Calculo 2 De dos variables_9na Edición - Ron Larson

SECCIÓN 13.1 Introducción a las funciones de varias variables 895
32. Para pensar Usar la función dada en el ejercicio 31.
a) Hallar el dominio y rango de la función.
b) Identificar los puntos en el plano xy donde el valor de la función
es 0.
c) ¿Pasa la superficie por todos los octantes del sistema de coordenadas
rectangular? Dar las razones de la respuesta.
En los ejercicios 33 a 40, dibujar la superficie dada por la función.
En los ejercicios 41 a 44, utilizar un sistema algebraico por computadora
para álgebra y representar gráficamente la función.
41. 42.
43. 44. f(x, y) x sen y
En los ejercicios 45 a 48, asociar la gráfica de la superficie con
uno de los mapas de contorno. [Los mapas de contorno están
marcados a), b), c) y d).]
a) b)
c) d)
45. 46.
47. 48.
En los ejercicios 49 a 56, describir las curvas de nivel de la función.
Dibujar las curvas de nivel para los valores dados de c.
En los ejercicios 57 a 60, utilizar una herramienta de graficación
para representar seis curvas de nivel de la función.
57. 58.
59. 60. h(x, y) 3 sen(x y)
gx, y
8
1 x 2 y 2 f x, y xy
f x, y x 2 y 2 2
y
x
−6
4
10
z
4
6
5
4
5
3 2 5
−2
x
y
z
f x, y cos x 2 2y 2
4
f x, y ln y x2
y
x
3
6
4
4
z
y
x
3
3
3
z
f x, y e 1x2 y 2
f x, y e 1x2 y 2
x
y
x
y
x
y
x
y
f x, y x 2 e xy2 z 1 12 144 16x2 9y 2
z y 2 x 2 1
Desarrollo de conceptos
61. ¿Qué es una gráfica de una función de dos variables? ¿Cómo
se interpreta geométricamente? Describir las curvas de nivel.
62. Todas las curvas de nivel de la superficie dada por
son círculos concéntricos. ¿Implica esto que la gráfica de f es un
hemisferio? Ilustrar la respuesta con un ejemplo.
63. Construir una función cuyas curvas de nivel sean rectas que
pasen por el origen.
z f x, y
32. Think About It Use the function given in Exercise 31.
(a) Find the domain and range of the function.
(b) Identify the points in the
plane at which the function
value is 0.
(c) Does the surface pass through all the octants of the rectangular
coordinate system? Give reasons for your answer.
In Exercises 33– 40, sketch the surface given by the function.
33. 34.
35. 36.
37. 38.
39.
40.
In Exercises 41–44, use a computer algebra system to graph the
function.
41. 42.
43. 44.
In Exercises 45–48, match the graph of the surface with one of
the contour maps. [The contour maps are labeled (a), (b), (c),
and (d).]
(a)
(b)
(c)
(d)
45. 46.
47. 48.
In Exercises 49–56, describe the level curves of the function.
Sketch the level curves for the given -values.
49.
50.
51.
52.
53.
54.
55.
56.
In Exercises 57–60, use a graphing utility to graph six level
curves of the function.
57. 58.
59. 60. h x, y 3 sin x y
g x, y
8
1 x 2 y 2 fx, y xy
f x, y x 2 y 2 2
c 0, ± 1 2, ±1, ± 3 2, ±2
f x, y ln x y ,
c ± 1 2 , ±1, ± 3 2 , ±2
f x, y x x2 y 2 ,
c 2, 3, 4, 1 2 , 1 3 , 1 4
fx, y e xy 2 ,
c ±1, ±2, . . . , ±6
f x, y
xy,
c 0, 1, 2, 3
f x, y 9 x 2 y 2 ,
c 0, 1, 2, 3, 4
z x 2 4y 2 ,
c 0, 2, 4, 6, 8, 10
z 6 2x 3y,
c 1, 0, 2, 4
z x y,
c
y
x
−6
4
10
z
4
6
5
4
5
3 2 5
−2
x
y
z
fx, y
cos
x 2 2y 2
4
fx, y ln y x 2
y
x
3
6
4
4
z
y
x
3
3
3
z
fx, y e 1 x2 y 2
fx, y e 1 x2 y 2 x
y
x
y
y
x
y
fx, y
x sin y
f x, y x 2 e xy 2 z
1
12 144 16x 2 9y 2
z y 2 x 2 1
f x, y
xy,
0,
x 0, y 0
x < 0 o y < 0
f x, y e x z
1
2 x 2 y 2
z x 2 y 2 g x, y
1
2 y
f x, y y2 f x, y 6 2x 3y
f x, y 4
xy-
13.1 Introduction to Functions of Several Variables 895
CAS
61. What is a graph of a function of two variables? How is it
interpreted geometrically? Describe level curves.
62. All of the level curves of the surface given by
are concentric circles. Does this imply that the graph of is
a hemisphere? Illustrate your answer with an example.
63. Construct a function whose level curves are lines passing
through the origin.
f
z
f x, y
WRITING ABOUT CONCEPTS
64. Considere la función para y
(a) Sketch the graph of the surface given by
(b) Make a conjecture about the relationship between the
graphs of y Explain your
reasoning.
(c) Make a conjecture about the relationship between the
graphs of y Explain your
reasoning.
(d) Make a conjecture about the relationship between
the graphs of y Explain your
reasoning.
(e) On the surface in part (a), sketch the graph of
z f x, x .
gx, y
1
2 fx, y .
f
gx, y f x, y .
f
gx, y f x, y 3.
f
f.
y 0.
x 0
f x, y
xy,
32. Think About It Use the function given in Exercise 31.
(a) Find the domain and range of the function.
(b) Identify the points in the
plane at which the function
value is 0.
(c) Does the surface pass through all the octants of the rectangular
coordinate system? Give reasons for your answer.
In Exercises 33– 40, sketch the surface given by the function.
33. 34.
35. 36.
37. 38.
39.
40.
In Exercises 41–44, use a computer algebra system to graph the
function.
41. 42.
43. 44.
In Exercises 45–48, match the graph of the surface with one of
the contour maps. [The contour maps are labeled (a), (b), (c),
and (d).]
(a)
(b)
(c)
(d)
45. 46.
47. 48.
In Exercises 49–56, describe the level curves of the function.
Sketch the level curves for the given -values.
49.
50.
51.
52.
53.
54.
55.
56.
In Exercises 57–60, use a graphing utility to graph six level
curves of the function.
57. 58.
59. 60. h x, y 3 sin x y
g x, y
8
1 x 2 y 2 fx, y xy
f x, y x 2 y 2 2
c 0, ± 1 2 , ±1, ± 3 2 , ±2
f x, y ln x y , c ± 1 2 , ±1, ± 3 2 , ±2
f x, y x x2 y 2 ,
c 2, 3, 4, 1 2 , 1 3 , 1 4
f x, y e xy 2 ,
c ±1, ±2, . . . , ±6
f x, y
xy,
c 0, 1, 2, 3
f x, y 9 x 2 y 2 ,
c 0, 1, 2, 3, 4
z x 2 4y 2 ,
c 0, 2, 4, 6, 8, 10
z 6 2x 3y,
c 1, 0, 2, 4
z x y,
c
y
x
−6
4
10
z
4
6
5
4
5
3 2 5
−2
x
y
z
f x, y
cos
x 2 2y 2
4
f x, y ln y x 2
y
x
3
6
4
4
z
y
x
3
3
3
z
fx, y e 1 x2 y 2
fx, y e 1 x2 y 2 x
y
x
y
y
x
y
fx, y
x sin y
f x, y x 2 e xy 2 z
1
12 144 16x 2 9y 2
z y 2 x 2 1
fx, y
xy,
0,
x 0, y 0
x < 0 o y < 0
fx, y e x z
1
2 x 2 y 2
z x 2 y 2 g x, y
1
2 y
fx, y y2 fx, y 6 2x 3y
f x, y 4
xy-
13.1 Introduction to Functions of Several Variables 895
CAS
61. What is a graph of a function of two variables? How is it
interpreted geometrically? Describe level curves.
62. All of the level curves of the surface given by
are concentric circles. Does this imply that the graph of is
a hemisphere? Illustrate your answer with an example.
63. Construct a function whose level curves are lines passing
through the origin.
f
z
f x, y
WRITING ABOUT CONCEPTS
64. Considere la función para y
(a) Sketch the graph of the surface given by
(b) Make a conjecture about the relationship between the
graphs of y Explain your
reasoning.
(c) Make a conjecture about the relationship between the
graphs of y Explain your
reasoning.
(d) Make a conjecture about the relationship between
the graphs of y Explain your
reasoning.
(e) On the surface in part (a), sketch the graph of
z f x, x .
gx, y
1
2 fx, y .
f
gx, y f x, y .
f
gx, y f x, y 3.
f
f.
y 0.
x 0
f x, y
xy,
Para discusión
64. Considerar la función
a) Trazar la gráfica de la superficie dada por f.
b) Conjeturar acerca de la relación entre las gráficas de f y
Explicar el razonamiento.
c) Conjeturar acerca de la relación entre las gráficas de f y
Explicar el razonamiento.
d) Conjeturar acerca de la relación entre las gráficas de f y
Explicar el razonamiento.
e) Sobre la superficie en el inciso a), trazar la gráfica de
32. Think About It Use the function given in Exercise 31.
(a) Find the domain and range of the function.
(b) Identify the points in the
plane at which the function
value is 0.
(c) Does the surface pass through all the octants of the rectangular
coordinate system? Give reasons for your answer.
In Exercises 33– 40, sketch the surface given by the function.
33. 34.
35. 36.
37. 38.
39.
40.
In Exercises 41–44, use a computer algebra system to graph the
function.
41. 42.
43. 44.
In Exercises 45–48, match the graph of the surface with one of
the contour maps. [The contour maps are labeled (a), (b), (c),
and (d).]
(a)
(b)
(c)
(d)
45. 46.
47. 48.
In Exercises 49–56, describe the level curves of the function.
Sketch the level curves for the given -values.
49.
50.
51.
52.
53.
54.
55.
56.
In Exercises 57–60, use a graphing utility to graph six level
curves of the function.
57. 58.
59. 60. h x, y 3 sin x y
g x, y
8
1 x 2 y 2 fx, y xy
f x, y x 2 y 2 2
c 0, ± 1 2, ±1, ± 3 2, ±2
f x, y ln x y ,
c ± 1 2 , ±1, ± 3 2 , ±2
f x, y x x2 y 2 ,
c 2, 3, 4, 1 2 , 1 3 , 1 4
fx, y e xy 2 ,
c ±1, ±2, . . . , ±6
f x, y
xy,
c 0, 1, 2, 3
f x, y 9 x 2 y 2 ,
c 0, 1, 2, 3, 4
z x 2 4y 2 ,
c 0, 2, 4, 6, 8, 10
z 6 2x 3y,
c 1, 0, 2, 4
z x y,
c
y
x
−6
4
10
z
4
6
5
4
5
3 2 5
−2
x
y
z
fx, y
cos
x 2 2y 2
4
fx, y ln y x 2
y
x
3
6
4
4
z
y
x
3
3
3
z
fx, y e 1 x2 y 2
fx, y e 1 x2 y 2 x
y
x
y
y
x
y
fx, y
x sin y
f x, y x 2 e xy 2 z
1
12 144 16x 2 9y 2
z y 2 x 2 1
fx, y
xy,
0,
x 0, y 0
x < 0 o y < 0
fx, y e x z
1
2 x 2 y 2
z x 2 y 2 gx, y
1
2 y
f x, y y2 fx, y 6 2x 3y
f x, y 4
xy-
13.1 Introduction to Functions of Several Variables 895
CAS
61. What is a graph of a function of two variables? How is it
interpreted geometrically? Describe level curves.
62. All of the level curves of the surface given by
are concentric circles. Does this imply that the graph of is
a hemisphere? Illustrate your answer with an example.
63. Construct a function whose level curves are lines passing
through the origin.
f
z
f x, y
WRITING ABOUT CONCEPTS
64. Considere la función para y
(a) Sketch the graph of the surface given by
(b) Make a conjecture about the relationship between the
graphs of y Explain your
reasoning.
(c) Make a conjecture about the relationship between the
graphs of y Explain your
reasoning.
(d) Make a conjecture about the relationship between
the graphs of y Explain your
reasoning.
(e) On the surface in part (a), sketch the graph of
z f x, x .
gx, y
1
2 fx, y .
f
gx, y f x, y .
f
gx, y f x, y 3.
f
f.
y 0.
x 0
f x, y
xy,
hink About It Use the function given in Exercise 31.
) Find the domain and range of the function.
) Identify the points in the plane at which the function
value is 0.
) Does the surface pass through all the octants of the rectangular
coordinate system? Give reasons for your answer.
ercises 33– 40, sketch the surface given by the function.
34.
36.
38.
ercises 41–44, use a computer algebra system to graph the
ion.
42.
44.
ercises 45–48, match the graph of the surface with one of
ntour maps. [The contour maps are labeled (a), (b), (c),
d).]
(b)
(d)
46.
47. 48.
In Exercises 49–56, describe the level curves of the function.
Sketch the level curves for the given -values.
49.
50.
51.
52.
53.
54.
55.
56.
In Exercises 57–60, use a graphing utility to graph six level
curves of the function.
57. 58.
59. 60. h x, y 3 sin x y
g x, y
8
1 x 2 y 2 fx, y xy
f x, y x 2 y 2 2
c 0, ± 1 2, ±1, ± 3 2, ±2
f x, y ln x y ,
c ± 1 2 , ±1, ± 3 2 , ±2
f x, y x x2 y 2 ,
c 2, 3, 4, 1 2 , 1 3 , 1 4
fx, y e xy 2 ,
c ±1, ±2, . . . , ±6
f x, y
xy,
c 0, 1, 2, 3
f x, y 9 x 2 y 2 ,
c 0, 1, 2, 3, 4
z x 2 4y 2 ,
c 0, 2, 4, 6, 8, 10
z 6 2x 3y,
c 1, 0, 2, 4
z x y,
c
y
x
−6
4
10
z
4
6
5
4
5
3 2 5
−2
x
y
z
fx, y
cos
x 2 2y 2
4
fx, y ln y x 2
y
x
3
6
4
4
z
y
3
3
3
z
fx, y e 1 x2 y 2
x, y e 1 x2 y 2 x
y
x
y
y
x
y
fx, y
x sin y
x, y x 2 e xy 2 z
1
12 144 16x 2 9y 2
y 2 x 2 1
x, y
xy,
0,
x 0, y 0
x < 0 o y < 0
x, y e x z
1
2 x 2 y 2
x 2 y 2 gx, y
1
2 y
x, y y2 fx, y 6 2x 3y
x, y 4
xy-
13.1 Introduction to Functions of Several Variables 895
61. What is a graph of a function of two variables? How is it
interpreted geometrically? Describe level curves.
62. All of the level curves of the surface given by
are concentric circles. Does this imply that the graph of is
a hemisphere? Illustrate your answer with an example.
63. Construct a function whose level curves are lines passing
through the origin.
f
z
f x, y
WRITING ABOUT CONCEPTS
64. Considere la función para y
(a) Sketch the graph of the surface given by
(b) Make a conjecture about the relationship between the
graphs of y Explain your
reasoning.
(c) Make a conjecture about the relationship between the
graphs of y Explain your
reasoning.
(d) Make a conjecture about the relationship between
the graphs of y Explain your
reasoning.
(e) On the surface in part (a), sketch the graph of
z f x, x .
g x, y
1
2 f x, y .
f
gx, y f x, y .
f
gx, y f x, y 3.
f
f.
y 0.
x 0
f x, y
xy,
Think About It Use the function given in Exercise 31.
(a) Find the domain and range of the function.
(b) Identify the points in the
plane at which the function
value is 0.
(c) Does the surface pass through all the octants of the rectangular
coordinate system? Give reasons for your answer.
xercises 33– 40, sketch the surface given by the function.
34.
36.
38.
xercises 41–44, use a computer algebra system to graph the
ction.
42.
44.
xercises 45–48, match the graph of the surface with one of
contour maps. [The contour maps are labeled (a), (b), (c),
(d).]
(b)
(d)
46.
47. 48.
In Exercises 49–56, describe the level curves of the function.
Sketch the level curves for the given -values.
49.
50.
51.
52.
53.
54.
55.
56.
In Exercises 57–60, use a graphing utility to graph six level
curves of the function.
57. 58.
59. 60. h x, y 3 sin x y
g x, y
8
1 x 2 y 2 fx, y xy
f x, y x 2 y 2 2
c 0, ± 1 2, ±1, ± 3 2, ±2
f x, y ln x y ,
c ± 1 2 , ±1, ± 3 2 , ±2
f x, y x x2 y 2 ,
c 2, 3, 4, 1 2 , 1 3 , 1 4
fx, y e xy 2 ,
c ±1, ±2, . . . , ±6
f x, y
xy,
c 0, 1, 2, 3
f x, y 9 x 2 y 2 ,
c 0, 1, 2, 3, 4
z x 2 4y 2 ,
c 0, 2, 4, 6, 8, 10
z 6 2x 3y,
c 1, 0, 2, 4
z x y,
c
y
x
−6
4
10
z
4
6
5
4
5
3 2 5
−2
x
y
z
f x, y
cos
x 2 2y 2
4
f x, y ln y x 2
y
x
3
6
4
4
z
y
x
3
3
3
z
fx, y e 1 x2 y 2
fx, y e 1 x2 y 2 x
y
x
y
y
x
y
fx, y
x sin y
f x, y x 2 e xy 2 z
1
12 144 16x 2 9y 2
z y 2 x 2 1
fx, y
xy,
0,
x 0, y 0
x < 0 o y < 0
fx, y e x z
1
2 x 2 y 2
z x 2 y 2 g x, y
1
2 y
fx, y y2 fx, y 6 2x 3y
f x, y 4
xy-
13.1 Introduction to Functions of Several Variables 895
61. What is a graph of a function of two variables? How is it
interpreted geometrically? Describe level curves.
62. All of the level curves of the surface given by
are concentric circles. Does this imply that the graph of is
a hemisphere? Illustrate your answer with an example.
63. Construct a function whose level curves are lines passing
through the origin.
f
z
f x, y
WRITING ABOUT CONCEPTS
64. Considere la función para y
(a) Sketch the graph of the surface given by
(b) Make a conjecture about the relationship between the
graphs of y Explain your
reasoning.
(c) Make a conjecture about the relationship between the
graphs of y Explain your
reasoning.
(d) Make a conjecture about the relationship between
the graphs of y Explain your
reasoning.
(e) On the surface in part (a), sketch the graph of
z f x, x .
gx, y
1
2 fx, y .
f
g x, y f x, y .
f
gx, y f x, y 3.
f
f.
y 0.
x 0
f x, y
xy,
. Think About It Use the function given in Exercise 31.
(a) Find the domain and range of the function.
(b) Identify the points in the
plane at which the function
value is 0.
(c) Does the surface pass through all the octants of the rectangular
coordinate system? Give reasons for your answer.
Exercises 33– 40, sketch the surface given by the function.
. 34.
. 36.
. 38.
.
.
Exercises 41–44, use a computer algebra system to graph the
nction.
. 42.
. 44.
Exercises 45–48, match the graph of the surface with one of
e contour maps. [The contour maps are labeled (a), (b), (c),
d (d).]
) (b)
) (d)
. 46.
47. 48.
In Exercises 49–56, describe the level curves of the function.
Sketch the level curves for the given -values.
49.
50.
51.
52.
53.
54.
55.
56.
In Exercises 57–60, use a graphing utility to graph six level
curves of the function.
57. 58.
59. 60. h x, y 3 sin x y
g x, y
8
1 x 2 y 2 fx, y xy
f x, y x 2 y 2 2
c 0, ± 1 2, ±1, ± 3 2, ±2
f x, y ln x y ,
c ± 1 2 , ±1, ± 3 2 , ±2
f x, y x x2 y 2 ,
c 2, 3, 4, 1 2 , 1 3 , 1 4
fx, y e xy 2 ,
c ±1, ±2, . . . , ±6
f x, y
xy,
c 0, 1, 2, 3
f x, y 9 x 2 y 2 ,
c 0, 1, 2, 3, 4
z x 2 4y 2 ,
c 0, 2, 4, 6, 8, 10
z 6 2x 3y,
c 1, 0, 2, 4
z x y,
c
y
x
−6
4
10
z
4
6
5
4
5
3 2 5
−2
x
y
z
fx, y
cos
x 2 2y 2
4
fx, y ln y x 2
y
x
3
6
4
4
z
y
x
3
3
3
z
fx, y e 1 x2 y 2
fx, y e 1 x2 y 2 x
y
x
y
y
x
y
fx, y
x sin y
f x, y x 2 e xy 2 z
1
12 144 16x 2 9y 2
z y 2 x 2 1
fx, y
xy,
0,
x 0, y 0
x < 0 o y < 0
fx, y e x z
1
2 x 2 y 2
z x 2 y 2 gx, y
1
2 y
f x, y y2 f x, y 6 2x 3y
f x, y 4
xy-
13.1 Introduction to Functions of Several Variables 895
61. What is a graph of a function of two variables? How is it
interpreted geometrically? Describe level curves.
62. All of the level curves of the surface given by
are concentric circles. Does this imply that the graph of is
a hemisphere? Illustrate your answer with an example.
63. Construct a function whose level curves are lines passing
through the origin.
f
z
f x, y
WRITING ABOUT CONCEPTS
64. Considere la función para y
(a) Sketch the graph of the surface given by
(b) Make a conjecture about the relationship between the
graphs of y Explain your
reasoning.
(c) Make a conjecture about the relationship between the
graphs of y Explain your
reasoning.
(d) Make a conjecture about the relationship between
the graphs of y Explain your
reasoning.
(e) On the surface in part (a), sketch the graph of
z f x, x .
gx, y
1
2 fx, y .
f
gx, y f x, y .
f
g x, y f x, y 3.
f
f.
y 0.
x 0
f x, y
xy,
qxp 10/27/08 12:05 PM Page 895
32. Think About It Use the function given in Exercise 31.
(a) Find the domain and range of the function.
(b) Identify the points in the
plane at which the function
value is 0.
(c) Does the surface pass through all the octants of the rectangular
coordinate system? Give reasons for your answer.
In Exercises 33– 40, sketch the surface given by the function.
33. 34.
35. 36.
37. 38.
39.
40.
In Exercises 41–44, use a computer algebra system to graph the
function.
41. 42.
43. 44.
In Exercises 45–48, match the graph of the surface with one of
the contour maps. [The contour maps are labeled (a), (b), (c),
and (d).]
(a)
(b)
(c)
(d)
45. 46.
47. 48.
In Exercises 49–56, describe the level curves of the function.
Sketch the level curves for the given -values.
49.
50.
51.
52.
53.
54.
55.
56.
In Exercises 57–60, use a graphing utility to graph six level
curves of the function.
57. 58.
59. 60. h x, y 3 sin x y
g x, y
8
1 x 2 y 2 fx, y xy
f x, y x 2 y 2 2
c 0, ± 1 2, ±1, ± 3 2, ±2
f x, y ln x y ,
c ± 1 2 , ±1, ± 3 2 , ±2
f x, y x x2 y 2 ,
c 2, 3, 4, 1 2 , 1 3 , 1 4
fx, y e xy 2 ,
c ±1, ±2, . . . , ±6
f x, y
xy,
c 0, 1, 2, 3
f x, y 9 x 2 y 2 ,
c 0, 1, 2, 3, 4
z x 2 4y 2 ,
c 0, 2, 4, 6, 8, 10
z 6 2x 3y,
c 1, 0, 2, 4
z x y,
c
y
x
−6
4
10
z
4
6
5
4
5
3 2 5
−2
x
y
z
f x, y
cos
x 2 2y 2
4
f x, y ln y x 2
y
x
3
6
4
4
z
y
x
3
3
3
z
fx, y e 1 x2 y 2
fx, y e 1 x2 y 2 x
y
x
y
y
x
y
fx, y
x sin y
f x, y x 2 e xy 2 z
1
12 144 16x 2 9y 2
z y 2 x 2 1
fx, y
xy,
0,
x 0, y 0
x < 0 o y < 0
fx, y e x z
1
2 x 2 y 2
z x 2 y 2 gx, y
1
2 y
f x, y y2 fx, y 6 2x 3y
f x, y 4
xy-
13.1 Introduction to Functions of Several Variables 895
CAS
61. What is a graph of a function of two variables? How is it
interpreted geometrically? Describe level curves.
62. All of the level curves of the surface given by
are concentric circles. Does this imply that the graph of is
a hemisphere? Illustrate your answer with an example.
63. Construct a function whose level curves are lines passing
through the origin.
f
z
f x, y
WRITING ABOUT CONCEPTS
64. Considere la función para y
(a) Sketch the graph of the surface given by
(b) Make a conjecture about the relationship between the
graphs of y Explain your
reasoning.
(c) Make a conjecture about the relationship between the
graphs of y Explain your
reasoning.
(d) Make a conjecture about the relationship between
the graphs of y Explain your
reasoning.
(e) On the surface in part (a), sketch the graph of
z f x, x .
gx, y
1
2 fx, y .
f
gx, y f x, y .
f
gx, y f x, y 3.
f
f.
y 0.
x 0
f x, y
xy,
1053714_1301.qxp 10/27/08 12:05 PM Page 895
32. Think About It Use the function given in Exercise 31.
(a) Find the domain and range of the function.
(b) Identify the points in the
plane at which the function
value is 0.
(c) Does the surface pass through all the octants of the rectangular
coordinate system? Give reasons for your answer.
In Exercises 33– 40, sketch the surface given by the function.
33. 34.
35. 36.
37. 38.
39.
40.
In Exercises 41–44, use a computer algebra system to graph the
function.
41. 42.
43. 44.
In Exercises 45–48, match the graph of the surface with one of
the contour maps. [The contour maps are labeled (a), (b), (c),
and (d).]
(a)
(b)
(c)
(d)
45. 46.
47. 48.
In Exercises 49–56, describe the level curves of the function.
Sketch the level curves for the given -values.
49.
50.
51.
52.
53.
54.
55.
56.
In Exercises 57–60, use a graphing utility to graph six level
curves of the function.
57. 58.
59. 60. h x, y 3 sin x y
g x, y
8
1 x 2 y 2 fx, y xy
f x, y x 2 y 2 2
c 0, ± 1 2, ±1, ± 3 2, ±2
f x, y ln x y ,
c ± 1 2 , ±1, ± 3 2 , ±2
f x, y x x2 y 2 ,
c 2, 3, 4, 1 2 , 1 3 , 1 4
fx, y e xy 2 ,
c ±1, ±2, . . . , ±6
f x, y
xy,
c 0, 1, 2, 3
f x, y 9 x 2 y 2 ,
c 0, 1, 2, 3, 4
z x 2 4y 2 ,
c 0, 2, 4, 6, 8, 10
z 6 2x 3y,
c 1, 0, 2, 4
z x y,
c
y
x
−6
4
10
z
4
6
5
4
5
3 2 5
−2
x
y
z
fx, y
cos
x 2 2y 2
4
fx, y ln y x 2
y
x
3
6
4
4
z
y
x
3
3
3
z
fx, y e 1 x2 y 2
fx, y e 1 x2 y 2 x
y
x
y
y
x
y
fx, y
x sin y
f x, y x 2 e xy 2 z
1
12 144 16x 2 9y 2
z y 2 x 2 1
fx, y
xy,
0,
x 0, y 0
x < 0 o y < 0
fx, y e x z
1
2 x 2 y 2
z x 2 y 2 gx, y
1
2 y
f x, y y2 f x, y 6 2x 3y
f x, y 4
xy-
13.1 Introduction to Functions of Several Variables 895
CAS
61. What is a graph of a function of two variables? How is it
interpreted geometrically? Describe level curves.
62. All of the level curves of the surface given by
are concentric circles. Does this imply that the graph of is
a hemisphere? Illustrate your answer with an example.
63. Construct a function whose level curves are lines passing
through the origin.
f
z
f x, y
WRITING ABOUT CONCEPTS
64. Considere la función para y
(a) Sketch the graph of the surface given by
(b) Make a conjecture about the relationship between the
graphs of y Explain your
reasoning.
(c) Make a conjecture about the relationship between the
graphs of y Explain your
reasoning.
(d) Make a conjecture about the relationship between
the graphs of y Explain your
reasoning.
(e) On the surface in part (a), sketch the graph of
z f x, x .
gx, y
1
2 fx, y .
f
gx, y f x, y .
f
gx, y f x, y 3.
f
f.
y 0.
x 0
f x, y
xy,