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CAPÍTULO 13

Page 908

Funciones de varias variables

13.3 Derivadas parciales n n n

Hallar y utilizar las derivadas parciales de una función de dos variables. Hallar y utilizar las derivadas parciales de una función de tres o más variables. Hallar derivadas parciales de orden superior de una función de dos o tres variables.

Mary Evans Picture Library

Derivadas parciales de una función de dos variables

JEAN LE ROND D’ALEMBERT (1717-1783) La introducción de las derivadas parciales ocurrió años después del trabajo sobre el cálculo de Newton y Leibniz. Entre 1730 y 1760, Leonhard Euler y Jean Le Rond d’Alembert publicaron por separado varios artículos sobre dinámica en los cuales establecieron gran parte de la teoría de las derivadas parciales. Estos artículos utilizaban funciones de dos o más variables para estudiar problemas de equilibrio, movimiento de fluidos y cuerdas vibrantes.

En aplicaciones de funciones de varias variables suele surgir la pregunta: ¿“Cómo afectaría al valor de una función un cambio en una de sus variables independientes”? Se puede contestar esta pregunta considerando cada una de las variables independientes por separado. Por ejemplo, para determinar el efecto de un catalizador en un experimento, un químico podría repetir el experimento varias veces usando cantidades distintas de catalizador, mientras mantiene constantes las otras variables como temperatura y presión. Para determinar la velocidad o la razón de cambio de una función f respecto a una de sus variables independientes se puede utilizar un procedimiento similar. A este proceso se le llama derivación parcial y el resultado se llama derivada parcial de f con respecto a la variable independiente elegida. DEFINICIÓN DE LAS DERIVADAS PARCIALES DE UNA FUNCIÓN DE DOS VARIABLES Si z 5 f sx, yd, las primeras derivadas parciales de f con respecto a x y y son las funciones fx y fy definidas por f sx 1 Dx, yd 2 f sx, yd Dx f sx, y 1 Dyd 2 f sx, yd fy sx, yd 5 lím lim Dy→0 Dy fxsx, yd 5 lím lim

Dx→0

siempre y cuando el límite exista. Esta definición indica que si z 5 f sx, yd, entonces para hallar fx se considera y constante y se deriva con respecto a x. De manera similar, para calcular fy , se considera x constante y se deriva con respecto a y. EJEMPLO 1

Hallar las derivadas parciales

Hallar las derivadas parciales fx y fy de la función f sx, yd 5 3x 2 x 2y 2 1 2x 3y. Solución

Si se considera y como constante y se deriva con respecto a x se obtiene

f sx, yd 5 3x 2 x 2y 2 1 2x 3y

Escribir la función original.

fxsx, yd 5 3 2

Derivada parcial con respecto a x.

2xy 2

1

6x 2y.

Si se considera x constante y se deriva con respecto a y obtenemos f sx, yd 5 3x 2 x 2y 2 1 2x 3y

Escribir la función original.

fysx, yd 5

Derivada parcial con respecto a y.

22x 2y

1

2x 3.

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SECCIÓN 13.3

Derivadas parciales

909

NOTACIÓN PARA LAS PRIMERAS DERIVADAS PARCIALES Si z 5 f sx, yd, las derivadas parciales fx y fy se denotan por ­ ­z f sx, yd 5 fxsx, yd 5 z x 5 ­x ­x y ­ ­z f sx, yd 5 fysx, yd 5 z y 5 . ­y ­y Las primeras derivadas parciales evaluadas en el punto sa, bd se denotan por ­z ­x

|

sa, bd

EJEMPLO 2

5 fxsa, bd

­z ­y

y

|

sa, bd

5 fysa, bd.

Hallar y evaluar las derivadas parciales

Dada f sx, yd 5 xe x y, hallar fx y fy, y evaluar cada una en el punto s1, ln 2d. 2

Solución z

Como

fxsx, yd 5 xe x ys2xyd 1 e x

(x0, y0, z0)

2

2y

Derivada parcial con respecto a x.

la derivada parcial de f con respecto a x en s1, ln 2d es fxs1, ln 2d 5 e ln 2s2 ln 2d 1 e ln 2 5 4 ln 2 1 2. Como fysx, yd 5 xe x ysx 2d 2

5 x3ex y 2

y

x

la derivada parcial de f con respecto a y en s1, ln 2d es

Plano: y = y0

­f 5 pendiente en la dirección x ­x

fys1, ln 2d 5 e ln 2 5 2.

Figura 13.29

z

Derivada parcial con respecto a y.

Las derivadas parciales de una función de dos variables, z 5 f sx, yd, tienen una interpretación geométrica útil. Si y 5 y0, entonces z 5 f sx, y0d representan la curva intersección de la superficie z 5 f sx, yd con el plano y 5 y0, como se muestra en la figura 13.29. Por consiguiente,

(x0, y0, z0)

fxsx0, y0d 5 lím lim

Dx→0

f sx0 1 Dx, y0d 2 f sx0, y0d Dx

representa la pendiente de esta curva en el punto sx0, y0, f sx0, y0 dd. Nótese que tanto la curva como la recta tangente se encuentran en el plano y 5 y0. Análogamente, fysx0, y0d 5 lim lím y

x

Plano: x = x0

­f 5 pendiente en la dirección y ­y Figura 13.30

Dy→0

f sx0, y0 1 Dyd 2 f sx0, y0d Dy

representa la pendiente de la curva dada por la intersección de z 5 f sx, yd y el plano x 5 x0 en sx0, y0, f sx0, y0dd, como se muestra en la figura 13.30. Informalmente, los valores ­fy­x y ­fy­y en sx0, y0, z0d denotan las pendientes de la superficie en las direcciones de x y y, respectivamente.

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CAPÍTULO 13

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Funciones de varias variables

Hallar las pendientes de una superficie en las direcciones de x y de y

EJEMPLO 3

Hallar las pendientes en las direcciones de x y de y de la superficie dada por f sx, yd 5 2

x2 25 2 y2 1 2 8

1 en el punto s 2, 1, 2d.

Solución

Las derivadas parciales de f con respecto a x y a y son

fxsx, yd 5 2x

fysx, yd 5 22y.

y

Derivadas parciales.

Por tanto, en la dirección de x, la pendiente es fx

112, 12 5 2 21

Figura 13.31a.

y en la dirección de y, la pendiente es fy

112, 12 5 22.

Figura 13.31b.

z

z

Superficie: 4

2 f(x, y) = − x − y2 + 25 2 8

(

( 12 , 1, 2 )

)

1 , 1, 2 2

y

2 3

2

Pendiente en la dirección de x: fx 1 , 1 = − 1 2 2

x

4

( )

a)

3

y

Pendiente en la dirección y:

x

( )

fy 1, 1 = −2 2

b)

Figura 13.31

EJEMPLO 4 Superficie: f(x, y) = 1 − (x − 1)2 − (y − 2)2

Hallar las pendientes de la superficie dada por

z 1

(1, 2, 1)

Hallar las pendientes de una superficie en las direcciones de x y de y

f sx, yd 5 1 2 sx 2 1d2 2 s y 2 2d 2

fx(x, y)

en el punto (1, 2, 1), en las direcciones de x y de y.

fy(x, y) 1 2 3 x

4

y

Solución

Las derivadas parciales de f con respecto a x y y son

fxsx, yd 5 22sx 2 1d

fysx, yd 5 22s y 2 2d.

y

Derivadas parciales.

Por tanto, en el punto (1, 2, 1), las pendientes en las direcciones de x y de y son fxs11, 2d 5 22s1 2 1d 5 0 Figura 13.32

como se muestra en la figura 13.32.

y

fys11, 2d 5 22s2 2 2d 5 0

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SECCIÓN 13.3

911

Derivadas parciales

Sin importar cuántas variables haya, las derivadas parciales se pueden interpretar como tasas, velocidades o razones de cambio. EJEMPLO 5

a

A = ab sen θ

θ

a sen θ

Derivadas parciales como velocidades o razones de cambio

El área de un paralelogramo con lados adyacentes a y b entre los que se forma un ángulo q está dada por A 5 ab sen q, como se muestra en la figura 13.33. a) Hallar la tasa o la razón de cambio de A respecto de a si a 5 10, b 5 20 y u 5

b

El área del paralelogramo es ab sen q Figura 13.33

p . 6

b) Calcular la tasa o la razón de cambio de A respecto de u si a 5 10, b 5 20 y u 5

p . 6

Solución a) Para hallar la tasa o la razón de cambio del área respecto de a, se mantienen b y q constantes y se deriva respecto de a para obtener ­A 5 b sen sin u ­a

Derivada parcial respecto a a.

­A p 5 20 sen sin 5 10. ­a 6

Sustituir a b y q.

b) Para hallar la tasa o la razón de cambio del área respecto de q, se mantiene a y b constantes y se deriva respecto de q para obtener ­A 5 ab cos u ­u

Derivada parcial respecto de q.

­A p 5 200 cos 5 100!3. ­u 6

Sustituir a, b y q.

Derivadas parciales de una función de tres o más variables El concepto de derivada parcial puede extenderse de manera natural a funciones de tres o más variables. Por ejemplo, si w 5 f sx, y, zd, existen tres derivadas parciales cada una de las cuales se forma manteniendo constantes las otras dos variables. Es decir, para definir la derivada parcial de w con respecto a x, se consideran y y z constantes y se deriva con respecto a x. Para hallar las derivadas parciales de w con respecto a y y con respecto a z se emplea un proceso similar. ­w f sx 1 Dx, y, zd 2 f sx, y, zd 5 fxsx, y, zd 5 lim lím Dx →0 ­x Dx ­w f sx, y 1 Dy, zd 2 f sx, y, zd 5 fysx, y, zd 5 lím lim Dy→0 ­y Dy ­w f sx, y, z 1 Dzd 2 f sx, y, zd 5 fzsx, y, zd 5 lím lim Dz→0 ­z Dz En general, si w 5 f sx1, x 2, . . . , xnd, hay n derivadas parciales denotadas por ­w 5 fxksx1, x2, . . . , xnd, ­xk

k 5 1, 2, . . . , n.

Para hallar la derivada parcial con respecto a una de las variables, se mantienen constantes las otras variables y se deriva con respecto a la variable dada.

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Funciones de varias variables

EJEMPLO 6

Hallar las derivadas parciales

a) Para hallar la derivada parcial de f sx, y, zd 5 xy 1 yz 2 1 xz con respecto a z, se consideran x y y constantes y se obtiene ­ fxy 1 yz 2 1 xzg 5 2yz 1 x. ­z b) Para hallar la derivada parcial de f(x, y, z) 5 z sen(xy 2 1 2z) con respecto a z, se consideran x y y constantes. Entonces, usando la regla del producto, se obtiene ­ ­ ­ fz sen sinsxy 2 1 2zdg 5 szd fsen fsinsxy 2 1 2zdg 1 sen sinsxy 2 1 2zd fzg ­z ­z ­z 5 szdfcossxy 2 1 2zdgs2d 1 sen sinsxy 2 1 2zd 5 2z cossxy 2 1 2zd 1 sen sinsxy 2 1 2zd. c) Para calcular la derivada parcial de f sx, y, z, wd 5 sx 1 y 1 zdyw con respecto a w, se consideran x, y y z constantes y se obtiene ­ x1y1z x1y1z 52 . ­w w w2

3

4

Derivadas parciales de orden superior Como sucede con las derivadas ordinarias, es posible hallar las segundas, terceras, etc., derivadas parciales de una función de varias variables, siempre que tales derivadas existan. Las derivadas de orden superior se denotan por el orden al que se hace la derivación. Por ejemplo, la función z 5 f sx, yd tiene las siguientes derivadas parciales de segundo orden. 1. Derivar dos veces con respecto a x:

1 2

­ ­f ­ 2f 5 2 5 fxx . ­x ­x ­x 2. Derivar dos veces con respecto a y:

1 2

­ ­f ­ 2f 5 2 5 fyy . ­y ­y ­y NOTA Observar que los dos tipos de notación para las derivadas parciales mixtas tienen convenciones diferentes para indicar el orden de derivación.

1 2

­ ­f ­f 5 Orden de derecha a ­y ­x ­y­x izquierda. 2

3. Derivar primero con respecto a x y luego con respecto a y:

1 2

­ ­f ­ 2f 5 5 fxy . ­y ­x ­y­x 4. Derivar primero con respecto a y y luego con respecto a x:

s fx dy 5 fxy

Orden de izquierda a derecha.

Se puede recordar el orden de ambas notaciones observando que primero se deriva con respecto a la variable más “cercana” a f. n

1 2

­ ­f ­ 2f 5 5 fyx . ­x ­y ­x­y Los casos tercero y cuarto se llaman derivadas parciales mixtas (cruzadas).

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SECCIÓN 13.3

EJEMPLO 7

Derivadas parciales

913

Hallar derivadas parciales de segundo orden

Hallar las derivadas parciales de segundo orden de f sx, yd 5 3xy 2 2 2y 1 5x 2y 2, y determinar el valor de fxys21, 2d. Solución Empezar por hallar las derivadas parciales de primer orden con respecto a x y y. fxsx, yd 5 3y 2 1 10xy 2

y

fysx, yd 5 6xy 2 2 1 10x 2y

Después, se deriva cada una de éstas con respecto a x y con respecto a y. fxxsx, yd 5 10y 2

y

fyysx, yd 5 6x 1 10x 2

fxysx, yd 5 6y 1 20xy

y

fyxsx, yd 5 6y 1 20xy

En s21, 2d, el valor de fxy es fxys21, 2d 5 12 2 40 5 228. NOTA En el ejemplo 7 las dos derivadas parciales mixtas son iguales. En el teorema 13.3 se dan condiciones suficientes para que esto ocurra. n

TEOREMA 13.3 IGUALDAD DE LAS DERIVADAS PARCIALES MIXTAS Si f es una función de x y y tal que fxy y fyx son continuas en un disco abierto R, entonces, para todo sx, yd en R, fxysx, yd 5 fyxsx, yd.

El teorema 13.3 también se aplica a una función f de tres o más variables siempre y cuando las derivadas parciales de segundo orden sean continuas. Por ejemplo, si w 5 f sx, y, zd y todas sus derivadas parciales de segundo orden son continuas en una región abierta R, entonces en todo punto en R el orden de derivación para obtener las derivadas parciales mixtas de segundo orden es irrelevante. Si las derivadas parciales de tercer orden de f también son continuas, el orden de derivación para obtener las derivadas parciales mixtas de tercer orden es irrelevante. EJEMPLO 8

Hallar derivadas parciales de orden superior

Mostrar que fxz 5 fzx y fxzz 5 fzxz 5 fzzx para la función dada por f sx, y, zd 5 ye x 1 x ln z. Solución Derivadas parciales de primer orden: fxsx, y, zd 5 ye x 1 ln z,

fzsx, y, zd 5

x z

Derivadas parciales de segundo orden (nótese que las dos primeras son iguales): 1 fxzsx, y, zd 5 , z

1 fzxsx, y, zd 5 , z

fzzsx, y, zd 5 2

x z2

Derivadas parciales de tercer orden (nótese que las tres son iguales): 1 fxzzsx, y, zd 5 2 2, z

1 fzxzsx, y, zd 5 2 2, z

fzzxsx, y, zd 5 2

1 z2

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CAPÍTULO 13

Página 914

Funciones de varias variables

13.3 Ejercicios Para pensar En los ejercicios 1 a 4, utilizar la gráfica de la superficie para determinar el signo de la derivada parcial indicada.

En los ejercicios 41 a 44, utilizar la definición de derivadas parciales empleando límites para calcular fx冇x, y冈 y fy冇x, y冈. y

41. f x, y

3x

2y

42. f x, y

y

44. f x, y

x2

z

43. f x, y

x

y

5 5

1. fx共4, 1兲

2. fy共⫺1, ⫺2兲

3. fy共4, 1兲

4. fx共⫺1, ⫺1兲

En los ejercicios 5 a 8, explicar si se debe usar o no la regla del cociente para encontrar la derivada parcial. No derivar. x y x2

y 1

6.

xy x x

2

8.

1

x x x2 2

2x

5y

3

x2y3

11. f x, y 13. z

x y

15. z

x2

17. z

e xy

3y 2

4xy

2 2y

10. f x, y

x2

12. f x, y

4x3y 2

14. z

2y 2 x

16. z

y3

18. z

ex

ln

x y

22. z

ln xy

23. z

ln x2

y2

24. z

ln

25. z

x2 2y

3y 2 x

26. z

x2

29. f x, y 31. z

cos xy

33. z

tan 2x

35. z

ey sen xy

y

senh 2x

39. f x, y 40. f x, y

1 dt

x

y

,

2,

2

1, 1 2,

2xy , 4x 2 5y 2

53. g共x, y兲 ⫽ 4 ⫺ x 2 ⫺ y 2

共1, 1, 2兲

4

2 1, 1

54. h共x, y兲 ⫽ x 2 ⫺ y 2

共⫺2, 1, 3兲

z

z 7 6 5

2

x

y

3

3 y

y2 x2

ln 2x

sen 5x cos 5y

36. z

cos x 2

2t y

arccos xy,

4

x

sen x

2y y2 2

cosh xy

x

2t x

50. f x, y

2

34. z

1 dt

x y

49. f x, y

y arctan , x

xy

y

t2

2,

1

32. z

38. z

3y

sen xy,

4

30. f x, y

y

48. f x, y

, 4 3

En los ejercicios 53 y 54, calcular las pendientes de la superficie en las direcciones de x y de y en el punto dado.

y y

x x

x2

28. g x, y

y2

y,

4 3 2

21. z

x

cos 2x

y

ye

e

47. f x, y

xy

0, 0

y x

20. z

27. h x, y

2y 2

2xy 2

x e

2

cos y,

1

19. z

2

,0

e

52. f x, y

En los ejercicios 9 a 40, hallar las dos derivadas parciales de primer orden. 9. f x, y

x

46. f x, y

51. f x, y

y 1 xy

y x

ey sen x,

45. f x, y

x

37. z

y

En los ejercicios 45 a 52, evaluar fx y fy en el punto dado.

−5

7.

y2

1 x

2

5.

2xy

1 dt

y2 y3

CAS

En los ejercicios 55 a 58, utilizar un sistema algebraico por computadora y representar gráficamente la curva en la intersección de la superficie con el plano. Hallar la pendiente de la curva en el punto dado. Superficie

Plano

Punto

55. z ⫽ 冪49 ⫺ x 2 ⫺ y 2

x⫽2

56. z ⫽ x 2 ⫹ 4y 2

y⫽1

57. z ⫽ 9x 2 ⫺ y 2

y⫽3

58. z ⫽ 9x 2 ⫺ y 2

x⫽1

共2, 3, 6兲 共2, 1, 8兲 共1, 3, 0兲 共1, 3, 0兲

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SECCIÓN 13.3 Partial Derivadas parciales 13.3 Partial Derivatives 13.3 Derivatives 13.3 Partial Partial Derivatives Derivatives 13.3 En los ejercicios 59 a the 64, calcular lasderivatives derivadas with parciales de InExercises Exercises 59–64, 64, find thefirst firstpartial partial derivatives with respect In 59– find respect In Exercises Exercises 59–64, 64, find find the the first first partial partial derivatives derivatives with with respect respect In 59– primer orden tox,x,y,y,and and to z.z. con respecto a x, y y z. to x, and z. x, y, y, and z. to In 59– 64, first In Exercises 59– 64, find the the first partial partial derivatives derivatives with with respect respect InExercises Exercises 59–sen 64,find the 59. H x, y, z sen x 2y 2y first 3z partial derivatives with respect 59. H x, y, z xfind 3z to and x, y, z. to and x, y, z. 59. H x, y, z sen x 2y 3z 59. H x, y, z sen x 2y 3z to x, y, and z. 22 3x2y2y 5xyz 5xyz 10yz 10yz2222 60. f f x,x,y,y,zz 3x 60. f x, x, y, 3x 5xyz 10yz 60. H fH y, zzzz 3x yy xx 5xyz 10yz 60. 59. x, y, sen 2y 3z 59. x, y, sen 2y 3z 59. H x, y, z sen x 2y 3z 7xz 7xz 2 2 2 2 2 2 2 7xz 2 7xz 61. wffwx, 62.222 ww xx 3x yy zz5xyz 61. 62. 60. y, 10yz 60. 22 w xx yy 61. w 62. w 61. 62. fwx, x, y, y, zzzxx22 3x 3xyy22yyy zz5xyz 5xyz 10yz 10yz 60. xx yy 7xz 7xz 2 2 2 7xz Fx,x,y,y,zzxx222 lnln 63. Fw 63. yyy222 xxxx222zzz222 yyyy222 zzzz22262. 61. w w 61. 62. F x, x, y, y, zzx ln ln 63. F 63. w w xx yy 61. 62. w x y 1 1 22 11222 zz222 x,y,y, y,zzz ln Gx,x, 64. G 64. F x y 63. F ln x y 63. 2 x, y, y, zz G x, 64. G 64. F ln11 x xx2222 y yy2222 z zz2222 63. zz 11 xx 11 yy 1 x, y, zzz65–70, G 64. x, y, G 64. x, y, G 64. 2 2 2and f , f , In Exercises 65–70, evaluate the givenpoint. point. 2 2 f f atat f , f , In Exercises evaluate the yyyfxfx2xx,, fyfyyyfz,z,xzand xxx 2evaluar 2f yfzffz at ,2and en elgiven punto dado. En los ejercicios 65111aevaluate 70, In Exercises Exercises 65–70, evaluate the given point. point. In 65–70, and the given y zzzat 3 22 yz2,2evaluate , 1,1,1,1,11ff ,, ff ,, and 65.Exercises f f x,x,y,y,zz 65–70, 65. xx33yz In In at the given point. yz ,evaluate , 1, 1, 1, 1, 11fxxx , fyyy , and x, y, y, zz 65–70, 65.Exercises ff x, xx3yz 65. In Exercises 65–70, evaluate and fffzzz at at the the given given point. point. 2,1,1,22 2xyz 3yz, 3yz, 66. f f x,x,y,y,zz xx22y22y3333 2xyz 2, 66. 3 2 f x, y, z x 2, 1, 2 y 2xyz 3yz, 66. 3 2 f x, y, z x y 2xyz 3yz, 2, 1, 2 66. 65. xxx3yz yz 2,,, 1, 1, 1, 65. x, y, 1, 1, 1, 111 65. fff x, x, y, y, zzz xx2 yz 3 x 2 3 x f x, y, z , 1, 1, 1 67. f x, y, z , 1, 1, 13yz, 67. xxyz 2, 66. x, y, 2, 1, 2xyz 66. x,y, y,zzzz yz 1,2xyz 1, 3yz, 1 67. ffff x, x, y, 1, 13yz, 67. x2yy,y, 3 1, 2, 1, 1, 222 2xyz 66. yz yz xxx xy xy1, ,,, xy 67. x, y, 67. xy1, 1, , 1, 1, 3,3,1111,1, 11 67. ffff x,x, x,y,y, y,zzzz ,1, 68. 68. yz x, y, y, zz xyz 3, 1, 1, 11 68. ff x, 68. xyz yy zz,, 3, xx yy zz xy xy xy 1, 68. x, y, 3, 1, 68. 3,0, 1, 68. ffff x,x, x,y,y, y,zzzz zxxzsen senyyxx zzy,,,y, , 3, 0, ,1,11 44 69. 69. x, y, y, zz senyxx z yy ,, 0, 0,22 ,, 44 69. ff x, zzx sen 69. 22 2 2 2 2 2 2 x,y,y, y,zzzz 3x22xx yy2y2y ,, 2z 2z0, 1, 442,2,11 70. ffff x,x, 3x ,2, 1, 70. zzz sen 69. x, y, sen 69. 3x x yy y , 2z 2z20, 1, 4 2, 2, 11 70. ff x, 3x ,, 22,,,1, 70. x, y, y, zz sen 0, 69. 2 22find 22 In Exercises Exercises 71–80, 80, find the2z four second partial derivatives. derivatives. In 71– four second partial fff x, y, 3x yyy222the 2, 70. x, y, 3x 1, 2, 70. 2,, In los Exercises 71–71 80, find the2z four second partial derivatives. In Exercises 80, the second derivatives. x,ejercicios y, zzz 71– 3xa2 find 2zfour , las1, 1, 2, 111partial 70. En 80, calcular cuatro derivadas parciales Observe thatthe thesecond second mixed partials areequal. equal. Observe that mixed partials are Observe that the second mixed partials are equal. Observe that the second mixed partials are equal. de segundo orden. Observar que lassecond derivadas parciales mixtas In Exercises 71– 80, find the four partial derivatives. In Exercises 71– 80, find the four second partial derivatives. In segundo partial 2 71– 80, find the four second 2orden z 3xy 3xy 3y2222 derivatives. 71. 72. zz are de son iguales. zExercises xx2222 equal. 3y 71. 72. 22 the Observe that second mixed partials Observe that the second mixed partials are equal. z 3xy z x 3y 71. 72. z 3xy z x 3y 71. 72. Observe that the second mixed partials are equal. 2xy 3y 3y2222 3x222y2y2222 yy4444 73. zz xx2222 2xy 74. zz xx4444 3x 73. 74. 22 2xy 22 z x 2xy 3y z x 3x222 yy 73. 74. z x 3y z x 3x yy 73. 74. z 3xy z x 3y 71. 72. z 3xy z x 3y 71. 72. 2 2 2 2 x xx 3y 71. zzz 3xy 72. zzz lnln 75. 76. xx222 yy222 yy2 2 75. 76. 22 x 22 44 x z ln y z x x y 2 2 75. 76. z ln y z y 75. 76. 2xy xxx 4 y y 3x 73. 74. 2xy 3y 3y 3x 2yyy 2 x x yyy444 73. 74. 3y 2 73. zzzzz exxexx x2xtan 74. zzzzz 2xe tany2xy y 3ye xx 2xeyy 3x3ye 77. 78. 77. 78. x 22 y tan y 22 3ye 2xe 77. zz ee xxtan 78. zz 2xe 3ye 77. 78. 75. 76. ln 75. 76. ln xxx yyyyy x 2 yyy 2 75. zzz 76. zzz ln x y xx y x y y xyyy arctan 79. zzzz cos 80. zzzz arctan xy 79. 80. 77. 78. ecos tan 2xe 3ye 77. 78. x tan y xy y arctan 79. zz ecos 80. zz 2xe xy arctan 79. 80. ecos tan 3ye x 2xe 77. 78. xx3ye xx yyy zzz cos zzall arctan 79. 80. cos xy arctan 79. 80. cos xy xy z values arctanofof 79.Exercises 80. all In Exercises 81–88,for forf f x,x,yy, ,find find values andyysuch such xxand In 81–88, xof x, yy ,, find In Exercises Exercises 81–88, 81–88, for for ff x, find all all values values xof and yy such such In x xx and thatfxfxx,x,yy 00and andfyfyx,x,yy 00simultaneously. simultaneously. that x, yy x, yy that ffxx x, and ffyy x, simultaneously. 00 and 00 simultaneously. that ,,, find xx and In 81–88, values x, y such In Exercises 81–88, for find all values of of and such En los ejercicios 81 afor 88, 2ffpara f(x, y), all encontrar todos los yvalores f x, x, yyy2x In Exercises Exercises find 22 xyffforx, 2yall values of x and y such 81. f f x,x, yyyy que xx81–88, xy y=y22y20y y2x 2y 81. 22 fand x, 00(x, that simultaneously. x, x, that simultaneously. de y, (x, y) f y) = 0 simultáneamente. x, xy y 2x 2y 81.x fffyffxxxx, xy y 2x 2y 81. y y x,yytal y xx0200 and x, y 0 f that and simultaneously. x y y xy yy2222 5x 5x yy 82. f f x,x,yy xx222 xy 82. x,yyyy xy yyyy222 2x 5x 2y y 82. ffff x, x, xxx222 xy xy 5x y2y 82. x 81. x, xy 2x 81. x,yyy xxx22 4xy xy yyy22 2x 81. f ffx,x, 4xy 4x 2y 16y 33 83. 4x 16y 83. 2 2 2 2 x, yyyy 4xy 22yy 5x4x 4x 16y 16y 33 83. ff x, xx22 xy 4xy 83. 82. x, xy 82. x,yyy xxxxx222 xy xy yyyyy222 5x 5x yyy 82. fffff x,x, x, xy 84. 84. 2 2 22 22 2 f x, y x xy y 2 84. f x, y x xy y 84. 83. x, 4xy 4x 16y 83. x, yyy 1xx1x 2 114xy 4xy yyy 2 4x 4x 16y 16y 333 83. fff x, 2 2 1 1 2 2 1 1 f x, y xy 85. f x, y xy 85. 84. x, xy 84. x, yyyy xxxx 2 yyxy xyyyy2 85. ffff x, x, 85. xy xy 84. xx yy 1 1 1 1 3 3 1 33 112xy x,yyyy 3x 3x 12xy 86. ffff x,x, yy33 86. xy 85. x, xy 85. 12xy 12xy 86. ff x, 86. x, yy xy yy33 85. x3x x3x y 22 y 22 2 2 xy y y y22 87. f f x,x,yy eexxxx22 xyxy 87. yy x,yyyy 3x x, ee 333 xy12xy 87. ffff x, 87. yy333 86. x, 3x 12xy 86. 2 2 f x, y 3x 86. y222222 11y 88. f f x,x,yy lnlnx22x22 12xy y 88. x xy y f x, y ln x y 1 x xy y f x, y ln x y 1 88. 88. eee x 2 xy y 2 87. x, 87. x, yyy 87. fff x, 22 use CAS In InExercises Exercises 89–92, use computer algebrasystem systemtotofind findthe the CAS 89–92, algebra f x, y ln yyy222 aaaacomputer 1computer 88. f x, y ln 1computer 88. CAS In CAS In Exercises Exercises 89–92, use algebra system system to to find find the the 89–92, algebra f x, y ln xxx 2 use 1 88. firstand andsecond secondpartial partialderivatives derivativesofofthe thefunction. function.Determine Determine first first and and second second partial partial derivatives derivatives of of the the function. function. Determine Determine first CAS CAS whether In Exercises 89–92, use aaa computer system the yto find In Exercises 89–92, use computer algebra system find the whether there exist values ofxxand and suchalgebraico that and los ejercicios 89 avalues 92, un sistema por compuyyalgebra 00and fxfxx,x,yto there exist of such that CAS En In Exercises useutilizar computer system the x, yyto find 00 and ffxx x, whether there89–92, exist values values of xx and and yyalgebra such that that and whether there exist of such first and second partial derivatives of the function. Determine fyfirst x, y 0 first and second partial derivatives of the function. Determine simultaneously. y hallar las derivadas parciales de primero y segundo x, y 0 ftadora simultaneously. partial derivatives of the function. Determine x, yyand second ffyyy x, 00 simultaneously. simultaneously. yy such whether exist values of that and x,deyy x y 0y0 tales whether there exist values of and such that fffxx x, and orden dethere la función. Determinar si existen valores whether exist of xxx and and such and 22 x x, yxx2 2 0yy sec x,yy that 25 25 90. fyf x, f fyyx,x, yythere xxsec yfy xvalues 89. 90. 22 f89. x, 0 simultaneously. f89. x, 0 simultaneously. que y simultáneamente. x x, y c 5 0 x, y c 5 0 f x, y x sec y f x, y 25 x y22 f x, y x sec y f x, y 25 x y 89. 90. 90. y y x y fy x, y 0 simultaneously. xy xx xy 22 22 xy xx xy x,yyyy ln x,yyyy 91. ffff x,x, 92. ffff x,x, 91. 92. xln 89. 90. x, xln sec x, 25 89. 90. 91. ff x, 92. ff x, 91. 92. x, yy xlnsec x, yy xx 25 25yy xxx 2 yyy 2 89. 90. x2222 yyy yy2222 xsec xx yy xx yy xxx xy xy xy ln 91. 92. x, ln x, 91. 92. x, yyy ln xx222 yy222 x, yyy 91. fff x, 92. fff x, x x x y x yyy

915 915 915 915 915

13.3 915 13.3 Partial Derivatives 915 13.3 Partial Partial Derivatives Derivatives 915 En los ejercicios 93 a 96, mostrar quepartial las derivadas parciales fxyy, , In Exercises 93–96, show that themixed mixed partial derivatives fxyy In Exercises 93–96, show that the derivatives ffxyy In Exercises 93–96, show that the mixed partial derivatives In Exercises 93–96, show that the mixed partial derivatives xyy,, fare yequal. fyyx son iguales. fmixtas fyyx, are andffxyy fyxy , ,and yxy equal. yxy yyx ffyxy and ffyyx are equal. equal. yxy,, and yyx are In Exercises 93–96, show In Exercises 93–96, show that that the the mixed mixed partial partial derivatives derivatives ffxyy xyy,, In93. Exercises 93–96, y,zzare xyz show that the mixed partial derivatives fxyy , f f x,x,y, xyz fff93. , f and equal. , f and are equal. x, y, y, xyz 93. ff x, xyz 93. yxy yyx yxy yyxzz yxy , and fyyx are equal. 3xy 4yz 4yz zz3333 94. f f x,x,y,y,zz xx2222 3xy 94. f 3xy 4yz 4yz zz x, y, z x 94. f 3xy x, y, z x 94. f x, y, z xyz 93. f x, y, z xyz 93. x x sen yz x,y,y, y,zzz eexyz 93. f ffx,x, 95. sen yz 95. x x x, y, y, zz sen yz 4yz 95. ff x, 95. 3xy xexxe222 sen 94. 3xyyz 4yz zz333 x, y, 94. x, y, y, zzz 94. fff x, 2z 3xy 4yz z 2z x 2z x 2z x, y, z f 96. x, y, z f 96. 95. x, y, yz 95. x, y, 96. ffff x, 96. x, y, y, zzzz xeexe x sen sen yz 95. y yz ysen xx yy 2z 2z 2z zz 96. x, y, 96. x, y, y,Equation 96. fff x, Laplace’s InExercises Exercises 97–100,97 show that thefunction function Laplace’s Equation 97–100, show that the Ecuación dez Laplace los ejercicios a 100, mostrar que la xxx In yyIn Laplace’s Equation Exercises 97–100, show show that the function function Laplace’s Equation Exercises 97–100, that the yInEn 2z x2 2 1 2 2z2 y2 2 2 0. 2 2z 2 5 0. satisfies Laplace’sla equation x 221 2z2­/ / zy/­x 0. satisfies Laplace’s equation función satisface ecuación de Laplace 1 ­ z ­y / 2 22z 2 / / 1 zz/ yy 0. 0. satisfies Laplace’s Laplace’s equation equation z/ xx 1 satisfies Laplace’s Equation In Exercises 97–100, show that the function Laplace’s Equation In Exercises 97–100, show that the function 1 y1y thaty ythe function Laplace’s Equation In Exercises 97–100, y 2 5xy e0.eyy2y sen 97. zz5Laplace’s 98. zz 2z2 12show 5xy sen xx x 97. 98. 121ee 22 22 1 5 dsin sen y2e e y2s satisfies equation 1 satisfies equation 2 98. 5xy e0. sen sen 5xy xx 97. zz Laplace’s 98. 97. 1zz 2zzz//22 yeyey2 2 e0. satisfies Laplace’s equation 2zzz// xxx98. yy y x 1 x 1 yy yy yy senxx exxxsen arctan senyy 99. zzzz5 e5xy 100. zzzz z arctan 99. 100. 1 eearctan 100. 97. 98. 97. 98. y 2arctan 2arctan e sen sen yy zz 5 99. zz e5xy 100. 99. 100. 5xy sen x 97. 98. xxeee xy sen 2 e xx y y y eeexxx sen arctan 99. 100. zzEquation arctan sen yyyIn 99. 100. arctan 99. zEquation 100. zzz101 Wave InExercises Exercises 101–104, show that thefunction function Wave 101–104, show that the Ecuación desen ondas En los ejercicios a 104, que la xxxmostrar Wave Equation In Exercises Exercises 101–104, show that the function function Wave Equation In 101–104, show that the 2 2 2 2 2 2zz t t2 2 22zz x2x2 . . 2 2 2 c satisfies the wave equation c satisfies the wave equation función the satisface la ecuación22zde // xx25 2 . c x­ z /­x c. /z// tondas t22 cc22­ z22z/z­t satisfies the wave equation equation . satisfies wave Wave Equation In Exercises 101–104, show that the function Wave Equation In Exercises 101–104, show that the function WavezzEquation In 101–104, that the function sen ct Exercises cos 4x 4ct4ct 101. 102. sen xx 2ct zccz222 zshow cos 101. 22z 102. 22 102. 2 22s.4x 4ct 2z 101. zzz 5the sen(x ct) 5 x4x satisfies wave zz/ cos xx4x satisfies the wave equation 2zz/ ttt102. 2 2 .. 1 sen ct cos 4x 4ct d 101. 102. sen xx equation ct 4ct 101. czz 2cos satisfies the wave equation / / ln ct sen ct sensenxxwx 103. 104. zz z sen sen 103. 104. senctct wct lnxsxxxx 1ctct ctd 103.zzzzz 5lnln 104. ln ct sen ct sen 103. 104. zz 5 sen sen xx 103. 104. 101. 102. sen ct cos 4x 4ct 101. 102. sen xxx ct ct cos 4x 4x 4ct 4ct 101. zzz sen 102. zzz cos Heat InExercises Exercises 105and and106, 106, show that thefunction Heat Equation In show that the Ecuación En los 105 ejercicios y 106, mostrar zEquation xxx calor ct zzz105 sen ct 103. zEquation ln ct sen ct sen xfunction 103. 104. Heat Equation In Exercises Exercises 105104. and 106, show that thexfunction function Heat In 105 and 106, show that the z ln lndel ct sen ct sen sen x que la 103. 104. x22225 satisfies the heatequation equation t t calor cc2222 2­z z22z/ //x­t . . c 2x­ 2z /­x 2c. zz/ /del satisfies the heat función satisface la ecuación 2 satisfies the the heat heat equation equation zz/ tt cc zz/ xx .. satisfies Heat Exercises that the Heat Equation In Exercises 105 and 106, show that the function function HeatEquation Equation xIn In Exercises105 105and and22106, 106,show show that xequation xx xthe function theat x t heat c106. xxx2t2e2tttsen ..sen satisfies the equation cc2 zz222zzz/ e5 satisfies the 2t xxequation zzz// ttt 106. x x e cos 105. 106. zzz 5 eee2t 105. sen . satisfies the heat 105. cos sin tcos t cosccc sencc c 105. zz ee cos 106. zz / ee sen 105. 106. cc cc x xxx x x eee ttt sen zzz ejercicios eee ttt cos 105. 106. cos sen 105. 106. EnExercises los 108, siwhether existe othere no función cos sen 105. 106. zzzwhether In Exercises 107cc107 andy 108, 108, determinar determine there exists In 107 and determine exists aa ccc una In Exercises Exercises 107 107c and and 108, 108, determine determine whether whether there there exists aa In exists f(x, y) con derivadas parciales Explicar el razonafunction with thegiven given partialdadas. derivatives. Explain your yy with f f x,x,las function the partial derivatives. Explain your x, yy with function ff x, with the the given given partial partial derivatives. derivatives. Explain Explain your your function miento. SiIfIf tal función existe, dar ungive ejemplo. In Exercises 107 108, whether there In Exercises 107 and 108, determine whether there exists reasoning. such afunction function exists, give anexample. example. reasoning. such exists, an In Exercises 107 aand 108, determine determine there exists exists aaa reasoning. If such such a function function exists, give givewhether an example. example. reasoning. If aand exists, an ff x, function the given x, function with the given partial partial derivatives. derivatives. Explain Explain your your x, yyy with function with the sen 3xgiven 2y ,f f x,x,give sen 3x 3xExplain 2y your 107. f f x,x,yfyIf 33sen 3x 2y ,partial yy derivatives. 22sen 2y 107. reasoning. function exists, reasoning. such function exists, an example. x, yyIf 3x 2y 2y x,give sen 3x 3x 2y 2y 107. xffxxx x, 3x ,, yffyyy x, yy an 22example. sen 107. reasoning. If such such33aasen asen function exists, give an example. 2x y,y,fyfyx,x,yy xx 4y 4y 108. fxfxx,x,yy 2x 108. fyy x, x, yy 2x33 sen y, f3x x, yy2y ,, xfxfy x, 4y 22 sen 108. ff x, 2x y, 4y 108. 107. x, 3x 2y 107. x, yyy 3 sen sen 3x 3x 2y 2y , fyy x, x, yyy 2 sen sen 3x 3x 2y 2y 107. fffxxxxx x, In Exercises 109 and 110, find the first partial derivative with In Exercises 109 and 110, find the first partial derivative with f f x, y 2x y, x, y x 4y 108. ffyy110, ffxx x, yy 109 2x y, x, yfind xx first 4y 108. EnExercises los ejercicios 109 y110, encontrar lapartial primera derivada parx, 2x y, x, y 4y 108. In Exercises 109 and 110, the first partial derivative with In and find the derivative with x y respect x.x. respect totorespecto cial con a x. x. respect to to x. respect In 109 110, the first In Exercises 109 and and 110,2z22find find the first partial partial derivative derivative with with In Exercises Exercises and z first partial derivative with y y 2 2222 zthe tan zeez 22 find 109. f f x,x, y,y,x. zz 109tan yy2z2110, 109. respect to respect to x, y,x. tan yy22zz eezz yy zz 109. ff x, tan 109. x.zz respect toy, yy 22y2 22 222 y 1 z x,y,y, y,zzzz xxtan senh 110. ffff x,x, senh 110. yyy222zzyzyeeezzyz2yy22yyy2222 yyzyzz 111zzz 109. x, y, 109. senh 110. ff x, xxtan senh 110. x, y, y, zz tan 109. z zz yyy yy222 22 yy 11 zz x, y, z x senh 110. f x, y, z x senh 110. f y 2 y 1 z senh 110. NGGz AABBxOOU UTT zCzzCOONNCCEEPPTTSS WWRRIfITTx,I INy,

WRRIITTIIN NGG AABBOOU UTT CCOON NCCEEPPTTSS W

111. Let Letf fbe beaafunction functionofoftwo twovariables variablesxxand andy.y.Describe Describethe the 111. fGbe x and y. Describe 111. Let beAAaaBBfunction function ofN two variables and y. Describe the the fG 111. of two variables W RRRIIIprocedure TTTLet IIIN OOOU TTT CCCO CCCfirst EEEfirst P TTTpartial Spartial W N U O N P SS xderivatives. procedure for finding the derivatives. for finding the W N G A B U O N P Desarrollo de conceptos procedure for for finding finding the the first first partial partial derivatives. derivatives. procedure fff be xxx and y. 111. Let aaasurface function of Describe the y. 111. Let be function of two variables and Describe the 112. Sketch representing function two variables f fofof 112. Sketch aasurface representing aafunction variables y.two 111. Let be function of two two variables variables and Describe the 112. Sketch surface representing apartial function of two variables 112. Sketch aa surface representing apartial function of two variables 111. procedure Sea f una función de dos variables x yderivatives. y. ffDescribir el procefor finding the first procedure for finding the first derivatives. andy.y.Use Use the sketchto togive give geometric interpretationsofof xxprocedure and the sketch geometric interpretations for finding the first partial derivatives. y. Use and y. Use the sketch to give geometric geometric interpretations of xxdimiento and the sketch give interpretations of para hallar lasto derivadas parciales de primer orden. 112. aaa surface 112. Sketch surface representing function of two variables and f f xxand f f y.y.representing 112. Sketch Sketch surface representing aaa function function fff of of two two variables variables f x f y. and f x f y. and 112. xxDibujar unathe superficie que represente una función f de of dos and Use to interpretations y. and Use the sketch to give interpretations of zgeometric 113. Sketch Sketch the graph ofaafunction function whosederivative derivative f f x,x,yy whose 113. the graph of xSketch y. andy. Use thesketch sketch togive givezgeometric geometric of x, yy interpretations 113. Sketch the graph of function whose derivative zz ff x, 113. the of aa function whose derivative variables xgraph y y. Utilizar la gráfica para dar una interpref x f and f x f y. and always negative and whose whose derivative derivative fyfy isis always always fxfxfisis always negative and x f y. and is always always negative and whose whose derivative ffyy is is always always ffxtación negative and derivative geométrica de ­fy­x y ­fy­y. x is 113. Sketch the x, 113. Sketch the graph of function whose derivative positive. x, yyy whose 113. positive. Sketch thegraph graphof ofaaafunction functionzzz fff x, whosederivative derivative positive. positive. zderivative 5 f sx, yd fcuya 113. ffxDibujar la gráfica de una función negative and whose always ffyy is is always always negative and whose is derivada always x is 114. Sketch Sketch thegraph graph function whose derivazz derivative f f x,x,yy whose 114. the ofofaafunction fSketch negative whose derivative always x is always y isderivax, yyfy sea 114. positive. the graph graph of aa and function whose derivazz ff x, 114. Sketch the of function whose derivanegativa y cuya derivada siempre pofx sea siempre positive. tives andfyfyare arealways alwayspositive. positive. tives fxfxand positive. tives and ffyy are are always always positive. positive. tives ffxx and sitiva. zzzsuchfffthat x, 114. graph x, 114. Sketch the graph of whose derivaxfunction fwhose 115. If athe function and yy such that and fderivafyx are are f f isis athe 115. IfSketch function ofofaaaxfunction and and x, yyyfxy 114. Sketch graph of of whose derivaxy yx xfunction 115. If function of xuna and suchz 5 that and are ff is yy such 115. If aa lafunction of and that ffyx are f sx,ffxy 114. tives Dibujar gráfica de función cuyas xyydand yx deriffwhat fis are always positive. tives and are always positive. ffxx and yywhat continuous, is the relationship between the mixed continuous, is the relationship between the mixed f tives and are always positive. ywhat is continuous, the relationship relationship between the the mixed mixed continuous, the vadasx fx y fywhat sean is siempre positivas. between fff is xxx and 115. If aaa function of 115. If is function of and such that and are partial derivatives? Explain. xy and yx are 115. partial If isderivatives? function Explain. of and yyy such such that that fffxy and fffyx partial derivatives? Explain. partial Explain. yx are x relationship 115. continuous, Si f esderivatives? una what función de y y tal que fxybetween y fyxxysonthe continuas, is the mixed continuous, what is the relationship between the mixed continuous, what is the relationship between the mixed ¿qué relación existe entre las derivadas parciales mixtas? partial partial derivatives? Explain. partial derivatives? derivatives? Explain. Explain. Explicar.

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60. f x, y, z 61. w

3x y x2

63. F x, y, z

y2

CAPÍTULO 13

Funciones de varias variables

916 916916Chapter 13 13 Functions of Several Variables Chapter of Several Variables Chapter 13 Functions Functions of Several Variables 916 Chapter 13 Functions of Several Variables

Functions of Several Variables 916 Chapter 13 Functions of Several Variables Chapter 13 Functions 916 Chapter 13 916 of Several ChapterVariables 13 Functions Functions of of Several Several Variables Variables

10yz

z2

ln x 2

62. w y2

94. f x, y 7xz x

95. f x, y y

96. f x, y

z2

1

64. G x, y, z

916

5xyz

1

x2

y2

z2

In Exercises 65–70, evaluate fx , fy , and fz at the given point. 3

2

Laplace’s E satisfies La 97. z

5

x yz , 1, 1, 1 65. f x, y, z 99. z e 123. 66. Distribución de La temperatura en cualquier 3 f x,123. y,Temperature z Temperature x2temperatura 2, 1, 2temperature yDistribution 2xyz 3yz, 123. Temperature The temperature at any point 123. Distribution The temperature at any point Para discusión Distribution The at any point 2 2 1.5y 2, C A PCSATPOSNTEO N EN E placa de acero es temperature sx, yd de una T 5 500 22 0.6x 123. punto Temperature Distribution The at 2 any point 2 where C A P S T O NCEA P S T O123. 221.5y in yx, a in steel plate is plate y x, 1.5y , where T is 500 0.6x ainsteel plate , 2,xwhere Tis T 500 0.6x0.6x x xWave Equa Temperature Distribution The temperature at donde anyx,point a steel ymedidos 1.5y 500 xplate 2 2 where y son en metros. En el punto (2, 3), hallar el x y in a steel is x, y 1.5y , T 500 0.6x x Find the four second partial derivatives of the function 116. Find the four second partial derivatives of the function 116. 116. Encontrar las cuatro segundas derivadas parciales de la fun116. Find the four second partial derivatives of the function f x, y, z , 1, 1, 1 67. 123. Temperature Distribution The temperature at any point 2 2 and yand are inpoint meters. At theAtpoint rates C A P S T116. O N EFind the four second x,partial 2, 3any ,2,find are measured in meters. the point find the the rates ymeasured 32, ,the a steel plate where ofisthe y in derivatives 1.5y , T function 500Distribution 0.6x 123. x and are measured in meters. At the point find rates y 3 , yz 123. Temperature The temperature at any Temperature Distribution The temperature at point satisfies th ritmo de la a, la recoDistribution The temperature any point 2respecto 22, T Opartial N E derivatives ofCCAthe and arecambio measured in temperatura meters. At the point find rates y de 3 at ,distancia SSfunction TTOOgiven N bygiven the second f by x, yby 2y the the second f x,f ysen x .xShow 2y 2y . Show dada por Mostrar quethat lasthat send Show second x, y xsensen .that a the steel plate is where x,123. y inTemperature 1.5y Tthe 500 0.6x xthe APPción NEEbygiven 2 of 2ofwhere 22 with 22 the ofin0.6x change of temperature with respect to change the temperature with respect to distances the distances and in At theplate pointis2, rates given Show that thea second f x, partial y sen xy are2ymeasured . of change of temperature respect to xxthe distances in steel x, ymeters. 1.5y , is T3 , find 500 x500 aaofsteel plate x, yychange 1.5y ,, where TT the 0.6x 116. that Findthe the four second derivatives the function rrida en la placa en las direcciones del eje x y y. xy in steel plate is where x, 1.5y 500 0.6x of of the temperature with respect to the distances mixed partial derivatives and f are equal. f mixed partial derivatives and f are equal. f gundas derivadas parciales mixtas f y f son iguales. Show second sen the x four 2y . second mixed partial derivatives and f are equal. f and are measured in meters. At the point find the rates y 2, 3 , Find partial derivatives of the function xy yx xy yx 116. Find the four second partial derivatives of the xyyx xy the f are x, y, zalong ,in At 3,in 1, 68.the 116. by Find theypartial foursen second partial derivatives ofsecond the function plate the directions of3 the axes. y-and moved the the plate the1the directions ofx-the x-rates y-axes. of temperature with respect to the distances mixed derivatives fof equal. fShow moved along plate directions ofand the and xy-axes. 101. z s and measured in meters. At point find the yyxarefunction 2, 3 xalong , the and measured in the yymoved 2,the ,, find xy and yx are given that the f x, x 2y .change and are measured in meters. theinpoint point find the rates 3 xmoved along the plate in the directions andthe y-axes. ymeters. zrates tives fby of change ofaxes. the temperature with respect to of the2, given f x,fyxy are equal. sen x 2y . Show given by that the second fthat x, yythe second sen xx 2y .. Show 124. Temperatura aparente UnaAt medida de ladistances percepción del xy and given by Show that the second f x, sen 2y moved along the plate in the directions of the and xyof change of the temperature with respect to the distances of change of the temperature with respect to the distances mixed partial derivatives fxy and fyx are equal. 124.change Apparent Temperature Awith measure of how hot weather feels feels 124. Apparent Temperature A respect measure ofthe how hothot weather 124. Temperature Ax-measure of how weather feels103. z l ofApparent the temperature moved along the plate inpor the directions of thepromedio and to axes. y-es mixed partial derivatives fxy and fyx arepartial equal.derivatives 124. of Apparent Temperature A measure of how hot weather feels mixed fyx are equal. ffxy and calor ambiental unas personas eldistances Índice de mixed partial derivatives moved Aalong the plate in the of and xyxy and fTemperature yx are equal. moved the in the directions of the and axes. xyan person isyperson Temperature Index.Index. AIndex. toaverage an average person Temperature A A 124. Apparent measure of how hot directions weather feels f x,along y, zthe zplate sen xinaxes. ,theisApparent 0,the ,Apparent 4Apparent 69. to to an average is the Temperature moved along the plate the directions of the and axes. xyMarginal Revenue A pharmaceutical corporation has two 117. Marginal Revenue A pharmaceutical corporation hashas twotwo temperatura toTemperature an average person is the Apparent Temperature aparente. Un modelo para índicefeels es Index. A 117. 117. Ingreso marginal UnaRevenue corporación farmacéutica tiene dos124. 117. Marginal A pharmaceutical corporation 2hoteste Apparent Aindex measure of weather Heat Equa 117. Marginal Revenue Atopharmaceutical corporation has two model this isfeels model for this index is how an average124. person is the Apparent Temperature Index. Afor model for this is of Apparent A offor how weather Apparent Temperature measure weather feels 124. Apparent Temperature A2index measure of1,how how2,hot hot weather plants that produce the same over-the-counter If x1 to xan plants thatthat produce the the same over-the-counter medicine. If124. model thishot index is A A pharmaceutical corporation has two plantas que producen laproduce misma medicina. Si x1 y medicine. x2Temperature son los xaverage plants same over-the-counter medicine. If1 measure 2Apparent 2 person is the Temperature Index. A feels 1 satisfies th f x, y, z 3x y 2z , 1 70. plants that produce the model same over-the-counter medicine. If x1 is the Apparent forcorporation this index isan A 5 0.885t 2 22.4h 1 1.20th 2 0.544 117. Marginal Revenue A pharmaceutical has two to average person Temperature Index. A to an average person is the Apparent Temperature Index. A person is 22.4h the22.4h Apparent Temperature andIfde are of produced 1at and 2, plant xand the numbers of produced 1plant and plant 2, 2,to an xthe xRevenue e same Revenue over-the-counter medicine. números unidades producidos enunits la planta 1atyplant enatlaplant planta 2, and are the numbers of units produced plant 1 and x2numbers A average 0.885t 22.4h 1.20th 0.544 A Ais0.885t 1.20th 0.544 0.885t 1.20th 0.544 Index. A arginal A 117. pharmaceutical has twounits 2 are Marginal A pharmaceutical corporation has two 1 index 117. Marginal Revenue A pharmaceutical corporation two andproduce the of units produced atmodel plant 1If and plant 2, is model for x2 are2corporation A this0.885t 22.4his 1.20th 0.544 xhas that thenumbers same over-the-counter medicine. for index model for this index 1this model for this index is respectively, then the total revenue for the product is given by respectively, then the total revenue for the product is given by snts of units producedthe atplants plant 1over-the-counter and plant 2, respectivamente, entonces el ingreso total del producto está respectively, then the total revenue for the product is given by A 0.885t 22.4h 1.20th 0.544 x that produce same medicine. If In Exercises 71–80, find the four second partial derivatives. x plants that produce the same over-the-counter medicine. If 1 donde esA where la temperatura aparente en in grados Celsius, A where t esCelsius, 1 plants that produce thetotal samerevenue over-the-counter medicine. If xby respectively, then the product and are the numbers of200x units produced 1 and x2 dado where is the temperature degrees t islathe 2 at plant 2plantis isapparent apparent temperature in degrees Celsius, A1.20th t is tthe 2forx2the 22,given is the apparent temperature in Celsius, degrees is the105. z e 22.4h 0.544 Athe 2When When Rpor 200x 200x 4x .210.885t x2,11 x41 A1.20th R 200x 200x 4x8x 8x x4x 4x . plant 4 0.885t revenue for theof product is given by Cuando When R 200x 4x 8x x 4x . 4 x dtotal the numbers units produced at plant 1 and plant 2, x2 are Observe that the second mixed partials are equal. 1 2 1 1 2 2 where is the apparent temperature in degrees Celsius, is the A t forma 2 2 1 2 1 1 2 2 and are the numbers of units produced at plant and x 1 2 1 1 2 1 A 22.4h 0.544 A 0.885t 22.4h 1.20th 0.544 temperatura del aire y es la humedad relativa dada en h 22200x and are the numbers of units produced at plant 1 and plant 2, x When R 200x 4x 8x x 4x . 4 x A 0.885t 22.4h 1.20th 0.544 2 1for A 1 product 2 apparent 2is given respectively, the total revenue the by la air and hand isand humidity in decimal air air temperature, the relative humidity in decimal htheishrelative where isthe the temperature the t istemperature, 2 temperature, is the relative humidity in decimal find (a) the marginal revenue for plant 1,1planx2and Rin x1, find (a) the marginal revenue plant 1, degrees x12, 12, R, Rx1,Celsius, When 4x12 8x 41product xthen xrespectively, = 4and y xthe 12, a) el(a) ingreso marginal para and find revenue for plant xthe 12, pectively, then total for given by 2= air temperature, and is the relativeCelsius, humidity in decimal hUMAP 2 encontrar then revenue for the product isfor given by 1x2 the4x 2 . revenue 2 1and (Fuente: The Journal) wherex1A, decimal. is the apparent temperature inThe degrees 2istotal 2 respectively, then the total revenue formarginal the product given by find (a) the marginal revenue for hplant 1, x21200x 12, R x1, 1humidity 2form. 2 t is the 2 When R 200x 4x 8x 4xrevenue 4the x2, form. (Source: The UMAP Journal) form. (Source: The UMAP Journal) air and isAis the relative in decimal (Source: UMAP Journal) z 3xy z x 3y 71. 72. In Exercis where is apparent temperature in degrees Celsius, is the 1 and 1 ingreso 122xmarginal 2temperature, 2 . para 1 t 2plant 22the where is the apparent temperature in degrees Celsius, is the A t 2 (b) marginal revenue for plant 2, . R x and (b) the marginal revenue for plant . R x he200x marginal revenue for R x , ta 1, y b) el la planta 2, and (b) the marginal for plant 2, R x . where the apparent temperature in degrees Celsius, t is the When 200x2 4x1 RR 8x1x2200x 4x x11 48x form. A is (Source: Journal) 200x 4x 4x xx1 2 424 1 air temperature, and theUMAP relative humidity in decimal 1112 .marginal 22 revenue 11x 22 222..R When h yisThe When 200x 4x 8x(Source: xplant 4x (b) for 2,The xR2. x2 Journal) 1revenue and xx2 . and findthe (a) the marginal for plant 1,UMAP 12,200x a) Hallar ­Ay­h sitthe t h5 308 ythhzwhen 530 0.80. form. 2­Ay­t 2is 4tand 2 decimal 20.80. 40.80. function f air temperature, and is the relative humidity in decimal 1, 1 h air temperature, and relative humidity in h (a) Find and when A t A h (a) Find and when and A t A h t 30 h forfind plant (a) Find and and A A 30 h 0.80. z x 2xy 3y x 3x y y 73. 74. air temperature, and is the relative humidity in decimal h devenue (a)2,theRmarginal revenue for plant 1, x2 12, , R x and find (a) the marginal revenue for plant 1, x 12, , R x 2118. Marginal Costs A company manufactures types oftypes 118. Marginal Costs A1 A company manufactures of of (Source: Marginal Costs company form. 22 118. and find (a)A the marginal revenue for manufactures plant 1,two xmarginal 12,Costs , types R two x11two (a) Find The A UMAP t and Journal) A h when t 30 and h 0.80. and (b)Costo the revenue for plant 2, R txdos 118. Marginal company manufactures two types of reasoning. (Source: The UMAP Journal) 2. form. form. (Source: The UMAP Journal) 118. marginal Una empresa fabrica tipos de estufas de (a) Find and when and A A h t 30 h 0.80. b) ¿Qué influye más sobre A, la temperatura del aire o la 2 2 form. (Source: The UMAP Journal) dcompany (b) the marginal revenue for plant 2, . R x and (b) the marginal revenue for plant 2, . R x (b) Which has a greater effect on air temperature or wood-burning stoves: a freestanding model and a fireplaceA, (b) Which has a greater effect on air temperature or wood-burning stoves: a freestanding model and a fireplaceA, manufactures two types of Which has a30greater effect on A, or wood-burning stoves: a freestanding and a fireplaceln xtemperature yah when zWhich y air temperature 76. 2 and (b) the marginal revenue for plant 2, R and x22. model (a) Find75. and t andx(b) AExplicar. 0.80. (b)Ahumedad? has greatert effect on hzA, air or wood-burning stoves: a manufactures freestanding model a offireplace118. Marginal Costs A company two types combustión de madera: el modelo autoestable y el modelo para (a) Find and andhumidity? A t Find 30humidity? 0.80. (a) and hh y 0.80. A tt hand AA Explain. hh when tt 30 Explain. insertinsert model. The cost function for producing freestanding xeffect model. The cost function producing xt freestanding (b)The Which has afor greater on temperature a freestanding model and Marginal a manufactures fireplaceA,A airh when humidity? Explain. insert model. function for producing x freestanding xA x (a) when arginal Costs A company two types of cost 118. Costs A company manufactures two types of 107. fx x, y zhas eaor tan yand 3ye z and 2xe 0.80. 77. Find 78.30temperature 118. Marginal Costs Acost company manufactures two types of humidity? Explain. insert model. The function for producing freestanding xpara (b) Which greater effect on A, air or wood-burning stoves: a ychimenea. freestanding model and a fireplaceinserción en una La función deiscosto producir andmodel fireplace-insert stoves isstoves yand fireplace-insert stoves is ystoves: humidity? Explain. stod-burning function for producing freestanding x wood-burning and fireplace-insert (b) Which has a greater effect on air temperature or stoves: a freestanding and a fireplaceA, 125. Ley de los gases ideales La ley de los gases ideales establece (b) Which has a greater effect on air temperature or a freestanding model and a fireplaceA, 125. Ideal Gas Law The Ideal Gas Law states that PV nRT, 125. Ideal Gas Law The Ideal Gas Law states that PV nRT, 125. Ideal Gas Law The Ideal Gas Law states that PV nRT, (b) Which has a greater effect on air temperature or wood-burning stoves: a freestanding model and a fireplaceA, and fireplace-insert stoves is y 108. fx x, y humidity? Explain. model. The cost function producingenx una freestanding x estufas autoestables y y for de inserción chimenea es 125. Ideal Gas Law The Ideal Gas Law states that yPV nRT, stoves is The cost insert humidity? Explain. ert model. function forCproducing freestanding x cost que elisnumber volumen, esofelmoles 5 nRT, Ppressure, zPVnRT, cos xy zisesthe 79. 80. humidity? insert model. The function for producing freestanding xIdeal where isExplain. pressure, ofnmoles ofmoles P Vesislavolume, nV where isPpressure, isVvolume, the number of of Pdonde Vpresión, narctan 125. Ideal Gas Law The Gas Law states that PV where is is volume, is the number of n humidity? Explain. insert model. The cost function for producing freestanding x 32 xy 175x 205y 1 050. C 32 xy 175x 205y 1 050. C 32 xy 175x 205y 1 050. and y fireplace-insert xof moles of where pressure, the PV numbernRT, P isThe V is volume, n isthat 125. Ideal Gas Law Ideal Gas Law states C5y32 32!xyxy1stoves 175xis 205y 050. d y205y fireplace-insert número de moles de gas, esGas una constante (la constante de RnRT, stoves is gas, is aRthat fixed constant (the gas constant), and isnRT, absolute Rgas, Tand isRThe ais fixed constant (the(the gasgas constant), isTabsolute T los where is1050. pressure, volume, the Ideal number of Law moles ofLaw V is Gas gas, aIdeal fixed constant constant), and is absoluteIn Exercis C 175x 1 205y and fireplace-insert stoves is1P11050. y fireplace-insert 125. Ideal Lawn isThe PV 125. Ideal states that PV 1 050. stoves is and 125.Gas Ideal Gas The Ideal Gas Law states that nRT, gas, isstates aLaw fixed constant (the gasLaw constant), and RGas TPVis where is pressure, is volume, is the number of moles ofabsolute P V n theFind marginal costs Ccosts xCand Cx and yCiswhen x when 80 gases) y T es temperatura absoluta. Mostrar que (a) (a) Find the the marginal costs x and y when xV isx volume, 80 C 32 xy(a) Find 175x 205y 1marginal 050. Cwhere C yconstant), 80Twhere temperature. Show that temperature. Show that gas, is a fixed constant (the gas and is absolute R temperature. Show that pressure, is the number of moles of P n respect to is pressure, is volume, is the number of moles of P V n f x, y , nfind x andofy such In Exercises 81–88, allnumber values of of moles Find xy the marginal costs C11 050. x and C y when x 80 pressure, is volume, the P isconstant Vfor 32 xy 175x 205y C 1Calcular 050. temperature. Show that 32 175x 205y gas, R iswhere a fixed (the gas constant),isand T is absolute C(a) xy 205y 050. s­Cy­x d fixed a) costos marginales y ­Cy­y cuando ylos 20. and175x y y 20. costs C x and C y when x32 and 80 and 20. temperature. Show that gas, is a constant (the gas constant), and is absolute R T gas, is a fixed constant (the gas constant), and is absolute R T f x, y 0 x, y 0 f that and simultaneously. and y 20. gas, is­V a Tfixed RTxShow ­T ­P P VTP constant (a) Find the marginal costs C x and C y when x 80 PV V y (the gas constant), and T is absolute temperature. y marginal yWhen 5 20.additional 5 109. f x, y T P V 5that 21. that 1. 1. 1. Find the marginal costs C and80 when C yadditional x production 80C xproduction (a) xxFind the costs and C yyis when xxwhich 80 Show that (b) When is required, which model ofmodel (b) required, of oftemperature. Show (b) When additional required, which marginal costs when Cistemperature. 80ofmodel temperature. Show T PCproduction V xis and ­P ­V ­T T P V and(a) y(b)Find 20. theadditional V P When production required, which model VT that T y2 2x 2y P 2 1. f x, y x xy 81. and T P V y 20. 1. and y 20. b) model Cuando requiera producción adicional, de results inresults theincost increasing at ¿qué a higher How stove results the cost increasing at modelo aathigher rate? How production is required, which ofse20. stove in T the cost increasing arate? higher rate? T How P V and ystove Vincreasing Pis required, 110. f x, y stove results in the cost at a Thigher How P Vrate? TT PP VV 1. Whenrate? additional production which model ofmodel? 2 estufa hará incrementar eldetermined costo con una tasa más alta? ¿Cócan this be determined from the cost model? can this be determined from the cost model? eWhen cost increasing a(b)higher How f x,126. yMarginal x1. xyUtility y2 The 5xde y function 82. Marginal Utility The utility fydx,es can this be from the cost utility function a a x,isf yx,a yis is 1. Marginal Utility The utility V126. T 126. P Uyisfuna 126. Utilidad marginal La función utilidad UUfunction 5Uffsx,x,U additionalatproduction is required, which model of (b) When additional production is required, which model of 1. (b)results When additional production required, which model of can this be determined fromatisthe cost model? 126. Marginal Utility The utility function a y V T P T P V stove in the cost increasing a higher rate? How Px, ymeasure V Tmeasure puede determinarse esto athe del modelo costo? mined the in cost of utility satisfaction) derived bypersona a by person 2 the 2 utility of4xy the utility (or (or satisfaction) a person 126. Marginal Utility Thecentury, utility function a measure U testf83. of the satisfaction) by a person medida (o(or satisfacción) que obtiene unaderived stovefrom results themodel? cost increasing at aPsychology higher rate? How stove results in the cost increasing at rate? How 119.mo Psychology in the century, an del intelligence test test 119. Psychology Early intwentieth twentieth century, an intelligence f x,isyde xutilidad y (or 4x 16y 3 derived Early inpartir the twentieth an intelligence results inEarly the cost increasing at aa higher higher rate? How measure oflaThe the utility satisfaction) derived by a person can Psychology thisstove be119. determined from the cost model? 119. Early in the twentieth century, an intelligence test 126.derived Marginal Utility utility function is aSuppose Uy.two f products x,x yand from the consumption of two products the y. from the consumption of two products and Suppose the the W R I T I N x y. measure of the utility (or satisfaction) by a person from the consumption of and Suppose x y. por el consumo de dos productos y Suponer que la función x can this be determined from the cost model? can this be determined from the cost model? 126. Marginal Utility The utility function is a products Ux2 f xy x, y ofyutility called the Stanford-Binet Test (more commonly known asknown theasIQ Utility The function aa called the the Stanford-Binet Testcost (more commonly known the IQ U ff Suppose x, the twentieth century, an intelligence test 2two called Stanford-Binet Test commonly as 126. the IQMarginal 119. Psicología Recientemente en el siglo xx se(more desarrolló una pruecan this be determined from the model? 126. Marginal Utility The utility function is the U2a y. x, yy is from theyutility consumption and f the x, 84. called the Stanford-Binet Test (more commonly known asproducts the IQ x and measure ofutilidad (or satisfaction) person 2 derived 2 2xby 2 2 2is1 119. Psychology Early in the twentieth century, an intelligence test utility function is U 5x xy 3y . utility function U 5x xy 3y . from the consumption of two Suppose the y. utility function is U 5x xy 3y . de es U 5 25x xy 2 3y . measure of the utility (or satisfaction) derived by a person 111. Let test) was developed. In this test, an individual’s mental age is M 2 2 measure of the utility (or satisfaction) derived by a person test) was developed. In this test, an individual’s mental age is M net Test (more commonly known as the IQ test) was developed. In century, thisde test,Stanford-Binet anintelligence individual’s(más mental age M ismeasure ba decentury, inteligencia llamada la Prueba ychology Early in the119. twentieth an intelligence test of the utility (or5x satisfaction) bythe a person Psychology Early in the twentieth an test utility function is U . 119. Psychology Early inIn the twentieth an intelligence test) was developed. this test, an century, individual’s mental M is3y 2.from the consumption of two Suppose xxyand 3y y.derived 2 age test 1 y. 1 products called the Stanford-Binet Test (more commonly known as the IQ utility function is U 5x xy from the consumption of two products and Suppose x proc divided by the individual’s chronological age and the quotient C from the consumption of two products and Suppose the divided by the individual’s chronological age and the quotient C x y. nled thisthe test, an individual’s mental age is M (a) Determine the marginal utility of product x. divided by the individual’s chronological age and the quotient C (a) Determine the marginal utility of product x. conocida como la prueba IQ). En esta prueba, una edad mental (a) Determine the marginal utility of product f x, y xy x. 85. Stanford-Binet Test (more commonly known asTest thechronological IQ a) la 5x utilidad delproduct producto x. called the Stanford-Binet (more as the from theisconsumption products the 2 of two x and y. called theby Stanford-Binet Test (more commonly commonly known asquotient the IQ IQ divided the individual’s age Cknown andMthe utility function xymarginal 3y 2. of (a) Determinar Determine x. Suppose ymarginal test)and was developed. In this an individual’s mental age is 2U x the 22 utility 22 utility function isof UIQ. 5xx.2 utility xy (b)function 3y . is multiplied bytest, 100. The result is cronológica the IQ. MIQ. is isMmultiplied by 100. The result isindividual’s the individual’s U 5x xy 3y . ual’s chronological the quotient is multiplied by 100. The result is the individual’s individual es dividida entre la edad individual C, (a) Determine the marginal utility product t) was developed. Inage thisC test, an individual’s mental age is M utility function is test) was developed. In this test, an individual’s mental age is Uthe marginal 5xthe xy 3yof .product 112. Ske Determine utility y. y. y. (b) (b) Determine the marginal utility of product Determine marginal utility of product test) was developed. InThe thisresult test, an individual’s is multiplied by 100. isage theCindividual’s IQ.age M is b) la del producto 3 utilidad 3product dividedIQ. by individual’s chronological the mental quotient (a) Determine the utilitymarginal ofyutility (b)Determinar the marginal of x. product y. y. fDetermine x,product y marginal 3x 12xy 86. The is individual’s the individual’s ydivided elthe cociente es multiplicado por 100. Eland resultado es elutility IQ indiidedresult by the chronological age and quotient Cindividual’s divided by the chronological age and the quotient C (a) Determine the marginal utility of x. marginal x an MtheM (a) Determine the utility of product x. M (b) Determine the marginal of product y. by the individual’s chronological age and the quotient C (a) Determine the marginal utility of product x. (c) When and should a person consume one x 2 y 3, (c) When and should a person consume one x 2 y 3, (c) When and should a person consume one x 2 y 3, is multiplied byM, 100. The is the individual’s IQ. MM, IQ 100 IQCby M, Cresult 100 IQ CThe 100 c) Si debe consumir una unidad másone de xthe 5 marginal 2 eyx 2y2 5 3, xy y 2y¿se of (b) Determine utility product y. (c) When and should a person consume x 3, vidual. multiplied by 100. The resultis is the individual’s IQ. f x, y 87. multiplied 100. result is the individual’s IQ. f IQ M, C 100 C C C is multiplied byC100. The(c) result is the IQ.3, should (b) yDetermine the marginal of product y. (b) Determine the marginal utility product y. more of of x orof y? y? y? unitproduct of product or one more unitproduct of product xone When a person utility consume one more x individual’s 2 and more unit of product or one more unit ofel product x more (b) producto Determine the marginal of product y.unit xunit oof una unidad más de producto y? Explicar M more unit or one more unit ofone product y? x1utility 2 product 2 (c) When and should a person consume x 2 y 3, M f x, y ln x y 88. M IQ M, C Find Find 100 113. Ske your reasoning. the partial derivatives ofunit respect and with IQofwith your reasoning. the100 partial derivatives with respect to and more ofIQ orxMone unitwith product xto y? reasoning. Find the partial derivatives ofproduct with respect tomore IQ M and M (c) When and should aExplain person consume oneshould 2M ywith 3,of(c) When and aa person consume one xx Explain 22Explain y your 3, razonamiento. M, C 100 IQ M, C Cthe (c) When and should person consume Explain your reasoning. Find partial derivatives of IQ with respect to M and with M, Cwith 100 more unit of product unit of product x ory one3, more y? one C with respect to IQ fx is C respect to Evaluate the partial derivatives at the C.to respect Evaluate theyour partial derivatives at point the point C. C. Explain reasoning. atives of IQ M and respect to Evaluate the partial derivatives at the point C more unit of product or one more unit of product x y? more unit of product or one more unit of product x y? CAS (d) Use a computer algebra system to graph the function. CAS (d) Use a computer algebra system to graph the function. CAS (d) Use a computer algebra system to graph the function. more of product or system one unit of product y? the respect to C. Evaluate the partial derivatives at the point CAS d) Utilizar un sistema algebraico pormore computadora y function. represenIn 89–92, use ax computer algebra system to find Explain your Find the partial derivatives ofand with respect toresult. with IQ M and CAS (d)Exercises Use unit areasoning. computer algebra to graph the pos and interpret the result. (Source: Adapted from 12, 10 and interpret the result. (Source: Adapted from 12, 10 ate the partial derivatives at the point interpret the (Source: Adapted from 12, 10 Explain your reasoning. nd the partial derivatives of Find with respect to CAS with IQ12, M parciales Explain your reasoning. the partial derivatives of respect to IQ M Interpret thela marginal utilities oflasproducts andx and x maryx and Interpret the the marginal utilities of of products y y (d) Use awith computer algebra system to graph the function. Interpret marginal utilities products Explain your Find the partial derivatives of with respect toAdapted and with IQde M aand andderivadas interpret the result. from 10Evaluate Encontrar las IQ (Source: con M ywith con gráficamente función. utilidades and second partial derivatives ofofthe the function. respect to C. the partial derivatives atrespecto the point Interpret thereasoning. marginal utilities products and x Determine y CASFourth (d) Usefirst axtar computer algebra system toInterpretar graph function. Bernstein/Clark-Stewart/Roy/Wickens, Psychology, Fourth Bernstein/Clark-Stewart/Roy/Wickens, Psychology, etpect thetoresult. (Source: from Bernstein/Clark-Stewart/Roy/Wickens, Psychology, Fourth the Adapted partial derivatives at the point C. Evaluate 114. Ske respect to Evaluate the partial derivatives at the point C. graphically. graphically. Interpret the marginal utilities of products and y graphically. respect to Evaluate the partial derivatives at the point C. CAS (d) Use a computer algebra system to graph the function. Bernstein/Clark-Stewart/Roy/Wickens, Psychology, Fourth CAS (d) Use a computer algebra system to graph the function. a C. Evaluar derivadas parciales en el punto ginales de productos x y of y system con una gráfica. x and yto 0 and whether exist values such CAS and interpret the las result. (Source: Adapted from (12, 10) 12, 10respecto (d) Use computer algebra graph thefx function. graphically. Interpret theathere marginal utilities of products and xthat yx, y Edition) Edition) wart/Roy/Wickens, Fourth Edition) the result. (Source: Adaptedthe from 2, 10 and interpretPsychology, tive and interpret result. (Source: Adapted from 12, 10 graphically. and interpret the result. (Source: Adapted from 12, 10 Interpret the marginal utilities of products and x y Edition) Interpret the marginal utilities of products and x y 127. Modeling Data Per capita consumptions (in gallons) of 127. Modeling Data Per capita consumptions (in gallons) of e interpretar el resultado. (Fuente: Adaptado de Bernstein/Clark127. Modeling Data Per capita consumptions (in gallons) of fModelo x,Interpret y matemático 0 Data simultaneously. Bernstein/Clark-Stewart/Roy/Wickens, Psychology, Fourth the marginal utilities of products and yof xconsumos yModeling graphically. 127. Per En capita consumptions (inlosgallons) 127. la tabla se muestran rnstein/Clark-Stewart/Roy/Wickens, Psychology, Fourth Bernstein/Clark-Stewart/Roy/Wickens, Psychology, Fourth 120. Marginal Productivity Consider the Cobb-Douglas produc120. Marginal Productivity Consider the Cobb-Douglas produc120. Marginal Productivity Consider the Cobb-Douglas producBernstein/Clark-Stewart/Roy/Wickens, Psychology, graphically. different typestypes of types milk in States from 1999 through different of milk inUnited the United States from 1999 through 127.Psicología, Modeling Data Per graphically. capita Fourth consumptions (in gallons) of different ofthe milk in the United States from 1999 through 115. If f Edition) graphically. 120.Steward/Roy/Wickens, Marginal Productivity Consider4a. theed.) Cobb-Douglas produc0.7 0.3 different types of milk in United States 1999 0.7 0.3 0.3 per (en galones) dethe diferentes tipos de leche en Estados 0.7 Edition) 127. Modeling Data Per capita consumptions (in gallons) ofofthrough tion tion function and . milk f x, different yf x,f y200x xWhen 1000 function andand yWhen 200x x x States 1000 yition) Consider the Cobb-Douglas producfcápita x, y2005 xshown sec y inshown fConsumption x,Consumption yfrom 25 x 2 flavored y 2 milk, 89. 90. tion function y. inWhen .the x,0.7ytypes 200x 1000 Edition) con 0.3y 2005 are theintable. Consumption ofgallons) flavored milk, are shown the table. flavored of United from 1999 through 2005 are in the table. of milk, 127. Modeling Data Per capita consumptions (in gallons) of tion function When and y . f x, y 200x x 1000 127. Modeling Data Per capita consumptions (in of 120. Productividad marginal Considerar la función deofproducción Marginal Productivity Consider the Cobb-Douglas produc127. Data1999 Perthe capita consumptions (in gallons) of y 2005 are shown in table. Consumption of milk, Unidos hasta 2005. Elfrom consumo deflavored leche light 0.7 0.3 120. types ofdesde milk in the United States 1999 through find (a) the marginal productivity labor, y Cobb-Douglas 500, flabor, x, fdifferent findfind (a) theare marginal productivity of labor, yProductivity and y . When 200x x Marginal 1000 (a) the marginal productivity of milk y 500, 500, fx, flavored x,Modeling arginal Productivity Consider the producpart 120. Consider the Cobb-Douglas producplain reduced-fat milk, and plain light and skim milks are plain reduced-fat milk, and plain light and skim milks are 2005 shown in the table. Consumption of milk, x xy plain reduced-fat milk, and plain light and skim milks are 0.7 0.3 120. Marginal Productivity Consider the Cobb-Douglas produc0.7 0.3 different types of in the United States from 1999 through find (a) the marginal productivity of labor, y 500, f x, different types of milk in the United States from 1999 through de f 200x sx,the yd marginal 5productivity y .ofSi x of 5 1capital, 000 y y. y f5 500, tionof function When y 200x . 0.7 fand x,the y(b)marginal x capital, 1000 different milk in the United States from 1999 through plain milk, andgrasa plainy92. light milks are descremada, leche baja en leche entera se representa f x,reduced-fat ytypes lnoftable. f flavored x,and y skim 91. 0.7 Cobb-Douglas 0.3 0.3 2005 are shown in represented the Consumption of milk, and (b) fand the marginal productivity y.light f x, and (b) productivity of capital, f y. 2 2 by 0.7 0.3 nmarginal functionproductivity When and y . f x, y 200xlabor, x 1000 tion function When and y . f x, y 200x x 1000 represented by the variables and respectively. x, y, z, represented by the variables and respectively. x, y, z, plain reduced-fat milk, and plain and skim milks are the variables and respectively. x, y, z, yytable. xin xrespectively. y milk, tion function When and y of . capital, f x, y productivity 200x xf are 2005 shown in table. Consumption ofbyflavored and (b) theproductividad marginal 2005 are shown the Consumption hallar a)(a) la marginal del trabajo, y b) la the plain find the marginal productivity of labor, y f 500, fy.1000 x, 2005 arevariables shown in the table. Consumption of flavored milk, represented variables andof por las x,the y plain z,milk, respectivamente. (Fuente: Dex, y,skim z,flavored reduced-fat milk, and light milks areU.S. productivity y. (a)capital, the marginal ofAbout labor, 500, find of f It x,be find (a) the marginal productivity ofnumber yy productivity 500, ff x,applicants x,y,to and (Source: U.S. Department of and Agriculture) 121. Think About ItAbout Let number oflabor, applicants a toand NLet (Source: U.S. Department of Agriculture) 121. Think It be thethe number of of applicants ato plain Nthe represented variables respectively. z, (Source: U.S. Department ofand Agriculture) 121. Think Let the aplain N by find (a) productivity labor, 500, plain reduced-fat light and skim milks are reduced-fat milk, and plain light skim milks are marginal del capital, ­fy­y. and121. (b)productividad the marginal productivity capital, fbe y. plain reduced-fat milk, and plain light and skim milks are (Source: U.S. Department of Agriculture) Think About It the Letmarginal be the number ofofapplicants tox,amilk,represented partment of Agriculture) N of by the variables and respectively. x, y, z, dt (b) thethe marginal productivity of capital, fp the y. pproductivity and (b) the of capital, ffhousing y. university, charge for food and housing at of theAgriculture) university, the charge for food and atuniversity, the university, U.S. Department number of applicants to amarginal N be university, the charge for food and housing at the p(Source: and (b) the marginal productivity of capital, y. represented by theuniversity, variables respectively. x, y, and by z, the represented variables x, represented by the and z, respectively. x, y, y, and z, respectively. university, charge fornumber food de and housing ata the university, p the (Source: U.S. Department of variables Agriculture) 121. Think About It tand Let be the of applicants toof auniversiNthe N 121. Para pensar el número and the tuition. is the a is function of and such that N p ofuna tand tuition. aisaspirantes function thatof tSea N p U.S. t asuch e forAbout food and at the university, and the tuition. a function and such that t N p t 1999 2000 2001 2002 2003 2004 2005 Año 1999 2000 2001 2002 2003 Año (Source: Department Agriculture) ink It housing Let N121. be the number of applicants to a 1999 2000 2001 2002 20032004 20042005 2005 Año (Source: U.S. Department of Agriculture) Think About It Let be number of applicants to N (Source: U.S. Department of Agriculture) 121. dad, Think About It Let be the number of applicants to a N andppthe the tuition. is a function of and such that tel N p t 1999 2000 2001 2002 2003 2004 2005 Año university, charge for food and housing at the university, costo y< alojamiento en la is universiand What information gained by N ppN