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914 CAPÍTULO 13 Funciones de varias variables
Para pensar En los ejercicios 1 a 4, utilizar la gráfica de la super-ficie para determinar el signo de la derivada parcial indicada.
1. 2.
3. 4.
En los ejercicios 5 a 8, explicar si se debe usar o no la regla delcociente para encontrar la derivada parcial. No derivar.
En los ejercicios 9 a 40, hallar las dos derivadas parciales deprimer orden.
En los ejercicios 41 a 44, utilizar la definición de derivadas par-ciales empleando límites para calcular y
En los ejercicios 45 a 52, evaluar y en el punto dado.
En los ejercicios 53 y 54, calcular las pendientes de la superficieen las direcciones de x y de y en el punto dado.
53. 54.
En los ejercicios 55 a 58, utilizar un sistema algebraico por compu-tadora y representar gráficamente la curva en la intersección dela superficie con el plano. Hallar la pendiente de la curva en elpunto dado.
Superficie Plano Punto
55.
56.
57.
58. �1, 3, 0�x � 1z � 9x2 � y 2
�1, 3, 0�y � 3z � 9x2 � y 2
�2, 1, 8�y � 1z � x2 � 4y 2
�2, 3, 6�x � 2z � �49 � x2 � y 2
yx 3 3
7
6
4
3
5
2
z
yx
2
4
2
z
��2, 1, 3��1, 1, 2�h�x, y� � x2 � y 2g�x, y� � 4 � x2 � y 2
fyfx
fy�x, y�.fx�x, y�
fx��1, �1�fy�4, 1�fy��1, �2�fx�4, 1�
y
x
5
5
2
−5
z
1133..33 Ejercicios
5. 6.
7. 8.y
xyx2 1x
xyx2 1
xx yx2 1y
x yx2 1
9. 10.
11. 12.
13. 14.
15. 16.
17. 18.
19. 20.
21. 22.
23. 24.
25. 26.
27. 28.
29. 30.
31. 32.
33. 34.
35. 36.
37. 38.
39.
40. f x, yy
x
2t 1 dtx
y
2t 1 dt
f x, yy
x
t 2 1 dt
z cosh xy2z senh 2x 3y
z cos x2 y 2z ey sen xy
z sen 5x cos 5yz tan 2x y
z sen x 2yz cos xy
f x, y 2x y3f x, y x2 y 2
g x, y ln x2 y 2h x, y e x2 y2
zxy
x2 y 2zx2
2y3y 2
x
z ln x yx y
z ln x2 y2
z ln xyz ln xy
z yey xz x2e2y
z ex yz exy
z y3 2xy2 1z x2 4xy 3y 2
z 2y2 xz x y
f x, y 4x3y 2f x, y x2y3
f x, y x2 2y 2 4f x, y 2x 5y 3
41. 42.
43. 44. f x, y1
x yf x, y x y
f x, y x2 2xy y 2f x, y 3x 2y
y
45.
46.
47.
48.
49.
50.
51.
52. 1, 1f x, y2xy
4x2 5y 2,
2, 2f x, yxy
x y,
1, 1f x, y arccos xy,
2, 2f x, y arctan yx,
f x, y sen xy, 2, 4
f x, y cos 2x y , 4
, 3
f x, y e x cos y, 0, 0
f x, y ey sen x, , 0
CAS
Larson-13-03.qxd 26/2/10 14:19 Página 914
SECCIÓN 13.3 Derivadas parciales 915
En los ejercicios 59 a 64, calcular las derivadas parciales deprimer orden con respecto a x, y y z.
En los ejercicios 65 a 70, evaluar fx, fy y fz en el punto dado.
En los ejercicios 71 a 80, calcular las cuatro derivadas parcialesde segundo orden. Observar que las derivadas parciales mixtasde segundo orden son iguales.
En los ejercicios 81 a 88, para f(x, y), encontrar todos los valoresde x y y, tal que fx(x, y) = 0 y fy(x, y) = 0 simultáneamente.
En los ejercicios 89 a 92, utilizar un sistema algebraico por compu-tadora y hallar las derivadas parciales de primero y segundoorden de la función. Determinar si existen valores de x y y talesque y simultáneamente.
En los ejercicios 93 a 96, mostrar que las derivadas parcialesmixtas fxyy, fyxy y fyyx son iguales.
Ecuación de Laplace En los ejercicios 97 a 100, mostrar que lafunción satisface la ecuación de Laplace
97. 98.
99. z 5 ex sen y 100.
Ecuación de ondas En los ejercicios 101 a 104, mostrar que lafunción satisface la ecuación de ondas
101. z 5 sen(x 2 ct) 102.
103. 104. z 5 sen wct sen wx
Ecuación del calor En los ejercicios 105 y 106, mostrar que lafunción satisface la ecuación del calor
105. 106.
En los ejercicios 107 y 108, determinar si existe o no una funciónf(x, y) con las derivadas parciales dadas. Explicar el razona-miento. Si tal función existe, dar un ejemplo.
En los ejercicios 109 y 110, encontrar la primera derivada par-cial con respecto a x.
z 5 e2t sin xc
z 5 e2t cos xc
z /t 5 c2x2z /x2c.
z 5 lnsx 1 ctdz 5 coss4x 1 4ctd
2z /t 2 5 c2x2z /x2c.
z 5 arctan yx
z 512sey 2 e2ydsin xz 5 5xy
2z /x2 1 2z /y2 5 0.
fyxx, yc 5 0fxxx, yc 5 0
In Exercises 59–64, find the first partial derivatives with respectto and
59.
60.
61. 62.
63.
64.
In Exercises 65–70, evaluate and at the given point.
65.
66.
67.
68.
69.
70.
In Exercises 71–80, find the four second partial derivatives.Observe that the second mixed partials are equal.
71. 72.
73. 74.
75. 76.
77. 78.
79. 80.
In Exercises 81–88, for find all values of and suchthat and simultaneously.
81.
82.
83.
84.
85.
86.
87.
88.
In Exercises 89–92, use a computer algebra system to find thefirst and second partial derivatives of the function. Determinewhether there exist values of and such that and
simultaneously.
89. 90.
91. 92.
In Exercises 93–96, show that the mixed partial derivatives and are equal.
93.
94.
95.
96.
Laplace’s Equation In Exercises 97–100, show that the functionsatisfies Laplace’s equation
97. 98.
99. 100.
Wave Equation In Exercises 101–104, show that the functionsatisfies the wave equation
101. 102.
103. 104.
Heat Equation In Exercises 105 and 106, show that the functionsatisfies the heat equation
105. 106.
In Exercises 107 and 108, determine whether there exists a function with the given partial derivatives. Explain yourreasoning. If such a function exists, give an example.
107.
108.
In Exercises 109 and 110, find the first partial derivative withrespect to
109.
110. f x, y, z x senh yz
y2 2 y 1 z
f x, y, z tan y2z ez2 y 2 z
x.
fy x, y x 4yfx x, y 2x y,
fy x, y 2 sen 3x 2yfx x, y 3 sen 3x 2y ,
f x, y
z e t sen xc
z e t cos xc
z/ t c2 2z/ x2 .
z sen ct sen xz ln x ct
z cos 4x 4ctz sen x ct
2z/ t 2 c2 2z/ x2 .
z arctan yx
z ex sen y
z 12 ey e y sen xz 5xy
2z/ x2 1 2z/ y2 0.
f x, y, z2z
x y
f x, y, z e x sen yz
f x, y, z x2 3xy 4yz z3
f x, y, z xyz
fyyxfyxy,fxyy,
f x, yxy
x yf x, y ln
xx2 y2
f x, y 25 x2 y 2f x, y x sec y
fy x, y 0fx x, y 0yx
f x, y ln x2 y 2 1
f x, y ex2 xy y2
f x, y 3x3 12xy y3
f x, y1x
1y
xy
f x, y x2 xy y2
f x, y x2 4xy y 2 4x 16y 3
f x, y x2 xy y2 5x y
f x, y x2 xy y2 2x 2y
fy x, y 0fx x, y 0yxf x, y ,
z arctan yx
z cos xy
z 2xey 3ye xz ex tan y
z ln x yz x2 y 2
z x4 3x2y 2 y4z x2 2xy 3y 2
z x2 3y2z 3xy2
1, 2, 1f x, y, z 3x2 y2 2z2,
0, 2
, 4f x, y, z z sen x y ,
3, 1, 1f x, y, zxy
x y z,
f x, y, zxyz
, 1, 1, 1
2, 1, 2f x, y, z x2y3 2xyz 3yz,
f x, y, z x3yz2, 1, 1, 1
fzfy,fx,
G x, y, z1
1 x2 y 2 z 2
F x, y, z ln x2 y 2 z 2
w7xz
x yw x2 y 2 z2
f x, y, z 3x2y 5xyz 10yz 2
H x, y, z sen x 2y 3z
z.y,x,
13.3 Partial Derivatives 915
CAS
111. Let be a function of two variables and Describe theprocedure for finding the first partial derivatives.
112. Sketch a surface representing a function of two variablesand Use the sketch to give geometric interpretations of
and
113. Sketch the graph of a function whose derivativeis always negative and whose derivative is always
positive.
114. Sketch the graph of a function whose deriva-tives and are always positive.
115. If is a function of and such that and arecontinuous, what is the relationship between the mixedpartial derivatives? Explain.
fyxfxyyxf
fyfx
z f x, y
fyfx
z f x, y
f y.f xy.x
f
y.xf
WRITING ABOUT CONCEPTS
1053714_1303_pg 915.qxp 10/30/08 8:30 AM Page 915
In Exercises 59–64, find the first partial derivatives with respectto and
59.
60.
61. 62.
63.
64.
In Exercises 65–70, evaluate and at the given point.
65.
66.
67.
68.
69.
70.
In Exercises 71–80, find the four second partial derivatives.Observe that the second mixed partials are equal.
71. 72.
73. 74.
75. 76.
77. 78.
79. 80.
In Exercises 81–88, for find all values of and suchthat and simultaneously.
81.
82.
83.
84.
85.
86.
87.
88.
In Exercises 89–92, use a computer algebra system to find thefirst and second partial derivatives of the function. Determinewhether there exist values of and such that and
simultaneously.
89. 90.
91. 92.
In Exercises 93–96, show that the mixed partial derivatives and are equal.
93.
94.
95.
96.
Laplace’s Equation In Exercises 97–100, show that the functionsatisfies Laplace’s equation
97. 98.
99. 100.
Wave Equation In Exercises 101–104, show that the functionsatisfies the wave equation
101. 102.
103. 104.
Heat Equation In Exercises 105 and 106, show that the functionsatisfies the heat equation
105. 106.
In Exercises 107 and 108, determine whether there exists a function with the given partial derivatives. Explain yourreasoning. If such a function exists, give an example.
107.
108.
In Exercises 109 and 110, find the first partial derivative withrespect to
109.
110. f x, y, z x senh yz
y2 2 y 1 z
f x, y, z tan y2z ez2 y 2 z
x.
fy x, y x 4yfx x, y 2x y,
fy x, y 2 sen 3x 2yfx x, y 3 sen 3x 2y ,
f x, y
z e t sen xc
z e t cos xc
z/ t c2 2z/ x2 .
z sen ct sen xz ln x ct
z cos 4x 4ctz sen x ct
2z/ t 2 c2 2z/ x2 .
z arctan yx
z ex sen y
z 12 ey e y sen xz 5xy
2z/ x2 1 2z/ y2 0.
f x, y, z2z
x y
f x, y, z e x sen yz
f x, y, z x2 3xy 4yz z3
f x, y, z xyz
fyyxfyxy,fxyy,
f x, yxy
x yf x, y ln
xx2 y2
f x, y 25 x2 y 2f x, y x sec y
fy x, y 0fx x, y 0yx
f x, y ln x2 y 2 1
f x, y ex2 xy y2
f x, y 3x3 12xy y3
f x, y1x
1y
xy
f x, y x2 xy y2
f x, y x2 4xy y 2 4x 16y 3
f x, y x2 xy y2 5x y
f x, y x2 xy y2 2x 2y
fy x, y 0fx x, y 0yxf x, y ,
z arctan yx
z cos xy
z 2xey 3ye xz ex tan y
z ln x yz x2 y 2
z x4 3x2y 2 y4z x2 2xy 3y 2
z x2 3y2z 3xy2
1, 2, 1f x, y, z 3x2 y2 2z2,
0, 2
, 4f x, y, z z sen x y ,
3, 1, 1f x, y, zxy
x y z,
f x, y, zxyz
, 1, 1, 1
2, 1, 2f x, y, z x2y3 2xyz 3yz,
f x, y, z x3yz2, 1, 1, 1
fzfy,fx,
G x, y, z1
1 x2 y 2 z 2
F x, y, z ln x2 y 2 z 2
w7xz
x yw x2 y 2 z2
f x, y, z 3x2y 5xyz 10yz 2
H x, y, z sen x 2y 3z
z.y,x,
13.3 Partial Derivatives 915
CAS
111. Let be a function of two variables and Describe theprocedure for finding the first partial derivatives.
112. Sketch a surface representing a function of two variablesand Use the sketch to give geometric interpretations of
and
113. Sketch the graph of a function whose derivativeis always negative and whose derivative is always
positive.
114. Sketch the graph of a function whose deriva-tives and are always positive.
115. If is a function of and such that and arecontinuous, what is the relationship between the mixedpartial derivatives? Explain.
fyxfxyyxf
fyfx
z f x, y
fyfx
z f x, y
f y.f xy.x
f
y.xf
WRITING ABOUT CONCEPTS
1053714_1303_pg 915.qxp 10/30/08 8:30 AM Page 915
In Exercises 59–64, find the first partial derivatives with respectto and
59.
60.
61. 62.
63.
64.
In Exercises 65–70, evaluate and at the given point.
65.
66.
67.
68.
69.
70.
In Exercises 71–80, find the four second partial derivatives.Observe that the second mixed partials are equal.
71. 72.
73. 74.
75. 76.
77. 78.
79. 80.
In Exercises 81–88, for find all values of and suchthat and simultaneously.
81.
82.
83.
84.
85.
86.
87.
88.
In Exercises 89–92, use a computer algebra system to find thefirst and second partial derivatives of the function. Determinewhether there exist values of and such that and
simultaneously.
89. 90.
91. 92.
In Exercises 93–96, show that the mixed partial derivatives and are equal.
93.
94.
95.
96.
Laplace’s Equation In Exercises 97–100, show that the functionsatisfies Laplace’s equation
97. 98.
99. 100.
Wave Equation In Exercises 101–104, show that the functionsatisfies the wave equation
101. 102.
103. 104.
Heat Equation In Exercises 105 and 106, show that the functionsatisfies the heat equation
105. 106.
In Exercises 107 and 108, determine whether there exists a function with the given partial derivatives. Explain yourreasoning. If such a function exists, give an example.
107.
108.
In Exercises 109 and 110, find the first partial derivative withrespect to
109.
110. f x, y, z x senh yz
y2 2 y 1 z
f x, y, z tan y2z ez2 y 2 z
x.
fy x, y x 4yfx x, y 2x y,
fy x, y 2 sen 3x 2yfx x, y 3 sen 3x 2y ,
f x, y
z e t sen xc
z e t cos xc
z/ t c2 2z/ x2 .
z sen ct sen xz ln x ct
z cos 4x 4ctz sen x ct
2z/ t 2 c2 2z/ x2 .
z arctan yx
z ex sen y
z 12 ey e y sen xz 5xy
2z/ x2 1 2z/ y2 0.
f x, y, z2z
x y
f x, y, z e x sen yz
f x, y, z x2 3xy 4yz z3
f x, y, z xyz
fyyxfyxy,fxyy,
f x, yxy
x yf x, y ln
xx2 y2
f x, y 25 x2 y 2f x, y x sec y
fy x, y 0fx x, y 0yx
f x, y ln x2 y 2 1
f x, y ex2 xy y2
f x, y 3x3 12xy y3
f x, y1x
1y
xy
f x, y x2 xy y2
f x, y x2 4xy y 2 4x 16y 3
f x, y x2 xy y2 5x y
f x, y x2 xy y2 2x 2y
fy x, y 0fx x, y 0yxf x, y ,
z arctan yx
z cos xy
z 2xey 3ye xz ex tan y
z ln x yz x2 y 2
z x4 3x2y 2 y4z x2 2xy 3y 2
z x2 3y2z 3xy2
1, 2, 1f x, y, z 3x2 y2 2z2,
0, 2
, 4f x, y, z z sen x y ,
3, 1, 1f x, y, zxy
x y z,
f x, y, zxyz
, 1, 1, 1
2, 1, 2f x, y, z x2y3 2xyz 3yz,
f x, y, z x3yz2, 1, 1, 1
fzfy,fx,
G x, y, z1
1 x2 y 2 z 2
F x, y, z ln x2 y 2 z 2
w7xz
x yw x2 y 2 z2
f x, y, z 3x2y 5xyz 10yz 2
H x, y, z sen x 2y 3z
z.y,x,
13.3 Partial Derivatives 915
CAS
111. Let be a function of two variables and Describe theprocedure for finding the first partial derivatives.
112. Sketch a surface representing a function of two variablesand Use the sketch to give geometric interpretations of
and
113. Sketch the graph of a function whose derivativeis always negative and whose derivative is always
positive.
114. Sketch the graph of a function whose deriva-tives and are always positive.
115. If is a function of and such that and arecontinuous, what is the relationship between the mixedpartial derivatives? Explain.
fyxfxyyxf
fyfx
z f x, y
fyfx
z f x, y
f y.f xy.x
f
y.xf
WRITING ABOUT CONCEPTS
1053714_1303_pg 915.qxp 10/30/08 8:30 AM Page 915
In Exercises 59–64, find the first partial derivatives with respectto and
59.
60.
61. 62.
63.
64.
In Exercises 65–70, evaluate and at the given point.
65.
66.
67.
68.
69.
70.
In Exercises 71–80, find the four second partial derivatives.Observe that the second mixed partials are equal.
71. 72.
73. 74.
75. 76.
77. 78.
79. 80.
In Exercises 81–88, for find all values of and suchthat and simultaneously.
81.
82.
83.
84.
85.
86.
87.
88.
In Exercises 89–92, use a computer algebra system to find thefirst and second partial derivatives of the function. Determinewhether there exist values of and such that and
simultaneously.
89. 90.
91. 92.
In Exercises 93–96, show that the mixed partial derivatives and are equal.
93.
94.
95.
96.
Laplace’s Equation In Exercises 97–100, show that the functionsatisfies Laplace’s equation
97. 98.
99. 100.
Wave Equation In Exercises 101–104, show that the functionsatisfies the wave equation
101. 102.
103. 104.
Heat Equation In Exercises 105 and 106, show that the functionsatisfies the heat equation
105. 106.
In Exercises 107 and 108, determine whether there exists a function with the given partial derivatives. Explain yourreasoning. If such a function exists, give an example.
107.
108.
In Exercises 109 and 110, find the first partial derivative withrespect to
109.
110. f x, y, z x senh yz
y2 2 y 1 z
f x, y, z tan y2z ez2 y 2 z
x.
fy x, y x 4yfx x, y 2x y,
fy x, y 2 sen 3x 2yfx x, y 3 sen 3x 2y ,
f x, y
z e t sen xc
z e t cos xc
z/ t c2 2z/ x2 .
z sen ct sen xz ln x ct
z cos 4x 4ctz sen x ct
2z/ t 2 c2 2z/ x2 .
z arctan yx
z ex sen y
z 12 ey e y sen xz 5xy
2z/ x2 1 2z/ y2 0.
f x, y, z2z
x y
f x, y, z e x sen yz
f x, y, z x2 3xy 4yz z3
f x, y, z xyz
fyyxfyxy,fxyy,
f x, yxy
x yf x, y ln
xx2 y2
f x, y 25 x2 y 2f x, y x sec y
fy x, y 0fx x, y 0yx
f x, y ln x2 y 2 1
f x, y ex2 xy y2
f x, y 3x3 12xy y3
f x, y1x
1y
xy
f x, y x2 xy y2
f x, y x2 4xy y 2 4x 16y 3
f x, y x2 xy y2 5x y
f x, y x2 xy y2 2x 2y
fy x, y 0fx x, y 0yxf x, y ,
z arctan yx
z cos xy
z 2xey 3ye xz ex tan y
z ln x yz x2 y 2
z x4 3x2y 2 y4z x2 2xy 3y 2
z x2 3y2z 3xy2
1, 2, 1f x, y, z 3x2 y2 2z2,
0, 2
, 4f x, y, z z sen x y ,
3, 1, 1f x, y, zxy
x y z,
f x, y, zxyz
, 1, 1, 1
2, 1, 2f x, y, z x2y3 2xyz 3yz,
f x, y, z x3yz2, 1, 1, 1
fzfy,fx,
G x, y, z1
1 x2 y 2 z 2
F x, y, z ln x2 y 2 z 2
w7xz
x yw x2 y 2 z2
f x, y, z 3x2y 5xyz 10yz 2
H x, y, z sen x 2y 3z
z.y,x,
13.3 Partial Derivatives 915
CAS
111. Let be a function of two variables and Describe theprocedure for finding the first partial derivatives.
112. Sketch a surface representing a function of two variablesand Use the sketch to give geometric interpretations of
and
113. Sketch the graph of a function whose derivativeis always negative and whose derivative is always
positive.
114. Sketch the graph of a function whose deriva-tives and are always positive.
115. If is a function of and such that and arecontinuous, what is the relationship between the mixedpartial derivatives? Explain.
fyxfxyyxf
fyfx
z f x, y
fyfx
z f x, y
f y.f xy.x
f
y.xf
WRITING ABOUT CONCEPTS
1053714_1303_pg 915.qxp 10/30/08 8:30 AM Page 915
In Exercises 59–64, find the first partial derivatives with respectto and
59.
60.
61. 62.
63.
64.
In Exercises 65–70, evaluate and at the given point.
65.
66.
67.
68.
69.
70.
In Exercises 71–80, find the four second partial derivatives.Observe that the second mixed partials are equal.
71. 72.
73. 74.
75. 76.
77. 78.
79. 80.
In Exercises 81–88, for find all values of and suchthat and simultaneously.
81.
82.
83.
84.
85.
86.
87.
88.
In Exercises 89–92, use a computer algebra system to find thefirst and second partial derivatives of the function. Determinewhether there exist values of and such that and
simultaneously.
89. 90.
91. 92.
In Exercises 93–96, show that the mixed partial derivatives and are equal.
93.
94.
95.
96.
Laplace’s Equation In Exercises 97–100, show that the functionsatisfies Laplace’s equation
97. 98.
99. 100.
Wave Equation In Exercises 101–104, show that the functionsatisfies the wave equation
101. 102.
103. 104.
Heat Equation In Exercises 105 and 106, show that the functionsatisfies the heat equation
105. 106.
In Exercises 107 and 108, determine whether there exists a function with the given partial derivatives. Explain yourreasoning. If such a function exists, give an example.
107.
108.
In Exercises 109 and 110, find the first partial derivative withrespect to
109.
110. f x, y, z x senh yz
y2 2 y 1 z
f x, y, z tan y2z ez2 y 2 z
x.
fy x, y x 4yfx x, y 2x y,
fy x, y 2 sen 3x 2yfx x, y 3 sen 3x 2y ,
f x, y
z e t sen xc
z e t cos xc
z/ t c2 2z/ x2 .
z sen ct sen xz ln x ct
z cos 4x 4ctz sen x ct
2z/ t 2 c2 2z/ x2 .
z arctan yx
z ex sen y
z 12 ey e y sen xz 5xy
2z/ x2 1 2z/ y2 0.
f x, y, z2z
x y
f x, y, z e x sen yz
f x, y, z x2 3xy 4yz z3
f x, y, z xyz
fyyxfyxy,fxyy,
f x, yxy
x yf x, y ln
xx2 y2
f x, y 25 x2 y 2f x, y x sec y
fy x, y 0fx x, y 0yx
f x, y ln x2 y 2 1
f x, y ex2 xy y2
f x, y 3x3 12xy y3
f x, y1x
1y
xy
f x, y x2 xy y2
f x, y x2 4xy y 2 4x 16y 3
f x, y x2 xy y2 5x y
f x, y x2 xy y2 2x 2y
fy x, y 0fx x, y 0yxf x, y ,
z arctan yx
z cos xy
z 2xey 3ye xz ex tan y
z ln x yz x2 y 2
z x4 3x2y 2 y4z x2 2xy 3y 2
z x2 3y2z 3xy2
1, 2, 1f x, y, z 3x2 y2 2z2,
0, 2
, 4f x, y, z z sen x y ,
3, 1, 1f x, y, zxy
x y z,
f x, y, zxyz
, 1, 1, 1
2, 1, 2f x, y, z x2y3 2xyz 3yz,
f x, y, z x3yz2, 1, 1, 1
fzfy,fx,
G x, y, z1
1 x2 y 2 z 2
F x, y, z ln x2 y 2 z 2
w7xz
x yw x2 y 2 z2
f x, y, z 3x2y 5xyz 10yz 2
H x, y, z sen x 2y 3z
z.y,x,
13.3 Partial Derivatives 915
CAS
111. Let be a function of two variables and Describe theprocedure for finding the first partial derivatives.
112. Sketch a surface representing a function of two variablesand Use the sketch to give geometric interpretations of
and
113. Sketch the graph of a function whose derivativeis always negative and whose derivative is always
positive.
114. Sketch the graph of a function whose deriva-tives and are always positive.
115. If is a function of and such that and arecontinuous, what is the relationship between the mixedpartial derivatives? Explain.
fyxfxyyxf
fyfx
z f x, y
fyfx
z f x, y
f y.f xy.x
f
y.xf
WRITING ABOUT CONCEPTS
1053714_1303_pg 915.qxp 10/30/08 8:30 AM Page 915
In Exercises 59–64, find the first partial derivatives with respectto and
59.
60.
61. 62.
63.
64.
In Exercises 65–70, evaluate and at the given point.
65.
66.
67.
68.
69.
70.
In Exercises 71–80, find the four second partial derivatives.Observe that the second mixed partials are equal.
71. 72.
73. 74.
75. 76.
77. 78.
79. 80.
In Exercises 81–88, for find all values of and suchthat and simultaneously.
81.
82.
83.
84.
85.
86.
87.
88.
In Exercises 89–92, use a computer algebra system to find thefirst and second partial derivatives of the function. Determinewhether there exist values of and such that and
simultaneously.
89. 90.
91. 92.
In Exercises 93–96, show that the mixed partial derivatives and are equal.
93.
94.
95.
96.
Laplace’s Equation In Exercises 97–100, show that the functionsatisfies Laplace’s equation
97. 98.
99. 100.
Wave Equation In Exercises 101–104, show that the functionsatisfies the wave equation
101. 102.
103. 104.
Heat Equation In Exercises 105 and 106, show that the functionsatisfies the heat equation
105. 106.
In Exercises 107 and 108, determine whether there exists a function with the given partial derivatives. Explain yourreasoning. If such a function exists, give an example.
107.
108.
In Exercises 109 and 110, find the first partial derivative withrespect to
109.
110. f x, y, z x senh yz
y2 2 y 1 z
f x, y, z tan y2z ez2 y 2 z
x.
fy x, y x 4yfx x, y 2x y,
fy x, y 2 sen 3x 2yfx x, y 3 sen 3x 2y ,
f x, y
z e t sen xc
z e t cos xc
z/ t c2 2z/ x2 .
z sen ct sen xz ln x ct
z cos 4x 4ctz sen x ct
2z/ t 2 c2 2z/ x2 .
z arctan yx
z ex sen y
z 12 ey e y sen xz 5xy
2z/ x2 1 2z/ y2 0.
f x, y, z2z
x y
f x, y, z e x sen yz
f x, y, z x2 3xy 4yz z3
f x, y, z xyz
fyyxfyxy,fxyy,
f x, yxy
x yf x, y ln
xx2 y2
f x, y 25 x2 y 2f x, y x sec y
fy x, y 0fx x, y 0yx
f x, y ln x2 y 2 1
f x, y ex2 xy y2
f x, y 3x3 12xy y3
f x, y1x
1y
xy
f x, y x2 xy y2
f x, y x2 4xy y 2 4x 16y 3
f x, y x2 xy y2 5x y
f x, y x2 xy y2 2x 2y
fy x, y 0fx x, y 0yxf x, y ,
z arctan yx
z cos xy
z 2xey 3ye xz ex tan y
z ln x yz x2 y 2
z x4 3x2y 2 y4z x2 2xy 3y 2
z x2 3y2z 3xy2
1, 2, 1f x, y, z 3x2 y2 2z2,
0, 2
, 4f x, y, z z sen x y ,
3, 1, 1f x, y, zxy
x y z,
f x, y, zxyz
, 1, 1, 1
2, 1, 2f x, y, z x2y3 2xyz 3yz,
f x, y, z x3yz2, 1, 1, 1
fzfy,fx,
G x, y, z1
1 x2 y 2 z 2
F x, y, z ln x2 y 2 z 2
w7xz
x yw x2 y 2 z2
f x, y, z 3x2y 5xyz 10yz 2
H x, y, z sen x 2y 3z
z.y,x,
13.3 Partial Derivatives 915
CAS
111. Let be a function of two variables and Describe theprocedure for finding the first partial derivatives.
112. Sketch a surface representing a function of two variablesand Use the sketch to give geometric interpretations of
and
113. Sketch the graph of a function whose derivativeis always negative and whose derivative is always
positive.
114. Sketch the graph of a function whose deriva-tives and are always positive.
115. If is a function of and such that and arecontinuous, what is the relationship between the mixedpartial derivatives? Explain.
fyxfxyyxf
fyfx
z f x, y
fyfx
z f x, y
f y.f xy.x
f
y.xf
WRITING ABOUT CONCEPTS
1053714_1303_pg 915.qxp 10/30/08 8:30 AM Page 915
In Exercises 59–64, find the first partial derivatives with respectto and
59.
60.
61. 62.
63.
64.
In Exercises 65–70, evaluate and at the given point.
65.
66.
67.
68.
69.
70.
In Exercises 71–80, find the four second partial derivatives.Observe that the second mixed partials are equal.
71. 72.
73. 74.
75. 76.
77. 78.
79. 80.
In Exercises 81–88, for find all values of and suchthat and simultaneously.
81.
82.
83.
84.
85.
86.
87.
88.
In Exercises 89–92, use a computer algebra system to find thefirst and second partial derivatives of the function. Determinewhether there exist values of and such that and
simultaneously.
89. 90.
91. 92.
In Exercises 93–96, show that the mixed partial derivatives and are equal.
93.
94.
95.
96.
Laplace’s Equation In Exercises 97–100, show that the functionsatisfies Laplace’s equation
97. 98.
99. 100.
Wave Equation In Exercises 101–104, show that the functionsatisfies the wave equation
101. 102.
103. 104.
Heat Equation In Exercises 105 and 106, show that the functionsatisfies the heat equation
105. 106.
In Exercises 107 and 108, determine whether there exists a function with the given partial derivatives. Explain yourreasoning. If such a function exists, give an example.
107.
108.
In Exercises 109 and 110, find the first partial derivative withrespect to
109.
110. f x, y, z x senh yz
y2 2 y 1 z
f x, y, z tan y2z ez2 y 2 z
x.
fy x, y x 4yfx x, y 2x y,
fy x, y 2 sen 3x 2yfx x, y 3 sen 3x 2y ,
f x, y
z e t sen xc
z e t cos xc
z/ t c2 2z/ x2 .
z sen ct sen xz ln x ct
z cos 4x 4ctz sen x ct
2z/ t 2 c2 2z/ x2 .
z arctan yx
z ex sen y
z 12 ey e y sen xz 5xy
2z/ x2 1 2z/ y2 0.
f x, y, z2z
x y
f x, y, z e x sen yz
f x, y, z x2 3xy 4yz z3
f x, y, z xyz
fyyxfyxy,fxyy,
f x, yxy
x yf x, y ln
xx2 y2
f x, y 25 x2 y 2f x, y x sec y
fy x, y 0fx x, y 0yx
f x, y ln x2 y 2 1
f x, y ex2 xy y2
f x, y 3x3 12xy y3
f x, y1x
1y
xy
f x, y x2 xy y2
f x, y x2 4xy y 2 4x 16y 3
f x, y x2 xy y2 5x y
f x, y x2 xy y2 2x 2y
fy x, y 0fx x, y 0yxf x, y ,
z arctan yx
z cos xy
z 2xey 3ye xz ex tan y
z ln x yz x2 y 2
z x4 3x2y 2 y4z x2 2xy 3y 2
z x2 3y2z 3xy2
1, 2, 1f x, y, z 3x2 y2 2z2,
0, 2
, 4f x, y, z z sen x y ,
3, 1, 1f x, y, zxy
x y z,
f x, y, zxyz
, 1, 1, 1
2, 1, 2f x, y, z x2y3 2xyz 3yz,
f x, y, z x3yz2, 1, 1, 1
fzfy,fx,
G x, y, z1
1 x2 y 2 z 2
F x, y, z ln x2 y 2 z 2
w7xz
x yw x2 y 2 z2
f x, y, z 3x2y 5xyz 10yz 2
H x, y, z sen x 2y 3z
z.y,x,
13.3 Partial Derivatives 915
CAS
111. Let be a function of two variables and Describe theprocedure for finding the first partial derivatives.
112. Sketch a surface representing a function of two variablesand Use the sketch to give geometric interpretations of
and
113. Sketch the graph of a function whose derivativeis always negative and whose derivative is always
positive.
114. Sketch the graph of a function whose deriva-tives and are always positive.
115. If is a function of and such that and arecontinuous, what is the relationship between the mixedpartial derivatives? Explain.
fyxfxyyxf
fyfx
z f x, y
fyfx
z f x, y
f y.f xy.x
f
y.xf
WRITING ABOUT CONCEPTS
1053714_1303_pg 915.qxp 10/30/08 8:30 AM Page 915
In Exercises 59–64, find the first partial derivatives with respectto and
59.
60.
61. 62.
63.
64.
In Exercises 65–70, evaluate and at the given point.
65.
66.
67.
68.
69.
70.
In Exercises 71–80, find the four second partial derivatives.Observe that the second mixed partials are equal.
71. 72.
73. 74.
75. 76.
77. 78.
79. 80.
In Exercises 81–88, for find all values of and suchthat and simultaneously.
81.
82.
83.
84.
85.
86.
87.
88.
In Exercises 89–92, use a computer algebra system to find thefirst and second partial derivatives of the function. Determinewhether there exist values of and such that and
simultaneously.
89. 90.
91. 92.
In Exercises 93–96, show that the mixed partial derivatives and are equal.
93.
94.
95.
96.
Laplace’s Equation In Exercises 97–100, show that the functionsatisfies Laplace’s equation
97. 98.
99. 100.
Wave Equation In Exercises 101–104, show that the functionsatisfies the wave equation
101. 102.
103. 104.
Heat Equation In Exercises 105 and 106, show that the functionsatisfies the heat equation
105. 106.
In Exercises 107 and 108, determine whether there exists a function with the given partial derivatives. Explain yourreasoning. If such a function exists, give an example.
107.
108.
In Exercises 109 and 110, find the first partial derivative withrespect to
109.
110. f x, y, z x senh yz
y2 2 y 1 z
f x, y, z tan y2z ez2 y 2 z
x.
fy x, y x 4yfx x, y 2x y,
fy x, y 2 sen 3x 2yfx x, y 3 sen 3x 2y ,
f x, y
z e t sen xc
z e t cos xc
z/ t c2 2z/ x2 .
z sen ct sen xz ln x ct
z cos 4x 4ctz sen x ct
2z/ t 2 c2 2z/ x2 .
z arctan yx
z ex sen y
z 12 ey e y sen xz 5xy
2z/ x2 1 2z/ y2 0.
f x, y, z2z
x y
f x, y, z e x sen yz
f x, y, z x2 3xy 4yz z3
f x, y, z xyz
fyyxfyxy,fxyy,
f x, yxy
x yf x, y ln
xx2 y2
f x, y 25 x2 y 2f x, y x sec y
fy x, y 0fx x, y 0yx
f x, y ln x2 y 2 1
f x, y ex2 xy y2
f x, y 3x3 12xy y3
f x, y1x
1y
xy
f x, y x2 xy y2
f x, y x2 4xy y 2 4x 16y 3
f x, y x2 xy y2 5x y
f x, y x2 xy y2 2x 2y
fy x, y 0fx x, y 0yxf x, y ,
z arctan yx
z cos xy
z 2xey 3ye xz ex tan y
z ln x yz x2 y 2
z x4 3x2y 2 y4z x2 2xy 3y 2
z x2 3y2z 3xy2
1, 2, 1f x, y, z 3x2 y2 2z2,
0, 2
, 4f x, y, z z sen x y ,
3, 1, 1f x, y, zxy
x y z,
f x, y, zxyz
, 1, 1, 1
2, 1, 2f x, y, z x2y3 2xyz 3yz,
f x, y, z x3yz2, 1, 1, 1
fzfy,fx,
G x, y, z1
1 x2 y 2 z 2
F x, y, z ln x2 y 2 z 2
w7xz
x yw x2 y 2 z2
f x, y, z 3x2y 5xyz 10yz 2
H x, y, z sen x 2y 3z
z.y,x,
13.3 Partial Derivatives 915
CAS
111. Let be a function of two variables and Describe theprocedure for finding the first partial derivatives.
112. Sketch a surface representing a function of two variablesand Use the sketch to give geometric interpretations of
and
113. Sketch the graph of a function whose derivativeis always negative and whose derivative is always
positive.
114. Sketch the graph of a function whose deriva-tives and are always positive.
115. If is a function of and such that and arecontinuous, what is the relationship between the mixedpartial derivatives? Explain.
fyxfxyyxf
fyfx
z f x, y
fyfx
z f x, y
f y.f xy.x
f
y.xf
WRITING ABOUT CONCEPTS
1053714_1303_pg 915.qxp 10/30/08 8:30 AM Page 915
Desarrollo de conceptos111. Sea f una función de dos variables x y y. Describir el proce-
dimiento para hallar las derivadas parciales de primer orden.
112. Dibujar una superficie que represente una función f de dosvariables x y y. Utilizar la gráfica para dar una interpre-tación geométrica de y
113. Dibujar la gráfica de una función cuya derivadasea siempre negativa y cuya derivada sea siempre po-
sitiva.
114. Dibujar la gráfica de una función cuyas deri-vadas y sean siempre positivas.
115. Si es una función de y tal que y son continuas,¿qué relación existe entre las derivadas parciales mixtas?Explicar.
fyxfxyyxf
fyfx
z 5 f sx, yd
fyfx
z 5 f sx, ydfyy.fyx
In Exercises 59–64, find the first partial derivatives with respectto and
59.
60.
61. 62.
63.
64.
In Exercises 65–70, evaluate and at the given point.
65.
66.
67.
68.
69.
70.
In Exercises 71–80, find the four second partial derivatives.Observe that the second mixed partials are equal.
71. 72.
73. 74.
75. 76.
77. 78.
79. 80.
In Exercises 81–88, for find all values of and suchthat and simultaneously.
81.
82.
83.
84.
85.
86.
87.
88.
In Exercises 89–92, use a computer algebra system to find thefirst and second partial derivatives of the function. Determinewhether there exist values of and such that and
simultaneously.
89. 90.
91. 92.
In Exercises 93–96, show that the mixed partial derivatives and are equal.
93.
94.
95.
96.
Laplace’s Equation In Exercises 97–100, show that the functionsatisfies Laplace’s equation
97. 98.
99. 100.
Wave Equation In Exercises 101–104, show that the functionsatisfies the wave equation
101. 102.
103. 104.
Heat Equation In Exercises 105 and 106, show that the functionsatisfies the heat equation
105. 106.
In Exercises 107 and 108, determine whether there exists a function with the given partial derivatives. Explain yourreasoning. If such a function exists, give an example.
107.
108.
In Exercises 109 and 110, find the first partial derivative withrespect to
109.
110. f x, y, z x senh yz
y2 2 y 1 z
f x, y, z tan y2z ez2 y 2 z
x.
fy x, y x 4yfx x, y 2x y,
fy x, y 2 sen 3x 2yfx x, y 3 sen 3x 2y ,
f x, y
z e t sen xc
z e t cos xc
z/ t c2 2z/ x2 .
z sen ct sen xz ln x ct
z cos 4x 4ctz sen x ct
2z/ t 2 c2 2z/ x2 .
z arctan yx
z ex sen y
z 12 ey e y sen xz 5xy
2z/ x2 1 2z/ y2 0.
f x, y, z2z
x y
f x, y, z e x sen yz
f x, y, z x2 3xy 4yz z3
f x, y, z xyz
fyyxfyxy,fxyy,
f x, yxy
x yf x, y ln
xx2 y2
f x, y 25 x2 y 2f x, y x sec y
fy x, y 0fx x, y 0yx
f x, y ln x2 y 2 1
f x, y ex2 xy y2
f x, y 3x3 12xy y3
f x, y1x
1y
xy
f x, y x2 xy y2
f x, y x2 4xy y 2 4x 16y 3
f x, y x2 xy y2 5x y
f x, y x2 xy y2 2x 2y
fy x, y 0fx x, y 0yxf x, y ,
z arctan yx
z cos xy
z 2xey 3ye xz ex tan y
z ln x yz x2 y 2
z x4 3x2y 2 y4z x2 2xy 3y 2
z x2 3y2z 3xy2
1, 2, 1f x, y, z 3x2 y2 2z2,
0, 2
, 4f x, y, z z sen x y ,
3, 1, 1f x, y, zxy
x y z,
f x, y, zxyz
, 1, 1, 1
2, 1, 2f x, y, z x2y3 2xyz 3yz,
f x, y, z x3yz2, 1, 1, 1
fzfy,fx,
G x, y, z1
1 x2 y 2 z 2
F x, y, z ln x2 y 2 z 2
w7xz
x yw x2 y 2 z2
f x, y, z 3x2y 5xyz 10yz 2
H x, y, z sen x 2y 3z
z.y,x,
13.3 Partial Derivatives 915
CAS
111. Let be a function of two variables and Describe theprocedure for finding the first partial derivatives.
112. Sketch a surface representing a function of two variablesand Use the sketch to give geometric interpretations of
and
113. Sketch the graph of a function whose derivativeis always negative and whose derivative is always
positive.
114. Sketch the graph of a function whose deriva-tives and are always positive.
115. If is a function of and such that and arecontinuous, what is the relationship between the mixedpartial derivatives? Explain.
fyxfxyyxf
fyfx
z f x, y
fyfx
z f x, y
f y.f xy.x
f
y.xf
WRITING ABOUT CONCEPTS
1053714_1303_pg 915.qxp 10/30/08 8:30 AM Page 915
sen
sen
Larson-13-03.qxd 3/12/09 18:48 Page 915
916 CAPÍTULO 13 Funciones de varias variables
117. Ingreso marginal Una corporación farmacéutica tiene dosplantas que producen la misma medicina. Si x1 y x2 son losnúmeros de unidades producidos en la planta 1 y en la planta 2,respectivamente, entonces el ingreso total del producto estádado por Cuandox1 = 4 y x2 = 12, encontrar a) el ingreso marginal para la plan-ta 1, y b) el ingreso marginal para la planta 2,
118. Costo marginal Una empresa fabrica dos tipos de estufas decombustión de madera: el modelo autoestable y el modelo parainserción en una chimenea. La función de costo para producirx estufas autoestables y y de inserción en una chimenea es
a) Calcular los costos marginales y cuandoy
b) Cuando se requiera producción adicional, ¿qué modelo deestufa hará incrementar el costo con una tasa más alta? ¿Có-mo puede determinarse esto a partir del modelo del costo?
119. Psicología Recientemente en el siglo xx se desarrolló una prue-ba de inteligencia llamada la Prueba de Stanford-Binet (másconocida como la prueba IQ). En esta prueba, una edad mentalindividual M es dividida entre la edad cronológica individual C,y el cociente es multiplicado por 100. El resultado es el IQ indi-vidual.
Encontrar las derivadas parciales de IQ con respecto a M y conrespecto a C. Evaluar las derivadas parciales en el punto (12, 10)e interpretar el resultado. (Fuente: Adaptado de Bernstein/Clark-Steward/Roy/Wickens, Psicología, 4a. ed.)
120. Productividad marginal Considerar la función de producciónde Cobb-Douglas Si x 5 1 000 y y 5 500,hallar a) la productividad marginal del trabajo, y b) laproductividad marginal del capital,
121. Para pensar Sea el número de aspirantes a una universi-dad, p el costo por alimentación y alojamiento en la universi-dad, y t el costo de la matrícula. N es una función de p y t talque y ¿Qué información se obtiene alsaber que ambas derivadas parciales son negativas?
122. Inversión El valor de una inversión de $1 000 al 6% deinterés compuesto anual es
donde I es la tasa anual de inflación y R es la tasa de impuestopara el inversor. Calcular y Determinar si la tasa de impuesto o la tasa de inflación es elmayor factor “negativo” sobre el crecimiento de la inversión.
123. Distribución de temperatura La temperatura en cualquierpunto de una placa de acero es donde y son medidos en metros. En el punto (2, 3), hallar elritmo de cambio de la temperatura respecto a la distancia reco-rrida en la placa en las direcciones del eje x y y.
124. Temperatura aparente Una medida de la percepción delcalor ambiental por unas personas promedio es el Índice detemperatura aparente. Un modelo para este índice es
donde es la temperatura aparente en grados Celsius, es latemperatura del aire y es la humedad relativa dada en formadecimal. (Fuente: The UMAP Journal)
a) Hallar y si y
b) ¿Qué influye más sobre A, la temperatura del aire o lahumedad? Explicar.
125. Ley de los gases ideales La ley de los gases ideales estableceque donde es la presión, es el volumen, es elnúmero de moles de gas, es una constante (la constante de losgases) y T es temperatura absoluta. Mostrar que
126. Utilidad marginal La función de utilidad es unamedida de la utilidad (o satisfacción) que obtiene una personapor el consumo de dos productos y Suponer que la funciónde utilidad es
a) Determinar la utilidad marginal del producto
b) Determinar la utilidad marginal del producto
c) Si y ¿se debe consumir una unidad más deproducto x o una unidad más de producto y? Explicar elrazonamiento.
d) Utilizar un sistema algebraico por computadora y represen-tar gráficamente la función. Interpretar las utilidades mar-ginales de productos x y y con una gráfica.
127. Modelo matemático En la tabla se muestran los consumosper cápita (en galones) de diferentes tipos de leche en EstadosUnidos desde 1999 hasta 2005. El consumo de leche light ydescremada, leche baja en grasa y leche entera se representapor las variables x, y y z, respectivamente. (Fuente: U.S. De-partment of Agriculture)
Un modelo para los datos está dado por
a) Hallar y
b) Interpretar las derivadas parciales en el contexto del problema.
zy
.zx
y 5 3,x 5 2
y.
x.
U 5 25x2 1 xy 2 3y 2.y.x
U 5 f sx, yd
TP
PV
VT
5 21.
RnVPPV 5 nRT,
h 5 0.80.t 5 308AyhAyt
htA
A 5 0.885t 2 22.4h 1 1.20th 2 0.544
yxT 5 500 2 0.6x2 2 1.5y2,sx, yd
VRs0.03, 0.28d.VIs0.03, 0.28d
Nyt < 0.Nyp < 0
N
fyy.
916 Chapter 13 Functions of Several Variables
117. Marginal Revenue A pharmaceutical corporation has twoplants that produce the same over-the-counter medicine. If and are the numbers of units produced at plant 1 and plant 2,respectively, then the total revenue for the product is given by
When and find (a) the marginal revenue for plant 1,and (b) the marginal revenue for plant 2,
118. Marginal Costs A company manufactures two types ofwood-burning stoves: a freestanding model and a fireplace-insert model. The cost function for producing freestandingand fireplace-insert stoves is
(a) Find the marginal costs and when and
(b) When additional production is required, which model ofstove results in the cost increasing at a higher rate? Howcan this be determined from the cost model?
119. Psychology Early in the twentieth century, an intelligence testcalled the Stanford-Binet Test (more commonly known as the IQtest) was developed. In this test, an individual’s mental age isdivided by the individual’s chronological age and the quotientis multiplied by 100. The result is the individual’s
Find the partial derivatives of with respect to and withrespect to Evaluate the partial derivatives at the point
and interpret the result. (Source: Adapted fromBernstein/Clark-Stewart/Roy/Wickens, Psychology, FourthEdition)
120. Marginal Productivity Consider the Cobb-Douglas produc-tion function When and
find (a) the marginal productivity of labor,and (b) the marginal productivity of capital,
121. Think About It Let be the number of applicants to auniversity, the charge for food and housing at the university,and the tuition. is a function of and such that
and What information is gained bynoticing that both partials are negative?
122. Investment The value of an investment of $1000 earning 6%compounded annually is
where is the annual rate of inflation and is the tax rate forthe person making the investment. Calculate and Determine whether the tax rate or the rateof inflation is the greater “negative” factor in the growth of theinvestment.
123. Temperature Distribution The temperature at any pointin a steel plate is where
and are measured in meters. At the point find the ratesof change of the temperature with respect to the distancesmoved along the plate in the directions of the and axes.
124. Apparent Temperature A measure of how hot weather feelsto an average person is the Apparent Temperature Index. Amodel for this index is
where is the apparent temperature in degrees Celsius, is theair temperature, and is the relative humidity in decimalform. (Source: The UMAP Journal)
(a) Find and when and
(b) Which has a greater effect on air temperature orhumidity? Explain.
125. Ideal Gas Law The Ideal Gas Law states that where is pressure, is volume, is the number of moles ofgas, is a fixed constant (the gas constant), and is absolutetemperature. Show that
126. Marginal Utility The utility function is ameasure of the utility (or satisfaction) derived by a personfrom the consumption of two products and Suppose theutility function is
(a) Determine the marginal utility of product
(b) Determine the marginal utility of product
(c) When and should a person consume onemore unit of product or one more unit of product Explain your reasoning.
(d) Use a computer algebra system to graph the function.Interpret the marginal utilities of products and graphically.
127. Modeling Data Per capita consumptions (in gallons) ofdifferent types of milk in the United States from 1999 through2005 are shown in the table. Consumption of flavored milk,plain reduced-fat milk, and plain light and skim milks are represented by the variables and respectively.(Source: U.S. Department of Agriculture)
A model for the data is given by
(a) Find and
(b) Interpret the partial derivatives in the context of the problem.
zy.
zx
z 0.92x 1.03y 0.02.
z,y,x,
yx
y?xy 3,x 2
y.
x.
U 5x2 xy 3y 2.y.x
U f x, y
TP
PV
VT
1.
TRnVP
PV nRT,
A,
h 0.80.t 30A hA t
htA
A 0.885t 22.4h 1.20th 0.544
y-x-
2, 3 ,yxT 500 0.6x2 1.5y2,x, y
VR 0.03, 0.28 .VI 0.03, 0.28
RI
V I, R 1 0001 0.06 1 R
1 I
10
N t < 0.N p < 0tpNt
pN
f y.f x,y 500,
x 1000f x, y 200x0.7y0.3.
12, 10C.
MIQ
IQ M, CMC
100
IQ.C
M
y 20.x 80C yC x
C 32 xy 175x 205y 1 050.
yx
R x2.R x1,x2 12,x1 4R 200x1 200x2 4x1
2 8x1x2 4x22.
x2
x1
116. Find the four second partial derivatives of the functiongiven by Show that the secondmixed partial derivatives and are equal.fyxfxy
f x, y sen x 2y .
CAPSTONE
Año 1999 2000 2001 2002 2003 2004 2005
x 1.4 1.4 1.4 1.6 1.6 1.7 1.7
y 7.3 7.1 7.0 7.0 6.9 6.9 6.9
z 6.2 6.1 5.9 5.8 5.6 5.5 5.6
CAS
1053714_1303.qxp 10/27/08 12:06 PM Page 916
f sx, yd 5 200x0.7y0.3.
y 5 20.x 5 80CyydsCyx
C 5 32!xy 1 175x 1 205y 1 1050.
916 Chapter 13 Functions of Several Variables
117. Marginal Revenue A pharmaceutical corporation has twoplants that produce the same over-the-counter medicine. If and are the numbers of units produced at plant 1 and plant 2,respectively, then the total revenue for the product is given by
When and find (a) the marginal revenue for plant 1,and (b) the marginal revenue for plant 2,
118. Marginal Costs A company manufactures two types ofwood-burning stoves: a freestanding model and a fireplace-insert model. The cost function for producing freestandingand fireplace-insert stoves is
(a) Find the marginal costs and when and
(b) When additional production is required, which model ofstove results in the cost increasing at a higher rate? Howcan this be determined from the cost model?
119. Psychology Early in the twentieth century, an intelligence testcalled the Stanford-Binet Test (more commonly known as the IQtest) was developed. In this test, an individual’s mental age isdivided by the individual’s chronological age and the quotientis multiplied by 100. The result is the individual’s
Find the partial derivatives of with respect to and withrespect to Evaluate the partial derivatives at the point
and interpret the result. (Source: Adapted fromBernstein/Clark-Stewart/Roy/Wickens, Psychology, FourthEdition)
120. Marginal Productivity Consider the Cobb-Douglas produc-tion function When and
find (a) the marginal productivity of labor,and (b) the marginal productivity of capital,
121. Think About It Let be the number of applicants to auniversity, the charge for food and housing at the university,and the tuition. is a function of and such that
and What information is gained bynoticing that both partials are negative?
122. Investment The value of an investment of $1000 earning 6%compounded annually is
where is the annual rate of inflation and is the tax rate forthe person making the investment. Calculate and Determine whether the tax rate or the rateof inflation is the greater “negative” factor in the growth of theinvestment.
123. Temperature Distribution The temperature at any pointin a steel plate is where
and are measured in meters. At the point find the ratesof change of the temperature with respect to the distancesmoved along the plate in the directions of the and axes.
124. Apparent Temperature A measure of how hot weather feelsto an average person is the Apparent Temperature Index. Amodel for this index is
where is the apparent temperature in degrees Celsius, is theair temperature, and is the relative humidity in decimalform. (Source: The UMAP Journal)
(a) Find and when and
(b) Which has a greater effect on air temperature orhumidity? Explain.
125. Ideal Gas Law The Ideal Gas Law states that where is pressure, is volume, is the number of moles ofgas, is a fixed constant (the gas constant), and is absolutetemperature. Show that
126. Marginal Utility The utility function is ameasure of the utility (or satisfaction) derived by a personfrom the consumption of two products and Suppose theutility function is
(a) Determine the marginal utility of product
(b) Determine the marginal utility of product
(c) When and should a person consume onemore unit of product or one more unit of product Explain your reasoning.
(d) Use a computer algebra system to graph the function.Interpret the marginal utilities of products and graphically.
127. Modeling Data Per capita consumptions (in gallons) ofdifferent types of milk in the United States from 1999 through2005 are shown in the table. Consumption of flavored milk,plain reduced-fat milk, and plain light and skim milks are represented by the variables and respectively.(Source: U.S. Department of Agriculture)
A model for the data is given by
(a) Find and
(b) Interpret the partial derivatives in the context of the problem.
zy.
zx
z 0.92x 1.03y 0.02.
z,y,x,
yx
y?xy 3,x 2
y.
x.
U 5x2 xy 3y 2.y.x
U f x, y
TP
PV
VT
1.
TRnVP
PV nRT,
A,
h 0.80.t 30A hA t
htA
A 0.885t 22.4h 1.20th 0.544
y-x-
2, 3 ,yxT 500 0.6x2 1.5y2,x, y
VR 0.03, 0.28 .VI 0.03, 0.28
RI
V I, R 1 0001 0.06 1 R
1 I
10
N t < 0.N p < 0tpNt
pN
f y.f x,y 500,
x 1000f x, y 200x0.7y0.3.
12, 10C.
MIQ
IQ M, CMC
100
IQ.C
M
y 20.x 80C yC x
C 32 xy 175x 205y 1 050.
yx
R x2.R x1,x2 12,x1 4R 200x1 200x2 4x1
2 8x1x2 4x22.
x2
x1
116. Find the four second partial derivatives of the functiongiven by Show that the secondmixed partial derivatives and are equal.fyxfxy
f x, y sen x 2y .
CAPSTONE
Año 1999 2000 2001 2002 2003 2004 2005
x 1.4 1.4 1.4 1.6 1.6 1.7 1.7
y 7.3 7.1 7.0 7.0 6.9 6.9 6.9
z 6.2 6.1 5.9 5.8 5.6 5.5 5.6
CAS
1053714_1303.qxp 10/27/08 12:06 PM Page 916
916 Chapter 13 Functions of Several Variables
117. Marginal Revenue A pharmaceutical corporation has twoplants that produce the same over-the-counter medicine. If and are the numbers of units produced at plant 1 and plant 2,respectively, then the total revenue for the product is given by
When and find (a) the marginal revenue for plant 1,and (b) the marginal revenue for plant 2,
118. Marginal Costs A company manufactures two types ofwood-burning stoves: a freestanding model and a fireplace-insert model. The cost function for producing freestandingand fireplace-insert stoves is
(a) Find the marginal costs and when and
(b) When additional production is required, which model ofstove results in the cost increasing at a higher rate? Howcan this be determined from the cost model?
119. Psychology Early in the twentieth century, an intelligence testcalled the Stanford-Binet Test (more commonly known as the IQtest) was developed. In this test, an individual’s mental age isdivided by the individual’s chronological age and the quotientis multiplied by 100. The result is the individual’s
Find the partial derivatives of with respect to and withrespect to Evaluate the partial derivatives at the point
and interpret the result. (Source: Adapted fromBernstein/Clark-Stewart/Roy/Wickens, Psychology, FourthEdition)
120. Marginal Productivity Consider the Cobb-Douglas produc-tion function When and
find (a) the marginal productivity of labor,and (b) the marginal productivity of capital,
121. Think About It Let be the number of applicants to auniversity, the charge for food and housing at the university,and the tuition. is a function of and such that
and What information is gained bynoticing that both partials are negative?
122. Investment The value of an investment of $1000 earning 6%compounded annually is
where is the annual rate of inflation and is the tax rate forthe person making the investment. Calculate and Determine whether the tax rate or the rateof inflation is the greater “negative” factor in the growth of theinvestment.
123. Temperature Distribution The temperature at any pointin a steel plate is where
and are measured in meters. At the point find the ratesof change of the temperature with respect to the distancesmoved along the plate in the directions of the and axes.
124. Apparent Temperature A measure of how hot weather feelsto an average person is the Apparent Temperature Index. Amodel for this index is
where is the apparent temperature in degrees Celsius, is theair temperature, and is the relative humidity in decimalform. (Source: The UMAP Journal)
(a) Find and when and
(b) Which has a greater effect on air temperature orhumidity? Explain.
125. Ideal Gas Law The Ideal Gas Law states that where is pressure, is volume, is the number of moles ofgas, is a fixed constant (the gas constant), and is absolutetemperature. Show that
126. Marginal Utility The utility function is ameasure of the utility (or satisfaction) derived by a personfrom the consumption of two products and Suppose theutility function is
(a) Determine the marginal utility of product
(b) Determine the marginal utility of product
(c) When and should a person consume onemore unit of product or one more unit of product Explain your reasoning.
(d) Use a computer algebra system to graph the function.Interpret the marginal utilities of products and graphically.
127. Modeling Data Per capita consumptions (in gallons) ofdifferent types of milk in the United States from 1999 through2005 are shown in the table. Consumption of flavored milk,plain reduced-fat milk, and plain light and skim milks are represented by the variables and respectively.(Source: U.S. Department of Agriculture)
A model for the data is given by
(a) Find and
(b) Interpret the partial derivatives in the context of the problem.
zy.
zx
z 0.92x 1.03y 0.02.
z,y,x,
yx
y?xy 3,x 2
y.
x.
U 5x2 xy 3y 2.y.x
U f x, y
TP
PV
VT
1.
TRnVP
PV nRT,
A,
h 0.80.t 30A hA t
htA
A 0.885t 22.4h 1.20th 0.544
y-x-
2, 3 ,yxT 500 0.6x2 1.5y2,x, y
VR 0.03, 0.28 .VI 0.03, 0.28
RI
V I, R 1 0001 0.06 1 R
1 I
10
N t < 0.N p < 0tpNt
pN
f y.f x,y 500,
x 1000f x, y 200x0.7y0.3.
12, 10C.
MIQ
IQ M, CMC
100
IQ.C
M
y 20.x 80C yC x
C 32 xy 175x 205y 1 050.
yx
R x2.R x1,x2 12,x1 4R 200x1 200x2 4x1
2 8x1x2 4x22.
x2
x1
116. Find the four second partial derivatives of the functiongiven by Show that the secondmixed partial derivatives and are equal.fyxfxy
f x, y sen x 2y .
CAPSTONE
Año 1999 2000 2001 2002 2003 2004 2005
x 1.4 1.4 1.4 1.6 1.6 1.7 1.7
y 7.3 7.1 7.0 7.0 6.9 6.9 6.9
z 6.2 6.1 5.9 5.8 5.6 5.5 5.6
CAS
1053714_1303.qxp 10/27/08 12:06 PM Page 916
916 Chapter 13 Functions of Several Variables
117. Marginal Revenue A pharmaceutical corporation has twoplants that produce the same over-the-counter medicine. If and are the numbers of units produced at plant 1 and plant 2,respectively, then the total revenue for the product is given by
When and find (a) the marginal revenue for plant 1,and (b) the marginal revenue for plant 2,
118. Marginal Costs A company manufactures two types ofwood-burning stoves: a freestanding model and a fireplace-insert model. The cost function for producing freestandingand fireplace-insert stoves is
(a) Find the marginal costs and when and
(b) When additional production is required, which model ofstove results in the cost increasing at a higher rate? Howcan this be determined from the cost model?
119. Psychology Early in the twentieth century, an intelligence testcalled the Stanford-Binet Test (more commonly known as the IQtest) was developed. In this test, an individual’s mental age isdivided by the individual’s chronological age and the quotientis multiplied by 100. The result is the individual’s
Find the partial derivatives of with respect to and withrespect to Evaluate the partial derivatives at the point
and interpret the result. (Source: Adapted fromBernstein/Clark-Stewart/Roy/Wickens, Psychology, FourthEdition)
120. Marginal Productivity Consider the Cobb-Douglas produc-tion function When and
find (a) the marginal productivity of labor,and (b) the marginal productivity of capital,
121. Think About It Let be the number of applicants to auniversity, the charge for food and housing at the university,and the tuition. is a function of and such that
and What information is gained bynoticing that both partials are negative?
122. Investment The value of an investment of $1000 earning 6%compounded annually is
where is the annual rate of inflation and is the tax rate forthe person making the investment. Calculate and Determine whether the tax rate or the rateof inflation is the greater “negative” factor in the growth of theinvestment.
123. Temperature Distribution The temperature at any pointin a steel plate is where
and are measured in meters. At the point find the ratesof change of the temperature with respect to the distancesmoved along the plate in the directions of the and axes.
124. Apparent Temperature A measure of how hot weather feelsto an average person is the Apparent Temperature Index. Amodel for this index is
where is the apparent temperature in degrees Celsius, is theair temperature, and is the relative humidity in decimalform. (Source: The UMAP Journal)
(a) Find and when and
(b) Which has a greater effect on air temperature orhumidity? Explain.
125. Ideal Gas Law The Ideal Gas Law states that where is pressure, is volume, is the number of moles ofgas, is a fixed constant (the gas constant), and is absolutetemperature. Show that
126. Marginal Utility The utility function is ameasure of the utility (or satisfaction) derived by a personfrom the consumption of two products and Suppose theutility function is
(a) Determine the marginal utility of product
(b) Determine the marginal utility of product
(c) When and should a person consume onemore unit of product or one more unit of product Explain your reasoning.
(d) Use a computer algebra system to graph the function.Interpret the marginal utilities of products and graphically.
127. Modeling Data Per capita consumptions (in gallons) ofdifferent types of milk in the United States from 1999 through2005 are shown in the table. Consumption of flavored milk,plain reduced-fat milk, and plain light and skim milks are represented by the variables and respectively.(Source: U.S. Department of Agriculture)
A model for the data is given by
(a) Find and
(b) Interpret the partial derivatives in the context of the problem.
zy.
zx
z 0.92x 1.03y 0.02.
z,y,x,
yx
y?xy 3,x 2
y.
x.
U 5x2 xy 3y 2.y.x
U f x, y
TP
PV
VT
1.
TRnVP
PV nRT,
A,
h 0.80.t 30A hA t
htA
A 0.885t 22.4h 1.20th 0.544
y-x-
2, 3 ,yxT 500 0.6x2 1.5y2,x, y
VR 0.03, 0.28 .VI 0.03, 0.28
RI
V I, R 1 0001 0.06 1 R
1 I
10
N t < 0.N p < 0tpNt
pN
f y.f x,y 500,
x 1000f x, y 200x0.7y0.3.
12, 10C.
MIQ
IQ M, CMC
100
IQ.C
M
y 20.x 80C yC x
C 32 xy 175x 205y 1 050.
yx
R x2.R x1,x2 12,x1 4R 200x1 200x2 4x1
2 8x1x2 4x22.
x2
x1
116. Find the four second partial derivatives of the functiongiven by Show that the secondmixed partial derivatives and are equal.fyxfxy
f x, y sen x 2y .
CAPSTONE
Año 1999 2000 2001 2002 2003 2004 2005
x 1.4 1.4 1.4 1.6 1.6 1.7 1.7
y 7.3 7.1 7.0 7.0 6.9 6.9 6.9
z 6.2 6.1 5.9 5.8 5.6 5.5 5.6
CAS
1053714_1303.qxp 10/27/08 12:06 PM Page 916
Para discusión116. Encontrar las cuatro segundas derivadas parciales de la fun-
ción dada por Mostrar que las se-gundas derivadas parciales mixtas fxy y fyx son iguales.
916 Chapter 13 Functions of Several Variables
117. Marginal Revenue A pharmaceutical corporation has twoplants that produce the same over-the-counter medicine. If and are the numbers of units produced at plant 1 and plant 2,respectively, then the total revenue for the product is given by
When and find (a) the marginal revenue for plant 1,and (b) the marginal revenue for plant 2,
118. Marginal Costs A company manufactures two types ofwood-burning stoves: a freestanding model and a fireplace-insert model. The cost function for producing freestandingand fireplace-insert stoves is
(a) Find the marginal costs and when and
(b) When additional production is required, which model ofstove results in the cost increasing at a higher rate? Howcan this be determined from the cost model?
119. Psychology Early in the twentieth century, an intelligence testcalled the Stanford-Binet Test (more commonly known as the IQtest) was developed. In this test, an individual’s mental age isdivided by the individual’s chronological age and the quotientis multiplied by 100. The result is the individual’s
Find the partial derivatives of with respect to and withrespect to Evaluate the partial derivatives at the point
and interpret the result. (Source: Adapted fromBernstein/Clark-Stewart/Roy/Wickens, Psychology, FourthEdition)
120. Marginal Productivity Consider the Cobb-Douglas produc-tion function When and
find (a) the marginal productivity of labor,and (b) the marginal productivity of capital,
121. Think About It Let be the number of applicants to auniversity, the charge for food and housing at the university,and the tuition. is a function of and such that
and What information is gained bynoticing that both partials are negative?
122. Investment The value of an investment of $1000 earning 6%compounded annually is
where is the annual rate of inflation and is the tax rate forthe person making the investment. Calculate and Determine whether the tax rate or the rateof inflation is the greater “negative” factor in the growth of theinvestment.
123. Temperature Distribution The temperature at any pointin a steel plate is where
and are measured in meters. At the point find the ratesof change of the temperature with respect to the distancesmoved along the plate in the directions of the and axes.
124. Apparent Temperature A measure of how hot weather feelsto an average person is the Apparent Temperature Index. Amodel for this index is
where is the apparent temperature in degrees Celsius, is theair temperature, and is the relative humidity in decimalform. (Source: The UMAP Journal)
(a) Find and when and
(b) Which has a greater effect on air temperature orhumidity? Explain.
125. Ideal Gas Law The Ideal Gas Law states that where is pressure, is volume, is the number of moles ofgas, is a fixed constant (the gas constant), and is absolutetemperature. Show that
126. Marginal Utility The utility function is ameasure of the utility (or satisfaction) derived by a personfrom the consumption of two products and Suppose theutility function is
(a) Determine the marginal utility of product
(b) Determine the marginal utility of product
(c) When and should a person consume onemore unit of product or one more unit of product Explain your reasoning.
(d) Use a computer algebra system to graph the function.Interpret the marginal utilities of products and graphically.
127. Modeling Data Per capita consumptions (in gallons) ofdifferent types of milk in the United States from 1999 through2005 are shown in the table. Consumption of flavored milk,plain reduced-fat milk, and plain light and skim milks are represented by the variables and respectively.(Source: U.S. Department of Agriculture)
A model for the data is given by
(a) Find and
(b) Interpret the partial derivatives in the context of the problem.
zy.
zx
z 0.92x 1.03y 0.02.
z,y,x,
yx
y?xy 3,x 2
y.
x.
U 5x2 xy 3y 2.y.x
U f x, y
TP
PV
VT
1.
TRnVP
PV nRT,
A,
h 0.80.t 30A hA t
htA
A 0.885t 22.4h 1.20th 0.544
y-x-
2, 3 ,yxT 500 0.6x2 1.5y2,x, y
VR 0.03, 0.28 .VI 0.03, 0.28
RI
V I, R 1 0001 0.06 1 R
1 I
10
N t < 0.N p < 0tpNt
pN
f y.f x,y 500,
x 1000f x, y 200x0.7y0.3.
12, 10C.
MIQ
IQ M, CMC
100
IQ.C
M
y 20.x 80C yC x
C 32 xy 175x 205y 1 050.
yx
R x2.R x1,x2 12,x1 4R 200x1 200x2 4x1
2 8x1x2 4x22.
x2
x1
116. Find the four second partial derivatives of the functiongiven by Show that the secondmixed partial derivatives and are equal.fyxfxy
f x, y sen x 2y .
CAPSTONE
Año 1999 2000 2001 2002 2003 2004 2005
x 1.4 1.4 1.4 1.6 1.6 1.7 1.7
y 7.3 7.1 7.0 7.0 6.9 6.9 6.9
z 6.2 6.1 5.9 5.8 5.6 5.5 5.6
CAS
1053714_1303.qxp 10/27/08 12:06 PM Page 916
916 Chapter 13 Functions of Several Variables
117. Marginal Revenue A pharmaceutical corporation has twoplants that produce the same over-the-counter medicine. If and are the numbers of units produced at plant 1 and plant 2,respectively, then the total revenue for the product is given by
When and find (a) the marginal revenue for plant 1,and (b) the marginal revenue for plant 2,
118. Marginal Costs A company manufactures two types ofwood-burning stoves: a freestanding model and a fireplace-insert model. The cost function for producing freestandingand fireplace-insert stoves is
(a) Find the marginal costs and when and
(b) When additional production is required, which model ofstove results in the cost increasing at a higher rate? Howcan this be determined from the cost model?
119. Psychology Early in the twentieth century, an intelligence testcalled the Stanford-Binet Test (more commonly known as the IQtest) was developed. In this test, an individual’s mental age isdivided by the individual’s chronological age and the quotientis multiplied by 100. The result is the individual’s
Find the partial derivatives of with respect to and withrespect to Evaluate the partial derivatives at the point
and interpret the result. (Source: Adapted fromBernstein/Clark-Stewart/Roy/Wickens, Psychology, FourthEdition)
120. Marginal Productivity Consider the Cobb-Douglas produc-tion function When and
find (a) the marginal productivity of labor,and (b) the marginal productivity of capital,
121. Think About It Let be the number of applicants to auniversity, the charge for food and housing at the university,and the tuition. is a function of and such that
and What information is gained bynoticing that both partials are negative?
122. Investment The value of an investment of $1000 earning 6%compounded annually is
where is the annual rate of inflation and is the tax rate forthe person making the investment. Calculate and Determine whether the tax rate or the rateof inflation is the greater “negative” factor in the growth of theinvestment.
123. Temperature Distribution The temperature at any pointin a steel plate is where
and are measured in meters. At the point find the ratesof change of the temperature with respect to the distancesmoved along the plate in the directions of the and axes.
124. Apparent Temperature A measure of how hot weather feelsto an average person is the Apparent Temperature Index. Amodel for this index is
where is the apparent temperature in degrees Celsius, is theair temperature, and is the relative humidity in decimalform. (Source: The UMAP Journal)
(a) Find and when and
(b) Which has a greater effect on air temperature orhumidity? Explain.
125. Ideal Gas Law The Ideal Gas Law states that where is pressure, is volume, is the number of moles ofgas, is a fixed constant (the gas constant), and is absolutetemperature. Show that
126. Marginal Utility The utility function is ameasure of the utility (or satisfaction) derived by a personfrom the consumption of two products and Suppose theutility function is
(a) Determine the marginal utility of product
(b) Determine the marginal utility of product
(c) When and should a person consume onemore unit of product or one more unit of product Explain your reasoning.
(d) Use a computer algebra system to graph the function.Interpret the marginal utilities of products and graphically.
127. Modeling Data Per capita consumptions (in gallons) ofdifferent types of milk in the United States from 1999 through2005 are shown in the table. Consumption of flavored milk,plain reduced-fat milk, and plain light and skim milks are represented by the variables and respectively.(Source: U.S. Department of Agriculture)
A model for the data is given by
(a) Find and
(b) Interpret the partial derivatives in the context of the problem.
zy.
zx
z 0.92x 1.03y 0.02.
z,y,x,
yx
y?xy 3,x 2
y.
x.
U 5x2 xy 3y 2.y.x
U f x, y
TP
PV
VT
1.
TRnVP
PV nRT,
A,
h 0.80.t 30A hA t
htA
A 0.885t 22.4h 1.20th 0.544
y-x-
2, 3 ,yxT 500 0.6x2 1.5y2,x, y
VR 0.03, 0.28 .VI 0.03, 0.28
RI
V I, R 1 0001 0.06 1 R
1 I
10
N t < 0.N p < 0tpNt
pN
f y.f x,y 500,
x 1000f x, y 200x0.7y0.3.
12, 10C.
MIQ
IQ M, CMC
100
IQ.C
M
y 20.x 80C yC x
C 32 xy 175x 205y 1 050.
yx
R x2.R x1,x2 12,x1 4R 200x1 200x2 4x1
2 8x1x2 4x22.
x2
x1
116. Find the four second partial derivatives of the functiongiven by Show that the secondmixed partial derivatives and are equal.fyxfxy
f x, y sen x 2y .
CAPSTONE
Año 1999 2000 2001 2002 2003 2004 2005
x 1.4 1.4 1.4 1.6 1.6 1.7 1.7
y 7.3 7.1 7.0 7.0 6.9 6.9 6.9
z 6.2 6.1 5.9 5.8 5.6 5.5 5.6
CAS
1053714_1303.qxp 10/27/08 12:06 PM Page 916
916 Chapter 13 Functions of Several Variables
117. Marginal Revenue A pharmaceutical corporation has twoplants that produce the same over-the-counter medicine. If and are the numbers of units produced at plant 1 and plant 2,respectively, then the total revenue for the product is given by
When and find (a) the marginal revenue for plant 1,and (b) the marginal revenue for plant 2,
118. Marginal Costs A company manufactures two types ofwood-burning stoves: a freestanding model and a fireplace-insert model. The cost function for producing freestandingand fireplace-insert stoves is
(a) Find the marginal costs and when and
(b) When additional production is required, which model ofstove results in the cost increasing at a higher rate? Howcan this be determined from the cost model?
119. Psychology Early in the twentieth century, an intelligence testcalled the Stanford-Binet Test (more commonly known as the IQtest) was developed. In this test, an individual’s mental age isdivided by the individual’s chronological age and the quotientis multiplied by 100. The result is the individual’s
Find the partial derivatives of with respect to and withrespect to Evaluate the partial derivatives at the point
and interpret the result. (Source: Adapted fromBernstein/Clark-Stewart/Roy/Wickens, Psychology, FourthEdition)
120. Marginal Productivity Consider the Cobb-Douglas produc-tion function When and
find (a) the marginal productivity of labor,and (b) the marginal productivity of capital,
121. Think About It Let be the number of applicants to auniversity, the charge for food and housing at the university,and the tuition. is a function of and such that
and What information is gained bynoticing that both partials are negative?
122. Investment The value of an investment of $1000 earning 6%compounded annually is
where is the annual rate of inflation and is the tax rate forthe person making the investment. Calculate and Determine whether the tax rate or the rateof inflation is the greater “negative” factor in the growth of theinvestment.
123. Temperature Distribution The temperature at any pointin a steel plate is where
and are measured in meters. At the point find the ratesof change of the temperature with respect to the distancesmoved along the plate in the directions of the and axes.
124. Apparent Temperature A measure of how hot weather feelsto an average person is the Apparent Temperature Index. Amodel for this index is
where is the apparent temperature in degrees Celsius, is theair temperature, and is the relative humidity in decimalform. (Source: The UMAP Journal)
(a) Find and when and
(b) Which has a greater effect on air temperature orhumidity? Explain.
125. Ideal Gas Law The Ideal Gas Law states that where is pressure, is volume, is the number of moles ofgas, is a fixed constant (the gas constant), and is absolutetemperature. Show that
126. Marginal Utility The utility function is ameasure of the utility (or satisfaction) derived by a personfrom the consumption of two products and Suppose theutility function is
(a) Determine the marginal utility of product
(b) Determine the marginal utility of product
(c) When and should a person consume onemore unit of product or one more unit of product Explain your reasoning.
(d) Use a computer algebra system to graph the function.Interpret the marginal utilities of products and graphically.
127. Modeling Data Per capita consumptions (in gallons) ofdifferent types of milk in the United States from 1999 through2005 are shown in the table. Consumption of flavored milk,plain reduced-fat milk, and plain light and skim milks are represented by the variables and respectively.(Source: U.S. Department of Agriculture)
A model for the data is given by
(a) Find and
(b) Interpret the partial derivatives in the context of the problem.
zy.
zx
z 0.92x 1.03y 0.02.
z,y,x,
yx
y?xy 3,x 2
y.
x.
U 5x2 xy 3y 2.y.x
U f x, y
TP
PV
VT
1.
TRnVP
PV nRT,
A,
h 0.80.t 30A hA t
htA
A 0.885t 22.4h 1.20th 0.544
y-x-
2, 3 ,yxT 500 0.6x2 1.5y2,x, y
VR 0.03, 0.28 .VI 0.03, 0.28
RI
V I, R 1 0001 0.06 1 R
1 I
10
N t < 0.N p < 0tpNt
pN
f y.f x,y 500,
x 1000f x, y 200x0.7y0.3.
12, 10C.
MIQ
IQ M, CMC
100
IQ.C
M
y 20.x 80C yC x
C 32 xy 175x 205y 1 050.
yx
R x2.R x1,x2 12,x1 4R 200x1 200x2 4x1
2 8x1x2 4x22.
x2
x1
116. Find the four second partial derivatives of the functiongiven by Show that the secondmixed partial derivatives and are equal.fyxfxy
f x, y sen x 2y .
CAPSTONE
Año 1999 2000 2001 2002 2003 2004 2005
x 1.4 1.4 1.4 1.6 1.6 1.7 1.7
y 7.3 7.1 7.0 7.0 6.9 6.9 6.9
z 6.2 6.1 5.9 5.8 5.6 5.5 5.6
CAS
1053714_1303.qxp 10/27/08 12:06 PM Page 916
In Exercises 59–64, find the first partial derivatives with respectto and
59.
60.
61. 62.
63.
64.
In Exercises 65–70, evaluate and at the given point.
65.
66.
67.
68.
69.
70.
In Exercises 71–80, find the four second partial derivatives.Observe that the second mixed partials are equal.
71. 72.
73. 74.
75. 76.
77. 78.
79. 80.
In Exercises 81–88, for find all values of and suchthat and simultaneously.
81.
82.
83.
84.
85.
86.
87.
88.
In Exercises 89–92, use a computer algebra system to find thefirst and second partial derivatives of the function. Determinewhether there exist values of and such that and
simultaneously.
89. 90.
91. 92.
In Exercises 93–96, show that the mixed partial derivatives and are equal.
93.
94.
95.
96.
Laplace’s Equation In Exercises 97–100, show that the functionsatisfies Laplace’s equation
97. 98.
99. 100.
Wave Equation In Exercises 101–104, show that the functionsatisfies the wave equation
101. 102.
103. 104.
Heat Equation In Exercises 105 and 106, show that the functionsatisfies the heat equation
105. 106.
In Exercises 107 and 108, determine whether there exists a function with the given partial derivatives. Explain yourreasoning. If such a function exists, give an example.
107.
108.
In Exercises 109 and 110, find the first partial derivative withrespect to
109.
110. f x, y, z x senh yz
y2 2 y 1 z
f x, y, z tan y2z ez2 y 2 z
x.
fy x, y x 4yfx x, y 2x y,
fy x, y 2 sen 3x 2yfx x, y 3 sen 3x 2y ,
f x, y
z e t sen xc
z e t cos xc
z/ t c2 2z/ x2 .
z sen ct sen xz ln x ct
z cos 4x 4ctz sen x ct
2z/ t 2 c2 2z/ x2 .
z arctan yx
z ex sen y
z 12 ey e y sen xz 5xy
2z/ x2 1 2z/ y2 0.
f x, y, z2z
x y
f x, y, z e x sen yz
f x, y, z x2 3xy 4yz z3
f x, y, z xyz
fyyxfyxy,fxyy,
f x, yxy
x yf x, y ln
xx2 y2
f x, y 25 x2 y 2f x, y x sec y
fy x, y 0fx x, y 0yx
f x, y ln x2 y 2 1
f x, y ex2 xy y2
f x, y 3x3 12xy y3
f x, y1x
1y
xy
f x, y x2 xy y2
f x, y x2 4xy y 2 4x 16y 3
f x, y x2 xy y2 5x y
f x, y x2 xy y2 2x 2y
fy x, y 0fx x, y 0yxf x, y ,
z arctan yx
z cos xy
z 2xey 3ye xz ex tan y
z ln x yz x2 y 2
z x4 3x2y 2 y4z x2 2xy 3y 2
z x2 3y2z 3xy2
1, 2, 1f x, y, z 3x2 y2 2z2,
0, 2
, 4f x, y, z z sen x y ,
3, 1, 1f x, y, zxy
x y z,
f x, y, zxyz
, 1, 1, 1
2, 1, 2f x, y, z x2y3 2xyz 3yz,
f x, y, z x3yz2, 1, 1, 1
fzfy,fx,
G x, y, z1
1 x2 y 2 z 2
F x, y, z ln x2 y 2 z 2
w7xz
x yw x2 y 2 z2
f x, y, z 3x2y 5xyz 10yz 2
H x, y, z sen x 2y 3z
z.y,x,
13.3 Partial Derivatives 915
CAS
111. Let be a function of two variables and Describe theprocedure for finding the first partial derivatives.
112. Sketch a surface representing a function of two variablesand Use the sketch to give geometric interpretations of
and
113. Sketch the graph of a function whose derivativeis always negative and whose derivative is always
positive.
114. Sketch the graph of a function whose deriva-tives and are always positive.
115. If is a function of and such that and arecontinuous, what is the relationship between the mixedpartial derivatives? Explain.
fyxfxyyxf
fyfx
z f x, y
fyfx
z f x, y
f y.f xy.x
f
y.xf
WRITING ABOUT CONCEPTS
1053714_1303_pg 915.qxp 10/30/08 8:30 AM Page 915
916 Chapter 13 Functions of Several Variables
117. Marginal Revenue A pharmaceutical corporation has twoplants that produce the same over-the-counter medicine. If and are the numbers of units produced at plant 1 and plant 2,respectively, then the total revenue for the product is given by
When and find (a) the marginal revenue for plant 1,and (b) the marginal revenue for plant 2,
118. Marginal Costs A company manufactures two types ofwood-burning stoves: a freestanding model and a fireplace-insert model. The cost function for producing freestandingand fireplace-insert stoves is
(a) Find the marginal costs and when and
(b) When additional production is required, which model ofstove results in the cost increasing at a higher rate? Howcan this be determined from the cost model?
119. Psychology Early in the twentieth century, an intelligence testcalled the Stanford-Binet Test (more commonly known as the IQtest) was developed. In this test, an individual’s mental age isdivided by the individual’s chronological age and the quotientis multiplied by 100. The result is the individual’s
Find the partial derivatives of with respect to and withrespect to Evaluate the partial derivatives at the point
and interpret the result. (Source: Adapted fromBernstein/Clark-Stewart/Roy/Wickens, Psychology, FourthEdition)
120. Marginal Productivity Consider the Cobb-Douglas produc-tion function When and
find (a) the marginal productivity of labor,and (b) the marginal productivity of capital,
121. Think About It Let be the number of applicants to auniversity, the charge for food and housing at the university,and the tuition. is a function of and such that
and What information is gained bynoticing that both partials are negative?
122. Investment The value of an investment of $1000 earning 6%compounded annually is
where is the annual rate of inflation and is the tax rate forthe person making the investment. Calculate and Determine whether the tax rate or the rateof inflation is the greater “negative” factor in the growth of theinvestment.
123. Temperature Distribution The temperature at any pointin a steel plate is where
and are measured in meters. At the point find the ratesof change of the temperature with respect to the distancesmoved along the plate in the directions of the and axes.
124. Apparent Temperature A measure of how hot weather feelsto an average person is the Apparent Temperature Index. Amodel for this index is
where is the apparent temperature in degrees Celsius, is theair temperature, and is the relative humidity in decimalform. (Source: The UMAP Journal)
(a) Find and when and
(b) Which has a greater effect on air temperature orhumidity? Explain.
125. Ideal Gas Law The Ideal Gas Law states that where is pressure, is volume, is the number of moles ofgas, is a fixed constant (the gas constant), and is absolutetemperature. Show that
126. Marginal Utility The utility function is ameasure of the utility (or satisfaction) derived by a personfrom the consumption of two products and Suppose theutility function is
(a) Determine the marginal utility of product
(b) Determine the marginal utility of product
(c) When and should a person consume onemore unit of product or one more unit of product Explain your reasoning.
(d) Use a computer algebra system to graph the function.Interpret the marginal utilities of products and graphically.
127. Modeling Data Per capita consumptions (in gallons) ofdifferent types of milk in the United States from 1999 through2005 are shown in the table. Consumption of flavored milk,plain reduced-fat milk, and plain light and skim milks are represented by the variables and respectively.(Source: U.S. Department of Agriculture)
A model for the data is given by
(a) Find and
(b) Interpret the partial derivatives in the context of the problem.
zy.
zx
z 0.92x 1.03y 0.02.
z,y,x,
yx
y?xy 3,x 2
y.
x.
U 5x2 xy 3y 2.y.x
U f x, y
TP
PV
VT
1.
TRnVP
PV nRT,
A,
h 0.80.t 30A hA t
htA
A 0.885t 22.4h 1.20th 0.544
y-x-
2, 3 ,yxT 500 0.6x2 1.5y2,x, y
VR 0.03, 0.28 .VI 0.03, 0.28
RI
V I, R 1 0001 0.06 1 R
1 I
10
N t < 0.N p < 0tpNt
pN
f y.f x,y 500,
x 1000f x, y 200x0.7y0.3.
12, 10C.
MIQ
IQ M, CMC
100
IQ.C
M
y 20.x 80C yC x
C 32 xy 175x 205y 1 050.
yx
R x2.R x1,x2 12,x1 4R 200x1 200x2 4x1
2 8x1x2 4x22.
x2
x1
116. Find the four second partial derivatives of the functiongiven by Show that the secondmixed partial derivatives and are equal.fyxfxy
f x, y sen x 2y .
CAPSTONE
Año 1999 2000 2001 2002 2003 2004 2005
x 1.4 1.4 1.4 1.6 1.6 1.7 1.7
y 7.3 7.1 7.0 7.0 6.9 6.9 6.9
z 6.2 6.1 5.9 5.8 5.6 5.5 5.6
CAS
1053714_1303.qxp 10/27/08 12:06 PM Page 916
916 Chapter 13 Functions of Several Variables
117. Marginal Revenue A pharmaceutical corporation has twoplants that produce the same over-the-counter medicine. If and are the numbers of units produced at plant 1 and plant 2,respectively, then the total revenue for the product is given by
When and find (a) the marginal revenue for plant 1,and (b) the marginal revenue for plant 2,
118. Marginal Costs A company manufactures two types ofwood-burning stoves: a freestanding model and a fireplace-insert model. The cost function for producing freestandingand fireplace-insert stoves is
(a) Find the marginal costs and when and
(b) When additional production is required, which model ofstove results in the cost increasing at a higher rate? Howcan this be determined from the cost model?
119. Psychology Early in the twentieth century, an intelligence testcalled the Stanford-Binet Test (more commonly known as the IQtest) was developed. In this test, an individual’s mental age isdivided by the individual’s chronological age and the quotientis multiplied by 100. The result is the individual’s
Find the partial derivatives of with respect to and withrespect to Evaluate the partial derivatives at the point
and interpret the result. (Source: Adapted fromBernstein/Clark-Stewart/Roy/Wickens, Psychology, FourthEdition)
120. Marginal Productivity Consider the Cobb-Douglas produc-tion function When and
find (a) the marginal productivity of labor,and (b) the marginal productivity of capital,
121. Think About It Let be the number of applicants to auniversity, the charge for food and housing at the university,and the tuition. is a function of and such that
and What information is gained bynoticing that both partials are negative?
122. Investment The value of an investment of $1000 earning 6%compounded annually is
where is the annual rate of inflation and is the tax rate forthe person making the investment. Calculate and Determine whether the tax rate or the rateof inflation is the greater “negative” factor in the growth of theinvestment.
123. Temperature Distribution The temperature at any pointin a steel plate is where
and are measured in meters. At the point find the ratesof change of the temperature with respect to the distancesmoved along the plate in the directions of the and axes.
124. Apparent Temperature A measure of how hot weather feelsto an average person is the Apparent Temperature Index. Amodel for this index is
where is the apparent temperature in degrees Celsius, is theair temperature, and is the relative humidity in decimalform. (Source: The UMAP Journal)
(a) Find and when and
(b) Which has a greater effect on air temperature orhumidity? Explain.
125. Ideal Gas Law The Ideal Gas Law states that where is pressure, is volume, is the number of moles ofgas, is a fixed constant (the gas constant), and is absolutetemperature. Show that
126. Marginal Utility The utility function is ameasure of the utility (or satisfaction) derived by a personfrom the consumption of two products and Suppose theutility function is
(a) Determine the marginal utility of product
(b) Determine the marginal utility of product
(c) When and should a person consume onemore unit of product or one more unit of product Explain your reasoning.
(d) Use a computer algebra system to graph the function.Interpret the marginal utilities of products and graphically.
127. Modeling Data Per capita consumptions (in gallons) ofdifferent types of milk in the United States from 1999 through2005 are shown in the table. Consumption of flavored milk,plain reduced-fat milk, and plain light and skim milks are represented by the variables and respectively.(Source: U.S. Department of Agriculture)
A model for the data is given by
(a) Find and
(b) Interpret the partial derivatives in the context of the problem.
zy.
zx
z 0.92x 1.03y 0.02.
z,y,x,
yx
y?xy 3,x 2
y.
x.
U 5x2 xy 3y 2.y.x
U f x, y
TP
PV
VT
1.
TRnVP
PV nRT,
A,
h 0.80.t 30A hA t
htA
A 0.885t 22.4h 1.20th 0.544
y-x-
2, 3 ,yxT 500 0.6x2 1.5y2,x, y
VR 0.03, 0.28 .VI 0.03, 0.28
RI
V I, R 1 0001 0.06 1 R
1 I
10
N t < 0.N p < 0tpNt
pN
f y.f x,y 500,
x 1000f x, y 200x0.7y0.3.
12, 10C.
MIQ
IQ M, CMC
100
IQ.C
M
y 20.x 80C yC x
C 32 xy 175x 205y 1 050.
yx
R x2.R x1,x2 12,x1 4R 200x1 200x2 4x1
2 8x1x2 4x22.
x2
x1
116. Find the four second partial derivatives of the functiongiven by Show that the secondmixed partial derivatives and are equal.fyxfxy
f x, y sen x 2y .
CAPSTONE
Año 1999 2000 2001 2002 2003 2004 2005
x 1.4 1.4 1.4 1.6 1.6 1.7 1.7
y 7.3 7.1 7.0 7.0 6.9 6.9 6.9
z 6.2 6.1 5.9 5.8 5.6 5.5 5.6
CAS
1053714_1303.qxp 10/27/08 12:06 PM Page 916
1 050.
Larson-13-03.qxd 3/12/09 18:48 Page 916
Franjas de Moiré
SECCIÓN 13.3 Derivadas parciales 917
128. Modelo matemático La tabla muestra el gasto en atenciónpública médica (en miles de millones de dólares) en compen-sación a trabajadores x, asistencia pública y y seguro médico delEstado z, del año 2000 al 2005. (Fuente: Centers for Medicareand Medicaid Services)
Un modelo para los datos está dado por
a) Hallar y
b) Determinar la concavidad de las trazas paralelas al plano xz.Interpretar el resultado en el contexto del problema.
c) Determinar la concavidad de las trazas paralelas al plano yz.Interpretar el resultado en el contexto del problema.
¿Verdadero o falso? En los ejercicios 129 a 132, determinar si ladeclaración es verdadera o falsa. Si es falsa, explicar por qué odar un ejemplo que demuestre que es falsa.
129. Si y entonces
130. Si entonces
131. Si entonces
132. Si una superficie cilíndrica tiene rectas generatricesparalelas al eje y, entonces
133. Considerar la función definida por
a) Hallar y para
b) Utilizar la definición de derivadas parciales para hallary
Sugerencia:
c) Utilizar la definición de derivadas parciales para hallary
d) Utilizando el teorema 13.3 y el resultado del inciso c),indicar qué puede decirse acerca de fxy o fyx.
134. Sea Hallar y
135. Mostrar la función
a) Hallar fx(0, 0) y fy(0, 0).
b) Determinar los puntos (si los hay) en los que fx(x, y) ono existe.
136. Considerar la función Mostrar que
PARA MAYOR INFORMACIÓN Para más información sobre esteproblema, ver el artículo “A Classroom Note on a NaturallyOccurring Piecewise Defined Function” de Don Cohen enMathematics and Computer Education.
fxsx, yd 5 5 4x3sx2 1 y2d1y3,
0,
sx, yd Þ s0, 0d
sx, yd 5 s0, 0d.
f sx, yd 5 sx2 1 y2d2y3.
fysx, yd
f sx, yd 5 sx3 1 y3d1y3.
fysx, yd.fxsx, ydf sx, yd 5 Ey
x
!1 1 t3 dt.
fyxs0, 0d.fxys0, 0d
fxs0, 0d 5 limDx→0
f sDx, 0d 2 f s0, 0d
Dx.43
fys0, 0d.fxs0, 0d
sx, yd Þ s0, 0d.fysx, ydfxsx, yd
f sx, yd 5 5xysx2 2 y2d , x2 1 y2
0,
sx, yd Þ s0, 0d
sx, yd 5 s0, 0d.
zyy 5 0.z 5 f sx, yd
2zyx
5 sxy 1 1de xy.z 5 exy,
szyxd 1 szyyd 5 f9sxdgsyd 1 f sxdg9syd.z 5 f sxdgsyd,
z 5 csx 1 yd.zyx 5 zyy,z 5 f sx, yd
2zy2.
2zx2
Léase el artículo “Moiré Fringes and the Conic Sections” de MikeCullen en The College Mathematics Journal. El artículo describecómo dos familias de curvas de nivel dadas por
y
pueden formar franjas de Moiré. Después de leer el artículo, escribirun documento que explique cómo se relaciona la expresión
con las franjas de Moiré formadas por la intersección de las dosfamilias de curvas de nivel. Utilizar como ejemplo uno de los mo-delos siguientes.
fx
?gx
1fy
?gy
gsx, yd 5 bfsx, yd 5 a
Mik
e C
ulle
nM
ike
Cul
len
lím
13.3 Partial Derivatives 917
128. Modeling Data The table shows the public medical expenditures (in billions of dollars) for worker’s compensation
public assistance and Medicare from 2000 through 2005.(Source: Centers for Medicare and Medicaid Services)
A model for the data is given by
(a) Find and
(b) Determine the concavity of traces parallel to the plane.Interpret the result in the context of the problem.
(c) Determine the concavity of traces parallel to the plane.Interpret the result in the context of the problem.
True or False? In Exercises 129–132, determine whether thestatement is true or false. If it is false, explain why or give anexample that shows it is false.
129. If and then
130. If then
131. If then
132. If a cylindrical surface has rulings parallel to the axis, then
133. Consider the function defined by
(a) Find and for
(b) Use the definition of partial derivatives to find and
Hint:
(c) Use the definition of partial derivatives to find and
(d) Using Theorem 13.3 and the result of part (c), what can besaid about or
134. Let Find and
135. Consider the function
(a) Find and
(b) Determine the points (if any) at which or fails to exist.
136. Consider the function Show that
fx x, y4x
3 x2 y2 1 3,
0,
x, y 0, 0
x, y 0, 0.
f x, y x2 y2 2 3.
fy x, yfx x, y
fy 0, 0 .fx 0, 0
f x, y x3 y3 1 3.
fy x, y .fx x, yf x, yy
x
1 t3 dt.
fyx?fxy
fyx 0, 0 .fxy 0, 0
fx 0, 0 límx→0
f x, 0 f 0, 0
x.
fy 0, 0 .fx 0, 0
x, y 0, 0 .fy x, yfx x, y
f x, yxy x2 y2
, x2 y2
0,
x, y 0, 0
x, y 0, 0.
z y 0.y-z f x, y
2zy x
xy 1 e xy.z exy,
z x z y f x g y f x g y .z f x g y ,
z c x y .z x z y,z f x, y
yz-
xz-
2zy2.
2zx2
z 1.2225x2 0.0096y2 71.381x 4.121y 354.65.
zy,x,
Año 2000 2001 2002 2003 2004 2005
x 24.9 28.1 30.1 31.4 32.1 33.5
y 207.5 233.2 258.4 281.9 303.2 324.9
z 224.3 247.7 265.7 283.5 312.8 342.0
FOR FURTHER INFORMATION For more information about thisproblem, see the article “A Classroom Note on a Naturally OccurringPiecewise Defined Function” by Don Cohen in Mathematics andComputer Education.
Read the article “Moiré Fringes and the Conic Sections” by MikeCullen in The College Mathematics Journal. The article describeshow two families of level curves given by
and
can form Moiré patterns. After reading the article, write a paperexplaining how the expression
is related to the Moiré patterns formed by intersecting the twofamilies of level curves. Use one of the following patterns as anexample in your paper.
fx
gx
fy
gy
g x, y bf x, y a
Moiré Fringes
S E C T I O N P R O J E C T
Mik
e C
ulle
nM
ike
Cul
len
1053714_1303.qxp 10/27/08 12:06 PM Page 917
13.3 Partial Derivatives 917
128. Modeling Data The table shows the public medical expenditures (in billions of dollars) for worker’s compensation
public assistance and Medicare from 2000 through 2005.(Source: Centers for Medicare and Medicaid Services)
A model for the data is given by
(a) Find and
(b) Determine the concavity of traces parallel to the plane.Interpret the result in the context of the problem.
(c) Determine the concavity of traces parallel to the plane.Interpret the result in the context of the problem.
True or False? In Exercises 129–132, determine whether thestatement is true or false. If it is false, explain why or give anexample that shows it is false.
129. If and then
130. If then
131. If then
132. If a cylindrical surface has rulings parallel to the axis, then
133. Consider the function defined by
(a) Find and for
(b) Use the definition of partial derivatives to find and
Hint:
(c) Use the definition of partial derivatives to find and
(d) Using Theorem 13.3 and the result of part (c), what can besaid about or
134. Let Find and
135. Consider the function
(a) Find and
(b) Determine the points (if any) at which or fails to exist.
136. Consider the function Show that
fx x, y4x
3 x2 y2 1 3,
0,
x, y 0, 0
x, y 0, 0.
f x, y x2 y2 2 3.
fy x, yfx x, y
fy 0, 0 .fx 0, 0
f x, y x3 y3 1 3.
fy x, y .fx x, yf x, yy
x
1 t3 dt.
fyx?fxy
fyx 0, 0 .fxy 0, 0
fx 0, 0 límx→0
f x, 0 f 0, 0
x.
fy 0, 0 .fx 0, 0
x, y 0, 0 .fy x, yfx x, y
f x, yxy x2 y2
, x2 y2
0,
x, y 0, 0
x, y 0, 0.
z y 0.y-z f x, y
2zy x
xy 1 e xy.z exy,
z x z y f x g y f x g y .z f x g y ,
z c x y .z x z y,z f x, y
yz-
xz-
2zy2.
2zx2
z 1.2225x2 0.0096y2 71.381x 4.121y 354.65.
zy,x,
Año 2000 2001 2002 2003 2004 2005
x 24.9 28.1 30.1 31.4 32.1 33.5
y 207.5 233.2 258.4 281.9 303.2 324.9
z 224.3 247.7 265.7 283.5 312.8 342.0
FOR FURTHER INFORMATION For more information about thisproblem, see the article “A Classroom Note on a Naturally OccurringPiecewise Defined Function” by Don Cohen in Mathematics andComputer Education.
Read the article “Moiré Fringes and the Conic Sections” by MikeCullen in The College Mathematics Journal. The article describeshow two families of level curves given by
and
can form Moiré patterns. After reading the article, write a paperexplaining how the expression
is related to the Moiré patterns formed by intersecting the twofamilies of level curves. Use one of the following patterns as anexample in your paper.
fx
gx
fy
gy
g x, y bf x, y a
Moiré Fringes
S E C T I O N P R O J E C T
Mik
e C
ulle
nM
ike
Cul
len
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PROYECTO DE TRABAJO
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