; t :::; 1.<br />
C2: x = 1 => dx =Ddt, y = t => dy = dt, D:::; t $ 2.<br />
C3: x= 1-t => dx = -dt, y=2-2t => dy = - 2dt, D$ t$ 1.<br />
o c 1<br />
(I,OJ<br />
X<br />
Thus fcxydx + x 2 y 3 dy = f xydx+x 2 y 3 dy<br />
C1 + C2 + C3<br />
= f 0<br />
1<br />
Ddt + f 0<br />
2<br />
t 3 dt + J; [-(1- t)(2 - 2t)- 2(1- t?(2 - 2t) 3 ] dt<br />
= D + [~t 4 ]~ + [t(1 - t? + ~( 1- t) 6 ]~ = 4- ¥ = ~<br />
(b) f c xydx +x 2 y 3 dy = ffv [:x (x 2 y 3 ) - /J~ (xy)] dA = f 0<br />
1<br />
J0 2 ~(2xy 3 - x)dydx<br />
rt [ 1 4 ] y=2z<br />
=<br />
d rt ( s<br />
Jo 2xy - xy 2 2) d y=O X= Jo 8x - X X = 4 f - 3 2 = 3'<br />
2<br />
5.<br />
y<br />
4 (2,4)<br />
The region D enclos<strong>ed</strong> by Cis given by {(x, y) I D :::; x:::; 2, x :::; y $ 2x }, so<br />
f c xy 2 dx + 2x 2 ydy = ffv [ :, (2xiy) - : 11<br />
(xy 2 )] dA<br />
~ I: I:"' ( 4xy - 2xy) dy dx<br />
= r2 [xl 2] y=2x d~<br />
J o Y y= x ""<br />
= f 2 3x 3 dx = !!.x 4 ] 2 = 12<br />
J o 4 o<br />
7. fc (v + e-fi) dx+ (2x +cosy 2 ) dy = Jfv [:x (2x + cosy 2 ) - :v· (v + e-fi)] dA<br />
= fol Jyfi (2 - 1) dx dy = fol (yl/2 - y2) dy = ~<br />
@ 2012 Ccngage Learning.. AU Rights Reserv<strong>ed</strong>. May nor be sc:a.nncd, copi<strong>ed</strong>, or dupliCDtctl, or post<strong>ed</strong> to n publicly .:JCC("SSiblc website. in whole or in JUrt.
314 0 CHAPTER 16 VECTOR CALCULUS<br />
9. fc y 3 dx- x 3 dy = ffv [:., (-x 3 )- :U (y 3 ) ] dA = ff 0 (- 3x 2 - 3y 2 ) dA = J:"" J:(-3r 2 ) rdrdO<br />
= - 3 J:" dO ..r: r 3 dr = -3(27!)(4) = - 2471"<br />
11. F (x, y) = (y cosx - xysinx, xy + xcos x) and the region D enclos<strong>ed</strong> by Cis given by<br />
{(x, y) I 0 ~ x ~ 2, 0 ~ y ~ 4- 2x}. Cis travers<strong>ed</strong> clockwise, so-C gives the positive orientation.<br />
fc F · dr = - J_ 0 (ycosx- xys.inx)dx + (x~ + xcosx) dy = -<br />
ffv [ :., (xy + xcosx) - :Y (ycosx- xysinx)] dA<br />
=- ffv(Y - xsin x + cosx -: cosx + xsin x)dA = - J 0<br />
2<br />
.{0<br />
4<br />
-<br />
2<br />
:r ydydx<br />
=- f 2 [ ly 2 ] v= 4 - 2 "' dx =- f 2 1(4 - 2x) 2 dx - - f 2 (B - Bx + 2x 2 ) dx- - (Bx- 4x 2 + ~x 3 ) 2<br />
Jo 2 v=O Jo 2 - Jo - a o<br />
= - (16 - 16 + 1 : - o) = - 1 6<br />
3<br />
13. F (x, y) = (y- cosy, x sin y) and the region D enclos<strong>ed</strong> by Cis the disk with radius 2 center<strong>ed</strong> at (3, -4).<br />
C is travers<strong>ed</strong> clockwise, so -C gives the positive orientation .<br />
.f 0<br />
F · dr = - J_ 0<br />
(y - cosy)dx + (xsiny) dy = - JJ 0<br />
[ :., (xsiny)- :u (y- cosy)] dA<br />
= - ffv(siny - 1 - sin y) dA = ffv dA = area of D .= 7r(2? = 47r<br />
15. Here C = C1 + C2 where<br />
01 can be parametriz<strong>ed</strong> as X = t, y = 1, - 1 ~ t ~ 1, and<br />
y<br />
C2 isgivenbyx=-t, y=2 - t 2 , - 1 ~t~ l.<br />
Then the line integral is<br />
. 1<br />
f y 2 e"' dx + x 2 ev dy = J_ 1<br />
[1· et + t 2 e · 0] dt<br />
~~ .<br />
+ J~ 1 [(2- t2 ? e-t( - 1) + (- t) 2 e 2 - t· (- 2t)] dt<br />
= J~ 1 [et - (2 - t 2 ) 2 e- t - 2t 3 e 2 2<br />
-t ] dt =-Be+ 4Be- 1<br />
-I 0<br />
X<br />
according to a CAS. The double integral is<br />
!<br />
. r (aQ aP) ; ·1 r2_,2 .<br />
. J D ax - 8y dA = - 111 (2xeY - 2ye"') dy dx = - Be + 4Be- 1 ' verifying Green's Theorem in this case.<br />
17. By Green's Theorem, W = fc F · dr = fc x(x + y) dx + xy 2 dy = J( 0<br />
(y 2 - x) dA where C is the path describ<strong>ed</strong> in the<br />
question and D is the triangle hound<strong>ed</strong> by C . So<br />
W = J 01<br />
J;-"'(y 2 - x) dy dx = f 0<br />
1<br />
[ty 3 - xy]~ = ~-"' dx = J; (i(1 - x) 3 - x(1 - x)) dx<br />
• = [- f.I ( 1 - x) 4 - ~ x 2 + ~ x 3 ] ~ = (- ~ + ft ) - (- f2) = - 1<br />
1 2<br />
19. Let C1 be the arch of the cycloid from (0, 0) to (27r, 0), which corresponds to 0 ~ t ~ 271", and let C2 be the segment from<br />
(271", 0) to (0, 0), so 02 is given by X = 271"..:. t, y = 0, 0 ~ t :::; 271". Then c = cl u 02 is travers<strong>ed</strong> clockwise, so - C is<br />
orient<strong>ed</strong> positively. Thus -C encloses the area under one arch of the cycloid and from (5) we have<br />
A = - f_ 0<br />
y dx = .{ 0 1<br />
y dx + f . 0 2<br />
y dx = f~" (1 -cos t)(l - cost) dt + f~" 0 (- dt)<br />
= J~" (1 - 2 cost+ cos 2 t) dt + 0 = (t- 2 sin t + ~t + ~sin 2t] ~" = 371"<br />
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SECTION 16.4 GREEN'S THEOREM 0 315<br />
21. (a) Using Equation 16.2.8, we write parametric equations of the lin_e segment as x = (1 - t)x1 + tx2, y = (1- t)y1 + ty 2 ,<br />
0 ~ t ~ 1. Then dx = (x2- xl) dt and dy = (y2- yl) dt, so<br />
1<br />
fc x dy- y dx = f 0 [(1 - t)x1 + tx2](y2- Yl) dt + [(1 - t)y1 + ty2](x2- x1) dt<br />
= J; (x1(Y2 - Yl) - y1(x2- xl) + t((y2- Y1)(x2- x1)-:- (x2 - x1)(y2- Y1)]) dt<br />
1<br />
= f 0 (x1y2- x2yl) dt = x~y2- X2Y1<br />
(b) We apply Green's Theorem to the path C = C1 U C2 U · · · U Cn, where Ci is the line segment that joins (xi , Yi) to<br />
(xi+l,Yi+l) fori= 1, 2, ..., n -1, and Gn is the linesegmentthatjoins (x,.,yn) to (xl,Yl). From (5),<br />
~ J~ x dy - y dx = JJ 0<br />
dA, where D is tite polygon bound<strong>ed</strong> by C. Therefore<br />
area of polygon= A( D)= ffv dA = ~ J~ xdy- ydx<br />
= ~ (J~ 1 x dy- ydx + fc 2<br />
xdy - ydx + .... + J~n-l xdy- ydx +fen xdy- ydx)<br />
To evaluate these integrals we use the formula from (a) to get<br />
A(D) = ~[(x1y2- X2Yl) + (X2Y3 - XaY2) + ' · · + (Xn-;-lYn- XnYn- 1) + (XnYl - XlYn)].<br />
(c) A = ~ [(0 · 1 - 2 · 0) + (2 · 3 - 1 · 1) + (1 · 2 - 0 · 3) + (0 · 1 - (-1) · 2) + ( ~ 1 · 0- 0 · 1)]<br />
= Ho+5+2+2) = ~<br />
23. We orient the quarter-circular region as shown in the figure.<br />
A = {7ra 2 sox = 1l'a! / 2<br />
fc x 2 dy and y = - 1ra! 12<br />
fc y 2 dx.<br />
y<br />
a<br />
Here c = c l + c2 + Cs where cl: X = t, y = 0, 0 ~ ·t ~ a;<br />
C2: x = a cost, y =a sin t, 0 ~ t. ~ ~;and<br />
c3: X = 0, y = a - t, 0 ~ t ~ a. Then<br />
fcx 2 dy = fc, x 2 dy + fc 2<br />
x 2 dy +.fc 3<br />
x 2 dy = foa Odt + J 12 0 " (acost) 2 (acost) dt + J; Odt<br />
= J~"/ 2 a 3 cos 3 tdt = a 3 J 0<br />
"/\1- sin 2 t) costdt = a 3 [sint -.i sin 3 t]; 12 = ~a~<br />
_ 1 J 2<br />
4a<br />
so x = 7ra2 /2 Jc x dy = 311'.<br />
c.<br />
a<br />
X<br />
fcy 2 dx = fc, y 2 dx + )~ 2 Y2 dx + fc 3<br />
Y 2 dx = foa 0 dt + f 12 " 0<br />
(a sin t) 2 (- a sin t) dt + Ja 0<br />
0 dt<br />
= J 0 " 12 (- a 3 sin 3 t) dt = - a 3 J 0 " 1 \ 1 - cos 2 t) sin tdt = -a 3 [ ~ cos 3 t- cos t]; 12 = - ~a 3 ,<br />
so 'ii = - 1l'a!/ 2<br />
£y2 dx = ~;.Thus (x,y) = ( ::. ::)·<br />
25. By Green's Theorem, - ~pfcy 3 dx = - kp J.f'v( - 3y 2 ) dA = ffv y 2 pdA = I ., and<br />
tPfcx 3 dy = ~p ff 0 (3x 2 ) dA = ffv x 2 pdA = Iu .<br />
27. As in Example 5, let C' be a counterclockwise-orient<strong>ed</strong> circle with center the origin and radius a, where a is chosen to<br />
be small enough so that C' lies inside C, and D the region bound<strong>ed</strong> by C and C'. Here<br />
=><br />
EJP = 2x(x 2 + y 2 ) 2 . - 2xy · 2(x 2 + y 2 ) · 2y = 2x 3 - 6x~ 2 and<br />
oy (x2 + y2)4 (x2 + y2)3<br />
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316 0 CHAPTER 16 VECTOR CALCULUS<br />
{)Q -2x(x 2 + y 2 ? - (y 2 - .x 2 ) • 2(x 2 + y 2 ) · 2x 2:~; 3 - 6xy 2 .<br />
'* -{) = ( 2 2 ) 4<br />
= a . Thus, as m the example,<br />
.. X X + y (x2 + y2)<br />
L P dx + Q dy + j_c, P dx + Q dy = I L ( ~~ -~:) dA = I L<br />
and J c F · dr = .fc, F · dr. We parametrize C' as r (t) =a cost i +a :sin t j , 0:::; t:::; 21r. Then<br />
1 1 d 1<br />
0 dA = 0<br />
2 " 2(acost)(asint)i+ (a 2 sin. 2 t - a 2 cos 2 t) j ( . . ·)<br />
F · dr = F · r =<br />
2<br />
· - a sm t 1 + a cost J dt<br />
c C' o (a 2 cos 2 t + a 2 sin 2 t)<br />
. 1 l2~ 1 l 2"<br />
= - (-cost sin 2 t - cos 3 t) dt = - (-cost sin 2 t- cost (1 - sin 2 t)) dt<br />
a o<br />
2<br />
"<br />
11 2 = -- ~ costdt = - -1 sint ] = 0<br />
a o a o<br />
.29. Since C is a simple clos<strong>ed</strong> path which doesn' t pass through or enclose the origin, there exists an open region that doesn't<br />
a o<br />
contain the origin but does contain D. Thus P = -y/(x 2 + y 2 ) and Q = xj(x 2 + y 2 ) have continuous partial derivatives on<br />
this open region containing D and we can apply Green's Theorem. But by Exercise 16.3.35(a), aPjay = aQj8x, so<br />
fcF · dr = ffv OdA ~ 0.<br />
. . & M<br />
31. Usmg the first part of(S), we have that I In dx dy = A(R) = Ion X dy. But X = g(u, v), and dy = au du + 8v dv,<br />
and we orient as by taking the positive direction to be that which corresponds, under the mapping, to the positive direction<br />
alongaR, so<br />
{ X dy = { g(u, ·u) ( 8 ah du + {)ah dv) = r g(u, v) {)ah du + g(u, v) Bah dv<br />
i on l as u v .los u v<br />
= ± .ffs [ :v. (g(u, v) g~) - :v (g(u, v) ~~) ) dA [using Green's Theorem in the uv-plane]<br />
= ± ff (!L9.. !ll! + g(u v) 0 2 " - !l.!l 81 ' - g(u v) 82 S Uu Dv > l:lul:lv l:l·v Vv. > 8vUu " ) dA [using the Chain Rule]<br />
= ± .ffs (~~ ~- Z~ ~) dA _[by the equality of mix<strong>ed</strong> partials] = ± Jj~ ~~::~\ dudv<br />
The sign is chosen to be positive if the orientation that we gave to 8S corresponds to-the usual positive orientation, and it is<br />
negative othe~ i se. In either case, since A(R) is positive, the sign chosen must be the same as the sign of ~i:: ~?.<br />
ThereforeA(R) = ll dxdy= lfsl~~::~~ ~dudv .<br />
16.5 Curl and Divergence<br />
j k<br />
1. (a) curlF = 'il x F = 8j 8x 8 / 8y 8/8z<br />
x + yz y + xz z + xy<br />
= [:Y(z + xy) ...:. :z(y + xz)] i - [:x(z + xy)- :z(x + yz)] j + [:x (y ~ xz) - :y (x + yz)] k<br />
= (x- x) i- (y - y) j + (z- z) k = 0<br />
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8 8 8<br />
(b)clivF = 'V ·iF=- (x+yz) + - (y+xz)+ - (z + xy)=1 + 1+1 =3<br />
0 X 0 y 0 Z<br />
j<br />
k<br />
3. (a) curlF = \1 x F = 8/8x 8/8y 8/8z = (ze"- 0) i- (yze"' - xye•)j + (0 - xe") k<br />
xyez 0 yze"'<br />
= ze"' i + (xye" - yze"' ) j - xez k<br />
8 8 8 .<br />
(b) clivF = 'V · F =ox (xye") + oy (0) + oz (yze'") = ye• + 0 + ye"' = y(e• + e"' )<br />
SECTION 16.5 CURL AND DIVERGENCE D 317<br />
5. (a) curlF = \1 x F =<br />
j<br />
8/8x 8/8y 8/ 8z<br />
X y z<br />
.jx2 + y2 + z2 .jx2 + y2 + z2<br />
k<br />
= ( 2<br />
?l<br />
X +y- + Z 2 ) 312<br />
[(- yz+ yz) i -(-xz +xz)j + (- xy+xy) k ]=O<br />
(b) div F = \1. F = ..E._ ( x ) + ~ ( Y ) +..E._ ( Z )<br />
ox .j x2 + y2 + z 2 8y .j x2 + y2 + z2 az .j x2 + y2 + z2<br />
x2 + y2 + z2 _ x2 x2 + y2 + z2 _ y2 x2 + y2 + z2 _ z 2 2x2 + 2 y2 + 2 z2 2<br />
= (x2 + y2 + z2)3/ 2 + (x2 + y2 + z2)3/ 2 + (x2 + y2 + z2)3/ 2 = (x2 + y2 + z 2)3/ 2 = -..jr-x=;2;=+=y:;:= 2 =+=z=;;: 2<br />
j<br />
k<br />
7. (a) curlF = \1 x F = 8/ ax 8/8y 8/ 8z = {0 - eV cosz) i - (e"' cosx - O)j + (0- e'" cosy) k<br />
e"' sin y eY sin z<br />
ez sin x<br />
= (-e 11 cosz, - e" cosx, -e"' cosy)<br />
(b) cliv F = \1 · F = ! ( e"' sin y) + ~ ( eV sin z) + ! ( e" sin x) = e'" sin y + eY sin z + e" sin x<br />
9. If the vector field is F = .Pi+ Q j + Rk, then we know R = 0. ln addition, the x-component of each vector ofF is 0, so<br />
8P 8P aP 8R 8R aR . . 8Q ·<br />
P = 0, hence - 0<br />
= -a = -a = - 0<br />
= - 0<br />
= -a = 0. Q decreases as y mcreases, so - 0<br />
< 0, but Q doesn't change<br />
X y Z X y Z y<br />
. h d' . . oQ oQ 0<br />
In t eX- Or Z · 1rectJons, SO ax = az = .<br />
. 8P 8Q 8R 8Q<br />
(a)cliv F = - +-+-a =0+ - + O< O<br />
0 X 8 y Z 0 y<br />
(b) curlF =(. oR - aQ) i + (oP- oR) j + (oQ - ap) k = (0 -O) i+ (0 - O)j + (0 - O)k = 0<br />
8y 8z az OX ax ay<br />
11. If the vector field is F = Pi + Q j + R k,' then we know R = 0. ln addition, the y-component of each vector ofF is 0, so<br />
8Q 8Q 8Q 8R aR 8R . . 8P .<br />
Q = 0, hence ox = 8y = az = ox = oy = az = 0. P mcreases as y mcreases, so oy > 0, but P doesn't change m<br />
. th d' . ap ap O<br />
e X- or Z - lreCtiOns, SO ax = az = .<br />
. ap aQ aR<br />
(a) d1v F = - + - + - = 0 + 0 + 0 = 0<br />
ax 8y 8z<br />
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318 0 CHAPTER 16 VECTOR CALCULUS<br />
(b) curlF = ({)R - {)Q) i + ({)P - {)R) j + ({)Q - {)p) k = (0- 0) i -t (0 - O) j + (o- {)P) k =-{)P k<br />
8y .{)z {)z 8x ax 8y 8y 8y<br />
: fJP 0 8Pk . . . . th . d" .<br />
S<br />
uy vy • ·<br />
mce -;:;- > , - .q,, IS a vector porntmg m e negative z- 1rect1on.<br />
13. curiF = "il x F = 8f8x 8f8y<br />
j<br />
k<br />
8f8z = (6xyz 2 - 6xyz 2 ) i - (~y 2 z 2 - 3y 2 z 2 )j + (2yz 3 - 2yz 3 ) k = 0<br />
and F is defin<strong>ed</strong> on all oflR 3 with component functions which have continuous partial derivatives, so by Theorem 4,<br />
F is conservativ~. Thus, there exists a· function f such that F = "il f. Then f., ( x, y , z) = y 2 z 3 implies<br />
f(x,y,z) = xy 2 z 3 + g(y,z) and f 11 (x,y,z) = 2xyz 3 + g 11 (y,z). But f 11 (x,y,z) = 2xyz 3 ,sog(y,z) = h(z) and<br />
f(x, y,z) = xy 2 z 3 + h(z). Thus fz(x, y, z) = 3xy 2 z 2 + h'(z) but f:(x, y, z) = 3xy 2 z 2 so· h(z) = K, a constant.<br />
Hence a potential function for F is f (x, y, z) = xy 2 z 3 + K.<br />
15. curl F = "il X F = 8f8x 8f8y 8f8z<br />
j<br />
3xy 2 z 2 2x 2 yz 3 3x 2 y 2 z 2<br />
= (6x 2 yz 2 - 6x 2 yz 2 ) i - (6xy 2 z 2 - 6xy 2 z) j .+ (4xyz 3 - 6xyz 2 ) k<br />
= 6xy 2 z(I- z) j + 2xyz 2 (2z- 3) k =1- 0<br />
so F is not conservative.<br />
17. cur!F = "il x F = 8f8x 8f8y 8f8z<br />
j<br />
k<br />
k<br />
F is defin<strong>ed</strong> on all oflll 3 , and the partial derivatives of the component functi~ns are continuous,'so F is conservative. Thus<br />
there exists a function f such that "il f =F. Then f,(x, y, z) = e 11 = implies f(x, y, z) = xe 11 = + g(y, z) =><br />
fu(x, y, z) = xzev= + g 11 (y, z). But / 11<br />
{x, y, z) = xze 11 =, so g(y, z) = h(z) and f(x, y, z) = xe 11 z + h(z).<br />
Thus f:(x, y, z) = xye 11 z + h'(z) but f:(x, y, z) = xye 11 = so h(z) =Kanda potential function for F is<br />
f(x, y, z) = xe 11 = + K .<br />
. 19. No. Assume there is such a G. Then div(curl G ) ='! (x sin y) + ~ (cosy) + ~ (z - xy) =sin y- siny + 1 f 0,<br />
uX uy uz<br />
which contradicts Theorem 11 .<br />
j<br />
k<br />
21 . curlF = 8f8x 8f8y 8f8z = {0- 0) i + {0- O)j + {0 - 0) k = 0. Hence F = f(x) i + g(y) j + h(z) k<br />
is irrotational.<br />
f(x) g(y) h(z)<br />
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SECTION 16.5 CURL AND DIVERGENCE 0 319<br />
ForExercises23-29, 1etF(x,y,z) = P1 i+ Qd + R1 ~and G(x,y,z) = P2 i + Q2j + R2 k.<br />
23. div(F +G)= div(Pt + P2 , Q1 + Qz,R1 + R 2 ) = o(P 1 0 + P2 ) + o(Q 1 0 + Qz )" + o(R1 / R 2 )<br />
• X y Z<br />
= oP1 + oPz + oQ1 + oQ2 + oR1 + oR2 = (oP1 + oQ1 + oR1) + ( oP2 + oQ2 + oR2)<br />
ox ox oy oy o z o z ox oy o z ox oy o z<br />
= div(P1 ,,Q1, R1) + div(Pz , Qz , Rz ) = div F + div G<br />
27. div(F X G) ='iJ · (F x G)=<br />
= [Q 1<br />
oR2 + R 2<br />
oQ1 _ Q 2 oR1 _ R 1 oQ2] _ [p 1<br />
oRz + Rz oP1 _ Pz oR1 _ R 1<br />
oPz]<br />
ox ox ox ox .oy oy ay ay<br />
+ [Pt oQ2 + Q 2<br />
oP1 _ p 2 oQ1. _ Q 1<br />
oP2 ]<br />
oz oz . oz oz<br />
= [p 2 (oR1 _ oQ1) +Q 2 (oP1 _ oR1) +Rz(oQ1 _ 8P1 )]<br />
oy oz oz ox OX ay<br />
_ [p 1<br />
(oRz _ oQz) -f_ Q 1<br />
(oP2 _ oRz) + R 1 (oQz ~ aP2 )]<br />
oy az az ox ox 8y<br />
= G · curl F - F · curl G<br />
j<br />
29. curl{ curl F) = 'iJ x ('iJ x F) = ajax a;ay<br />
k<br />
a;az<br />
Now let's consider grad(div F ) - 'iJ 2 F and compare with the above.<br />
(Note that 'iJ 2 F is defin<strong>ed</strong> 'on page J J 19 [ET I 095).)<br />
[continu<strong>ed</strong>]<br />
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320 0 CHAPTER 16 VECTOR CALCULUS<br />
d(d . F) n 2p-[(EPH+fiQ1+fiR1)·+(8 2 P1 +8 2 Ch 8 2 R1)· (8 2 P1. 8 2 Q1 8 2 1<br />
R1)k]<br />
gra IV - V - OX2 OXOY OXOZ<br />
oyox oy2 + [)y[)z J + , OZOX + Ozoy + OZ 2<br />
_ [(8 2 P1 + 8 2 P1 + B 2 P1) i + (8 2 Q1 + 8 2 Q1 + 8 2 Q1) j<br />
8x 2 8y 2 8z 2 8x 2 8y2 8z 2<br />
Then applying Clairaut's Theorem to reverse the order of differentiation in the second partial derivatives as ne<strong>ed</strong><strong>ed</strong> and<br />
comparing, we have curl curl F = grad div F - \l 2 F as desir<strong>ed</strong>.<br />
31. (a)\lr =V.Jx2+y2 + z2= x i + y j + z k = xi + yj+zk =~<br />
. .Jx2+y2+z2 .Jx2+y2+z2 .Jx2+y2+z2 .Jx2+y2 + z2 r<br />
j<br />
k<br />
(b)\lxr= :x :V :z =l:V(z) - !(v)]i + [:z(x) - :x(z)]j + [:x(y)-:V(x)]k = O<br />
X y Z<br />
(c) \l (.!) = \l ( 1 ) .<br />
r .J x2 + y2 + z2 ·<br />
1<br />
1<br />
1<br />
-<br />
(2x) ·<br />
(2y)<br />
(2z)<br />
2 .J x2 + y2 + z2 . 2 .J x2 + y2 + z2 . 2 V x2 + y2 + z2<br />
--~~~~~---1 - J - k<br />
x2 + y2 + z2 x2 + y2 + z2 x2 + y2 + z~<br />
x i+ y j +z k r<br />
(x2 + y2 + z2)3/2 =-r3<br />
(d) \lin r = \lln(x2 + y2 + z2)1/2 = ~ \lln(x2 + y2 + z2)<br />
x . y • z k x i + y j +z k r<br />
--::----::-----::- 1 + J + = = -<br />
x2 + y2 + z2 x2 + y2 + z2 x2 + y2 + z2 x2 + y2 + z2 7.2<br />
33. By (13), fc f(\lg) · n ds = ffv div(f\lg) dA = JJ'o!f div(\lg) + \lg · \l f] dA by Exercise 25. But div(\lg) = \l 2 g.<br />
Hence ffv f\l 2 gdA = fc f(Vg) · nds - ffv \lg · \lf dA.<br />
35. Let f(x, y) = 1. Then V f = 0 and Green's first identity (see Exercise 33) says<br />
ffv \l 2 gdA = f c (Vg) · n ds - ffv 0 · \lgdA =><br />
ffv \l 2 gdA = J~ \lg · n ds. But g is harmonic on D, so<br />
V 2 g = 0 =><br />
fc\lg· nds = OandfcDngds = fc(\lg· n)ds =O.<br />
37. (a) We know thatw = vjd, and from the diagram sin (;I= d/r => v = d!JJ = (sinO)rw = lw x rl. But vis perpendicular<br />
to both wand r, so that v = w x r.<br />
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SECTION 16.6 PARAMETRIC SURFACES AND THE;IR AREAS 0 321<br />
j<br />
k<br />
(b) From(a), v=w x r= 0 0 w =( O ·z -wy)i + (wx~ O ·z) j+ ( O · y -x· O ) k=-wyi+ wx j<br />
X y Z<br />
(c) cur] v = "il X v = ajax a jay ajaz<br />
j<br />
k<br />
-wy wx 0<br />
= [~ (0) - ~ (wx)] i + [~ (- wy) - ~ (o)] j + [~ (wx)- ~ (-wy)] k<br />
ay az az ax ax ay<br />
= [w - (- w)] k = 2wk = 2w<br />
39. For any continuous function f on iR 3 , define a vector field G ( x , y, z) = (g(x, y , z) , 0, 0) where g( x, y, z ) = f ox f ( t, y, z) dt.<br />
a a a a x<br />
Then div G = -a (g(x ,.y, z)) +- a (0) +- a (0) = ~ f 0<br />
f (t , y , z) dt = f (x, y, z) by the Fundamental Theorem of<br />
X y Z v X .<br />
Calculus. Thus every continuous function f on R 3 is the divergence of some vector field.<br />
16.6 Parametric Surfaces and Their Areas<br />
1. P (7, 10, 4) lies on the parametric surface r ( u, v) = (2u + 3v , 1 + 5u - v , 2 + u + v) if and only if there are values for u<br />
and v where 2u + 3v = 7, 1 + 5u - v = 10, and 2 + u + v = 4. But solving the first two equations simultaneously gives<br />
u = 2, v = 1 and t~ese values do not satisfy the third equation, so P does not He on the surface.<br />
Q(5, 22, 5) lies on the surface if2u + 3v = 5, 1 + 5u- v = 22, and 2 + u + v = 5 for some values ofu and v. Solving the<br />
first two equations simultaneously gives u = 4, v = - 1 and these values satisfy the third equation, so Q lies on the surface.<br />
3. r(u ,v) = (u + v ) i + (3 - v) j + (1 + 4u + 5v) k = (0,3, 1} + u (1,0, 4) + v(1, -1, 5). From Example 3, we recognize<br />
this as a vector equation of a pl.ane through the point (0, 3, 1) and containing vectors a = {1, 0, 4) and b = (1, - 1, 5). If we<br />
j k<br />
wish to find a more conventional equation for the plane, a normal vector to the plane is ~ X b = 1 0 4 =4i - j-k<br />
1 - 1 5<br />
and an equation of the plane is 4(x- 0)- (y- 3) :._ (z- 1) = 0 or 4x- y- z = - 4.<br />
5. r(s, t ) = (s, t , e - s 2 ) , so the corresponding parametric equations for the surface are X = s, y = t, z = t 2 - s 2 . For any<br />
point (x, y , z) on the surface, we have z = y 2 - x 2 . With no restrictions on the parameters, the surface is z = y 2 - x 2 , which<br />
we recognize as a hyperbolic paraboloid.<br />
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322 0 CHAPTER 16 VECTOR CALCULUS<br />
The surf~e has parametric equations x = u 2 , y = v 2 , z = u + v, --; 1 ~ u ~ 1, -1 ~ v ~ 1.<br />
In Maple, the surface can be graph<strong>ed</strong> by entering<br />
pl ot3d([u"2,v"2,u+v) ,u=-l.. l,v=-l .. l);.<br />
In Mathematica we use the Parametr icPlot3D command.<br />
If we keep u constant at u 0 , x = uij, a constant, so the<br />
corresponding grid curves must be the curves parallel to the<br />
yz-plane. If vis constant, we have y = vij, a constant, so these<br />
grid curves are the curves parallel to the xz-plane.<br />
9. r (u,v) = (ucosv,usinv,u 5 ).<br />
The surface has pararnetrid equations x = u cos v, y = u sin v,<br />
z = u 5 , -1 ~ u ~ 1, 0 ~ v ~ 21r. Note that ifu = u 0 is constant<br />
then z· = u8 is constant and x = uo cos v, y = Uo sin v describe a<br />
vconstant<br />
circle in x, y of radius luo 1. so the corresponding grid curves are<br />
circles parallel to the xy-plane. If v = vo, a constant, the parametric<br />
equations become x = u cos vo, y = u sin vo, z = u 5 . Then<br />
y = (tan v 0 )x, so these are the grid curves we see that lie in vertical<br />
planes y = kx through the z-axis.<br />
11. x = sinv, y = cosusin4v, z = sin2usin4v, 0 ~ u ::; 21!', - ~ ::; v::; ~ -<br />
Note that if v = vo is constant, then x = sin vo is constant, so the<br />
corresponding grid curves must be parallel to the yz-plane. These<br />
are the vertically orient<strong>ed</strong> grid curves we see, each shap<strong>ed</strong> like a<br />
"figure-eight." When u = uo is held constant, the parametric<br />
equations become x = sin v, y = cos u 0 sin 4v,<br />
z = sin 2u 0 sin 4v. Since z is'a constant multiple of y, the<br />
corresponding grid curves are the curves contain<strong>ed</strong> in planes<br />
z = ky that pass through the x-axis.<br />
13. r( '1.1;• v) = u cos v i + u sin v j + v k. The parametric equations for the surface are x = u cos v, y = u sin v, z = v. We look at<br />
the grid curves first; if we fix v, then x andy parametrize a straight line in the plane z = v which intersects the z-axis. Ifu is<br />
held constant, the projection onto the xy-plane is circular; with z = v, each grid curve is a helix. The surface is a spiraling<br />
ramp, graph IV.<br />
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SECTION 16.6 PARAMETRIC SURFACES AND THEIR AREAS 0 323<br />
15. r ( u., v) = sin v i+ cos U. sin 2v j +sin u. sin 2v k. Parametric equations for the surface are x = sin v, y = cos u. sin 2v,<br />
z = sin u. sin 2v. lfv = vo is fix<strong>ed</strong>, then x = sin vo is constant, andy= {sin 2vo) cosu and z = (sin 2v 0 ) sin u. describe a<br />
circle of radius !sin 2<strong>vol</strong>, so each-corresponding grid curve is a circle contain<strong>ed</strong> in the vertical plane x =sin v 0 parallel to the<br />
yz-plane. The only possible surface is graph II. The grid curves we see running lengthwise along the surface correspond to<br />
holding u constant, in which casey = (cosu.o) sin 2v, z =(sin uo) sin 2v ~ z =(tan uo)y, so each grid curve lies in a<br />
plane z = ky that includes the x-axis.<br />
17. x = cos 3 u cos 3 v, y·= sin 3 u cos 3 v, z = sin 3 v. lfv = vo is held constant then z = sin 3 v 0 is constant, so the<br />
corresponding grid curve lies in a horizontal plane. Several of the graphs exhibit horizontal grid curves, but the curves for this<br />
surface are neither circles nor straight lines, so graph Ill is the only possibility. (ln fact, the horizontal grid curves here are<br />
members of the family x = acos 3 u, y = asin 3 u. and are call<strong>ed</strong> astroids.) The vertical grid curves we see on the surface<br />
correspond to u = uo held constant, as then we have x = cos 3 uo cos 3 v, y = sin 3 u.o cos 3 v so the corresponding grid curve<br />
lies in the vertical plane y = (tan 3 uo)x through the z-axis.<br />
19. From Example 3, parametric equations for the plane through the point (0, 0, 0) that contains the vectors a = (1, -1, O) and<br />
b = (0, 1, -1) are x = 0 + u(1) + v(O) = u., y = 0 + u.( - 1) + v(1) = v - u., z = 0 + u.(O) + v( - 1) = - v.<br />
21. Solving the equation for x gives x 2 = 1 + y 2 + -!z 2 ~ x = J1 + y 2 + tz 2 • (We choose the positive root since we want<br />
the part of the hyperboloid that corresponds to x ~ 0.) lfwc let y and z be the parameters, parametric equations are y = y,<br />
z=z, x= J1+y 2 +i·z 2 •<br />
23. Since the cone intersects the sphere in the circle x 2 + y 2 = 2, z = .../2 and we want the portion of the sphere above this, we<br />
can parametrize the surface as x = x, y = y, z = v' 4 - x 2 - y 2 where x 2 + y 2 ~ 2.<br />
Alternate solution: Using spherical coordinates, X = 2 sin ¢ cos e, y = 2 sin¢ sine, z = 2 cos¢ where 0 s;
324 0 CHAPTER 16 VECTOR CALCULUS<br />
31. (a) Replacing cos u. by sin u. and sin u. by cos u. gives parametric equations<br />
x = (2 + sin v ) sin u., y = (2 +sin v) cos u., z = u. +cos v. From the graph, it<br />
appears that the direction of the spiral is revers<strong>ed</strong>. We can verify this observation by<br />
noting that the projection of the spiral grid curves onto the xy-plane, given by<br />
x = (2 + sin v) sin u., y = (2 + sin v) cos u., z = 0, draws a circle in the clockwise<br />
direction for each value of v. The original equations, on the other hand, give circular<br />
projections drawn in the counterclockwise direction. The equation for z is identical in<br />
both surfaces, so as z increases, these grid curves spiral up in opposite directions for<br />
the two surfaces.<br />
(b) Replacing cos u. by cos 2u. and sin u. by sin 2u. gives parametric equations<br />
x = (2 + sin v) cos 2u., y = (2 +sin v) sin 2u., z = u. + cos v. From the graph, it<br />
appears that the number of coils in the surface. doubles within the same parametric<br />
domain. We can verify this observation by noting that the projection of the spiral grid<br />
curves onto the xy-plane, given by x = (2 +sin v) cos 2u., y = (2 + sin v) sin 2u,<br />
z = 0 (where v is constant), complete circular re<strong>vol</strong>utions for 0 ~ u. ~ 1r while the<br />
original surface requires 0 ~ u. ~ 21r for a complete re<strong>vol</strong>ution. Thus, the new<br />
surface winds around twice as fast as the original surface, and since the equation for z<br />
is identical in both surfaces, we observe twice as many circular coils in the same<br />
z-interval.<br />
33. r ( u., v) = ( u. + v) i + 3u. 2 j + ( u. - v) k.<br />
r u = i + 6u.j + k and r, = i - k, so r,. x r , = - 6u. i + 2 j - 6u k. Since the point (2, 3, 0) corresponds to u. = 1, v = 1, a<br />
normal vector to the surface at (2, 3, 0) is -6 i + 2j - 6 k , and an equation of the tangent plane is -6x + 2y- 6z = - 6 or<br />
3x- y + 3z = 3.<br />
35. r (u.,v)=u.cosv i +u.sinvj +v k => r(1 , ~) = (~.4. i )·<br />
r u = cos v i + sin v j and r , = -u sin v i + u. cos v j + k , so a normal vector to the surface at the point ( ~ ,<br />
4 , ~) is<br />
r u ( 1, f) x r , ( 1, i') = ( ~ i + 4 j) x (-4 i + ~ j + k) = 4 i - ~ j + k. Thus an equation of the ta~gen t plane at<br />
(t .4,f )is4(x ·- t) - ~(y-4) + 1 (z -~ )=0 or ~x - h +z =~ .<br />
37. r (u,v)=u 2 i +2u.sinvj +ucosv k => r (1,0)=(1,0, 1).<br />
ru = 2u. i + 2sinvj + cosv k and r , = 2u.cosvj - usinv k , 2<br />
so a normal vector to the surface at the. point (1, 0, 1) is<br />
r u(1, 0) X r,(1, 0) = (2 i + k ) X (2j) = - 2 i + 4 k.<br />
Thus an equation of the tangent plane at (1, 0, 1) is<br />
- 2(x- 1) + O(y- 0) + 4(z - 1) = 0 or -x + 2z = 1.<br />
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SECTION 16.6 PARAMETRIC SURFACES AND THEIR AREAS D 325<br />
39. The surfaceS is given by z = f(x, y) = 6- 3x - 2y which intersects the xy-plane in the line 3x + 2y = 6, soD is the<br />
triangular region given by { (x, y ) I 0:::; x:::; 2, 0 :::; v $ 3 - ~x }. By Formula 9, the surface area of S is<br />
I<br />
A (S)= !fv 1+ (~Y + (~Y dA<br />
= ffv J1 + ( - 3) 2 + (- 2)~ dA = Ji4J.(v dA = v'I4A(D ) = v'I4 (~ · 2 · 3) = 3 y'l4.<br />
41 . Here we can write z = f(x, y) = i --:- i x- jy and Dis the disk x 2 + y 2 $ 3, so by Fonnula 9 the area of the surface is<br />
A(S) =<br />
1 ~v<br />
1 + (:~r + (:~r dA 1 ~v = J1 + (-t/ + ( - ~) 2 dA = -41 1 ~v dA<br />
= :lP A(D) = fl · 1r(y'3) 2 = ViA1r<br />
43. z = f (x, y) = ~(x 3 1 2 + y 3 1 2 ) and D = {(x, y) I 0 $ x $ 1, 0 $ y $ 1 }. Then f, = x 1 1 2 , f v = y 1 1 2 and<br />
A(S) = ffv J1 + (..fi) 2 + (.jfj) 2 dA = f 0<br />
1<br />
J; y'1 + x + ydydx<br />
= fo 1 [t Cx + y + 1) 312 J::: dx = j f 0<br />
1<br />
[(x+ 2)<br />
312 - (x + 1)<br />
3 1<br />
2 ] dx<br />
= j [Hx + 2)5/2 _ ~(x + 1)5/ 2]: = Ji(35/2 _ 2s; 2 _ 2s~ 2 + 1) = A-( 3 1>/2 _ 2112 + 1)<br />
45. z = J(x, y) = xy with x 2 + y 2 $ 1, so f, = y, fu = x =><br />
A(S ) = ffv Jl + Y 2 + X 2 dA = J;"' J; -.fr2TI rdrd(} = J:"' [ ~ (r 2 + I?' 2J::: d(}<br />
= J;"' i(2 V2 - 1) d8 = 2 ; (2 .j2 - 1)<br />
47. A parametric representation of the surface is x = x, y = 4x + z 2 , z = z with 0 :::; x :::; 1: 0 :::; z :::; 1.<br />
Hence r :x: x r : = (i + 4 j ) X (2z j + k) = 4 i - j + 2z k.<br />
Note: In general, ify = f(x, z ) then r , ~ r: = :~ i-j + ¥z k and A (S) = JL 1 + ( :~y + ( ~~y dA. Then<br />
A(S) = J; J; ) 17 + 4z 2 1<br />
dxdz = J 0<br />
) 17 + 4z 2 dz<br />
= %(z ) 17 + 4z 2 + ¥ lni2z + v'4z 2 + 17 I)] ~ = 4I + 1 ,t(ln(2 + v'2I) - ln v'I7]<br />
49. r ., = (2u, v , 0), r , = (0, u, v), and r u x r ,. = (v 2 , - 2uv, 2u 2 ). Then<br />
1<br />
A(S) = ffv lr u x r vl dA = J 0<br />
J; .../v 4 + 4u 2 v 2 + 4u 4 dv du = J; J; J(v 2 + 2u2)2 dv du<br />
1<br />
= J J; 0 + (v2 2u 2 ) dv du = J; (tv 3 + 2u 2 1<br />
vJ ::~ du = J 0 (J + 4u 2 ) du = [ ~u + ~ u 3 ]~ = 4<br />
51. From Equation 9 we have A(S) = ffv J l + (1, ) 2 + Uu) 2 dA. But if Jf,J :::; 1 and If vi :::; 1 then 0 :::; (f.,? :::; 1,<br />
0 $ U u) 2 $ 1 => 1 $ 1 + U:~Y + Cfu) 2 $ 3 => 1 $ J 1 + (!:,) 2 + Uu) 2 $ v'3. By Property 15.3 .1 1,<br />
ffv1dA $ ffvJl+(f,.)2+(f 11 )2 dA $ ffv ../3dA => A(D) $ A(S)$../3A(D) => .<br />
1rR 2 $ A(S) $ ../31rR 2 •<br />
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326 D CHAPTER 16 VECTOR CALCULUS<br />
53. z = f(x , y) = e- "' 2 -"2<br />
with x<br />
2<br />
+ y 2 :=:; 4.<br />
A(S) = IIo J 1 + ( -2xe->= 2 -Y 2 ) 2 + ( - 2ye-"' 2 - v 2 ) 2 dA = IIo y'1 + 4(x2 + y2)e-2(>= 2 +u 2 > dA<br />
= I:1< I: v'1 + 4r 2 e - 2 r 2 r dr dB = I:" dB I: r y'1 + 4r2e- 2r 2 dr = 27r I: r V1 + 4r 2 e- 2r 2 dr ~ 13.9783<br />
Using the Midpoint Rule with f(x, y) =<br />
4x 2 + 4y 2<br />
1 + ( 1 2 2 ) 4<br />
. +x +Y<br />
4x 2 + 4y 2<br />
1 + (1 2 2)4 dydx.<br />
+x +y<br />
, m = 3, n = 2 we have<br />
3 2<br />
A(S) ~ 2: 2: f(xi, 'il; ) ~A = 4 (!(1, 1) + /(1, 3) + /(3, 1) + / (3, 3) + /(5, 1) + / (5, 3)] ~ 24.2055<br />
i = lj = l<br />
(b) Using a CAS we have A(S) = 1614<br />
1 + (l 4 x 2 :<br />
4 Y:)4 dy dx ~ 24.2476. This agrees with the estimate in part (a)<br />
+x +y . -<br />
to the first decimal place.<br />
57. z = 1 + 2x + 3y + 4y 2 , so<br />
Us ing a CAS, we have<br />
or !!! v'I4 + ll Jn n ~+ V'J!<br />
8 16 3 0 + 70 .<br />
59. (a) x = a sin u cosv, y = bsin u sin v, z = ccos u => (b) ~~<br />
2<br />
IZfi<br />
71.<br />
z 0<br />
and s ince the ranges of u and v ~ sufficient to generate the entire graph,<br />
the parametric equations represent an ellipsoid.<br />
(c) From the parametric equations (with a = 1, b = 2, and c = 3),<br />
- 2<br />
- 2<br />
0<br />
y<br />
'lii!<br />
~<br />
-<br />
we calculate r u = cosucosv i + 2cos usinvj - 3sin u k and<br />
r, =-s in usin v i + 2sin ucos v j. So r,. x r, = 6 sin 2 t! cos v i + 3 sin 2 'usinvj + 2 sin u cos u k , and the surface<br />
61. To find the region D : z = x 2 + y 2 implies z + z 2 = 4z or z 2 - 3z = 0. Thus z = 0 or z = 3 are the planes where the<br />
surfaces intersect. But x 2 + y 2 + z 2 = 4z implies x 2 + y 2 + (z -<br />
2) 2 = 4. so 'z = 3 intersects the upper hemisphere.<br />
Thus (z - 2) 2 = 4 - x 2 - y 2 or z = 2 + J 4 - x 2 - y 2 • Therefore D is the region inside the c~c le x 2 + y 2 + (3 - 2) 2 = 4,<br />
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that is, D = {( x, y) I x 2 + y 2 :
0<br />
328 0 CHAPTER 16 VECTOR CALCULUS<br />
16.7 Surface Integrals<br />
1. The faces of the box in the planes x = 0 and x = 2 have surface area 24 and centers (0, 2, 3), (2, 2, 3)0 The faces in y = 0 and<br />
y = 4 have surface area 12 and centers (1, 0, 3), (1, 4, 3), and the faces in z = 0 and z = 6 have area 8 and centers (1, 2, 0),<br />
(1, 2, 6)0 For each face we take the point P;j to beth~ center of the face and f(x, y, z) = e -ool(x+u+z), so by Definition I,<br />
ffs f(x, y, z) dS ~ [f(O, 2, 3)](24) + [!(2, 2, 3)](24) + [f(l, b, 3)](12)<br />
+ [f(l, 4, 3)](12) + [f(l, 2, 0)](8) + [f(l, 2, 6)](8)<br />
0<br />
30 We can use the xz- and yz-planes to divide H into four patches of equal size, each with surface area equal to ~ the surface<br />
area of a sphere with radius VSO, so D.S = H4)7r( v'50) 2 = 257l'. Then (±3, ±4, 5) are sample points in the four patches,<br />
and using a Riemann sum as in Definition l, we have<br />
ffH f(x, y , z) dS ~ j(3, 4, 5) D.S + f(3, -4, 5) !:!.S + !( -3, 4, 5) !:!.S + f( -3, -4, 5) D.S<br />
= (7 + 8 + 9 + 12}(257!') = 9007l' ~ 2827<br />
So r( u, v) = ( u + v) i + ( u - v) j + (1 + 2u + v) k, 0 ~ u ~ 2, 0 ~ v ~ 1 and<br />
0<br />
y'3 2 + 1 2 + (- 2)2 = -1140 Then by Formula2,<br />
r,. x r 11 = (i + j + 2k) X (i -j + k) = 3i+j - 2k ==? iru X rvl =<br />
ff 8<br />
(x + y + z) dS = ffv(u + v + u- v +1 + 2u + v) lru x r vl dA = f 0<br />
1<br />
f~(4u + v + 1)<br />
0<br />
-/I4dudv<br />
= .;I4 J; [2u 2 + uv + u] ::~ dv = .;I4 J; (2v + 10} dv = .;I4 [v 2 + lOv] ~ = 11 v'l4<br />
7o r(u,v) = (ucosv,usinov,v), 0 ~ u ~ 1, 0 ~ v ~ 7l' and<br />
ru X rv = (cosv,sinv, O) X (-'-usinv,ucosv, 1) = (sinv, - cosv,u ) · =?<br />
!ru X rv! = Vsin 2 v + cos 2 v + u 2 = ..)u 2 + 1. Then<br />
ffsoY dS = ffv (usin v) lru x r vl dA = J; j~" (usinv)<br />
0<br />
= [ t(u 2 + 1?/ 2 ] : [- cosv]~ = t(2 3 / 2 - 1) 0 2 = H2v'2 - 1)<br />
8z<br />
9. z = 1 + 2x + 3y so ox = 2 and oy = 30 Then by Fonnula 4,<br />
8z<br />
vu 2 + 1 dv du = J; u..Ju 2 + 1 du J 0<br />
" sin v dv<br />
= -/14 J:j J;(x2y + 2X3?i + 3x2y2) dy dx = -/14 J; [~ x2y2 + x3y2 + x2y3]~=~ dx<br />
= v'l4 J;(10x 2 + 4x 3 ) dx = v'l4 [ 4fx 3 + x 4 ] ~ = 171 v'f.1<br />
110 An equation of the plane through the points (1, 0, 0), (0, -2, 0), and (0, 0, 4) is 4x- 2y + z = 4, so Sis the region in the<br />
plane z = 4 - 4x + 2y over D = {(x, y) I 0 ~ x ~ 1,2x- 2 ~ y ~ 0}. Thus by Formula4,<br />
ffs X dS = ffv x y'( - 4) 2 + (2) 2 1<br />
+ 1 dA = ...J2f J; J 2 °x_ 2 x dy dx = v'2f f 0 [xy]~~~x- 2<br />
= v'2f J 01<br />
(- 2x 2 + 2x) dx = v'2f [-ix 3 + x 2 ]~ = v'2f (- i + 1) = 4J.<br />
dx<br />
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SECTION 16.7 SURFACE INTEGRALS 0 329<br />
13. S is the portion of the cone z 2 = x 2 + y 2 for 1 $ z $ 3, or equivalently, Sis the part of the surface z = ...jx 2 + y2 over the<br />
region D = { (x, y) Jl $ x 2 + y 2 $ 9 }. Thus<br />
r---------~----------~--<br />
·Jfs x2z2dS= Jfv x 2 (x 2 +y 2 )<br />
(~Y +(~Y + l dA<br />
15. Using x and z as parameters, we have r (x, z) = x i+ (x 2 + z 2 )j + z k 1 x 2 + z 2 :::; 4. Then<br />
r x x r : = (i + 2xj ) X (2zj + k) = 2x i - j +2z kand Jr, X r, J = J4x 2 + 1 + 4z 2 = ) 1 +4(x 2 +z 2 ). Thus<br />
JJ (x 2 + z 2 ))1 + 4(x 2 + z 2 ) dA = J;,. J; r 2 J 1 + 4r 2 rdrdO = J;,. dB J; r 2 v'1 + 4r2 r dr<br />
ffs ydS =<br />
x2 +z2~ ~ · .<br />
[let u = 1 + 47' 2 =><br />
7' 2 = t(u- 1) and idu = 1' dr]<br />
= 27T J:7 t(u- l )JU. id·u = ft1T ft17 (u3/2 ...:. ul/2) du<br />
= ...!.. 7T (1u5/2 - 1u3/2)17 = ...!..7T(1(17)5 /2- 1(17? /2 - 1 + 1) = 2:.. (391 Jl7 + 1)<br />
16. 5 3 1 16 :; 3 5 3 60<br />
17. Using spherical coordinates and Example 16.6.1 0 we have r ( ¢,B) = 2 sin 4> cos B i + 2 sin 4> sin B j + 2 cos 4> k and<br />
Jrq, x r~l = 4 sin¢. Then ffs(x 2 z + y 2 z) dS = .{ 0<br />
2 " J; 12 (4sin 2 1p)(2 cos¢)( 4 sin 4>) d¢ dB = 167T sin 4 ¢] ~ 1 2 = 167T.<br />
19. S is given by r (u,v) = ui + co~vj + sinv k, 0 $ u $ 3, 0 $ v $ 1Tj2. Then<br />
r u X rv = i X (- sin v j +cosvk ) = - cosvj-sinv kand Jr,. X rvl = ) cos 2 v +sin 2 v = 1, so<br />
.[[ 5<br />
(z + x 2 y) dS = J; 12 J;(sinv + u 2 co~v)(l) dudv = f~.,. 12 (3sin v + 9 cos v) dv<br />
= [- 3cosv + 9sin v] ~/ 2 = 0 + 9 + 3-0 = 12<br />
21. From Exercise 5, r (u, v) = (u + v) i + (u - ·u) j + (1 + 2u + v) k, 0 $ u $ 2, 0 $ v $ 1, and r u x r., = 3 i + j - 2 k.<br />
Then<br />
F (r (u, v)) = (1 + 2u + v)e
330 0 CHAPTER 16 VECTOR CALCULUS<br />
25. F(x, y, z) =xi - z j + y k, z = g(x, y) = .j4- x 2 - y 2 and Dis the quarter disk<br />
{ (x, y) I 0 ~ x ~ 2, 0 ~ y ~ v'4 - x 2 }. 8 has downward orientation, so by Fonnula 10,<br />
!1(<br />
x2<br />
= - - v' 4 - x 2 - y 2 y )<br />
· + y dA<br />
D v' 4 - x2 - y2 . v' 4 - x2 - y2<br />
= - IIv x 2 (4- (x 2 +y 2 ))- 1 1 2 dA = - I 0<br />
..,. 12 I 0 2 (r cos B?(4- r 2 )- 1 1 2 rdr dB<br />
·= - .[ 0<br />
..,.1 2 cos 2 B d8 I~ r 3 (4- r·2)- 1 1 2 dr [let u = 4- r 2 =? r 2 = 4 - u and-~ du = rdr]<br />
= - I 0<br />
..,. 12 (~ + t cos 28) d8 I~ -~(4 - u)(u)- 1 1 2 du<br />
= - [~8 + t sin28]~ 12 (-~) [8v'U - iu 3 1 2 ]: = -t(-~)( -16 + ¥) = -~1!'<br />
27. Let 8 1 be the paraboloid y = x 2 + z 2 , 0 ~ y ~ 1 and 82 the disk x 2 + z 2 ~ 1, y = 1. Since 8 is a clos<strong>ed</strong><br />
surface, we use the outward orientation.<br />
On S 1 : F (r (x, z)) = (x 2 + z 2 ) j - z k and r , x r = = 2x i - j + 2z k (si~ce the j-component must be negative on S1). Then<br />
fis<br />
1<br />
F · dS =<br />
II [-(x 2 .+ z 2 )- 2z 2 ] dA = -I~..,. I;(r 2 + 2r 2 sin 2 8) r dr d8<br />
o;2 +=2 $1<br />
= -I~..,. I; r 3 (1 + 2sin 2 8) dr d() = - ]~..,. (1 + 1-cos 28) dB-I 0<br />
1<br />
r 3 dr<br />
= - [28- t sin 28]~..,. [~r·4 ]~ = - 41!' . ~ = -1r<br />
OnS2: F(r(x, z))= j -zkandr.., x r , =j. Thenifs F ·dS= JI (1)dA=1r.<br />
Hence'Jis F · dS = - 1r + 1r = 0.<br />
2 .,2 + ;:2 $ 1<br />
29. Here 8 consists of the six faces of the cube as label<strong>ed</strong> in the figure. On 81:<br />
F = i + 2yj + 3z k, rv x rz = i and IIs 1<br />
F · dS = I~1 }~ 1 dydz = 4;<br />
82: F =xi + 2j + 3z k, r= x r, = j and IIs 2<br />
F · dS = I~ 1 I~ 1<br />
2dxdz = 8;<br />
8s: F = x i +2yj + 3k, r , x r v·= kand fis 3<br />
F · dS = I~ 1 I~ 1 3dxdy = 12;<br />
84 : F = -i +2yj +3zk,r: x r v = -i and IIs 4<br />
F · dS = 4;<br />
85: F = x i- 2j+3zk, r., x r: ~ - j andiJ~ 5<br />
F · dS = 8;<br />
86: F =xi+ 2yj- 3k, r 11 x r., = -k and J/~ 0 F · dS = J~ 1 j~ 1 3dxdy = 12.<br />
6<br />
Hence ffs F · dS = I: Jfs. F · dS = 48.<br />
i=l 1.<br />
31. HereS consists of four surfaces: S1 , the top surface (a portion ofth~ circular cylinder y 2 + z 2 = 1); S2, the bottom surface<br />
(a portion of the xy-plane); Ss, the front half-disk in the plane X= 2, and s 4 . the back half-disk in the plane X = 0.<br />
ll.
SECTION 16.7 SURFACE INTEGRALS 0 331<br />
On 8 1 : The surface is z = J1 - y 2 for 0 S x S 2, - 1 S y S 1 with upward orientation, so<br />
f F . dS = 1 ~ !<br />
1<br />
[-x 2 (0) - y 2 ( - h) + z 2 2 1<br />
] dy dx = 1 / ( h + 1 - y 2 ) dy dx<br />
! "<br />
ls1 o -1 1 -y 2 . · o - 1 1-y 2 .<br />
On 8 2 : The surface is z = 0 with downward orientation, so<br />
On 8 3 : The surface is x = 2 for - 1 S y S 1, O.S z S yf1- y 2 , orient<strong>ed</strong> in the positive x-direction. Regarding y and z as<br />
parameters, we have r 11 x r .., = i and<br />
On 8 4 : The surface is x = 0 for -1 :::; y :::; 1, 0 $ z $ yf1 - y 2 , orient<strong>ed</strong> in the negative x-direction. Regarding y and z as<br />
parameters, we use - (r 11<br />
x r z) = -i and<br />
ff 54<br />
F · dS = f~ 1 fo~ x 2 dzdy = f~ 1 fo~ (0) dz dy = 0<br />
Thus ffs F · dS = 1 + 0 + 211" + 0 = 211" + ~·<br />
33. z = xeY => 8zj8x = e 11 , 8zj8y = xe 11 , so by Formula 4, a CAS gives<br />
JJ 5<br />
(x 2 +y 2 +z 2 )d8 = J 01<br />
f 1 0<br />
(x 2 +y 2 +x 2 e 211 )Je 2 Y +x 2 e2v + 1dxdy ~ 4.5822.<br />
35. We use Formula 4 with z = 3- 2x 2 - y 2 => 8zj8x = ·- 4x, 8zj8y = -2y. The boundaries of the region<br />
3- 2x 2 - y 2 ~ 0 are -/i $ X s ji and -J3- 2x 2 $ y $ J3- 2.7: 2 • so we use a CAS (with precision r<strong>ed</strong>uc<strong>ed</strong> to<br />
seven or fewer digits; otherwise the calculation may take a long time) to calculate<br />
37. If 8 is given byy = h(x,z), then 8 is also the level surface f(x,y,z) = y - h(x,z) = 0.<br />
Vf(x,y,z) - h., i+j-h: k d . h · 1 h · h 1ft N ·<br />
n = I~/( )I = , an - n IS t e umt norma t at pomts tot e e . ow we proce<strong>ed</strong> as m the<br />
v x,y,z y'h~+l+h~<br />
derivation of (1 0), using Formula 4 to evaluate<br />
where D is the projection of 8 onto the xz-plane. Therefore I Is F · dS = I i ( P ~~ - Q + R ~~) dA.<br />
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332 0 CHAPTER 16 VECTOR CALCULUS<br />
39. m =· ffs }( dS = }( · 47r(~a 2 ) = 21ra 2 K; by symmetry M xz = lvf 11 z = 0, and<br />
M:cv = Jj~ z }( dS = }( f~" f 0 1f12 (a cos ¢)(a 2 sin¢) d¢ d() = 2711< a 3 [- ~cos 2¢] ~ 12 = 1r }( a 3 •<br />
Hence (x,y,z) = (0,0, ~ a).<br />
(b) 1:: = ffs(x 2 + y 2 )(10- Jx 2 +y 2 ) dS = ff (x 2 + y 2 )(10- jx 2 +y 2 ) ../2dA<br />
. 1 s :r 2 + v 2 s 16 •<br />
I<br />
43. The rate of flow through the cylinder is the flux JJ~ pv · n dS = ffs pv · dS . We use the parametric representation<br />
r (u,v) = 2cosui + 2 sinuj + v k for S, where 0 $ u $ 27r, 0 $ v $ 1, so r ,. = - 2sinu i + 2cosuj, r v = k , and the .<br />
outward orientation is given by ru x r v = 2 cos u i + 2 sin u j . Then<br />
1<br />
ffs pv · dS = p ]~" f 0<br />
(vi+ 4sin 2 u j + 4 cos 2 u k) · (2cosui+ 2sinuj ) dvdu<br />
2<br />
= p f 0 " . J: (2v casu+ 8sin 3 u) dv du = p J~" (casu+ 8 sin 3 u) du<br />
= p[sin u + 8(-i)(2 + sin 2 u) casu] ~" = 0 kg/s<br />
45. S consists of the hemisphere 8 1 given by z = ja 2 - x 2 - y 2 and the disk 8 2 given by 0 $ x 2 + y 2 $ a 2 , z = 0.<br />
On 81: E = asin¢ cosB i +a sin ¢ sinBj + 2acos¢k ,<br />
T q, x To = a 2 sin 2 ¢ cosB i +a 2 sin 2 sinl'.lj +a 2 sin ¢ costpk. Thus<br />
ffs 1 E · dS = .r;" f 0 " 1 \a 3 sin 3 q) + 2a 3 sin
SECTION 16.8 STOKES' THEOREM 0 333<br />
16 . 8 Stokes' Theorem<br />
1. Both Hand Pare orient<strong>ed</strong> piecewise-smooth surfaces that are bound<strong>ed</strong> by the simple, clos<strong>ed</strong>, smooth curve x 2 + y 2 = 4,<br />
z = 0 (which we can take to be orient<strong>ed</strong> positively for both surfaces). Then Hand P satisfy the hypotheses of Stokes'<br />
Theorem, so by (3) we know JJ H curl F · dS = fc F · dr = J j~ curl F · dS (where Cis the boundary curve).<br />
3. The paraboloid z = x 2 + y 2 intersects the cylinder x 2 + y 2 = 4 in the circle x 2 + y 2 = 4, z = 4. This boundary curve C<br />
should be orient<strong>ed</strong> in the counterclockwise direction when view<strong>ed</strong> from above, so a vector equation of C is<br />
r (t) = 2 cost i + 2sin t j + 4k, 0 ~ t ~ 271". Then r'(t) = -2 sin t i + 2 cos tj,<br />
F (r (t)) = (4cos 2 t)(16) i + (4 sin 2 t)(16) j + (2 cos t)(2 sin t)(4) k = 64 cos 2 t i + 64sin 2 t j + 16 sin t cost k ,<br />
and by Stokes' Theorem,<br />
Jf 5<br />
cur1F · dS = fc F · dr = J:"lr F (r (t)) · r'(t) dt = J:"(-128cos 2 t sint + 128sin 2 t qost + O)dt<br />
= 128 [~ cos 3 t + t sin 3 t] ~,. = 0<br />
5. C is the square in the plane z = -1. Rather than evaluating a line integral around C we can use Equation 3:<br />
J fs, curl F · dS = fc F · dr = jf 52<br />
curl F · dS where 81 is the original cube without the bottom and 8 2 is the bottom face<br />
of the cube. curlF = x 2 z i + (xy- 2xyz) j + (y- xz) k. For 82, we choose n = k so that C has the same orientation for<br />
both surfaces. Then curl F · n = y - xz = x +yon 82, where z = - 1. Thus ff 52<br />
curl F · dS = f~<br />
1 .C 1<br />
(x + y) dx dy = 0<br />
so ffs, curlF · dS = 0.<br />
7. curl F = - 2z i-2xj-2y k and we take the surfat<strong>ed</strong>. or post<strong>ed</strong> to n publicly actc-ssiblewcbsito. in whole or in part.
334 D CHAPTER 16 VECTOR CALCULUS<br />
(b) (c) One possible parametrization is x = 3 cost, y = 3 sin t,<br />
z = 1 - 3 cos t - 3 sin t, 0 ~ t ~ 271'.<br />
2<br />
-2<br />
13. The boundary curve Cis the circle x 2 + y 2 = 16, z = 4 orient<strong>ed</strong> in the clockwise direction as view<strong>ed</strong> from above (since Sis<br />
orient<strong>ed</strong> downward). We can parametrize C by r(t) = 4costi - 4sintj + 4k, 0 ~ t ~ 21r, and then<br />
r'(t) = - 4sint i-4costj. Thus F (r(t)) = 4sint i + 4cos t j - 2 k , F (r (t)) · r ' (t) = -16sin 2 t- 16cos 2 t = -16, and<br />
fcF · dr = J; .,. F(r(t)) · r'(t) dt = J;"(-16) dt = -16 (27r) = -327!'<br />
Now curl F = 2 k, and the projection D of Son the xy-plane is the disk x 2 + y 2 ~ 16, so by Equation 16.7.10 with<br />
z = g(x, y) = .Jx 2 + y 2 [and multiplying by - 1 for the downward orientation] we have<br />
ffs curl F · dS = - ffv( -0 - 0 + 2) dA = -2 · A(D) = -2 · 7r(4 2 ) = - 327!'<br />
15. The boundary curve C is the circle x 2 + z 2 = 1, y = 0 orient<strong>ed</strong> in the counterclockwise direction as view<strong>ed</strong> from the positive<br />
y-axis. Then C can be describ<strong>ed</strong> by r(t) = cost i - sin t k, 0 ~ t ~ 21r, and r' (t) = -sin t i- cost k. Thus<br />
F (r (t)) = -sint j +costk, F (r (t)) · r ' (t) = - cos 2 t, and .fc F · dr = J;"(-cos 2 t).dt = -~t- ~ sin2t]~.,. = -1r.<br />
Now curl F = -i- j - k , and S can be parametriz<strong>ed</strong> (see Example 16.6.1 0) by<br />
r (rfJ, 9) = sin rfJ cos9i+ sin rfJ sinBj + cosq'> k,O ~ B ~ 7!',0 ~ rp ~ 7!'. Then<br />
r .p x ro = sin 2 4> cos B i + sin 2 4> sin B j +sin 1p cos rp k and<br />
JJ 5<br />
curlF·dS= JJ cur!F·(r,.;x ro)dA=f 0<br />
" J 0<br />
" (-sin 2 ¢cosB - sin 2 ¢sinB - sin¢cos¢)dBd¢<br />
~ +~ S l .<br />
17. lt is 'easier to use Stokes' Theorem than to compute the work directly. LetS be the planar region enclos<strong>ed</strong> by the path of the<br />
particle, so Sis the portion of the plane z = h for 0 ~ x ~ 1, 0 ~ y ~ 2, with upward orientation.<br />
curl F =By i + 2z j + 2y k and<br />
fcF · dr = ff 5<br />
curlF · dS = ff 0<br />
[-By (0) - 2z (~ )<br />
= f 0<br />
1<br />
J; ~Y dy dx = J; [1Y 2 ]~=~ dx = f 0<br />
1<br />
3 dx = 3<br />
+ 2y] dA = J 0<br />
1<br />
J 0<br />
2<br />
(2y- h) dydx<br />
19. Assume S is center<strong>ed</strong> at the origin with radius a and let Ht and H 2 be the upper and lower hemispheres, respectively, of S.<br />
I<br />
F · dr by Stokes' Theorem. But C1 is the<br />
Then .ff 5<br />
curlF · dS = JJH 1<br />
curlF · dS + JJH 2<br />
curlF · dS = J~ 1<br />
F · dr + J~ 2<br />
circle x 2 + y 2 = a 2 orient<strong>ed</strong> in the counterclockwise direction while C 2 is the same circle orient<strong>ed</strong> in the clockwise direction.<br />
Hence fc 2<br />
F · dr = - fc 1<br />
F · dr so f fs curl F · dS = 0 as desir<strong>ed</strong>.<br />
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dS<br />
SECTION 16.9 THE DIVERGENCE THEOREM 0 335<br />
16.9 The Divergence Theorem<br />
1. div F = 3 + x + 2x = 3 + 3x, so<br />
J I IE div F dV = J: J: f 0<br />
1<br />
(3x + 3) dx dy dz = ~ (notice the triple integral is<br />
three times the <strong>vol</strong>ume of the cube plus three times X).<br />
To compute Jfs F · dS , on<br />
S 1 : n = i, F = 3 i +yj + 2z k, and Jf 51 F · dS = ffsi 3 dS = 3;<br />
X<br />
s2: F = 3x i + X j + 2xz k , n = j and I fs2 F<br />
0<br />
= I f s2 X dS = ~;<br />
S3: F = 3xi + xyj + 2x k, n = k and f f 53<br />
F · dS = fj 53<br />
2xdS = 1;<br />
S4 : F = 0 , ffs,, F · dS = 0; S:;: F = 3xi +2x k , n = - j and ff 56<br />
F · dS = ffs<br />
5<br />
OdS = 0;<br />
S 6 : F = 3xi + xyj , n = - kand JJ 56<br />
F · dS = JJ 56<br />
OdS = 0. Thus ff 5<br />
F · dS = ~ -<br />
3. div F = 0 + 1 + 0 = 1, so JJJ E div F dV = JJJ E l dV = V(E) = ~ 7T · 4 3 = 2 ~ 6 7T. S is a sphere of radius 4 center<strong>ed</strong> at<br />
the origin which can be parametriz<strong>ed</strong> by r (¢>, 0) = (4sin ¢>cos 0, 4 sin sin 0, 4 cos¢>), 0 $ ¢ $ 1r, 0 $ 0 $ 27T (similar to<br />
Example 16.6.10). Then<br />
r .p x r 9 = (4 cos ¢>cos 0, 4 cos sine, - 4sin ) x (- 4sin sin8,4 sin cos 0, 0)<br />
= (16sin 2 ¢>cos 0, 16 sin 2 ¢sin 0, 16 cos ¢>sin¢)<br />
and F (r (¢,0)) = (4cos¢,4sin ¢sin 0,4sincos0). Thus<br />
F · (r .p x r o) = 64 cos ¢sin 2 cos 0 + 64sin 3 ¢sin 2 B + 64cos ¢sin 2 ¢>cos() = 128 cos ¢sin 2 q> cos () + 64 sin 3 ¢> sin 2 ()<br />
and<br />
'<br />
f fs F · dS = JJ.; F · {r ,p x r o) dA = J~" J; (128cos ¢sin 2 ¢>cosO + 64sin 3 sin 2 8) ddB<br />
= J~" [ 1 ~ 8 sin 3 ¢cos0+64(- ~{2+sin 2 )cos¢)sin 2 8J !:~ dO<br />
0<br />
f27r 256 ° 2 0 d() = ill (l.() - l 28] 2 " - Jo<br />
ill<br />
3 Sill 3 2 4 Sill 0 - 3 7T<br />
5. div F = /}., (xye") + !Ju (xy 2 z 3 ) + //:: ( - ye" ) = ye" + 2xyz 3 - ye" = 2xyz 3 , so by the Divergence Theorem,<br />
Jfs F · dS = JJJ E div F dV = f 0<br />
3 j0<br />
2 f0<br />
1 2xyz 3 dzdydx = 2 J~ xdx J 0<br />
2<br />
ydy J 0<br />
1<br />
z 3 dz<br />
= 2 (~ x 2 ]~ [h 2 ]~ (tz"]~ = 2 (~) (2) {i) = ¥<br />
7. div F = 3y 2 + 0 + 3z 2 , so using cylindrical coordinates with y = r cos B; z = r sin B, x = x we have<br />
ffs F · dS = J/J E(3y 2 + 3z 2 ) dV = J~" J: f~ 1 {3r 2 cas 2 B + 3r 2 sin 2 B) r dx dr dB<br />
~ 3 f 0<br />
2<br />
" dB ] 0<br />
1 r<br />
3<br />
dr f~ 1 dx = 3{27r) (i) (3) = 9 2 "<br />
9. div F = 2x sin y - x sin y - x sin y = 0, so by the Divergence Th~orem , J f s F · dS = J J EO dV = 0.<br />
11. div F = y 2 + 0 + x 2 = x 2 + y 2 so<br />
Jj~ F · dS = JJJ E(x 2 + y 2 ) dV = J;" f~ .J;~ r 2 • r dz dr d(J ~ J~" .[ 0<br />
2 r 3 (4- r 2 ) dr dB<br />
= J~" dB J~ (4r 3 - r 5 ) dr = 27r [r 4 - 'kr 6 ) ~ = ~7T<br />
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336 0 CHAPTER 16 VECTOR CALCULUS<br />
13. F ( x, y , z) = x J x 2 + y 2 + z 2 i + yJ x 2 + y 2 + z 2 j + z J x 2 + y 2 + z 2 k , so<br />
div F = x. ~(x2 + y2 + z2)-112(2x) + (x2 + y2 + z2)112 + y. t(x2 + y2 + z2)-112(2y) + (x2 + y2 + z2)1/2<br />
Then<br />
+ z . ~(x2 + y2 + z2) - 112(2z) + (x2 + y2 + z2)1/2<br />
= (x2 + y2 + z2)-112 [x2 + (x2 + y2 !- z2) + y2 + (x2 + y2 + z2) + z2 + (x2 + y2 + z2)]<br />
4(x2 + y2 + z2)<br />
= = 4Jx2+y2+z2.<br />
Jx2 +y2 +z2<br />
!!L<br />
j·Tr/7:12,-11<br />
= 4J.x 2 + y2 + z2 dV = 4.J(li . p 2 sin¢ dp dB d¢<br />
• E 0 0 0<br />
= f 0<br />
1r 12 sin¢d¢J;"' d1J J;4p 3 dp = [-cos¢] ~ 12 [B]~Tr (p 4 ) ~ = (1) (27r) (1) = 21r<br />
' 4 4 )<br />
15. .[J s F ·dS=JJJ -/3-x 2 dV=J 1 J 1 f 2 -"' -u· v'3-x2dzdydx = 341 .J2+lllsin- 1 E - 1 - 1 Jo ao 20<br />
(:4 a<br />
17. For 81 we haven = - k, so F · n = F · ( -k) = - x 2 z- y 2 = -y 2 (since z = 0 on 81). So if Dis the unit disk, we get .<br />
.f Is 1<br />
F · dS = .ff 51<br />
F · n dS = I J'v (-y 2 2<br />
) dA = - f 1r It r·2 0<br />
( sin 2 fJ) r dr dB = - {1r. Now since Sz is clos<strong>ed</strong>, we can use<br />
the Divergence Theorem. Since div F = :., ( z 2 x) + tu ( iv 3 + tan z) + tz ( x 2 z + y 2 ) = z 2 + y 2 + x 2 , we use spherical<br />
coordinates to g~t JJ 52<br />
F · dS = I.JJ E divF dV = J 02<br />
1r Io1r! 2 I: p 2 · p 2 sindpddO =~'IT. Finally<br />
19. The vectors that end near P1 are longer than the vectors that start near P1, so the net flow is inward near Pt and div F(Pl) is<br />
negative. The vectors that end near P2 are shorter than the vectors that start near P2, so the net flow is outward near P2 and<br />
21.<br />
div F(P2) is positiye.<br />
5<br />
,,,,,,,,,/<br />
' ' ' \ 1 I f I I /<br />
-5~-----+------45<br />
From the graph it appears that for points above the x-axis, vectors starting near a<br />
particular point are longer than vectors ending there, so divergence is positive.<br />
The opposite is true at points below the x-axis, where divergence is negative.<br />
F(x,y) = (xy, x +y 2 ) =} divF = : , (xy)+ :Y (x+y 2 ) =y+2y =Jy.<br />
I \ ' ' '<br />
/ ~ I I I \ \ \ '<br />
/'/If \\\'\<br />
-5<br />
Thus div F > 0 for y > 0, and div F < 0 for y < 0.<br />
. X x i+yj+ z k 8 ( X ) (x 2 +y 2 +z 2 )-3x 2 . . . .<br />
23. Smcc - 3 = ( 2 2 2 ) 312 and - (<br />
8 2 2 2 ) 31 ., = ( 2<br />
.,<br />
lxl X + y + Z X X + y + Z - X + y- + Z 2 ) 512 With Similar expressiOns<br />
for ..!!_ ( y ) and ..!!_ ( z ) , we have<br />
8y (x2 + y2 + z2)3/2 8z (x2 + y2 + z2)3/2<br />
(<br />
x ) 3(x2 + y2 + z2)- 3{x2 + y2 + z2) .<br />
div -<br />
1<br />
l 3 = .. 12 · = 0, except at {0, 0, 0) where it is undefin<strong>ed</strong>.<br />
x<br />
(x2+y2+z2)"<br />
25 . .f Is a · n dS = J I IE diva dV = 0 since div a = 0.<br />
© 2012 Ccngagc Learning. AJI Rights Reserv<strong>ed</strong>. Mny not be scann<strong>ed</strong>. copi<strong>ed</strong>, or duplicat<strong>ed</strong>. or post<strong>ed</strong> t~ a publicly accessible wc.-bsitc. in whole or in part
CHAPTER 16 REVIEW 0 337<br />
27. JI 5<br />
curlF · dS =III Ediv(curlF) dV = 0 by Theorem 16.5. 11.<br />
29. ff 5<br />
(f'Vg) · n dS = JJJ E div(f'Vg) dV = fff E(f'V 2 g + 'Vg · 'V f) dV by Exercise 16.5.25.<br />
, 31. If c = c1 i + c2 j + C3 k is an arbitrary constant vector, we define F = fc = fc1 i + fc2 j + fc3 k. Then<br />
. . af af af . · .<br />
0<br />
c and the Divergence Theorem says .ffs F dS = f f f E dJV F dV =><br />
dtv F = dtv fc = ax Ct + f)y C2 + f)z C3 = 'V f<br />
0<br />
Ifs F · n dS = IIJ E 'V f · cdV. In particular, ifc = i then Jfs f i · n dS = JJJ E 'Vf · i dV =><br />
Jfs fn1 d~ = jj L :~ dV (where n ~ n 1 i + n2 j + n3 k). Similarly, ifc =j we have Jfs fn2 dS = JJL :~ dV,<br />
and c = k gives /Is f na dS = j J L :~ dV. Then<br />
.fJ~ f n dS = (ff 8 /n1 dS) i + (Jf 8 fn2 dS) j + (ff 5 jn3 dS) k<br />
= (/I L :~ dV) i + (/I L ~; dV) j + (/I L ~~ dV) k = I I L ( :~ i + ~ j + ~~ k) dV<br />
= JJJ E 'V f dV as desir<strong>ed</strong>.<br />
16 Review<br />
CONCEPT CHECK<br />
1. See Definitions 1 and 2 in Section 16.1. A vector field can represent, for example, the wind velocity at any location in space,<br />
the spe<strong>ed</strong> and direction of the ocean current at any location, or the force vectors of Earth's gravitational field at a location in<br />
space.<br />
2. (a) A conservative vector field F Is a vector field which is the gradient of some scalar function f.<br />
(b) The function f in part (a) is call<strong>ed</strong> a potential function for F , that is, F = 'V f .<br />
3. (a) See Definition 16.2.2.<br />
(b) We normally evaluate the line integral using Formula 16:2.3.<br />
(c) The mass ism = fc p (x, y) ds, and the center of mass is (x, Y) where x = ~ fc xp (x, y) ds, y = ~ fc yp (x, y) ds.<br />
(d) See (5) and (6) in Section 16.2 for plane curves; we have similar definitions when 0 is a space curve<br />
· [see the equation prec<strong>ed</strong>ing (10) in Section 16.2).<br />
(e) For plane curves, see Equations '16.2.7. We have similar results for space curves<br />
[see the equation prec<strong>ed</strong>ing (10) in Section 16.2 ).<br />
4. (a) See Definition 16.2. 13.<br />
(b) If F is a force field, J~ F · dr represents ti1e work done by Fin moving a particle along the curve 0 . .<br />
(c) Ic F ·dr=J 0<br />
Pdx+Qdy+Rdz<br />
5. See Theorem 16.3.2.<br />
, 0<br />
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338 0 CHAPTER 16 VECTOR CALCULUS<br />
6. (a) fc F · dr is independent of path if the line integral has the same value for any two curves that have the same initial and<br />
tenninal points.<br />
(b) See Theorem 16.3.4.<br />
7. See the statement of Green's Theorem on page.ll08 [ET 1084].<br />
8. See Equations 16.4.5.<br />
8R 8Q) . ( 8P 8R) . ( 8Q 8P) .<br />
( 8y 8z 8z 8x ax 8y .<br />
9. (a)curlF= --- •+ - - - J + - - - k=V x F<br />
(c) For curl F , see the discussion accompanying Figure 1 on page 111 8 [ET 1094] as well as Figure 6 and the accompanying<br />
discussion on page 1150 [ET 1126]. For div F, see the discussion following Example 5 on page 1119 [ET 1095] as well as<br />
the discussion prec<strong>ed</strong>ing (8) on page 1157 [ET 1133].<br />
10. See Theorem 16.3.6; see Theorem 16.5.4.<br />
11. (a) See (I) and (2) and the accompanying discussion in Section 16.6; See Figure 4 and· the accompanying discussion on<br />
page 1124 [ET 1100].<br />
(b) See Definition 16.6.6.<br />
(c) See Equation 16.6.9.<br />
12. (a) See (1) in Section 16.7.<br />
(b) We normally evaluate the surface integral using Fonnula 16.7.2.<br />
(c) See Formula 16.7.4.<br />
(d) The mass ism = Jf 5<br />
p(x,y,z)dS and the center of mass is (x,y,z) wherex = ~ JJ 8<br />
xp(x,y,z) dS,<br />
y = ~ Ifs yp(x, y, z ) dS, z = ~ .ffs zp(x, y , z) dS.<br />
13. (a) See Figures 6 and 7 and the accompanying discussion in Section 16.7. A Mtibius strip is a nonorientable surface; see<br />
Figures 4 and 5 and the accompanying discussion on page 1139 [ET 111 5].<br />
(b) See Definition 16.7.8.<br />
(c) See Formula 16.7.9.<br />
(d) SeeFonnula 16.7.10.<br />
14. See the statement of Stokes' Theorem on page 1146 [ET I J 22].<br />
15. See the statement of the Divergence Theorem on page 11 53 [ET J 129].<br />
16. In each theorem, we have an integral of a "derivative" over a region on the left side, while the right side in<strong>vol</strong>ves the values of<br />
the original function only on the boundary of the region.<br />
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CHAPTER 16 REVIEW 0 339<br />
TRUE-FALSE QUIZ<br />
1. False; div F is a scalar field.<br />
3. True, by Theorem 16.5.3 and the fact that div 0 = 0.<br />
5. False. See Exercise 16.3.35. (But the assertion is true if D is simply-connect<strong>ed</strong>; see Theorem 16.3.6.)<br />
7. False. For example, div(y i) = 0 = div(xj) but y i =I x j.<br />
9. True. See Exercise 16.5.24.<br />
11. True. Apply the Divergence Theorem and use the fact that div F = 0.<br />
. . .<br />
EXERCISES<br />
1. (a) Vectors starting on C point in roughly the direction opposite to C, so the tangential component F · T is negative.<br />
Thus J c F · dr = j~ F · T ds is negative.<br />
(b) The vectors that end near Pare shorter than the vectors that start near P, so the net flow is outward near P and·<br />
div F(P) is positive.<br />
3. fc y zcosx ds = J 0<br />
71' (3cos t) (3sint) cost .j(1) 2 + ( -3sint)2 + (3cos t)2 dt = J 0<br />
rr(9cos 2 t sin t)v'IO dt<br />
= 9 JlO ( - t cos 3 t)]~ = - 3 VfO ( -2) = 6 VfO<br />
5. fc y3 dx + x2 dy = J~1 [y3( -2y) + (1 - y2)2] dy = J~1 (-y4 - 2!/+ 1) dy<br />
= [- !y5 - l y3 + y] 1 = _ ! - 1 + 1-! - 1 + 1 = ~<br />
5 3 - 1 5 3 5 3 15<br />
7.C:x=1+2t => dx=2dt,y = 4t => dy = 4dt,z=-1+3t => dz = 3dt,O~t~l.<br />
fc xydx + y 2 dy + yzdz = f 0<br />
1<br />
[(1 + 2t)(4t)(2) + (4t) 2 (4) + (4t)(-1 + 3t)(3)] dt<br />
9. F (r(t)) = e- t i + t 2 ( -t)j + (t 2 + t 3 ) k, r'(t) = 2t i + 3e j - k and<br />
f. F · dr = rt (2te-t - 3t5 - (t 2 + t 3 ) ) dt = [-2te-t - 2e-t - !t 6 - !t 3 - !t 4 ] 1 = ll - 1<br />
c Jo 2 3 4 o 12 c ·<br />
11. : 11<br />
[(1 + xy)e"'"] = 2xe"' 11 + x 2 ye"' 11 = :, [e" + x 2 e"' 11 ] and the domain ofF is JR?, so F is conservative. Thus there<br />
exists a function f sue~ that F = 'V f. Then fu(x, y) = e 11 + x 2 e"' 11 implies f(x , y) = e 11 + xe"'ll + g(x) and then<br />
f,(x, y) = xye"' 11 + e"' 11 + g'(x) = (1 + xy)e"'ll + g'(x). But J,(x, y) = (1 + xy)e"' 11 , so g'(x) = 0 => g(x) = K.<br />
Thus f (x, y) = e 11 + xe"'" + [(is a potential function for F .<br />
13. Since tu (4x 3 y 2 - 2xy 3 ) = Bx 3 y- 6xy 2 = :, (2x 4 y-3x 2 y 2 + 4y 3 ) and the domain ofF is IR?, F is conse~ative.<br />
Furthermore f(x, y) = x 4 y 2 - x 2 y 3 + y 4 is a potential function for F . t = 0 corresponds to the point (0, 1) and t = 1<br />
corresponds to (1, 1), so fc F · dr = /(1, 1) - /(0,.1) = 1 - 1 = 0.<br />
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340 0 CHAPTER 16 VECTOR CALCULUS<br />
15. C1: r (t)=t i +t2 j , -l ~t~ l;<br />
C2: r (t) = :-ti + j, - 1 ~ t ~ 1.<br />
y<br />
cl<br />
. (- l , l ) ot--~~+---..., ( 1 . 1 )<br />
Then<br />
Ic xl dx- x 2 ydy = f 1<br />
(t 5 - 2t 5 )dt + J~ 1 tdt<br />
Using Green's Theorem, we have<br />
j~ xy 2 dx - x 2 ydy = Ji [<br />
~ [-.!t6)1 + [.!t2]1 = 0<br />
6 -1 2 -1 '<br />
:x (-x 2 y) - :y (xy 2 1<br />
) ] dA = Ji (-2xy- 2xy) dA = /_ - 4xydydx<br />
11:<br />
= I~ 1 (-2xy 2 J~::2 dx = I~ 1 (2x 5 - 2x)dx = [ ~x 6 - x 2 t 1<br />
= 0<br />
X<br />
17. Ic x 2 ydx- xy 2 dy. = II [ tfx ( -xy 2 )- %v (x 2 y)] dA = .f[ ( - y 2 - x 2 ) dA = - I;" I; r 3 dr dB= -81r<br />
:r2 + y2 ~ 4 :z:2 + y2 ~ 4<br />
19. If we assume there is such a vector field G, then div( curl G) = 2 + 3z - 2xz. But div( curl F) = 0 for all vector fields F.<br />
Thus such a G cannot exist.<br />
.21. For any piecewise-smooth simple clos<strong>ed</strong> plane curve C bounding a region D, we can apply Green's Theorem to<br />
F(x, y) = f(x) i + g~y) j to get Ic f(x) dx + g(y) dy = IIo [ /!"' g(y) - /! 11<br />
f(x)] dA = Jj 0 0 dA = 0.<br />
. '<br />
23. \7 2 f = 0 means that~:{ + ~:; = 0. Now ifF= fv i- f x j and C is any clos<strong>ed</strong> path in D , then applying Green's<br />
Theorem, we get<br />
Ic F · dr= Ic fvdx- f x dY = IIo [:x (-fx)- /! 11<br />
Uv)] dA<br />
= -.f[ 0 (fxx + f.uv) dA = - IJ'v 0 dA = 0<br />
Therefore the line integral is independent of path, by Theorem 16.3.3.<br />
25. z = f (x, y) = x 2 + 2y with 0 ~ x ~ 1, 0 ~ y ~ 2x. Thus<br />
27. z = f(x, y) = x 2 + y 2 with 0 ~ x 2 + y 2 ~ 4 so r., x r 11 = -2x i - 2y j + k (using upward orientation). Then<br />
II 5<br />
z dS= II (x 2 +y 2 )J4x 2 +4y 2 +ldA<br />
;r2 + y2 ~ 4<br />
(Substitute u = 1 + 4r 2 and use tables.)<br />
29. Since the sphere bounds a simple solid region, the Divergence Theorem applies and<br />
IIsF · dS =III E divFdV = f.[f E(z- 2) dV = I.ff EzdV- 2III EdV<br />
= O [ odd<br />
fu~ction in =. ]<br />
_ 2 . V(E) = _ 2 . h( 2 ? = _Q17r'<br />
and E os symmctnc 3 3<br />
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CHAPTER 16 REVIEW D 341<br />
Alternate solution: F(r(¢,B)) = 4sin¢ cosB cos¢ i - 4 sin¢ sinBj + 6sin¢ cosB k,<br />
r q, x ro = 4sin 2 ¢ cos.B i + 4sin 2 cos B. Then<br />
2<br />
J fs F · dS = f " 0<br />
f " 0<br />
(16 sin 3 ¢ cos¢ cos 2 B - 16 sin 3 ¢ sin 2 B + 24 sin 2 ¢ cos¢ cos B) d¢ dB<br />
= J:" ~( - 16sin 2 B) dB = - 64 3<br />
1r<br />
31. Since curlF = 0 , JJ 8<br />
(curlF) · dS = 0. We parametrize C: r (t) = costi +sint j , 0::; t::; 21r and<br />
f c F · dr = J:" (- cos 2 t sin t + sin 2 t cost) dt = 1 cos 3 t + t sin 3 t] ~,. = 0.<br />
33. The surface. is given by x + y + z = 1 or z = 1 - x- y, 0 ::; x::; 1, 0 ::; y::; 1- x and r, x r 11 = i + j + k. Then<br />
fcF · dr = ffs curlF · dS = ffv(- y i - z j - x k ) · (i + j + k) dA = ffv(-1)dA =-(area of D) =- ~.<br />
35. JJJ E divFdV = JJJ 3dV = 3(<strong>vol</strong>ume of sphere)= 47r. Then<br />
,2 + 11 2 + z2.,; 1<br />
F (r(¢, B)) · (r .p x r o) = sin 3
-0 PROBLEMS PLUS<br />
1. Let 81 be the portion ofD(8) between 8(a) and 8, and let 881 be its boundary. Also let 8L be the lateral surface of 81 [that<br />
is, the surface of 8 1 except 8 and 8(a)]. Applying the Divergence Theorem we have/" {<br />
But<br />
r ·3n<br />
d8 = { { { \l. r 3<br />
dV.<br />
J 8S1 r J J J s 1 r<br />
"il · :3 = \ :x 1 :y'!) · \ (x2 + y2x+ z2)3/2' (x2, + y2y + z2)3/2' (x2 + y./+ z2//2)<br />
(x2 + 1/ + z2 - 3x2) + (x2 + y2 + z2- 3y2) + (~2 + y2 + z2 - 3z2)<br />
- - o<br />
- (x2 + y2 + z2)5/2 -<br />
=> j" { r ' 3<br />
° d8 = !"{ 0 dV = 0. On 'the other hand, notice that for the surfaces of 881 other than S(a) and 8,<br />
l os , r J l s,<br />
r · n = O =><br />
{ r ·3n d8 =><br />
l os, r Js r Js(a) r JsL r i s r Js(a) r<br />
0 = !"{ r ·3n dS = !"{ r ·3n dS + !"{ r ·an d8 + !"{ r ·sn d8 = !"{ r ·3n d8 + !"<br />
!Is !"1 - r · . - n r · n r r<br />
3<br />
d8 = - -<br />
3 - d8. Notice that on S(a), r =a => n = - - =-- and r · r = r 2 = a 2 , so<br />
s 1 S(a) r r a .<br />
that - / " r r ·3n d8 = !" r r ·4r d8 = !" r a: d8 = ~ !" r dS = area of2S (a) = ID(8) 1.<br />
J s(a) r . Js(a) a J s(a) a a Js(a) a<br />
! "{<br />
Therefore ID(S)I = Js ~ r · n dS.<br />
3. The given line i ntegra l ~ .{ 0 (bz - CIJ) dx +(ex- az ) dy + (ay - bx) dz can be express<strong>ed</strong> as fc F · dr if we define the vector<br />
field F by F (x, y, z) =Pi + Q j + R k = ~(bz- CIJ) i + Hex- az) j + ~(ay- bx) k. Then define 8 to be the planar<br />
interior ofC, so Sis an orient<strong>ed</strong>, smooth surface. Stokes' Theorem says .fc F · dr = .ffs curlF · dS = JJ~ curl F . n d8.<br />
Now<br />
curl F = ( ~: -<br />
~~) i + (a:; - ~=) j + ( ~~ - ~:) k<br />
= (~a+ ~a) i + (~b+ tb) j + (~c+ tc) k = a i +bj + ck = n<br />
so curlF · n = n · n = lnl 2 = 1, hence Jfs curl F · n dS = Jfs dS w hich is simply the su rface area of S. Thus,<br />
J~ F · dr = ~ j~(bz- cy) dx +(ex- az) dy + (ay .- bx) dz is the plane area.enclos<strong>ed</strong> by C.<br />
5. (F ·\l) G= (p~:.rc+Ql:y+R l !)cP2 i +Q2j+R2 k)<br />
= (p 1<br />
8P2 + Q 1<br />
8P2 + R 1<br />
8P2) i + (PI 8Q2 + Q 1<br />
8Q2 + R 1<br />
8Q2) j<br />
ax By 8z OX 8y az<br />
= (F · \JP2) i + (F · \JQ2) j + (F · \JR2) k.<br />
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344 0 CHAPTER 16 PROBLEMS PLUS<br />
Similarly, (G · V) F = (G · V Pt) i + (G · VQt)j + (G · V Rl) k. Then<br />
F xcurlG =<br />
8R2j8y - 8Q2/8z 8P2/Bz - 8R2/8x 8Q2j8x- 8P2/8y<br />
= (Q 1<br />
8Q2 _ Q 1<br />
8P2 _ Rt 8P2 + R t BR2) i + (R 1<br />
BR2 _ Rt 8Q2 _ Pt 8Q2 + Pt BP2) j<br />
ax By Bz Bx By Bz Bx By<br />
k<br />
+ (p 1<br />
8P2 _ Pt 8R2 _ Q 1<br />
BR2 + Q 1<br />
BQ2 ) k<br />
8z ox By Bz<br />
and<br />
Then<br />
and<br />
Hence<br />
(F · V) G + F x curl G + ( G · V) F + G x curl F<br />
= [(p 1<br />
8P2 + p 2<br />
BPt ) + (Q 1<br />
BQ2 +Q 2<br />
BQt) + (Rt BR2 + R2 BRt)] i<br />
~ ~ ~ ~ ~ ~<br />
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or duplicat<strong>ed</strong>, or p
17 D SECOND-ORDER DIFFERENTIAL EQUATIONS<br />
17.1 Second-Order Linear Equations<br />
1. The auxiliary equation is r 2 - r- 6 = 0 => (r- 3)(r + 2) = 0 => r = 3, r = -2. Then by {8) the general s?lution<br />
isy = c1e 3 "' +cze- 2 "'.<br />
3. The auxiliary equation is r 2 + 16 = 0 => r = ± 4i. Then by ( 11 ) the general solution is<br />
y = e 0 "'(c1 cos4x + cz sin4x) = c1 cos4x + c2 sin4~.<br />
5. The auxiliary equation is 9r 2 - 12r· + 4 = 0 => (3r- 2) 2 = 0 => r = ~·Then by (10), the general solution is<br />
_1· The auxiliary equation is 2r·2 - r = r(2r - 1) = 0 => r = 0, r = t, soy = c1e 0 "' + cze"/ 2 = c1 + c 2 e"'1 2 .<br />
4± v'-36 .<br />
9. The auxiliary equation is r 2 - 4r + 13 = 0 => r =<br />
~ 2 ± 3t, soy = e 2 "'( c1 cos 3x + c 2 sin 3x ).<br />
11. The auxiliary equation is 2r 2 + 2r - 1 = 0 => r = - 2 ~ v'I2 = - ~ ± v;, so<br />
y = c 1 e( - l/2+VJ/2)t + cze( - l /2- VJ/2)t.<br />
2<br />
13. The auxiliary equation is 100r 2 + 200r + 101 = 0 => r =<br />
P = e-t (c 1 cos ( iot) + c2 sin U 0<br />
t)] .<br />
- 200 ± V - 400 1 .<br />
::;= -1 ± Tiit, so<br />
200<br />
15. The auxiliary equation is 5r 2 - 2r- 3 = (5r + 3)(r - 1) = 0 => r = _ 1 5,<br />
10<br />
r = 1, so the general solution is y = c 1 e- 3 x/S + c 2 e"' . We graph the basic<br />
solutions f(x) = e- S:r/ 5 , g(x) = e"' as well as y = e- 3 x/S + 2e"',<br />
y = e- 3 x /f> - e"', and.y = - 2e- 3 "'/ 5 - e"'. Each solution consists of a single<br />
continuous curve that approaches either 0 or ±oo as x _, ± oo.<br />
17. r 2 - 6r + 8 = (r - 4)(r- 2) = 0, so r = 4, r· = 2 and the general solution is y = c1e 4 "' + c 2 e 2 "'. Then<br />
·y' = 4c1e 4 "' + 2cze 2 "', so y(O) = 2 => c1 + cz = 2 and y'(O) = 2 => 4cl + 2cz = 2, giving c1 = - 1 and c 2 = 3.<br />
Thus the solution to the initial-value problem is y = 3e 2 "' -<br />
e 4 "'.<br />
19. 9r 2 + 12r + 4 = (3r + 2? = 0 => r =- ~and the general solution is y = c1e- 2 "'/ 3 + c 2 xe- 2 "'1 3 . Then y(O) = 1 =><br />
c1 = 1 and, since y' = - fr cle- 2 "' 13 + Cz (1 - ~x) e- 2 "'1 3 , y'(O) = 0 => - ~c1 + cz = 0, so c2 = t and the solution to<br />
the initial-value problem is y = e- 2 "'1 3 + ~xe- 2 "'/3.<br />
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3.c6 0 CHAPTER 17 SECOND-ORDER DIFFERENTIAL EQUATIONS<br />
21. r 2 - 6r + 10 = 0 => r = 3 ± i and the_ general solution is y = e 3 "'(c1 cosx + c2 sinx). Then 2 = y(O)·= c1 and<br />
3 = y' (0) = c2 + 3ct => c2 = -3 and the solution to the initial-value problem is y = e 3 "' (2 cos x - 3 sin x ).<br />
23. r 2 - r- 12 = (r- 4)(r + 3) = 0 => r = 4, r = - 3 and the general solution is y = c1e 4 "' + c 2 e- 3 "'. Then<br />
0 = y(1) = c1e 4 + c2e- 3 and 1 = y' (1) = 4cte 4 - 3c2e- 3 so Ct = te- 4 , c2 = -te 3 and the solution to the initial-value<br />
25. r 2 + 4 = 0 => r = ±2i and the general solution is y = c1 cos 2x + c2 sin 2x. Then 5 = y(O) = c1 and 3 = y(1r / 4) = c2,<br />
so the solution of the boundary-value problem is y = 5 cos 2x + 3 sin 2x.<br />
27. r 2 + 4r + 4 = (r + 2) 2 = 0 => r = -2 and the general solution is y = c1e- 2 "' + c2xe- 2 "' . Then 2 = y(O) = c1 and<br />
0 = y(1) = c1e- 2 + c2e- 2 so c2 = -2, and the solution of the boundary-value problem is y = 2e- 2 "' - 2xe- 2 "'.<br />
29. r 2 - 7' := r(r- 1) = 0 => r = 0, r = 1 and the general solution is y = c 1 + c2e"'. Then 1 = y(O) = c 1 + c2<br />
e -2 1 e-2 ~<br />
and 2 = y(1) = Ct + c2e so c1 = --, c2 = --. The solution of the boundary-value problem is y = --+ --.<br />
e-1 e - 1 . e-1 e- 1<br />
31. r 2 + 4T + 20 = 0 => r = - 2 ± 4i and the general solution is y = e- 2 "' (ct cos4x + c2 sin 4x). But 1 = y(O) = c1 and<br />
2 = y( 1r) = c1 e _ 2 ,. => Ct = 2e 2 ,., so there is no solution.<br />
33. (a) Case 1 (A = 0): y" + AY = 0 => y" = 0 which has an auxiliary equation r 2 = 9 => r = 0 => y = Ct + c2x<br />
where y(O) = 0 and y(L) = 0. Thus, 0 = y(O) = Ct and 0 = y(L) = c2L => Ct = c2 = 0. Thus y = 0.<br />
Case 2 (A< 0): y" + Ay = 0 has auxiliary equation r 2 = ->.. => r = ±..J=X [distinct and real since A< 0] =><br />
y = c 1 e.;::x"' + c2e- ..r-x"' where y(O) = 0 and y(L) = 0. Thus 0 = y(O) = Ct + c2 (*)and<br />
0 = y(L) = Cte.;=xL + c2e- .,..CXL {t).<br />
Multiplying(*) b~ e.;=xL and subtracting (t) gives c2 ( e.,..CXL - e- ..r-xL) ·= 0 => c2 = 0 and thus Ct = 0 from(*).<br />
Thus y = 0 for the cases >.. = 0 and A < 0.<br />
(b) y" + A 'I/ = 0 has an auxiliary equation r 2 . + A = 0 => r = ±i ,;>.. => y = c 1 cos ,;>.. x + c:2 sin,;>.. x where<br />
y(O) = 0 and y(L) = 0. Thus, 0 = y(O) = Ct and 0 = y(L) = c2 sin JXL since Ct = 0. Since we cannot have a trivial<br />
solution, c2 i' 0 and thus sin,;>.. L = 0 => ,;>.. L = n1r where n is an integer => A = n 2 1r 2 J L 2 and<br />
y = c2 sin( n1rx / L) where n is an integer.<br />
35. (a) r 2 - 2r + 2 = 0 => r = 1 ± i and the general solution is y = e"' (c 1 cosx + c2 sin x). lfy(a) = c and y(b) = d then<br />
ea. (c1 cos a+ c2 sin a) = c => Ct cos a+ c2 sin a= ce-a and eb (c 1 cosb + c2 sin b)= d =><br />
Ct cosb + c2 sin b = de- b. This gives a linear system in c1 and c2 which has a unique solution if the lines are not parallel.<br />
If the lines are not vertical or horizontal, we have parallel lines if cos a = .k cos band sin a = k sin b for some nonzero<br />
@) 2012 Cengage Le:uning. All Rights Rcscn'cd. Mny noc. be sc:umcd. copi<strong>ed</strong>. or dupJicatL-d, or post<strong>ed</strong> 10 a pub,h::ly accessible \vebsile, in whole or in JXUt.
cos a sin a<br />
constant k or --b = k = ---:--b<br />
'cos<br />
SID<br />
=><br />
sina<br />
cosa<br />
sinb<br />
cosb<br />
SECTION 17.2 NONHOMOGENEOUS LINEAR EQUATIONS 0 347<br />
=> tan a = tan b => b - a = mr, n any integer. (Note that<br />
none of cos a, cos b, sin a, sin b are zero.) If the lines are both horizontal then cos a = cos b = 0 ::::? b - a = mr, and<br />
similarly vertical lines means sin a = sin b = 0 =><br />
b- a = mr. Thus the systerit has a unique solution if b - a t= mr.<br />
(b) The linear system has no solution if the lines are parallel but not identical. From part (a) the lines are parallel if<br />
b - a = mr. If the lines are not horizontal, they are identical if ce-a = kde-b ::::?<br />
ce-a =k=~<br />
de-b cosb<br />
c b cos a . sin a .<br />
- = ea- --b. (If d = 0 then c = 0 also.) If they are honzontal then cos b = 0, but k = ---:--b also (and SID b t= 0) so<br />
d ~ ~<br />
. c a bsina h th h I . 'fb d c ..J. a- bcosa .<br />
we reqmre -d = e - ---:--b. T us . e system as no so ullon 1 - a = n1r an -d r e --b unless cos b = 0, m<br />
SID COS '<br />
. h c _J. a- bsina<br />
w h IC case d r e sin b .<br />
(c) The linear system has infinitely many solution if the lines are identical (and necessarily parallel). From part (b) this occurs<br />
b d<br />
. c a-bcosa ni b 0 0 hi<br />
cos<br />
h c a- bsina<br />
sm<br />
when - a = n1r an -d = e --b u ess cos = , m w c case -d = e ---:--b.<br />
=><br />
17.2 Nonhomogeneous Linear Equations<br />
1. The auxiliary equation is r 2 - 2r - 3 = (r- 3)(r + 1) = 0 ::::? r = 3, r = - 1, so the complementary solution is<br />
Yc(x) = c1e 3 "' + c2e-"'. We try the particular sc;>lution yp(x) = Acos2x + Bsin2x, so<br />
y~ = - 2A sin 2x + 2B cos 2x andy~ = - 4A cos 2x - 4B sin 2x. Substitution into the differential equation gives<br />
( -4Acos2x - 4Bsin2x)- 2( -2Asin2x + 2B~os2x)- 3(Acos2x + Bsin2x) = cos2x =><br />
(-7A- 4B) cos2x + (4A -7B) sin2x = cos2x. Then -7A - 4B = 1 and 4A - 7B = 0 => A =- :,., and<br />
B = - 6 ~. Thus the general solution is y( x) = Yc( x) + YP ( x) = c1 e 3 "' + c2e - x -<br />
7<br />
65 cos 2x - 6<br />
; sin 2x.<br />
3. The auxiliary equation is r 2 + 9 = 0 with roots r = ± 3i, so the complementary solution is Yc(x) = c1 cos 3x + c 2 sin 3x.<br />
Try the particular solution yp(x) = Ae- 2 "', soy~= - 2Ae- 2 "' andy~ = 4Ae- 2 "' . Substitution into the differential equation<br />
gives 4Ae- 2 "' + 9(Ae- 2 "') = e- 2 "' or 13Ae- 2 "' = e- 2 "'. Thus 13A = 1 ;> A= f3 and the general solution is<br />
5. The auxiliary equation is r 2 - 4r + 5 = 0 with roots r = 2 ± i, so the complementary ·solution is<br />
Yc(x) = e 2 "'(c1 cosx + c2 sin x). Try yp (x) = Ae-"', soy~= - Ae-"' andy~ = Ae-"'. Substitution gives<br />
Ae-:r- 4( -Ae-"') + 5(Ae-x) = e-"' => lOAe-"' = e-"' ::::? A = • 1<br />
Thus the general solution is<br />
10<br />
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348 0 CHAPTER 17 SECOND-ORDER DIFFERENTIAL EQUATIONS ,<br />
7. The a~xi liary equation is r 2 + 1 = 0 with roots 1' = ±i, so the complementary solution is Yc(x) = C1 cosx + c 2 sinx.<br />
For y"-+: y = e"' try yp 1 (x) = Ae"'. Then y~ 1 = y; 1<br />
= Ae"' and substitution gives Ae"' +'Ae"' = e"' => A=~.<br />
so yp 1 (x) = ~ e"'. For y" + y = x 3 try y 112 (x) = Ax 3 + Bx 2 + Cx + D. Then y~2 = 3Ax 2 + 2Bx + f:! and<br />
y; 2<br />
= 6Ax + 2B. Substituting, we have 6Ax + 2B + Ax 3 + Bx 2 + Cx + D = x 3 , so A = 1, B = 0,<br />
6A + C.= 0 => C = - 6, and 2B + D = 0 => D = 0. Thus Yv 2 ( x) = x 3 - 6x and the general solution is<br />
y(x) = Yc(x) + Yp 1 (x) + YP 2 (x) = c1 cosx + C2 sinx + ~e"' + x 3 - 6x. But 2 = y(O) = c1 + 4 =><br />
c1 = ~ and 0 = y' (0) = c2 + ~ - 6 => c2 .= ¥. Th11s the solution to the initial-value problem is<br />
y(x) = ~ cosx + ¥ sinx + %e"' + x 3 - 6x.<br />
9. The auxiliary equation is r 2 - r = 0 with roots r = 0, r = 1 so the complementary solution is Yc(x) = c 1 + c2 e"'.<br />
Try yp(x) = x(Ax + B)e"' so that no term in YP is a solution of the complementary equation. Then<br />
y~ = (Ax 2 + (2A + B)x + B)e"' andy;= (Ax 2 + (4A + B )x + (2A + 2B))e"'. Substitution into the differential equation<br />
gives (Ax 2 + (4A + B)x + (2A + 2B))e"'- (Ax 2 + (2A + B )x + B)e"' = xe"' => (2Ax + (2A + B))c"' = xe"' =><br />
A =~. B = -1. Thus yp(x) = (!x 2 -<br />
x)e"' and ~he general solution is y(x) = c1 + c2e"' + Gx 2 - x)e"'. But<br />
2 = y(O) = c 1 + c 2 and 1 = y' (0) = c2 - 1, so c2 = 2 and c1 = 0. The solution to the initial-value problem is<br />
y(x) = 2e"' + (~x 2 - x)e"' = e"'(!x 2 - x + 2)'.<br />
11. The auxiliary equation is r 2 + 3r + 2 = (r + 1)(r + 2) = 0, so r = - 1, r = - 2 and Yc(x) = c1e- "' + c2e- 2 "'.<br />
Try YP = A cos x + B sin x => y~ = - A sin x + B cos x, y; = - A cos x - B sin x . Substituting into the differential<br />
equation gives (- A cosx - B sin x) + 3( - A sinx + B cos x) + 2(Acosx + B sinx) = cosx or<br />
(A + 3B) cos x + ( -3A + B) sin x = cos x. Then solving the equations<br />
A + 3B ·= 1, - 3A + B = 0 gives A = 1<br />
10<br />
, B = fo and the general<br />
solution is y(x ) = c1e-"' -j- c2e- 2 "' + 1<br />
10<br />
cos x + fo sinx. The graph<br />
shows yp and several other solutions. Notice that all solutions are<br />
asymptotic to YP as x -+ oo. Except for y 11 , all solutions approach either oo<br />
or -oo as x -+ - oo.<br />
-3 ~~~7-?~~~~~ s<br />
Yp<br />
- 3<br />
13. Here Yc(x) = cre 2 "' + c2e-"', and a trial solution is 1h•(x) = (Ax+ B)e"' cosx + (Cx + D )e"' sin x.<br />
15. Here Yc(x) = c1e 2 "' + c2e"'. For y"- 3y' + 2y = e"' try y 1 , 1<br />
(x) = Axe"' (since y = Ae"' is a solution of the complementary<br />
equation) and fo ~ i/' - 3y' + 2y = sin x try yp 2 (x) = B cos x + C sin x. Thus a trial solution is<br />
Yp (x) = Yv 1 (x) + Yp 2 (x) =Axe"' + B cosx + Csinx.<br />
17. Since Yc(x) = e- x (c1 cos 3x + c2 sin 3x) we try Yv(x) = x(Ax 2 + Bx +C) e- x cos 3x + x(Dx 2 +Ex+ F) e-x sin 3x<br />
(so that no term ofyp is a solution of the complementary equation).<br />
@ 2012 Ccngagc lcmnjng. All Rights Rcser\'<strong>ed</strong>. May not be scann<strong>ed</strong>. copi<strong>ed</strong> 2 or duplicat<strong>ed</strong>, or post<strong>ed</strong> to a publicly accessible w
SECTION 17.2 NONHOMOGENEOUS LINEAR EQUATIONS D 349<br />
.Note: Solving Equations (7) and (9) in The Mettiod of Variation of Parameters gives<br />
and<br />
We will use these equations rather than resolving the system in each of the remaining exercises in this section.<br />
19. (a) Here4r 2 + 1 = 0 => r = ±tiand Yc(x) = Clcos(tx) + czsin(~x). Wetryaparticularsolution ofthe form<br />
Yv(x) = Acosx + Bsinx =><br />
y~ = -Asinx + Bcosx and v;: = - Acosx - Bsin x. Then the equatio~<br />
4y" + y = cosx becomes 4( - Acosx- Bsinx) + (Acosx + Bsinx) = cosx or<br />
-3Acosx- 3Bsinx = cosx => A= - 1, B = 0. Thus, Yv(x) = - % cosx and the general solution is<br />
(b) From (a) we know that Yc(x) = c1 cos~+ cz sin~· Setting Yl =cos~. Yz = sin~, we have<br />
I cosxcos~ 1 "' "'1 ·2x "'<br />
andu 2 =<br />
1<br />
= 2 cos (2 · 2 )cos 2 = 2 (1 -2sill 2 )cos 2 . Then<br />
4 ·2<br />
( ) J (} • X 2 X • "') dx X + 2 3 Z<br />
U 1<br />
d<br />
X = 2 Sill 2 - COS 2 Sill 2 = - COS 2<br />
'j COS 2 an<br />
U2 () X =<br />
l x · 2x<br />
J( x) d · :r. 2 · 3x Tl<br />
2 COS 2 - Sill 2 COS 2<br />
X = SID 2 - ~ SID 2 . lUS<br />
yp(x) = (-cos~ + ~ cos 3 ~)cos~ + (s in~ - i sin 3 ~)sin~ = - ( cos 2 ~ - sin 2 ~) + j ( cos 4 ~ - sin 4 ~)<br />
= - cos {2 · ~) + i ( cos 2 ~ + sin 2 ~ ) ( cos 2 ~ - sin 2 ~ ) = -cos x + j cos x = - i cos x<br />
and the general solution is y(x) = Yc(x) + y 1 ,(x) = c1 c os~ . + Cz sin~- t cos x.<br />
21. (a) r 2 - 2r + 1 = (r -1) 2 = 0 => r· = 1, S? the complementary solution is Yc(x) = c1e'" + czxex. A particular solution<br />
is of the form Yv(x) = Ae 2 "'. Thus 4Ae 2 "'- 4Ae 2 "' + Ae 2 "' = e 2 "' => Ae 2 "' = e 2 :r: => A= 1 => Yv(x) = e 2 "'.<br />
So a general solution is y(x) = Yc(x) + Yv(x) = c1ex + c2xe"' + e 2 "'.<br />
(b) From (a), Yc(x) = c1e"' + czxex, so set Yl = ex, Yz = xex. Then, Y1Y~ - YzY~ = e 2 "' (1 + x) - xe 2 "' = e 2 "' and so<br />
u~ = - xe"' => u1 (x) = - J xe:r: dx = - (x - 1)e"' [by parts] and u~ = e"' => u2(x) = J e"' dx = e"'. Hence<br />
yp (x ) = (1- x)e 2 "' + xe 2 "' = e 2 "' and the general solution is y(x) = Yc.(x) + yp(x) = c 1 e"' + czxe"' + e 2 "'.<br />
23. As in Example 5, Yc(x) = c1 sin x + cz cosx, so set Y1 = sinx, Yz = cos x . Then Y1Y~ - YzYi =-sin 2 x- cos 2 x = - 1,<br />
so ui =<br />
sec 2 x cosx<br />
_ 1<br />
= secx => ul(x) = J sccxdx =ln(secx + tan x)for. O
350 0 CHAPTER 17 SECOND-ORDER DIFFERENTlAL EQUATIONS<br />
o: 2o: 1 1 3x 1 -e 2 x<br />
25. Yt = e , Y2 = e and Yt Y2 - Y2Y1 = e . So Ut = ( 1<br />
) 3 =<br />
· +e- o: e"'<br />
e- x .<br />
--- and<br />
1 + e-x<br />
U2 (x) = I 3 ex 2 dx = m(e"' + 1 ) -e-o: = ln(1 + e- "') - e-x. Hence<br />
e"'+e"'<br />
ex<br />
yp(x) =ex 1nn .+e-x) + e 2 "'[ln(1 +e-x)- e-"'J and the general solution is<br />
y(x) = (c1 + ln(1 + e- "')]e"' + [c2 - e- x + ln(1 + e- ")]e 2 "'.<br />
27. r 2 - 2r + 1 = (r -1) 2 = 0 => r = 1 so Yc(x) = Ct e"' + c2xe"'. Thus Yt =ex, Y2 = xe"' and<br />
xe'c · e"' /(1 + x 2 )<br />
ex<br />
x<br />
+ x<br />
Y I Y~- y2y~ = e"'(x + 1)ex- xe"'e"' = e 2 "'. Sou~=-<br />
2<br />
= - - 1<br />
-- 2<br />
~<br />
Ut = -<br />
I x 1 ( 2 ) 1 e"' · e"' /(1 + x2 ) 1 I 1 d -I d<br />
1 + x 2 dx = - 2<br />
In 1 + x , u2 = e 2 "' = 1<br />
+ x 2 ~ U2 =<br />
1 + x 2 x = tan x an<br />
yp(x) = - 4e"' ln(1 +x 2 ) +xe"'tan- 1 x. Hence the general solution isy(x) = e"' [c1 + c2x- ~ ln(1 + x 2 ) +xtan- 1 x].<br />
17.3 Applications of Second-Order Differential Equations<br />
1. By Hooke's Law k(0.25) = 25 so k = 100 is the spring constant and the differential equation is 5x" + 100x = 0.<br />
The auxiliary equation is 5r 2 + 100 = 0 with roots r = ±2 J5 i , so the general solution to the differential equation is<br />
x(t) = c 1 cos(2 V5 t) + c2 sin(2 V5 t). We are given that x(O) = 0.35 ~ c1 = 0.35 and x 1 (0) = 0 ~<br />
2 J5 c2 = 0 => c2 = 0, so the position of the mass after t seconds is x(t) = 0.35 cos(2 J5 t).<br />
3. k(0.5) = 6 or k = 12 is the ·spring constant, so the initial-value problem is 2x" + 14x 1 + 12x = 0, x(O) = 1, x 1 (0) = 0.<br />
The general solution is x(t ) = c1e-Gt + c2e-t. But 1 = x(O) = c1 + c2 and 0 = x 1 (0) = -6c1 - c2. Thus the position is<br />
given by :i:(t) = - ~e- 6 t<br />
+ ~e-t.<br />
5. For critical damping we ne<strong>ed</strong> c 2 - 4mk = 0 or m = c 2 /(4k) = ·14 2 /(4 · 12) = ~ kg.<br />
2 . d<br />
7. We are given m = 1, k = 100, x(O) = -0.1 and x 1 (0) = 0. From (3), the differential equation is ~t~ + c d; + 100x = 0<br />
with auxiliary equation r 2 + cr + 100 = 0.<br />
If c = 10, we have two complex roots r = - 5 ± 5 v'3 i, so the motion is underdamp<strong>ed</strong> and the solution is<br />
x =e-st [ct cos(5 v'3t) + c 2 sin(5 v'3 t)J. Then -0.1 = x(O) = c1 and 0 = x 1 (0) = 5 v'3c2 - 5ct<br />
. - 1<br />
~ C2 - "i:ii"7s,<br />
sox=e-st [--;0.1cos(5v'3t)- 10 J:isin(5v'3t)].<br />
® 20J2 Cengoge Learning. All Rights Reserv<strong>ed</strong>. May not be scann<strong>ed</strong>, copi<strong>ed</strong>, or dupllco.lcd. or po~icd to a publicly nce:e.ssiblc w.:bsite, in "hole or in part.
SECTION 17.3 APPLICATIONS OF SECOND-ORDER DIFFERENTIAL EQUATIONS 0 351<br />
If c = 15, we again have underdamping since the auxi liary equation has roots r = - lf ± fL:{f-i. The general solution is<br />
x = e- 15 t/'2 [c1 cos(fL:{f-t) + c2 sin( 2flt) ], so - 0.1 = x (O) = c1 and 0 = x'(O) = fL:{f-c2 - lf c1 => c2 = - 10<br />
3<br />
77<br />
.<br />
Thus x - e-lSt/ 2 [-o 1 cos( 5 .fit) - ~ sin(!b(J_t)]<br />
- · 2 10 v7 2 ·<br />
For c = 20, we have equal roots r 1 = r2 = -10, so the oscillation is critically damp<strong>ed</strong> and the solution is<br />
x = (c1 + c2t)e- 10 t_ Then -0.1 = x(O) = c 1 and 0 = x'(O) = -10c1 + c2 => c2 = -1, sox= ( -0.1- t)e- 101 .<br />
If c = 25 the auxiliary equation has roots r1 = --,-5, r2 = ...:.20, so we have overdamping and the solution is<br />
X= c1e-st + C2e- 20 t. Then -0.1 = x(O) = Cl + C2 and 0 = x'(O) = -5cl- 20c2 => C1 =- { 5<br />
and C2 = fo•<br />
If c = 30 we have roots r = -15 ± 5 v's, so the motion is<br />
overdamp<strong>ed</strong> and the solution is x = c1 e( -IS+ 5 v'5 )t + c2e( - 15 -<br />
5<br />
v'5 )t.<br />
0.02<br />
c = 10<br />
c = 15<br />
Then -0.1 = x(O) = C1 + c2 and<br />
O=x'(O) = (-15+5v's)c1+(-15-5v's)c2 =><br />
-5 - 3 y'5 d -5 + 3 v'5<br />
c1 = 100 an c2 = 100 , so<br />
_ ( - s- 3 fi) (- 15+ s v'S)t + ( -s ± 3 v'5) e< -1s - & fi)t<br />
x- wo e 100 · ·<br />
-0.11<br />
9. The differential equation is mx" + kx = Fo coswot and wo "1- w = .jkfm. Here the auxiliary equation is mr 2 + k = 0<br />
with roots ±.jklmi = ±wi so Xc(t) = c1 coswt + C2 sinwt. Since wo "1- w, try x 1 ,(t) = Acoswot + B sinwot.<br />
Then we ne<strong>ed</strong> (m) ( -w5) (A coswot + B sinwot) + k(Acoswot + Bsinwot) = Fo coswot or A(k- mw~) = Fo and<br />
B(k - ~w5) = 0. Hence B = 0 and A = k Fo 2 = ( ;"0 2 ) since w2 = k. Thus the motion of the mass is given<br />
· - mw 0 m w - w 0 m<br />
11. From Equation 6, x(t) = f(t) + g(t) where f(t) = c1 coswt + c2 sinwt and g(t) = ( ;o 2 ) cosw 0 t. Then f<br />
mw - w 0<br />
is periodic, with period :.:r, and if w "1- wo, g is periodic with period :,~.<br />
~ = !..<br />
wo 1 J wo<br />
' => a = bw where a and b are non-zero integers. Then<br />
If :' 0<br />
is a rational number, then we can say<br />
'x(t +a·:;-)= f(t +a· 2 ;) + g(t +a· :.:r) = f(t) + g(t + :~ · :.:r) = f(t) + g(t + b - :,~) = j(t) + g(t) = x(t)<br />
so x(t) is periodic.<br />
13. Here the initial-value problem for the charge is Q" + 20Q' + 500Q = 12, Q(O) = Q' (0) = 0. Then<br />
Qc(t) = e- lOt(c1 cos 20t + c2 sin 20t) and try Qp (t) = A => 500A = 12 or A = 1 ~ 5 .<br />
The general solution is Q(t) = e-IOt(c1 cos 20t + c2 sin 20t) + 1 ~ 5 • But 0 = Q(O) = c1 + 1 ~ 6 and<br />
@ 2012 Ccngage Learning. All Rights Rescr\'t:d. May not be scann<strong>ed</strong>. co(\i<strong>ed</strong>, or duplicat<strong>ed</strong>. or poSt<strong>ed</strong> 10 o publicly ucccssiblc website. in whole or in part.
352 D CHAPTER 17 SECOND-ORDER DIFFERENTIAL EQUATIONS<br />
Q' (t) = I (t) = e-lot[( -10c1 + 20c2) cos 20t + ( - lOc2- 20ct) sin 20t) but 0 = Q'(O) = - 10c1 + 20c~. Thus the charge<br />
is Q(t) = - 2 ~ 0 e- tot(6 cos 20t + 3 sin 20t) + 1 ~ 5<br />
and the current is J(t) = e-lOt (~)sin 20t.<br />
15. As in Exercise 13, Qc(t) = e-tot(ci. cos 20t + c2 sin20t) but E(t) = 12sin lOt so try<br />
Qp(t) = A cos lOt+ B sin lOt. Substituting into the differential equation gives<br />
(-lOOA + 200B + 500A)cos lOt+ (-lOOB- 200A + 500B) sin lOt = ·12sin lOt =><br />
400A + 200B = 0 and 400B - 200A = 12. Thus A= - 2 ~ 0 , B = 1 ~ 5 and the general solution is<br />
Q(t) = e- lOt(c1 cos 20t + c2 sin 20t) - 2 ~ 0 cos lOt+ 1 ~ 5 sin lOt. But 0 = Q(O) = c1 - 2 ~ 0 so c1 = 2~ 0 .<br />
Also Q' (t) = fs sin lOt+ 2<br />
65<br />
cos lOt+ e-lOt[( - lOc1 + 20c2) cos 20t + ( -lOcz - 20ct) sin 20t] and<br />
0 = Q' (0) = fg - lOc1 + 20c2 so c2 = - 5 ~ 0 . Hence the charge is given by<br />
Q(t) = e-wt ( 2 ~ 0 cos 20t- 5 ~ 0 sin 20t] -<br />
2 ~ 0 cos lOt + 1 ~ 5 sin lOt.<br />
11. x(t)=Acos(wt +8) ¢:? x(t)=A[coswtcos8-sinwtsin5] ¢:? x(t)=A(~coswt+~sinwt)where<br />
cos 8 = c1 / A and sin 8 = -c2/ A ¢:? x(t) = c1 cos wt + c2 sin wt. [Note that cos 2 8 + sin 2 8 = 1 => c~ + c~ = A 2 .)<br />
17.4 Series Solutions<br />
00 00<br />
1. Let y(x) = 2: C,:.x" . Then y'(x) = 2: nenx"- 1 and the given equation, y' - y = 0, becomes<br />
n=O ·<br />
n=l<br />
00 00 00 00<br />
2: nenxn-l- 2: CnX" = 0. Replacing n by n + 1 in the first sum gives 2: (n + l)cn+tX"- L enx" = 0, so<br />
n=l n=O n=O n=O<br />
00<br />
2: [(n + l)Cn+I- en]x" = 0. Equating coefficients gives (n + l )cn+l - en= 0, so the recursion relation is<br />
n=O<br />
oo oo · oon<br />
in general, en= ~.Thus, the solution is y(x) = 2: cnx'' = 2: c~ x" = eo 2: ; = eoe"' .<br />
n. n=O n=O n. n=O n.<br />
00 00 00<br />
3. Assuming y(x) = 2: enx", we have y'(x) = 2: ncnx"- 1 = 2: (n + l)Cn+tx" and<br />
n = O n =l n=O<br />
~ 00 00 00<br />
-x 2 y =- I:: Cnx"+ 2 =- I:: Cn- 2x". Hence, the equation y' = x 2 y becomes I:: (n + l )cn+tx"- 2: Cn- 2x" = 0<br />
n=O n=2 n=O n=2<br />
or Ct + 2c2x + f: [(n + l)cn+I - Cn-2) x" = 0. Equating coefficients gives Ct = c2 = 0 and Cn+t = Cn- 2 1<br />
n=2 n+<br />
for n = 2, 3, ... . But c1 = 0, so C
SECTION 17.4 SERIES SOLUTIONS 0 353<br />
~ ~ ~<br />
5. Let y (x) = I: c.,x" '* y' (x) = I: nenx"- 1 and y" (x) = I: (n + 2){n + l)cn+2X". The differential equation<br />
n = O n=l n =O<br />
00 . 00 00 00<br />
becomes L: (n + 2){n + 1)cn+2x" + x I: nc.,.x"- 1 + L.: CnX 11 = 0 or L.: [(n + 2){n + 1)cn+2 + nen + en]x" = 0<br />
n=O n=l n = O n=O<br />
[since n~1 nenx" = n~O ncnx"] . Equating coefficients gives (n + 2)(n + 1)Cn+2 + (n + 1)Cn = 0, thus the<br />
. I . . - (n + l )c,. Cn 0 1 2 Th th<br />
recurs1onreat1ontscn+2= (n+ 2<br />
)(n+l) =--n-+- 2<br />
,n=,, , .... en eeven<br />
coe ffi ctents . are gtven . b y c2 = - Co C2 co C4 Co d . 1<br />
2 , C4 = - 4 = 2 . , co= -6 =-~,an m genera,<br />
4<br />
n Co (- 1)"Co . C1 C3 C1 C5 C1<br />
c2n = (-1)<br />
=<br />
• Theoddcoeffi.ctents are C3 = - - , cs = - - = - ,<br />
2 2 1<br />
c7 = - - = - --- ,<br />
2 · 4 · · · · · n n n. 3 5 3 · 5 7 3 · 5 · 7<br />
d<br />
. I ( l )n Cl . (-2)nn!ct Th I . .<br />
an m genera, C2n+l = -<br />
(Z ) = (<br />
3 5 7<br />
2<br />
l )l . e so utton IS<br />
· · ····· n+ 1 n+ .<br />
00 (-1)" ~<br />
00<br />
{-2)n n l<br />
1 {x) = c 0 "' - - x-" + c 1 "' x 2 "+<br />
y<br />
1<br />
n=O L- 2 11 n.<br />
'<br />
n L- =O {2 n + 1)1 ·<br />
00 00 00 00<br />
7. Let y (x) = L.: CnX" :::} y' (x) = I.: ncnx"- 1 = I.: (n + l)Cn+lx".and y" (x) = L.: (n + 2){n + l )c,+2xn. Then<br />
n=O n=l n=O n=O<br />
00 00 00 00<br />
(x-1)y"(x) = L (n+2){n+l )Cn+2Xn+l _ L.: (n+2)(n+l)cn+2X" = L n(n+1)Cn+1x"- I.: (n+2)(n+l )cr•+2x".<br />
n = O n=O n =l n =O<br />
DO 00<br />
Since L.: n(n + 1)cn+1x" = L.: n(n + 1)Cn+J.'t", the differential equation becomes<br />
n = l<br />
n=O<br />
DO 00 00<br />
L n(n + l)Cn+lX"- L.: (n + 2)(n + 1)Cn+2X" + 2: (n + 1)Cn+!X" = 0 :::}<br />
n=O n=O n =D<br />
00 00<br />
L: [n(n + 1)Cn+l - (n + 2)(n + l )Cn+2 + (n + 1)cn+1]x" = 0 or 2: [(n + 1) 2 cn+I - (n + 2)(n + 1)Cn+2]x" = 0.<br />
u=O<br />
Equating coefficients gives (n + 1) 2 c,,+I - (n + 2)(n + 1)cn+2 = 0 for n = 0, 1, 2, .... Then the recursion relation is<br />
n=O<br />
in general en = Ct , n = 1, 2, 3, .... Thus the solution is y(x) =Co + c1 f xn. Note that the solution can be express<strong>ed</strong> as<br />
n<br />
n =1 n<br />
co- c1 ln{l - x) for JxJ < 1.<br />
00 00<br />
9. Let y(x) = 2::.: c .. x".-Then -xy'(x) = - x L.: nc,.xn-I<br />
~ ~<br />
2: nc.,x" =- L.: ncnx",<br />
n=O n=l n= l n =O<br />
~<br />
y"(x) = L.: (n + 2){n + l )cn+2X" , and the equation y"- xy' - y = 0 becomes<br />
n = O<br />
00<br />
L: [(n + 2)(n + 1)Cn+2- nen - en]x" = 0. Thus, the recursio11 relation is<br />
n = O<br />
© 2012 Cengage L
354 0 CHAPTER 17 SECOND-ORDER DIFFERENTIAL EQUATIONS<br />
nc,.. + Cn c,..(n + 1 ) . Cn ~ 0 1 2 ·o fth · d' · · {0) 1 8<br />
Cn+2=(n+ 2<br />
)(n + 1 )=(n+ 2<br />
){n+ 1<br />
)=n+ 2<br />
.orn=, , , .. . . neo eg•vencon•tlons •sy =. ut<br />
y(O) = n~o c,..(O)" = ~ + 0 + 0 + .. · =eo •. so Co= 1. Hence, C2 = ~ = ~· C4 = ~ = /<br />
4 , C6 = ~ = 2 . ~ . 6 , ... ,<br />
C2n = 2<br />
! 1<br />
• The other given condition is y' (0) = 0. But y' {0) = f ncn(O)n-l = c1 + 0 + 0 + · · · = C1, so C1 = 0.<br />
n. n=l<br />
By the recursion relation, ca = ~ = 0 ~ cs = 0, .. . '. C2n+l = 0 for n = 0, 1, 2, . . .. Thus, the solution to the initial-value<br />
/<br />
I<br />
00 00 00 x2n 00 (x2/2)n 2<br />
problem is y(x) = L; c,..xn = L; C2nX 2 n = L; - ,.. 2 1 = L; - - 1 - = e"' / 2 .<br />
n =O . n=O n=O n. n = O n.<br />
00 00 00 00 00<br />
11. Assuming that y(x) = L; cnx", we have xy = x L; c,..xn = L; c,..xn+l, x 2 y' = x 2 L; nc,,xn- l = L; ncnxn+t,<br />
n=O n=O n=O n=l n=O<br />
y"(x)= f: n(n-1)c,..x"- 2 = f: (n+3){n + 2)Cn+ax"+ 1 [replace n with n + 3]<br />
n=2 n=- 1<br />
= 2c2 + f (n + 3){n + 2)Cn+ax"+l,<br />
n =O<br />
00<br />
and the equation y" + x 2 y' + xy = 0 becomes 2c2 + L; [(n + 3)(n + 2)Cn+a + ncn + c,..J xn+l = 0. So c2 = 0 and the<br />
n =O<br />
(n + l)c,..<br />
( )( ) , n = 0, 1, 2, .. . . But Co = y(O) = 0 = c2 and by the<br />
n+3 n+2<br />
recursion relation, Can = can+2 ~ 0 for n = 0, 1, 2, .... Also, c1 = y' (0) = 1, so C4 = - :~~ = - 4<br />
~ 3<br />
,<br />
17 Review<br />
CONCEPT CHECK<br />
1. (a) ay" +by' + cy = 0 where a, b, and care constants.<br />
(c) If the auxiliary equation has two distinct real roots r 1 and r·2, the solution is y = c 1 ert"' + c2er 2 "'. If the roots are real and<br />
equal, the solution is'!/ = c 1 er"' + c2xer"' where r is the common root. If the roots are complex, we can write r1 = a+. i/3<br />
and r2 = a- i/3, and the solution is y = ea"'(c1 cos/3x + c2 sin/3x).<br />
2. (a) An initial-value problem consists of finding a solution y of a second-order differential equation that also satisfies given<br />
conditions y(xo) =Yo andy' (xo) = '!/I. where yo and '!/1 are constants.<br />
© 2012 CcnCJige Lcnming. All RighiS Rcscr.'Cd. Mfty nol bo: sciUUICd, copi<strong>ed</strong>. or duplicat<strong>ed</strong>. or post<strong>ed</strong> to a publicly accessible website, in "itolc or in pan.
CHAPTER 17 REVIEW 0 355<br />
(b) A boundary-value problem consists. of finding a solution y of a second-order differential equation that al~o satisfies given<br />
boundary conditions y(xo) =Yo and y(x1) = Yl·<br />
3. (a) ay" +In/ + CIJ = G{x) where a, b, and care constan ts and G is a continuous function.<br />
(b) The complementary equation is the relat<strong>ed</strong> homogeneous equation ay" +by'+ cy = 0. If we find the general solution Yc<br />
of the complementary equation and YP is any particular solution of the original differential equation, then the general<br />
solution of the original differential equation is y(x) = Yr>(x) + Yc(x).<br />
(c) See Examples 1-5 and the associat<strong>ed</strong> discussion in Section 17.2.<br />
(d) See the discussion on pages 11 77- 1179 [ ET 1153-11 55].<br />
4. Second-order linear differential equations can be us<strong>ed</strong> to describe the motion of a vibrating spring or to analyze an electric<br />
circuit; see the discussion in Section 17 .3.<br />
5. See Example I and the prec<strong>ed</strong>ing discus.sion in Section 17 .4..<br />
1. True. See Theorem 17.1.3.<br />
TRUE-FALSE QUIZ<br />
3. True. cosh x and sinh x are linearly independent solutions of this linear homogeneous equation.<br />
EXERCISES<br />
·1. The auxiliary equation is 4r 2 - 1 = 0 => (2r + 1}(2r - 1) = 0 => r = ±t. Then the general solution<br />
3. The auxiliary equation is r 2 + 3 = 0 => r = ±J3 i. Then the general solution is y = c1 cos ( J3 x) + c 2 sin ( J3 x).<br />
5. r 2 - 4r + 5 = 0 => r = 2 ± i, so Yc (x) = e 2 "'{c 1 cosx + c 2 sin x ). Try YP (x) = Ae 2 "' => y~ = 2Ae 2 "'<br />
and y~ = 4Ae 2 "'. Substitution into the differential equation gives 4Ae 2 "' - 8Ae 2 "' + 5Ae 2 "' =-e 2 "' => A = 1 and<br />
the general solution is y( x) = e 2 "' ( c1 cos x + c2 sin x) + e 2 "'.<br />
7. r 2 -21·+ l=O => r =landyc(x) = cle"' + c2xe"'. Tryy 1 ,(x)=(Ax +B)cosx + (Cx+D) sin x =><br />
y~ = (C- Ax- B) sinx +(A+ Cx +D) cos x andy~= (2C - B- Ax)-cosx + ( -2A- D- Cx) sinx. Substitution<br />
gives (-2Cx + 2C- 2A- 2D)cosx + (2Ax - 2A + 2B - 2C}sinx =xcosx => A= 0, B = C = D = - t.<br />
The general solution is y( x) = c1 e"' + c2 xe"' - ! cos x - !(x + 1) sin x.<br />
9. r 2 - r- 6 = 0 . => r = -2, r = 3 and Yc(x) = c 1e __:_ 2 "' + C2 e 3 "'. For y" - y'- 6y = 1, try Yr>t (x) = A. Then<br />
y~ 1 (x) = y~ 1 (x) = 0 and substitution into the differential equation gives A= -i- For y"- y'- 6y = e- 2 :r try<br />
© 2012 Ccngogc Learning. All Righl5 Rescn-.d. May nol be S
356 D CHAPTER 17 SECOND-ORDER DIFFERENTIAL EQUATIONS<br />
yp 2<br />
(x) = Bxe- zx [since y = B e- 2 x satisfies the complementary equation). Then y~ = (B- 2Bx)e- 2 "' and<br />
y~ = (4Bx- 4B)e- 2 x, and substitution gives - 5Be- 2 "' = e- 2 "' '* · B = -k· The general solution then is<br />
y(x) = c1e- 2 x + Cze 3 "' + yp 1<br />
(x) + yp 2<br />
(x) = c1e- 2 x + cze 3 x - ~- ixe- 2 "'.<br />
11. The auxiliary equation is r 2 + 6r = 0 and the general solution is y(x) = c1 + c 2 e- 6 x = k1 + k 2 e- 6
(c) The differential equation y 11 + k 2 y = 0 has auxiliary equation r 2 + k 2 = 0. (This is the r of Section 17.1,<br />
CHAPTER 17 REVIEW D 357<br />
not the r measuring distance from the earth's center.) The roots of the auxiliary equation are ± ik, so by ( II) in<br />
Section 17.1' the general solution of our differential equation for t is y( t) := Cl cos kt + C2 sin kt. It follows that<br />
y' (t) = -c1k sin kt + c2k cos kt. Now y (0) = Randy' (0) = 0, so c1 = Rand c2k = 0. Thus y(t) = Rcos kt and<br />
y'(t) = - kRsinkt. This is simple harmonic motion (see Section 17.3) with amplitude R, frequency~. and phase angle 0.<br />
The period is T = 21r j k. R Rj 3960 mi = 3960 · 5280 ft and g = 32 ft/s 2 , so k = v'i(R Rj 1.24 x 10- 3 s- 1 and<br />
T = 27r/k Rj 5079 s Rj 85 min.<br />
(d) y(t) = O # coskt = O # kt = !+1rnfor someintegern => y'(t) = - kRsin(f+7rn)=±kR.Thusthe<br />
particle passes through the center of the earth with ~pe<strong>ed</strong> kR Rj 4.899 mi/s Rj 17,600 mi/ h.<br />
© 2012 Ccngage L""ming. All Rights Resc" ·<strong>ed</strong>. May not be scann<strong>ed</strong>. copi<strong>ed</strong>. or duplicnlcd, or post<strong>ed</strong> lo a pubficly accessible wcbsilc, in whole or in p:lrt.
0 APPENDIX<br />
Appendix H Complex Numbers<br />
1. (5- 6i) + (3 + 2i) = (5 + 3) + ( -6 + 2)i = 8 + ( -4)i = 8 - 4i<br />
3. (2 + 5i)(4- i) = 2(4) +2(-i) + {5i)(4) + (5i)(- i) = 8 - 2i + 20i - 5i 2 = 8 + 18i- 5(-1)<br />
5. 12 + 7i = 12 - 7i<br />
= 8 + 18i + 5 = 13 + 18i<br />
1+4i 1 +4i 3 - 2i 3 - 2i+ 12i-8(-1) 11 + 10i 11 10 .<br />
7· 3 + 2i = 3 + 2i . 3 - 2i = 3 2 + 22 = 13 = 13 + 13 ~<br />
1 1 1-i 1-i 1- i 1 1 .<br />
9 -- = -- . - - = = -- = - - - t<br />
. 1 +i 1+i 1- i 1 -(-1) 2 2 2<br />
11 . i 3 = i 2 . i = ( - 1 )i = -i<br />
13. yC25 = J25 i = 5i<br />
15. 12 - 5i = 12 + 15i and 112- 15il = J12 2 + ( -5)2 = )144 + 25 = Jl69 = 13<br />
17. - 4i = 0 - 4i = 0 + 4i = 4i and l-4il = J0 2 + ( -4) 2 = VW = 4<br />
19. 4x 2 + 9 = 0 # 4x 2 = - 9 # X<br />
2- _Q<br />
- 4<br />
21. By the quadratic formula, x 2 + 2x + 5 = 0 # x =<br />
- 2 ± J2 2 - 4(1)(5) -2 ± v'-16 -2 ± 4i .<br />
2 ( 1 ) = 2<br />
· · -1 ± y'1 2 - 4(1)(2) - 1 ±A 1 · v'7<br />
= - 2 ± 2i.<br />
23. By the quadratic formula, z 2 + z + 2 = 0 # z =<br />
2 ( 1 ) = 2<br />
=<br />
2<br />
= -1 ± 2t.<br />
25. For z = -3 + 3i, r = J( -3) 2 + 3 2 = 3 V2 and tan 8 = ! 3<br />
= -1 => 8 = 3 ; (since z lies in the second quadrant).<br />
Therefore, - 3 + 3i = 3 J2 (cos 3 4 ,. + i sin 3 ;).<br />
27. For z = 3 + 4i, r = )3 2 + 4 2 = 5 and tan 8 = ~ => 9 = tan - l ( ~) (since z lies in the first quadrant). Therefore,<br />
3 +4i = 5[cos(tan- 1 ~) +isin(tan- 1 1)].<br />
29. For z = .J3 -H, r = J ( v'3) 2 + 1 2 = 2 and tan 9 = -ja => 8 = % => z = 2 (cos -if + i sin -if).<br />
Forw=l+v'3i, r=2andtanB=v'3 => 8 =~ => -w =2(cosi + isin~).<br />
Therefore, zw = 2 · 2 [cos ( ~ + ~) + i sin ( i + ~)] = 4 (cos i + i sin i),<br />
z / w = H cos ( i - i) + i sin (% - i)] = cos (- %) + i sin (-f), and 1 = 1 + Oi = 1 (cos 0 + i sin 0) =><br />
® 2012 Ccngnge Learning. All Rights Reserv<strong>ed</strong>. May not be SCDJm<strong>ed</strong>, copi<strong>ed</strong>, or duplicat<strong>ed</strong>, or posa<strong>ed</strong> loa publicly ac.cc.ssiblc website. in "hole or in p3r1.. 359
360 D APPENDIX H COMPLEX NUMBERS<br />
1/ z = ~ [cos(O- ~) + i si~(O- ~ )] = ~[cos( - i) + i sin(-~) ] . For 1/z; we could al ~o use the fonnula that prec<strong>ed</strong>es<br />
Example 5 to obtain 1/ z = H cos "if - i sin i).<br />
31. For z = 2¥'3- 2i, r = J(2 v'3) 2 + (-2) 2 = 4 and tan6 = '27a = -~ => 6 = -~ =><br />
z = 4(cos( --jr} + isin( -i)J. Forw = - 1 + i, r = -/2, tanB = ! 1<br />
= -1 => 6 = 3 ; =><br />
w = ..J2 (cos 3 4 " + isin 3 ;). Therefore, zw = 4 V2 [cos(-{f + 3 ;) + i sin(-~+ 3 ;)] = 4 ..J2 (cos i; + isin i;),<br />
z/w- 4 [cos(-2I- 3 ")+isin(-2I - 3<br />
71')] -<br />
4<br />
- 72 [cos(- 11 71')+isin(- 11 71' )] 2 f2(cos 13 "+isin 13 6 4 6 4 - 72 12 12 - v ~ 12 . . 12<br />
")<br />
'<br />
and<br />
1/z = i(cos(-{f) - isin(-{f)] = Hcos "if+ isin"lf).<br />
33. For z = 1 + i, r = ..j2 and tan B = t = 1 => B = "i => z = ..j2 (cos "i + isin f). So by De Moivre's Tlworem,<br />
(1 +_i) 20 = [..J2 (cos "i +isinf)J 20 = (2 1 1 2 ? 0 (cos 20 ~71' + isin 20 ~") = 2 10 (cos51l' + isin51l')<br />
= 2 10 [-1 +i(O)] = -2 10 = - 1024<br />
So by De Moivre's Theorem,<br />
(2 v'3 + 2i) 5 = [4(cos "if+ isin ~)) 5 = 4 5 (cos 5 6 "+ i sin ~G11') = 1024[ -4 + ~i] = -512 v'3 + 512i.<br />
37. 1 = 1 + Oi = 1 (cosO+ isinG). Using Equation 3 with 7' = 1, n = 8, and 6 = 0, we have<br />
Wk- 1<br />
- 1/8[ (0+2k1l') .. (0+2k1l')J- k1l' .. k1l' · -<br />
cos<br />
8<br />
+ ~sm<br />
8<br />
-cos 4<br />
+ ~sm<br />
4<br />
, where k- 0, 1, 2, ... , 7.<br />
w 0 = 1(cosO+isinO) = 1, w1 = 1(cosf +isinf) = "7z + 72i,<br />
w2 = 1 (cos ~ + i sin ~) = i, w 3 = 1 (cos 3 ; + i sin a;.) = - "7z + 72 i,<br />
•<br />
1m<br />
•<br />
w4 = 1(cos1l'+isinir) = -1, w5 = 1(cos 5 ;<br />
+isin 5 ;) = -72 -72i,<br />
0<br />
Re<br />
w 6 = 1(cos a;+ isin a;) = ...:i, w 7 = 1(cos 7 ; + isin 7 ;) = 72 - 72i<br />
•<br />
•<br />
39. i = 0 + i = 1 (cos~ + i sin~). Using Equation 3 with r = 1, n = 3, and 6 = %, we have<br />
Wk = 1 1 / 3 COS<br />
[ (<br />
J!. 2 + 2k1l') + i sin ( ~ - + q 2k1l')] , where ' k = 0, 1, 2.<br />
3<br />
wo = (cos i + i sin i) = 4 + ~i<br />
WI = (cos· r,,. + i sin 5 ") - - v'3 + l i<br />
6 6 - 2 2<br />
w2 = ,(cos 0 6 " + i sin 9 ;) == - i<br />
1m<br />
•<br />
0<br />
-i<br />
•<br />
Re<br />
41 . Using Euler'sfonnula (6) withy=~. we have ei71'/ 2 =cos~+ isin ~ = 0 + 1i = i.<br />
© 2012 Cengnge Lenming. All Rights Reserve~ . May not be: scanncJ, copi
APPENDIX H COMPLEX NUMBERS 0 361<br />
43. Using Euler's formula (6) with y= i· we have ei"/ 3 = cos i + isin i = ~ + '!] i.<br />
47. Taker = 1 and n = 3 in De Moivre'sTheorem to get<br />
(l(cos (} + i sin IJ)] 3 = 1 3 (cos 39 + i sin31J)<br />
(cos IJ + isin 0) 3 = cos 31J + i sin30<br />
cos 3 (} + 3(cos 2 fJ)(i sin 0) + 3(cos IJ)(isln 1J? + (i sin IJ) 3 =cos 38 + i s_in3fJ<br />
cos 3 1J + (3 cos 2 f) sin fJ)·i - 3 cos IJ sin 2 ()- (sin 3 fJ)i =cos 31J + i sin 31J<br />
( cos 3 1J - 3 sin 2 f) cos IJ) + (3 sin f) cos 2 f) - sin 3 1J)i = cos 30 + i ~in 3(}<br />
Equating real and imaginary parts gives cos 30 = cos~ fJ - 3 sin 2 () cos IJ and sin 3B = 3 sin IJ cos 2 (} - sin 3 e.<br />
49. F (x) = er:r: = e<br />
F'(x) = (eax cosbx)' + i(e