Calculo 2 De dos variables_9na Edición - Ron Larson

13 Ejercicios de repaso
En los ejercicios 1 y 2, trazar la gráfica de la superficie de nivel
f(x, y, z) = c en el valor dado de c.
3. Conjetura Considerar la función f(x, y) = x 2 + y 2 .
a) Trazar la gráfica de la superficie dada por f.
b) Conjeturar la relación entre las gráficas de f y g(x, y) =
f(x, y) + 2. Explicar el razonamiento.
c) Conjeturar la relación entre las gráficas de f y g(x, y) =
f(x, y – 2). Explicar el razonamiento.
d) Sobre la superficie en el inciso a), trazar las gráficas de
z = f(1, y) y z = f(x, 1).
4. Conjetura Considerar la función
a) Trazar la gráfica de la superficie dada por f.
b) Conjeturar la relación entre las gráficas de f y g(x, y) = f(x + 2, y).
Explicar el razonamiento.
c) Conjeturar la relación entre las gráficas de f y g(x, y) = 4 –
f(x, y). Explicar el razonamiento.
d) Sobre la superficie en el inciso a), trazar las gráficas de z =
f(0, y) y z = f(x, 0).
En los ejercicios 5 a 8, utilizar un sistema algebraico por computadora
y representar gráficamente algunas de las curvas de nivel
de la función.
5. 6.
7. 8.
En los ejercicios 9 y 10, utilizar un sistema algebraico por computadora
y representar gráficamente la función.
9. 10.
En los ejercicios 11 a 14, hallar el límite y analizar la continuidad
de la función (si existe).
En los ejercicios 15 a 24, hallar todas las primeras derivadas parciales.
25. Para pensar Dibujar una gráfica de una función
cuyas derivadas y sean siempre negativas.
26. Hallar las pendientes de la superficie en las
direcciones x y y en el punto .
En los ejercicios 27 a 30, hallar todas las segundas derivadas parciales
y verificar que las segundas derivadas parciales mixtas son
iguales.
Ecuación de Laplace
En los ejercicios 31 a 34, mostrar que la
función satisface la ecuación de Laplace
En los ejercicios 35 y 36, hallar la diferencial total.
37. Análisis de errores Al medir los lados de un triángulo rectángulo
se obtienen los valores de 5 y 12 centímetros, con un posible
error de centímetro. Aproximar el error máximo posible al
calcular la longitud de la hipotenusa. Aproximar el error porcentual
máximo.
38. Análisis de errores Para determinar la altura de una torre, el
ángulo de elevación a la parte superior de la torre se midió desde
un punto a 100 pies
pie de la base. La medida del ángulo da
33°, con un posible error de 1°. Suponer que el suelo es horizontal,
para aproximar el error máximo al determinar la altura
de la torre.
39. Volumen Se mide un cono circular recto. Su radio y su altura
son 2 y 5 pulgadas, respectivamente. El posible error de
medición es – 1 8 de pulgada. Aproximar el error máximo posible en
el cálculo del volumen.
40. Superficie lateral Aproximar el error en el cálculo de la superficie
lateral del cono del ejercicio 39. (La superficie lateral está
dada por A rr 2 h 2 .
± 1 2
1
2
2, 0, 0
z x 2 ln y 1
f y
f x
z f x, y
gx, y y 1 x
f x, y e x2 y 2
f x, y
x
x y
f x, y x 2 y 2
f x, y ln xy
f x, y e x2 y 2
978 CAPÍTULO 13 Funciones de varias variables
1.
2. c 0
f x, y, z 4x 2 y 2 4z 2 ,
c 2
f x, y, z x 2 y z 2 ,
f x, y 1 x 2 y 2 .
11. 12.
13. 14. lím
x, y → 0, 0
x 2 y
x 4 y 2
lím
x, y → 0, 0
y
xe y2
1 x 2 lím
x, y → 1, 1
xy
x 2 y 2
lím
x, y → 1, 1
xy
x 2 y 2
15. 16.
17. 18. z ln x 2 y 2 1
z e y e x f x, y
xy
x
y
f x, y
e x cos y
z
x 2 +
2 z
y 2 0.
gx, y cos x 2y
h x, y x sen y y cos x
hx, y
x
x
y
fx, y 3x 2 xy 2y 3 2, 0, 0
y-
x-
z x 2 ln y 1
f y
f x z f x, y
ux, t
c sen akx cos kt
u x, t
ce n2 t sen nx
f x, y, z
1
1 x 2 y 2 z 2
fx, y, z
z arctan y x
w x 2 y 2 z 2
gx, y
xy
x 2 y 2
z ln x 2 y 2 1
z e y e x fx, y
xy
x
y
fx, y
e x cos y
lím
x, y → 0, 0
x 2 y
x 4 y 2
lím
x, y → 0, 0
y
xe y2
1 x 2 lím
x, y → 1, 1
xy
x 2 y 2
lím
x, y → 1, 1
xy
x 2 y 2 gx, y y 1 x
fx, y e x2 y 2 fx, y
x
x
y
fx, y x 2 y 2 fx, y ln xy
f x, y e x2 y 2
z f x, 0 .
z
f 0, y
gx, y 4 fx, y .
f
gx, y f x 2, y .
f
f.
fx, y 1 x 2 y 2 .
z f x, 1 .
z
f 1, y
gx, y f x, y 2.
f
gx, y f x, y 2.
f
f.
f x, y x 2 y 2 .
c 0
f x, y, z 4x 2 y 2 4z 2 ,
c 2
f x, y, z x 2 y z 2 ,
c.
f x, y, z
c
CAS
CAS
In Exercises 1 and 2, sketch the graph of the level surface
at the given value of
1.
2.
3. Conjecture Consider the function
(a) Sketch the graph of the surface given by
(b) Make a conjecture about the relationship between the
graphs of and Explain your
reasoning.
(c) Make a conjecture about the relationship between the
graphs of and Explain your
reasoning.
(d) On the surface in part (a), sketch the graphs of
and
4. Conjecture Consider the function
(a) Sketch the graph of the surface given by
(b) Make a conjecture about the relationship between the
graphs of and Explain your
reasoning.
(c) Make a conjecture about the relationship between the
graphs of and Explain your
reasoning.
(d) On the surface in part (a), sketch the graphs of
and
In Exercises 5–8, use a computer algebra system to graph
several level curves of the function.
5. 6.
7. 8.
In Exercises 9 and 10, use a computer algebra system to graph
the function.
9. 10.
In Exercises 11–14, find the limit and discuss the continuity of
the function (if it exists).
11. 12.
13. 14.
In Exercises 15–24, find all first partial derivatives.
15. 16.
17. 18.
19. 20.
21.
22.
23. 24.
25. Think About It Sketch a graph of a function
whose derivative is always negative and whose derivative is
always negative.
26. Find the slopes of the surface in the and
directions at the point .
In Exercises 27–30, find all second partial derivatives and
verify that the second mixed partials are equal.
27. 28.
29. 30.
Laplace’s Equation
In Exercises 31–34, show that the function
satisfies Laplace’s equation
31. 32.
33. 34.
In Exercises 35 and 36, find the total differential.
35. 36.
37. Error Analysis The legs of a right triangle are measured to be
5 centimeters and 12 centimeters, with a possible error of
centimeter. Approximate the maximum possible error in
computing the length of the hypotenuse. Approximate the
maximum percent error.
38. Error Analysis To determine the height of a tower, the angle
of elevation to the top of the tower is measured from a point 100
feet foot from the base. The angle is measured at with
a possible error of
Assuming that the ground is
horizontal, approximate the maximum error in determining the
height of the tower.
39. Volume A right circular cone is measured, and the radius and
height are found to be 2 inches and 5 inches, respectively. The
possible error in measurement is
inch. Approximate the
maximum possible error in the computation of the volume.
40. Lateral Surface Area Approximate the error in the computation
of the lateral surface area of the cone in Exercise 39. The
lateral surface area is given by A r r 2 h 2 .
1
8
1.
33 ,
± 1 2
1
2
z
xy
x 2 y 2
z
x sen xy
z
e y sen x
z
y
x 2 y 2 z x 3 3xy 2
z x 2 y 2
2 z
x 2 +
2 z
y 2 0.
gx, y cos x 2y
h x, y x sen y y cos x
hx, y
x
x
y
fx, y 3x 2 xy 2y 3 2, 0, 0
y-
x-
z x 2 ln y 1
f y
f x z f x, y
ux, t
c sen akx cos kt
u x, t
ce n2 t sen nx
fx, y, z
1
1 x 2 y 2 z 2
fx, y, z
z arctan y x
w x 2 y 2 z 2
gx, y
xy
x 2 y 2
z ln x 2 y 2 1
z e y e x fx, y
xy
x
y
fx, y
e x cos y
lím
x, y → 0, 0
x 2 y
x 4 y 2
lím
x, y → 0, 0
y
xe y2
1 x 2 lím
x, y → 1, 1
xy
x 2 y 2
lím
x, y → 1, 1
xy
x 2 y 2 gx, y y 1 x
fx, y e x2 y 2 fx, y
x
x
y
fx, y x 2 y 2 fx, y ln xy
f x, y e x2 y 2
z f x, 0 .
z
f 0, y
gx, y 4 fx, y .
f
gx, y f x 2, y .
f
f.
fx, y 1 x 2 y 2 .
z f x, 1 .
z
f 1, y
gx, y f x, y 2.
f
gx, y f x, y 2.
f
f.
fx, y x 2 y 2 .
c 0
f x, y, z 4x 2 y 2 4z 2 ,
c 2
f x, y, z x 2 y z 2 ,
c.
f x, y, z
c
978 Chapter 13 Functions of Several Variables
13 REVIEW EXERCISES See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
CAS
CAS
In Exercises 1 and 2, sketch the graph of the level surface
at the given value of
1.
2.
3. Conjecture Consider the function
(a) Sketch the graph of the surface given by
(b) Make a conjecture about the relationship between the
graphs of and Explain your
reasoning.
(c) Make a conjecture about the relationship between the
graphs of and Explain your
reasoning.
(d) On the surface in part (a), sketch the graphs of
and
4. Conjecture Consider the function
(a) Sketch the graph of the surface given by
(b) Make a conjecture about the relationship between the
graphs of and Explain your
reasoning.
(c) Make a conjecture about the relationship between the
graphs of and Explain your
reasoning.
(d) On the surface in part (a), sketch the graphs of
and
In Exercises 5–8, use a computer algebra system to graph
several level curves of the function.
5. 6.
7. 8.
In Exercises 9 and 10, use a computer algebra system to graph
the function.
9. 10.
In Exercises 11–14, find the limit and discuss the continuity of
the function (if it exists).
11. 12.
13. 14.
In Exercises 15–24, find all first partial derivatives.
15. 16.
17. 18.
19. 20.
21.
22.
23. 24.
25. Think About It Sketch a graph of a function
whose derivative is always negative and whose derivative is
always negative.
26. Find the slopes of the surface in the and
directions at the point .
In Exercises 27–30, find all second partial derivatives and
verify that the second mixed partials are equal.
27. 28.
29. 30.
Laplace’s Equation
In Exercises 31–34, show that the function
satisfies Laplace’s equation
31. 32.
33. 34.
In Exercises 35 and 36, find the total differential.
35. 36.
37. Error Analysis The legs of a right triangle are measured to be
5 centimeters and 12 centimeters, with a possible error of
centimeter. Approximate the maximum possible error in
computing the length of the hypotenuse. Approximate the
maximum percent error.
38. Error Analysis To determine the height of a tower, the angle
of elevation to the top of the tower is measured from a point 100
feet foot from the base. The angle is measured at with
a possible error of
Assuming that the ground is
horizontal, approximate the maximum error in determining the
height of the tower.
39. Volume A right circular cone is measured, and the radius and
height are found to be 2 inches and 5 inches, respectively. The
possible error in measurement is
inch. Approximate the
maximum possible error in the computation of the volume.
40. Lateral Surface Area Approximate the error in the computation
of the lateral surface area of the cone in Exercise 39. The
lateral surface area is given by A r r 2 h 2 .
1
8
1.
33 ,
± 1 2
1
2
z
xy
x 2 y 2
z
x sen xy
z
e y sen x
z
y
x 2 y 2 z x 3 3xy 2
z x 2 y 2
2 z
x 2 +
2 z
y 2 0.
gx, y cos x 2y
h x, y x sen y y cos x
hx, y
x
x
y
fx, y 3x 2 xy 2y 3 2, 0, 0
y-
x-
z x 2 ln y 1
f y
f x z f x, y
u x, t
c sen akx cos kt
u x, t
ce n2 t sen nx
f x, y, z
1
1 x 2 y 2 z 2
f x, y, z
z arctan y x
w x 2 y 2 z 2
g x, y
xy
x 2 y 2
z ln x 2 y 2 1
z e y e x fx, y
xy
x
y
fx, y
e x cos y
lím
x, y → 0, 0
x 2 y
x 4 y 2
lím
x, y → 0, 0
y
xe y2
1 x 2 lím
x, y → 1, 1
xy
x 2 y 2
lím
x, y → 1, 1
xy
x 2 y 2 gx, y y 1 x
fx, y e x2 y 2 fx, y
x
x
y
fx, y x 2 y 2 fx, y ln xy
f x, y e x2 y 2
z f x, 0 .
z
f 0, y
gx, y 4 fx, y .
f
gx, y f x 2, y .
f
f.
fx, y 1 x 2 y 2 .
z f x, 1 .
z
f 1, y
gx, y f x, y 2.
f
gx, y f x, y 2.
f
f.
fx, y x 2 y 2 .
c 0
f x, y, z 4x 2 y 2 4z 2 ,
c 2
f x, y, z x 2 y z 2 ,
c.
f x, y, z
c
978 Chapter 13 Functions of Several Variables
13 REVIEW EXERCISES See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
CAS
CAS
In Exercises 1 and 2, sketch the graph of the level surface
at the given value of
1.
2.
3. Conjecture Consider the function
(a) Sketch the graph of the surface given by
(b) Make a conjecture about the relationship between the
graphs of and Explain your
reasoning.
(c) Make a conjecture about the relationship between the
graphs of and Explain your
reasoning.
(d) On the surface in part (a), sketch the graphs of
and
4. Conjecture Consider the function
(a) Sketch the graph of the surface given by
(b) Make a conjecture about the relationship between the
graphs of and Explain your
reasoning.
(c) Make a conjecture about the relationship between the
graphs of and Explain your
reasoning.
(d) On the surface in part (a), sketch the graphs of
and
In Exercises 5–8, use a computer algebra system to graph
several level curves of the function.
5. 6.
7. 8.
In Exercises 9 and 10, use a computer algebra system to graph
the function.
9. 10.
In Exercises 11–14, find the limit and discuss the continuity of
the function (if it exists).
11. 12.
13. 14.
In Exercises 15–24, find all first partial derivatives.
15. 16.
17. 18.
19. 20.
21.
22.
23. 24.
25. Think About It Sketch a graph of a function
whose derivative is always negative and whose derivative is
always negative.
26. Find the slopes of the surface in the and
directions at the point .
In Exercises 27–30, find all second partial derivatives and
verify that the second mixed partials are equal.
27. 28.
29. 30.
Laplace’s Equation
In Exercises 31–34, show that the function
satisfies Laplace’s equation
31. 32.
33. 34.
In Exercises 35 and 36, find the total differential.
35. 36.
37. Error Analysis The legs of a right triangle are measured to be
5 centimeters and 12 centimeters, with a possible error of
centimeter. Approximate the maximum possible error in
computing the length of the hypotenuse. Approximate the
maximum percent error.
38. Error Analysis To determine the height of a tower, the angle
of elevation to the top of the tower is measured from a point 100
feet foot from the base. The angle is measured at with
a possible error of
Assuming that the ground is
horizontal, approximate the maximum error in determining the
height of the tower.
39. Volume A right circular cone is measured, and the radius and
height are found to be 2 inches and 5 inches, respectively. The
possible error in measurement is
inch. Approximate the
maximum possible error in the computation of the volume.
40. Lateral Surface Area Approximate the error in the computation
of the lateral surface area of the cone in Exercise 39. The
lateral surface area is given by A r r 2 h 2 .
1
8
1.
33 ,
± 1 2
1
2
z
xy
x 2 y 2
z
x sen xy
z
e y sen x
z
y
x 2 y 2 z x 3 3xy 2
z x 2 y 2
2 z
x 2 +
2 z
y 2 0.
g x, y cos x 2y
h x, y x sen y y cos x
h x, y
x
x
y
f x, y 3x 2 xy 2y 3 2, 0, 0
y-
x-
z x 2 ln y 1
f y
f x z f x, y
ux, t
c sen akx cos kt
u x, t
ce n2 t sen nx
fx, y, z
1
1 x 2 y 2 z 2
fx, y, z
z arctan y x
w x 2 y 2 z 2
g x, y
xy
x 2 y 2
z ln x 2 y 2 1
z e y e x fx, y
xy
x
y
fx, y
e x cos y
lím
x, y → 0, 0
x 2 y
x 4 y 2
lím
x, y → 0, 0
y
xe y2
1 x 2 lím
x, y → 1, 1
xy
x 2 y 2
lím
x, y → 1, 1
xy
x 2 y 2 gx, y y 1 x
fx, y e x2 y 2 fx, y
x
x
y
fx, y x 2 y 2 fx, y ln xy
f x, y e x2 y 2
z f x, 0 .
z
f 0, y
gx, y 4 fx, y .
f
gx, y f x 2, y .
f
f.
fx, y 1 x 2 y 2 .
z f x, 1 .
z
f 1, y
gx, y f x, y 2.
f
gx, y f x, y 2.
f
f.
fx, y x 2 y 2 .
c 0
f x, y, z 4x 2 y 2 4z 2 ,
c 2
f x, y, z x 2 y z 2 ,
c.
f x, y, z
c
978 Chapter 13 Functions of Several Variables
13 REVIEW EXERCISES See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
CAS
CAS
In Exercises 1 and 2, sketch the graph of the level surface
at the given value of
1.
2.
3. Conjecture Consider the function
(a) Sketch the graph of the surface given by
(b) Make a conjecture about the relationship between the
graphs of and Explain your
reasoning.
(c) Make a conjecture about the relationship between the
graphs of and Explain your
reasoning.
(d) On the surface in part (a), sketch the graphs of
and
4. Conjecture Consider the function
(a) Sketch the graph of the surface given by
(b) Make a conjecture about the relationship between the
graphs of and Explain your
reasoning.
(c) Make a conjecture about the relationship between the
graphs of and Explain your
reasoning.
(d) On the surface in part (a), sketch the graphs of
and
In Exercises 5–8, use a computer algebra system to graph
several level curves of the function.
5. 6.
7. 8.
In Exercises 9 and 10, use a computer algebra system to graph
the function.
9. 10.
In Exercises 11–14, find the limit and discuss the continuity of
the function (if it exists).
11. 12.
13. 14.
In Exercises 15–24, find all first partial derivatives.
15. 16.
17. 18.
19. 20.
21.
22.
23. 24.
25. Think About It Sketch a graph of a function
whose derivative is always negative and whose derivative is
always negative.
26. Find the slopes of the surface in the and
directions at the point .
In Exercises 27–30, find all second partial derivatives and
verify that the second mixed partials are equal.
27. 28.
29. 30.
Laplace’s Equation
In Exercises 31–34, show that the function
satisfies Laplace’s equation
31. 32.
33. 34.
In Exercises 35 and 36, find the total differential.
35. 36.
37. Error Analysis The legs of a right triangle are measured to be
5 centimeters and 12 centimeters, with a possible error of
centimeter. Approximate the maximum possible error in
computing the length of the hypotenuse. Approximate the
maximum percent error.
38. Error Analysis To determine the height of a tower, the angle
of elevation to the top of the tower is measured from a point 100
feet foot from the base. The angle is measured at with
a possible error of
Assuming that the ground is
horizontal, approximate the maximum error in determining the
height of the tower.
39. Volume A right circular cone is measured, and the radius and
height are found to be 2 inches and 5 inches, respectively. The
possible error in measurement is
inch. Approximate the
maximum possible error in the computation of the volume.
40. Lateral Surface Area Approximate the error in the computation
of the lateral surface area of the cone in Exercise 39. The
lateral surface area is given by A r r 2 h 2 .
1
8
1.
33 ,
± 1 2
1
2
z
xy
x 2 y 2
z
x sen xy
z
e y sen x
z
y
x 2 y 2 z x 3 3xy 2
z x 2 y 2
2 z
x 2 +
2 z
y 2 0.
gx, y cos x 2y
h x, y x sen y y cos x
hx, y
x
x
y
fx, y 3x 2 xy 2y 3 2, 0, 0
y-
x-
z x 2 ln y 1
f y
f x z f x, y
ux, t
c sen akx cos kt
u x, t
ce n2 t sen nx
f x, y, z
1
1 x 2 y 2 z 2
fx, y, z
z arctan y x
w x 2 y 2 z 2
gx, y
xy
x 2 y 2
z ln x 2 y 2 1
z e y e x fx, y
xy
x
y
fx, y
e x cos y
lím
x, y → 0, 0
x 2 y
x 4 y 2
lím
x, y → 0, 0
y
xe y2
1 x 2 lím
x, y → 1, 1
xy
x 2 y 2
lím
x, y → 1, 1
xy
x 2 y 2 gx, y y 1 x
fx, y e x2 y 2 fx, y
x
x
y
fx, y x 2 y 2 fx, y ln xy
f x, y e x2 y 2
z f x, 0 .
z
f 0, y
gx, y 4 fx, y .
f
gx, y f x 2, y .
f
f.
fx, y 1 x 2 y 2 .
z f x, 1 .
z
f 1, y
gx, y f x, y 2.
f
gx, y f x, y 2.
f
f.
f x, y x 2 y 2 .
c 0
f x, y, z 4x 2 y 2 4z 2 ,
c 2
f x, y, z x 2 y z 2 ,
c.
f x, y, z
c
978 Chapter 13 Functions of Several Variables
13 REVIEW EXERCISES See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
CAS
CAS
In Exercises 1 and 2, sketch the graph of the level surface
at the given value of
1.
2.
3. Conjecture Consider the function
(a) Sketch the graph of the surface given by
(b) Make a conjecture about the relationship between the
graphs of and Explain your
reasoning.
(c) Make a conjecture about the relationship between the
graphs of and Explain your
reasoning.
(d) On the surface in part (a), sketch the graphs of
and
4. Conjecture Consider the function
(a) Sketch the graph of the surface given by
(b) Make a conjecture about the relationship between the
graphs of and Explain your
reasoning.
(c) Make a conjecture about the relationship between the
graphs of and Explain your
reasoning.
(d) On the surface in part (a), sketch the graphs of
and
In Exercises 5–8, use a computer algebra system to graph
several level curves of the function.
5. 6.
7. 8.
In Exercises 9 and 10, use a computer algebra system to graph
the function.
9. 10.
In Exercises 11–14, find the limit and discuss the continuity of
the function (if it exists).
11. 12.
13. 14.
In Exercises 15–24, find all first partial derivatives.
15. 16.
17. 18.
19. 20.
21.
22.
23. 24.
25. Think About It Sketch a graph of a function
whose derivative is always negative and whose derivative is
always negative.
26. Find the slopes of the surface in the and
directions at the point .
In Exercises 27–30, find all second partial derivatives and
verify that the second mixed partials are equal.
27. 28.
29. 30.
Laplace’s Equation
In Exercises 31–34, show that the function
satisfies Laplace’s equation
31. 32.
33. 34.
In Exercises 35 and 36, find the total differential.
35. 36.
37. Error Analysis The legs of a right triangle are measured to be
5 centimeters and 12 centimeters, with a possible error of
centimeter. Approximate the maximum possible error in
computing the length of the hypotenuse. Approximate the
maximum percent error.
38. Error Analysis To determine the height of a tower, the angle
of elevation to the top of the tower is measured from a point 100
feet foot from the base. The angle is measured at with
a possible error of
Assuming that the ground is
horizontal, approximate the maximum error in determining the
height of the tower.
39. Volume A right circular cone is measured, and the radius and
height are found to be 2 inches and 5 inches, respectively. The
possible error in measurement is
inch. Approximate the
maximum possible error in the computation of the volume.
40. Lateral Surface Area Approximate the error in the computation
of the lateral surface area of the cone in Exercise 39. The
lateral surface area is given by A r r 2 h 2 .
1
8
1.
33 ,
± 1 2
1
2
z
xy
x 2 y 2
z
x sen xy
z
e y sen x
z
y
x 2 y 2 z x 3 3xy 2
z x 2 y 2
2 z
x 2 +
2 z
y 2 0.
gx, y cos x 2y
h x, y x sen y y cos x
hx, y
x
x
y
fx, y 3x 2 xy 2y 3 2, 0, 0
y-
x-
z x 2 ln y 1
f y
f x z f x, y
ux, t
c sen akx cos kt
u x, t
ce n2 t sen nx
fx, y, z
1
1 x 2 y 2 z 2
fx, y, z
z arctan y x
w x 2 y 2 z 2
gx, y
xy
x 2 y 2
z ln x 2 y 2 1
z e y e x fx, y
xy
x
y
fx, y
e x cos y
lím
x, y → 0, 0
x 2 y
x 4 y 2
lím
x, y → 0, 0
y
xe y2
1 x 2 lím
x, y → 1, 1
xy
x 2 y 2
lím
x, y → 1, 1
xy
x 2 y 2 gx, y y 1 x
fx, y e x2 y 2 fx, y
x
x
y
fx, y x 2 y 2 fx, y ln xy
f x, y e x2 y 2
z f x, 0 .
z
f 0, y
gx, y 4 fx, y .
f
gx, y f x 2, y .
f
f.
fx, y 1 x 2 y 2 .
z f x, 1 .
z
f 1, y
gx, y f x, y 2.
f
gx, y f x, y 2.
f
f.
fx, y x 2 y 2 .
c 0
f x, y, z 4x 2 y 2 4z 2 ,
c 2
f x, y, z x 2 y z 2 ,
c.
f x, y, z
c
978 Chapter 13 Functions of Several Variables
13 REVIEW EXERCISES See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
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