Calculo 2 De dos variables_9na Edición - Ron Larson

SECCIÓN 13.6 13.6 13.6 Directional Derivadas Derivatives direccionales and and y Gradients gradientes 943
943
13.6 Directional Derivatives and Gradients 943
Investigación
Investigation In En In Exercises los ejercicios 47 47 and and 47 y 48, 48, (a) (a) use utilizar use the the la graph gráfica to
to
para estimate Investigation estimar the the las components In componentes Exercises of of the the 47 del and vector 48, in in en the (a) the la use direction dirección the graph of of de the
the la to
tasa maximum estimate máxima rate the rate de components of of incremento increase of the en the la function vector función at in at the the en el direction given punto point. dado. of (b)
(b) the
Find b) Find maximum Hallar the the el gradiente rate of increase at at en the the el punto point the y and function and compararlo compare at the con it given it with el with estimado point. your
your (b)
del estimate Find inciso the in in a). part gradient part c) (a). ¿En (a). (c) (c) qué at In In the what dirección what point direction decrece and compare would más the the rápido it function with la función?
decreasing estimate Explicar. in at at part the the (a). greatest (c) In rate? what Explain. direction would the function be
your be
be
1
47.
decreasing fx, y 10x 1
f y
at 1
10 x
the 2 greatest
3xy y
rate? 2 , , 48.
Explain.
fx, fx, yy 2y 47. 2 yx,
1
f x, y
x 2 3xy y 2 , 48. fx, y 2y x,
1
1
47. 1, f1, 22
x, 2y
10 x 2 3xy y 2 , 48. 1, 1, fx, 22
2 y 2y x,
1, 2 z
1, z 2
3
z
z
z
1
1
1
3
3
1 2
1
1
3
3 y
3
3
y
x
x 3
1
2
2
3
x
2
3
3
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CAS
CAS49. 49. Investigación Investigation Considerar the the la function función
CAS 49. fx, Investigation
fx, yy x 2 y 2 Consider the function
fx, y x 2 y 2
at
fx,
en the
y
el punto point
x
4, 2 y
3, 2 1053714_1306.qxp at the point 10/27/08 4, 3, 77.
712:08 .
. PM Page 943
(a) (a) at Utilizar Use the Use point a a un computer sistema 4, 3, 7 algebraico .
por system computadora to to graph para the the dibujar surface la
(a) superficie represented Use a dada computer by by por the the esa function. algebra función. system to graph the surface
(b) (b) Determinar Determine represented la the the by derivada the directional function. direccional derivative D u f4, D u f3
4, como 3 as función
function Determine de , of of donde ,
, the where udirectional u ucos cos cos i derivative
i isen
sin sen sen j. Utilizar j.
j. DUse a u
a
u f 4, 3
as a
(b) Use f 4, a un computer 3sistema
as a
algebraico function system por of computadora , to where to graph uthe the para cos function representar i on on the the j.
interval gráficamente Use a computer 0, 0, 2 2 . la
.
(c) (c)
función Approximate algebra en system el intervalo the the to zeros graph zeros
0, of the 2.
of the function the function the in in interval part part (b) (b) 0, and
2 and.
c) (c) Aproximar interpret Approximate each each los in ceros in the the de zeros context la función of of of the the the del function problem. inciso in b) e part interpretar (b) and
(d) (d)
cada Approximate interpret uno en each el the contexto the in critical the context del numbers
problema. of the of of the problem.
the function in in part part (b)
(b)
d) Aproximar and los each números in the críticos de of la the función Investigation (d) and Approximate interpret In each Exercises the critical in the 47 context numbers and of 48, of the the (a) problem. function use del the inciso graph part b) (b) e to
estimate (e) (e)
interpretar Find
Find and the interpret f fcomponents 4, cada 4, 3each uno 3 and in and the el of contexto
explain the vector its its of del the in relationship
problema. problem. the direction to to your
your of the
maximum e) (e) Hallar answers Find f4, rate in in fpart of 4, part 3 increase (d).
(d). 3 y explicar and in explain the su function relación its relationship con the las given respuestas to point. your (b)
Find (f) (f)
del Use Use answers the inciso a a gradient computer
d). in part at (d).
algebra
the point system
and to to graph
compare the the level level
it with curve
your
estimate f )(f) Utilizar of of Use the the
in a part function computer sistema (a). f
(c)
fat algebraico at the In the
what level level system por direction c ccomputadora 7. to 7. On graph On
would this this the para the curve, level representar
the the of gráficamente the vector
function graph curve
be
decreasing at
function
the in in
greatest the la the fcurva at direction the
rate? de level nivel Explain. of
of c de f f 4, 7.
4, función On 3,
3, this and and fcurve, en state state el nivel graph its
its
c 7. En esta 1 to curva, the level representar 47. 48.
gráficamente 1
f relationship x, the y vector fx, y
el vector en la
10 x 2 to in the 3xy level direction y curve. 2 , of f 4, 3, and
50. dirección de
f 4, 3, the y establecer su relación
2y x, state its
50. Investigation relationship con la curva
de
1,
nivel.
2
Consider to the level the curve.
function 1, 2
50. Investigation8y
Consider the function
50. fx, Investigación y
z
8y
Considerar la función
1 x 2 y 2. z
fx, y
1 x 8y 2 y
fx, y
2. 1 8y
(a) fx, y
1 x 2
1 x 2 2 y y 2 that 2. (a) Analytically verify that the the level level curve of
of fx, fx, y
yat at the the level
level
(a) c cAnalytically 2
2is is a a circle. verify that the level curve 1 of fx, y at the level
(b) a) (b)
Verificar At At c the the 2point
is point
analíticamente a circle. 3, 3, 2
2 on on
que the the
la level curva de nivel for de f(x, cy) para 2,
el nivel the c = 2 es un círculo. 3 y
level curve for which
c 2,
the 1 of the rate
y
3(b) sketch At the the point vector 3, showing 2 on the level direction curve of for the which greatest c rate 2,
3
b) En of of sketch el increase punto the 3, of vector of the the 2
showing sobre function. la curva (To the (To direction print print de nivel an an of para enlarged the greatest cual copy copy c = of rate
of 2,
x
dibujar the el vector go the que apunta en dirección
2
the of increase graph, go of to the function. website www.mathgraphs.com.)
(To print an de enlarged mayor copy tasa ofo
(c) (c)
ritmo At At the the the
de graph, point
incremento
point go 3, 3, the 2
de
2website on la on the función.
the level www.mathgraphs.com.)
level 3
curve, sketch a a vector
c) (c) En such such At el that punto the that the point
the 3, directional Generated 23, sobre 2by Maple
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CAS
(d) (d)
cuya Use Use
derivada a a computer
direccional algebra
sea system
0.
CAS such that the directional derivative to to is graph 0. the the surface to
to
CAS CAS49. d) (d) Utilizar Investigation verify Use your a your un computer sistema answers Consider algebraico in in parts parts the system function
(a)–(c). por computadora to graph the para surface representar
gráficamente la superficie y verificar las respuestas a
to
fx,
verify
y
your
x
los incisos a) 2 answers
y
a c).
2 in parts (a)–(c).
at the point 4, 3, 7 .
(a) Use a computer algebra system to graph the surface
represented by the function.
(b) Determine the directional derivative D f 4, 3 as a
y
y
1
1
z
z
z
2
2
y
y
In En In Exercises los ejercicios 51–54, a find find 54, a hallar a normal un vector normal to to the the a level level la curva curve de
f nivel fx, In x, y
Exercises yfx, c
cat
y at P. P. 51–54, c en P. find a normal vector to the level curve
51.
f x,
fx,
y
y
c at
6
P.
51. f x, y 6 2x 2x 3y
3y 52.
52. fx, f x, y y x 2 x 2 y y 2
2
51. c cfx, 6,
6, y P P0, 6 0, 0
02x 3y 52. c cfx, 25,
25, y P Px3, 2 3, 4
4y 2
c 6, P 0, 0
c 25, Px
53. f x, y xy
54. f x, y
x3, 4
53. f x, y
xy
54.
f x, y x 2 x 2 xy y 2
53. c c fx, y
3, 3, P xy
P 1, 1, 3
3
54. fx, 1
c 2 , y
2
1
c P 1, 1
2 ,
Px 1, 2 1 y 2
In En In Exercises los c ejercicios 3, 55–58, P (a) 1, a (a) 58, 3
1
c 2
find find a) the encontrar the gradient el gradiente of of the the , P 1, 1
function de la función at
at P,
P,
(b) en (b) In find P, find Exercises b) a a encontrar unit unit 55–58, normal un (a) vector find to to the normal the the gradient level level unitario curve of the fpara fx, x, function y
yla curva c
cat
at P,
de P,
(c) nivel (c) (b) find find f(x, the a the y) unit = tangent c normal P, line c) line encontrar vector the the to level the la level recta level tangente curve f fx, f x, yx, ya yla ccurva cat
c
at at P,
de P,
and nivel and (c)(d) find (d) f(x, sketch y) the = tangent cthe the level P, level y line d) curve, trazar to the la unit level curva unit normal curve de nivel, f vector, x, el yvector and and c at the uni-
the P,
tario tangent and (d) normal line line sketch in in y the
la the xy- recta xy- level plane. tangente curve, the en el unit plano normal xy. vector, and the
55.
tangent
fx, y
line
4x
in 2 the
y
xy-plane.
55. fx, f y 4x 2 y
56.
56. fx, fx, f y y x x y 2
y 2
55. c cfx, 6, 6, yP P2, 2, 4x 10
10
2 y
56. c cfx, 3, 3, yP P4, 4, x 1
1y 2
57. 57. fx, fx, fc y y6, P3x 3x 2, 2 2 102y 2y 2 2 58.
58. fx, fx, fc y y3, P9x 9x 4, 2 2 14y 4y 2
2
57. c cfx, 1, 1, yP P1, 1, 3x 1
1
2 2y 2 58. c cfx, 40, 40, y P P2, 9x 2, 2 1
1 4y 2
c 1, P 1, 1
c 40, P 2, 1
WRITING ABOUT CONCEPTS
Desarrollo de conceptos
59. 59. WRITING Define the the ABOUT derivative CONCEPTS of of the the function z
z f fx, x, y
y in in the
the
59. 59. Definir direction Define la uthe uderivada derivative cos cos i i la sen of sen función the j.
j. function z f x, zy
en f la x, ydirección
in the
60. de
60. Write direction u a cos a paragraph ui cos sen sin ij.
describing sen j. the the directional derivative
60. 60. of Redactar of Write the the function a paragraph párrafo f
fin in the the que describing direction describa u uthe la derivada cos cos directional i i direccional sen sen derivative j
j
when
(a) la (a) of función the 0function 0and fand en (b)
(b) la f in dirección the 90 90 direction .
. de u u cos cos i i sen sin sen j cuando j when
61. a)
61. Define (a) the the 0
y gradient and b) (b) of of a a function 90 . of of two two variables. State State the
the
61. 61. Definir properties Define el the of gradiente of the the gradient. de of una a function función of de two dos variables. variables. State Dar the las
62. propiedades
62. Sketch properties the the of del graph the gradiente. of gradient.
of a a surface and and select a a point point P
Pon on the
the
62. 62. Dibujar surface. Sketch la the Sketch gráfica graph a a de vector of una a in surface in superficie the
the xy-
xy- and plane y select elegir giving un a point the punto the direction PP sobre the
of la of surface. superficie. steepest 13.6 Sketch ascent Directional
Dibujar a on vector the un the Derivatives
vector in surface the en xy- at
P. el plane P. plano
and giving Gradients
xy que the indique direction la
943
63. dirección
63. Describe of steepest the de the ascent mayor relationship ascenso the of of surface the sobre the gradient at la P. superficie to the level en P.
to the level curves
63. 63. of Describir of Describe a a surface la the relación given relationship by by zdel z gradiente f fof x, x, the y y .
. gradient con las to curvas the level de nivel curves de
In una Exercises of a superficie surface 51–54, given dada by find por zza fnormal fx, x, y y. . vector to the level curve
f x, y c at P.
CAPSTONE
51. fx, y 6 2x 3y 52. fx, y x 2 y 2
64. Para 64. CAPSTONE
discusión
Consider the fx, y 9 x 2 y 2 .
c 6,
the
P 0, function
0
fx, y 9 x 2 c y 25,
2 .
P 3, 4
64. (a) Considerar (a) Sketch the la the función function
graph of
of f fx, fin yin the the 9first first x 2 octant y 2 and .
and plot plot the
x
the
53. a) (a) fx, Trazar point
point Sketch y 1, la xy 1, 2, gráfica the 2, 4
4graph on on the de the of f en surface. f in el the primer 54. first octante fx, y y and
x
graficar 2 plot
y 2 the el
(b) (b) c punto Find Find point
3, (1, D1, u 2, P f2, 1, 2 ,
u cos i
1
sen j, for
u f4) 1, 4
4.
1, sobre 2 on 3
, the la where superficie. surface.
u cos i
c 2 ,
sen j,
for
P 1, 1
b) (b) Encontrar Find DD u 4. f u f1, 22,
, where donde u cos
i sen sin j. j, para for
In (c) Exercises (c) q Repeat = –p/4. part 55–58, part 4. (b) (b) for (a) for find the 3.
3. gradient of the function at P,
(b) (d) c) (d) find (c) Repetir Find
Find Repeat a unit fel f1, normal part inciso 1, 2
2and
(b) and b) for vector para f f1, 1, q 2 to = 2 . the p/3.
. level curve f x, y c at P,
(c)
(e) d)
find
Find
the
a unit
tangent line
u
to the level
to f
curve
1, 2 and
f x, y c at P,
(e) (d) Encontrar Find Find
a unit f f1, vector 2 2 and y
uf1, orthogonal f 22.
.
to
f 1, 2
and calculate
and (d) sketch
u f 1, 2 .
the level
the
curve, the unit normal
of
vector,
the and the
D tangent
e) (e) Encontrar Find
line u f 1, a unit 2 . Discuss the geometric meaning of the result.
in the
vector u
xy-plane.
unitario orthogonal u ortogonal f 1, 2para and f1, calculate 2 y
calcular D u f 1, D2 u f1, . Discuss 2. Discutir the geometric el significado meaning geométrico of the result. del
55. fx, resultado. y 4x 2 y
56. fx, y x y 2
65. 65. Temperature The at the point x, y
c 6, P 2, Distribution 10
The temperature c 3, at Pthe 4, point 1
x, y
65. on on Temperature a a metal plate plate Distribution The temperature at the point x, y
57. fx, y 3x 58. fx, y 9x
65. Distribución
x
de 2 2y
temperatura 2 La temperatura en el punto 2 4y
on a metal plate is
(x, 2
y)
T
de cuna x 2 placa 1, x
T P
y 2. 1, metálica 1 es
c 40, P 2, 1
x 2 x
y
T
2. x
Find TWRITING the x 2
x 2 y
y 2. 2. ABOUT of CONCEPTS
Find the direction of greatest increase in in heat heat from from the the point
point
3, 59. Find 3, 4 4 . Define . the direction the derivative of greatest of the increase function in heat z from f x, ythe in point the
Hallar
direction
la dirección
u cos
de mayor
i sen
incremento
j.
de calor en el punto
3, 4 .
(3, 4).
60. Write a paragraph describing the directional derivative
of the function f in the direction u cos i sen j when
(a) 0 and (b) 90 .
0
90.
61. Define the gradient of a function of two variables. State the