Exercicios resolvidos James Stewart vol. 2 7ª ed - ingles

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(p+r)2 SECTION 14o5 THE CHAIN RULE D 209 l 19o w / I~ r s r /\ /\ 1\ X J X y X J 210 z = x 4 + x 2 y, x = s + 2t - u, y = stu 2 => w = f(r, s, t), r = r(x, y), s = s(x, y), t = t(x, y) => ow ow or ow as ow at aw aw ar aw as ow at fu = &ofu+asax+8tax·~=aroy+asay + Bt~ oz = az ox+ az.ay = ( 4 xa + 2xy)( 1 ) + (x2)(tu2), OS OX OS ay as ' 0 oz = oz ax+ az ay = (4x3 + 2xy)(2) + (x2)(su2) m &m ~m · oz = az ax+ az ay = (4x3 + 2xy)( - 1} + (x2)(2stu). au ax au ay a·u - When s = 4, t = 2, and u = 1 we have x = 7 and y = 8, so az = (1484)(1} + {49)(2} = 1582, aaz = {1484) (2) + (49){4) = 3164, aaz = (1484)(-1) + {49)(16} = -700. as t u 23o w = xy + yz + zx, x = r cos 0, y = r sin 8, z = r8 => aw = awax + aawaay + aawaaz ~(y +z}(cos8} + (x+z)(sin8}+(y+x) (O), . ar axar yr zr aw = aw ax+ aaw aay + aaw aaz8 = (y + z)(- rsinO) + (~ + z )(rcosO) + (y + x}(r·)o ao ox o8 . y 8 z aw 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 ~~ = (2 + 1r)( - 2) + (0 + 1r)(O) + (2 + 0)(2) = -2rro .. 25o N = p + q, p = u + vw, q = v + uw, r = w + uv =9 p+r aN = aN ap + aN aq + aN ar. au ap au aq au 8r au ' = (p + r){1)- (p + q){1) ( 1 ) + (p + r)(1) - (p + q)(O) (w) + (p + r)(O) - (p + q)(1) (v) (p+r)2 (p + r)2 (p + r)2 = (r- q) + (p+r)w - (p+q)v (p+r)2 oN = aN ap + aN oq + aN or = r - q (w) + p + ro ( 1 ) + -(p + q) (u) = (r - q)w + (p + r} - (p + q)u' av op ov 8q 8v or ov 0 (p +r) 2 . (p+1')2 (p+ r)2 aN _ aN ap + 8N aq + 8N !!!.._ = ro - q (v) + p+r (u)+ -(p +q) ( 1 ) = (r- q)v +(p+r)u- (p + q) ow - ap ow aq ow or OW (p + r )2 (p + r) 2 (p + r) 2 (p + r )2 · aN -1 + (24)(4) - (25)(3) 20 5 When u = 2, v = 3, and w = 4 we have p = 14, q = 11, and r = ~0, so ou = . ( 24 )2 . = 576 = 144 , aN _ (-1)(4)+24-(25)(2) _ -30 __ ~ d 8N = · (- 1 )(3) + (24)(2)-25 =~ = ~ av - (24)2 - 576 - 96' an aw (24)2 576 144 ° © 2012 Ceng;~ gc LCJming. AJI Rights Reserved. Mn)' nol be scanned. co ph.-d, or duplicated, or posted lo o publicly accessible websitt:, in whole or in p;~rt .
210 0 CHAPTER 14 PARTIAL DERIVATIVES 27. y cosx = x 2 + y 2 , so let F(x, y) = ycosx - x 2 - y 2 = 0. Then by Equation 6 dy dx -y sin x - 2x 2x + y sin x = --:-----'"---- cos x - 2y cosx- 2y . and [2xy - (1 + y 2 ) (1 + x-iy 2 )]/(1 + x 4 y 2 ) _ (1 + y 2 )(1 + x 4 y 2 ) - 2xy (x2 - 2xy(1 + x4y 2 ))/(1 + x4y 2 ) - x 2 - 2xy(1+ x4y2) - 1 + x4y2 + y2 + x4y4 - 2xy - x 2 - 2xy - 2x 5 y3 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 f)z = .:_ Fx = _ 2x = -~ and oz = _ Fy = _ 4y = _ 2y ox Fz 6z 3z oy Fz QZ 3z . _ • oz F. . -yz yz 33. e· = x yz, so let F(x, y, z) = e· - xyz = 0. Then~ = - -F = --.-- = --- and . ux 'z e· - xy ez - xy 8z oy F 11 - xz xz - Fz = --e=---x-y = -ez ___ x_y 35. Since x andy are eac~ functions oft, T(x, y) is a function oft, so by the Chain Rule, ~ = ~: + ~~~;.After dx 1 3 seconds, x = .Jl+t = .,;r+3 = 2, y = 2 + ~ t = 2 + H 3) = 3, -d = .J1+t t 2 1 + t 1 -.!. and dy -.!. 2 v'1 + 3 - 4, dt - 3 . Then~~ = Tx(2, 3) ~~ + Ty(2, 3) ~; = 4(t) + 3(~ ) = 2. Thus the temperature is rising at a rate of2°C/ s. ac ac . 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. According to the graph, the diver is experiencing a temperature of approximately 12.5°C at t = 20 minutes, so . 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 dD 1 dT 1 . dC 80 dT 80 dD 1 1 . dt ~ '2 and (jj ~ - 10 . Then, by the Cham Rule, (jj = &T (jj + &Ddt ~ (3.36) (-Iii) + (0.016) ( 2) r::::: -0.33. Thus the speed of sound experienced by the diver is decreasing at a rate of approximately 0.33 m/s per minute. 39. (a) V = l!wh, so by the Chain Rule, 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. dt 81! dt aw dt &h dt dt dt dt © 2012 Ccngagc Learning. All Rights RescrYcd. May not be scanned. copied, or duplicated. or, posted toP publicly accessible website, in whole or in part.
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- STUDENT SOLUTIONS MANUAL for STEW
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.. BROOKS/COLE ~ I ~~r CENGAGE Lear
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D ABBREVIATIONS AND SYMBOLS CD cu D
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viii o CONTENTS 12.4 The Cross Prod
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10 D PARAMETRIC EQUATIONS AND POLAR
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SECTION 10.1 CURVES DEFINED BY PARA
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SECTION 10.1 CURVES DEFINED BY PARA
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SECTION 10.2 CALCULUS WITH PARAMETR
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SECTION 10.3 POLAR COORDINATES 0 13
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SECTION 10 .~ POLAR COORDINATES 0 1
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SECTION 10.4 A~~S AND LENGTHS IN PO
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SECTION 10.4 AREAS AND LENGTHS IN P
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SECTION 10.4 AREAS AND LENGTHS IN P
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SECTION 10.5 CONIC SECTIONS 0 27 5.
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SECTION 10.5 CONIC SECTIONS 0 29 35
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x2 y2 y2 a:2 _ a2 b 61. ;_2 - - = 1
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SECTION 10.6 CONIC SECTIONS IN POLA
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CHAPTER 10 REVIEW 0 35 the length o
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CHAPTER 10 REVIEW 0 37 EXERCISES 1.
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CHAPTER 10 REVIEW 0 39 25. x = t +
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CHAPTER 10 REVIEW 0 41 2 2 . 45. ~
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0 PROBLEMS PLUS l lt sin u dx cost
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11 . D INFINITE SEQUENCES AND SERIE
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SECTION 11.1 SEQUENCES 0 47 35. a,.
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SECTION 11.1 SEQUENCES D 49 71. Sin
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SECTION 11.2 SERIES 0 51 ak+l- a1.:
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SECTION 11.2 SERIES 0 53 13. n s.,
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SECTION 11.2 SERIES 0 55 47. F~r th
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. . (1+ c)- 2 scn es and set 1t equ
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SECTION 11.3 THE INTEGRAL TEST AND
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, SECTION 11.3 THE INTEGRAL TEST AN
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SECTION 11.4 THE COMPARISON TESTS D
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SECTION 11.5 ALTERNATING SERIES 0 6
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SECTION 11.5 ALTERNATING SERIES D 6
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SECTION 11.6 ABSOLUTE CONVERGENCE A
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SECTION 11.6 ABSOLUTE CONVERGENCE A
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17 lim I an+l I= SECTION 11.7 STRAT
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SECTION 11.8 POWER SERIES 0 75 l 00
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SECTION 11.8 POWER SERIES 0 n (b) I
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SECTION 11.9 REPRESENTATIONS OF FUN
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SECTION 11.9 REPRESENTATIONS OF FUN
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SECTION 11.10 TAYLOR AND MACLAURIN
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61 _ x_ x · sin x - x- tx 3 + 1~
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SECTION 11.11 APPLICATIONS OF TAYLO
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CHAPTER 11 REVIEW 0 97 J'(xn)(xn -
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CHAPTER 11 REVIEW 0 99 oo f (n) (O)
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CHAPTER 11 REVIEW 0 101 l 23. Consi
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49./- 1 - dx = -ln{4- x) + C and 4-
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D PROBLEMS PLUS 1. It would be far
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CHAPTER 11 PROBLEMS PLUS D 107 l 9.
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CHAPTER 11 PROBLEMS PLUS 0 109 x x
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112 0 CHAPTER 12 VECTORS AND THE GE
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120 D CHAPTER 12 VECTORS AND THE GE
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136 D CHA~TER 12 VECTORS AND THE GE
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D PROBLEMS PLUS 1. Since three-dime
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l CHAPTER 12 PROBLEMS PLUS 0 149 Eq
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13 D VECTOR FUNCTIONS 13.1 Vector F
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SECTION 13.1 VECTOR FUNCTIONS AND S
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SECTION 13.1 VECTOR FUNCTIONS AND S
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SECTION 15.5 APPLICATIONS OF DOUBLE
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SECTION 15.5 APPLICATIONS OF DOUBLE
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SECTION 1505 APPLICATIONS OF DOUBLE
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-I SECTION 15.6 SURFACE AREA 0 267
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SECTION 15.7 TRIPLE INTEGRALS 0 269
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SECTION 15.7 TRIPLE INTEGRALS D 271
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Therefore E = { (x, y, z) I -2 ~ X~
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43. I,. = foL foL foL k(y2 + z2)dz.
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SECTION 15.8 TRIPLE INTEGRALS IN CY
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M xv = I~1f I: I:2 6 - 3 r 2 (zK) r
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SECTION 15.9 TRIPLE INTEGRALS IN SP
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SECTION 15.9 TRIPLE INTEGRALS IN SP
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(b) The wedge in question is the sh
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SECTION 15.10 CHANGE OF VARIABLES I
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CHAPTER 15 REVIEW 0 289 15 Review C
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CHAPTER 15 REVIEW 0 291 l 9. The vo
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CHAPTER 15 REVIEW D 293 33. Using t
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49. Since u = x- y and v = x + y, x
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298 D CHAPTER 15 PROBLEMS PLUS To e
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300 0 CHAPTER 15 PROBLEMS PLUS 13.
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16 0 VECTOR CALCULUS 16.1 Vector Fi
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SECTION 16.2 LINE INTEGRALS 0 305 2
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SECTION 16.2 LINE INTEGRALS D 309 3
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SECTION 16.3 THE FUNDAMENTAL THEORE
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SECTION 16.4 GREEN'S THEOREM D 313
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SECTION 16.4 GREEN'S THEOREM 0 315
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8 8 8 (b)clivF = 'V ·iF=- (x+yz) +
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SECTION 16.5 CURL AND DIVERGENCE 0
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SECTION 16.6 PARAMETRIC SURFACES AN
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that is, D = {( x, y) I x 2 + y 2 :
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SECTION 16.7 SURFACE INTEGRALS 0 32
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SECTION 16.8 STOKES' THEOREM 0 333
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dS SECTION 16.9 THE DIVERGENCE THEO
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CHAPTER 16 REVIEW 0 337 27. JI 5 cu
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CHAPTER 16 REVIEW 0 339 TRUE-FALSE
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CHAPTER 16 REVIEW D 341 Alternate s
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344 0 CHAPTER 16 PROBLEMS PLUS Simi
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3.c6 0 CHAPTER 17 SECOND-ORDER DIFF
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348 0 CHAPTER 17 SECOND-ORDER DIFFE
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352 D CHAPTER 17 SECOND-ORDER DIFFE
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356 D CHAPTER 17 SECOND-ORDER DIFFE
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0 APPENDIX Appendix H Complex Numbe
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APPENDIX H COMPLEX NUMBERS 0 361 43
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Exercicios resolvidos James Stewart vol. 2 7ª ed - ingles

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