University Calculus: Early Transcendentals 4th

Joel Hass, Christopher Heil, Przemyslaw Bogacki

Chapter 13

Partial Derivatives

Educators

TE

Problem 1

Find the specific function values.
$$f(x, y)=x^{2}+x y^{3}$$
a. $f(0,0)$
b. $f(-1,1)$
c. $f(2,3)$
d. $f(-3,-2)$

TE
Thomas E.
Numerade Educator

Problem 2

Find the specific function values.
$$f(x, y)=\sin (x y)$$
a. $f\left(2, \frac{\pi}{6}\right)$
b. $f\left(-3, \frac{\pi}{12}\right)$
c. $f\left(\pi, \frac{1}{4}\right)$
d. $f\left(-\frac{\pi}{2},-7\right)$

TE
Thomas E.
Numerade Educator

Problem 3

Find the specific function values.
$$f(x, y, z)=\frac{x-y}{y^{2}+z^{2}}$$
a. $f(3,-1,2)$
b. $f\left(1, \frac{1}{2},-\frac{1}{4}\right)$
c. $f\left(0,-\frac{1}{3}, 0\right)$
d. $f(2,2,100)$

TE
Thomas E.
Numerade Educator

Problem 4

Find the specific function values.
$$f(x, y, z)=\sqrt{49-x^{2}-y^{2}-z^{2}}$$
a. $f(0,0,0)$
b. $f(2,-3,6)$
c. $f(-1,2,3)$
d. $f\left(\frac{4}{\sqrt{2}}, \frac{5}{\sqrt{2}}, \frac{6}{\sqrt{2}}\right)$

TE
Thomas E.
Numerade Educator

Problem 5

Find and sketch the domain for each function.
$$f(x, y)=\sqrt{y-x-2}$$

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Problem 6

Find and sketch the domain for each function.
$$f(x, y)=\ln \left(x^{2}+y^{2}-4\right)$$

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Problem 7

Find and sketch the domain for each function.
$$f(x, y)=\frac{(x-1)(y+2)}{(y-x)\left(y-x^{3}\right)}$$

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Problem 8

Find and sketch the domain for each function.
$$f(x, y)=\frac{\sin (x y)}{x^{2}+y^{2}-25}$$

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Problem 9

Find and sketch the domain for each function.
$$f(x, y)=\cos ^{-1}\left(y-x^{2}\right)$$

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Problem 10

Find and sketch the domain for each function.
$$f(x, y)=\ln (x y+x-y-1)$$

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Problem 11

Find and sketch the domain for each function.
$$f(x, y)=\sqrt{\left(x^{2}-4\right)\left(y^{2}-9\right)}$$

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Problem 12

Find and sketch the domain for each function.
$$f(x, y)=\frac{1}{\ln \left(4-x^{2}-y^{2}\right)}$$

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Problem 13

Find and sketch the level curves $f(x, y)=c$ on the same set of coordinate axes for the given values of $c .$ We refer to these level curves as a contour map.
$$f(x, y)=x+y-1, \quad c=-3,-2,-1,0,1,2,3$$

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Problem 14

Find and sketch the level curves $f(x, y)=c$ on the same set of coordinate axes for the given values of $c .$ We refer to these level curves as a contour map.
$$f(x, y)=x^{2}+y^{2}, \quad c=0,1,4,9,16,25$$

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Problem 15

Find and sketch the level curves $f(x, y)=c$ on the same set of coordinate axes for the given values of $c .$ We refer to these level curves as a contour map.
$$f(x, y)=x y, \quad c=-9,-4,-1,0,1,4,9$$

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Problem 16

Find and sketch the level curves $f(x, y)=c$ on the same set of coordinate axes for the given values of $c .$ We refer to these level curves as a contour map.
$$f(x, y)=\sqrt{25-x^{2}-y^{2}}, \quad c=0,1,2,3,4$$

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Problem 17

In Exercise (a) find the function's domain, (b) find the function's range, (c) describe the function's level curves, (d) find the boundary of the function's domain, (e) determine whether the domain is an open region, a closed region, or neither, and (f) decide whether the domain is bounded or unbounded.
$$f(x, y)=y-x$$

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Problem 18

In Exercise (a) find the function's domain, (b) find the function's range, (c) describe the function's level curves, (d) find the boundary of the function's domain, (e) determine whether the domain is an open region, a closed region, or neither, and (f) decide whether the domain is bounded or unbounded.
$$f(x, y)=\sqrt{y-x}$$

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Problem 19

In Exercise (a) find the function's domain, (b) find the function's range, (c) describe the function's level curves, (d) find the boundary of the function's domain, (e) determine whether the domain is an open region, a closed region, or neither, and (f) decide whether the domain is bounded or unbounded.
$$f(x, y)=4 x^{2}+9 y^{2}$$

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Problem 20

In Exercise (a) find the function's domain, (b) find the function's range, (c) describe the function's level curves, (d) find the boundary of the function's domain, (e) determine whether the domain is an open region, a closed region, or neither, and (f) decide whether the domain is bounded or unbounded.
$$f(x, y)=x^{2}-y^{2}$$

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Problem 21

In Exercise (a) find the function's domain, (b) find the function's range, (c) describe the function's level curves, (d) find the boundary of the function's domain, (e) determine whether the domain is an open region, a closed region, or neither, and (f) decide whether the domain is bounded or unbounded.
$$f(x, y)=x y$$

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Problem 22

In Exercise (a) find the function's domain, (b) find the function's range, (c) describe the function's level curves, (d) find the boundary of the function's domain, (e) determine whether the domain is an open region, a closed region, or neither, and (f) decide whether the domain is bounded or unbounded.
$$f(x, y)=y / x^{2}$$

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Problem 23

In Exercise (a) find the function's domain, (b) find the function's range, (c) describe the function's level curves, (d) find the boundary of the function's domain, (e) determine whether the domain is an open region, a closed region, or neither, and (f) decide whether the domain is bounded or unbounded.
$$f(x, y)=\frac{1}{\sqrt{16-x^{2}-y^{2}}}$$

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Problem 24

In Exercise (a) find the function's domain, (b) find the function's range, (c) describe the function's level curves, (d) find the boundary of the function's domain, (e) determine whether the domain is an open region, a closed region, or neither, and (f) decide whether the domain is bounded or unbounded.
$$f(x, y)=\sqrt{9-x^{2}-y^{2}}$$

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Problem 25

In Exercise (a) find the function's domain, (b) find the function's range, (c) describe the function's level curves, (d) find the boundary of the function's domain, (e) determine whether the domain is an open region, a closed region, or neither, and (f) decide whether the domain is bounded or unbounded.
$$f(x, y)=\ln \left(x^{2}+y^{2}\right)$$

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Problem 26

In Exercise (a) find the function's domain, (b) find the function's range, (c) describe the function's level curves, (d) find the boundary of the function's domain, (e) determine whether the domain is an open region, a closed region, or neither, and (f) decide whether the domain is bounded or unbounded.
$$f(x, y)=e^{-\left(x^{2}+y^{2}\right)}$$

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Problem 27

In Exercise (a) find the function's domain, (b) find the function's range, (c) describe the function's level curves, (d) find the boundary of the function's domain, (e) determine whether the domain is an open region, a closed region, or neither, and (f) decide whether the domain is bounded or unbounded.
$$f(x, y)=\sin ^{-1}(y-x)$$

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Problem 28

In Exercise (a) find the function's domain, (b) find the function's range, (c) describe the function's level curves, (d) find the boundary of the function's domain, (e) determine whether the domain is an open region, a closed region, or neither, and (f) decide whether the domain is bounded or unbounded.
$$f(x, y)=\tan ^{-1}\left(\frac{y}{x}\right)$$

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Problem 29

In Exercise (a) find the function's domain, (b) find the function's range, (c) describe the function's level curves, (d) find the boundary of the function's domain, (e) determine whether the domain is an open region, a closed region, or neither, and (f) decide whether the domain is bounded or unbounded.
$$f(x, y)=\ln \left(x^{2}+y^{2}-1\right)$$

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Problem 30

In Exercise (a) find the function's domain, (b) find the function's range, (c) describe the function's level curves, (d) find the boundary of the function's domain, (e) determine whether the domain is an open region, a closed region, or neither, and (f) decide whether the domain is bounded or unbounded.
$$f(x, y)=\ln \left(9-x^{2}-y^{2}\right)$$

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Problem 31

Show level curves for six functions. The graphs of these functions are given on the next page (items a-f), as are their equations (items g $-1$). Match each set of level curves with the appropriate graph and the appropriate equation.
(Graph cant copy)

a. (Graph cant copy)
b. (Graph cant copy)
c. (Graph cant copy)
d. (Graph cant copy)
e. (Graph cant copy)
f. (Graph cant copy)
g. $z=-\frac{x y^{2}}{x^{2}+y^{2}}$
h. $z=y^{2}-y^{4}-x^{2}$
i. $z=(\cos x)(\cos y) e^{-\sqrt{x^{2}+y^{2}} / 4}$
j. $z=e^{-y} \cos x$
k. $z=\frac{1}{4 x^{2}+y^{2}}$
l. $z=\frac{x y\left(x^{2}-y^{2}\right)}{x^{2}+y^{2}}$

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Problem 32

Show level curves for six functions. The graphs of these functions are given on the next page (items a-f), as are their equations (items g $-1$). Match each set of level curves with the appropriate graph and the appropriate equation.
(Graph cant copy)

a. (Graph cant copy)
b. (Graph cant copy)
c. (Graph cant copy)
d. (Graph cant copy)
e. (Graph cant copy)
f. (Graph cant copy)
g. $z=-\frac{x y^{2}}{x^{2}+y^{2}}$
h. $z=y^{2}-y^{4}-x^{2}$
i. $z=(\cos x)(\cos y) e^{-\sqrt{x^{2}+y^{2}} / 4}$
j. $z=e^{-y} \cos x$
k. $z=\frac{1}{4 x^{2}+y^{2}}$
l. $z=\frac{x y\left(x^{2}-y^{2}\right)}{x^{2}+y^{2}}$

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Problem 33

Show level curves for six functions. The graphs of these functions are given on the next page (items a-f), as are their equations (items g $-1$). Match each set of level curves with the appropriate graph and the appropriate equation.
(Graph cant copy)

a. (Graph cant copy)
b. (Graph cant copy)
c. (Graph cant copy)
d. (Graph cant copy)
e. (Graph cant copy)
f. (Graph cant copy)
g. $z=-\frac{x y^{2}}{x^{2}+y^{2}}$
h. $z=y^{2}-y^{4}-x^{2}$
i. $z=(\cos x)(\cos y) e^{-\sqrt{x^{2}+y^{2}} / 4}$
j. $z=e^{-y} \cos x$
k. $z=\frac{1}{4 x^{2}+y^{2}}$
l. $z=\frac{x y\left(x^{2}-y^{2}\right)}{x^{2}+y^{2}}$

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Problem 34

Show level curves for six functions. The graphs of these functions are given on the next page (items a-f), as are their equations (items g $-1$). Match each set of level curves with the appropriate graph and the appropriate equation.
(Graph cant copy)

a. (Graph cant copy)
b. (Graph cant copy)
c. (Graph cant copy)
d. (Graph cant copy)
e. (Graph cant copy)
f. (Graph cant copy)
g. $z=-\frac{x y^{2}}{x^{2}+y^{2}}$
h. $z=y^{2}-y^{4}-x^{2}$
i. $z=(\cos x)(\cos y) e^{-\sqrt{x^{2}+y^{2}} / 4}$
j. $z=e^{-y} \cos x$
k. $z=\frac{1}{4 x^{2}+y^{2}}$
l. $z=\frac{x y\left(x^{2}-y^{2}\right)}{x^{2}+y^{2}}$

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Problem 35

Show level curves for six functions. The graphs of these functions are given on the next page (items a-f), as are their equations (items g $-1$). Match each set of level curves with the appropriate graph and the appropriate equation.
(Graph cant copy)

a. (Graph cant copy)
b. (Graph cant copy)
c. (Graph cant copy)
d. (Graph cant copy)
e. (Graph cant copy)
f. (Graph cant copy)
g. $z=-\frac{x y^{2}}{x^{2}+y^{2}}$
h. $z=y^{2}-y^{4}-x^{2}$
i. $z=(\cos x)(\cos y) e^{-\sqrt{x^{2}+y^{2}} / 4}$
j. $z=e^{-y} \cos x$
k. $z=\frac{1}{4 x^{2}+y^{2}}$
l. $z=\frac{x y\left(x^{2}-y^{2}\right)}{x^{2}+y^{2}}$

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Problem 36

Show level curves for six functions. The graphs of these functions are given on the next page (items a-f), as are their equations (items g $-1$). Match each set of level curves with the appropriate graph and the appropriate equation.
(Graph cant copy)

a. (Graph cant copy)
b. (Graph cant copy)
c. (Graph cant copy)
d. (Graph cant copy)
e. (Graph cant copy)
f. (Graph cant copy)
g. $z=-\frac{x y^{2}}{x^{2}+y^{2}}$
h. $z=y^{2}-y^{4}-x^{2}$
i. $z=(\cos x)(\cos y) e^{-\sqrt{x^{2}+y^{2}} / 4}$
j. $z=e^{-y} \cos x$
k. $z=\frac{1}{4 x^{2}+y^{2}}$
l. $z=\frac{x y\left(x^{2}-y^{2}\right)}{x^{2}+y^{2}}$

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Problem 37

Display the values of the functions in two ways: (a) by sketching the surface $z=f(x, y)$ and (b) by drawing an assortment of level curves in the function's domain. Label each level curve with its function value.
$$f(x, y)=y^{2}$$

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Problem 38

Display the values of the functions in two ways: (a) by sketching the surface $z=f(x, y)$ and (b) by drawing an assortment of level curves in the function's domain. Label each level curve with its function value.
$$f(x, y)=\sqrt{x}$$

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Problem 39

Display the values of the functions in two ways: (a) by sketching the surface $z=f(x, y)$ and (b) by drawing an assortment of level curves in the function's domain. Label each level curve with its function value.
$$f(x, y)=x^{2}+y^{2}$$

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Problem 40

Display the values of the functions in two ways: (a) by sketching the surface $z=f(x, y)$ and (b) by drawing an assortment of level curves in the function's domain. Label each level curve with its function value.
$$f(x, y)=\sqrt{x^{2}+y^{2}}$$

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Problem 41

Display the values of the functions in two ways: (a) by sketching the surface $z=f(x, y)$ and (b) by drawing an assortment of level curves in the function's domain. Label each level curve with its function value.
$$f(x, y)=x^{2}-y$$

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Problem 42

Display the values of the functions in two ways: (a) by sketching the surface $z=f(x, y)$ and (b) by drawing an assortment of level curves in the function's domain. Label each level curve with its function value.
$$f(x, y)=4-x^{2}-y^{2}$$

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Problem 43

Display the values of the functions in two ways: (a) by sketching the surface $z=f(x, y)$ and (b) by drawing an assortment of level curves in the function's domain. Label each level curve with its function value.
$$f(x, y)=4 x^{2}+y^{2}$$

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Problem 44

Display the values of the functions in two ways: (a) by sketching the surface $z=f(x, y)$ and (b) by drawing an assortment of level curves in the function's domain. Label each level curve with its function value.
$$f(x, y)=6-2 x-3 y$$

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Problem 45

Display the values of the functions in two ways: (a) by sketching the surface $z=f(x, y)$ and (b) by drawing an assortment of level curves in the function's domain. Label each level curve with its function value.
$$f(x, y)=1-|y|$$

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Problem 46

Display the values of the functions in two ways: (a) by sketching the surface $z=f(x, y)$ and (b) by drawing an assortment of level curves in the function's domain. Label each level curve with its function value.
$$f(x, y)=1-|x|-|y|$$

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Problem 47

Display the values of the functions in two ways: (a) by sketching the surface $z=f(x, y)$ and (b) by drawing an assortment of level curves in the function's domain. Label each level curve with its function value.
$$f(x, y)=\sqrt{x^{2}+y^{2}+4}$$

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Problem 48

Display the values of the functions in two ways: (a) by sketching the surface $z=f(x, y)$ and (b) by drawing an assortment of level curves in the function's domain. Label each level curve with its function value.
$$f(x, y)=\sqrt{x^{2}+y^{2}-4}$$

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Problem 49

Find an equation for, and sketch the graph of, the level curve of the function $f(x, y)$ that passes through the given point.
$$f(x, y)=16-x^{2}-y^{2}, \quad(2 \sqrt{2}, \sqrt{2})$$

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Problem 50

Find an equation for, and sketch the graph of, the level curve of the function $f(x, y)$ that passes through the given point.
$$f(x, y)=\sqrt{x^{2}-1}, \quad (1,0)$$

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Problem 51

Find an equation for, and sketch the graph of, the level curve of the function $f(x, y)$ that passes through the given point.
$$f(x, y)=\sqrt{x+y^{2}-3}, \quad(3,-1)$$

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Problem 52

Find an equation for, and sketch the graph of, the level curve of the function $f(x, y)$ that passes through the given point.
$$f(x, y)=\frac{2 y-x}{x+y+1}, \quad(-1,1)$$

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Problem 53

Sketch a typical level surface for the function.
$$f(x, y, z)=x^{2}+y^{2}+z^{2}$$

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Problem 54

Sketch a typical level surface for the function.
$$f(x, y, z)=\ln \left(x^{2}+y^{2}+z^{2}\right)$$

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Problem 55

Sketch a typical level surface for the function.
$$f(x, y, z)=x+z$$

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Problem 56

Sketch a typical level surface for the function.
$$f(x, y, z)=z$$

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Problem 57

Sketch a typical level surface for the function.
$$f(x, y, z)=x^{2}+y^{2}$$

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Problem 58

Sketch a typical level surface for the function.
$$f(x, y, z)=y^{2}+z^{2}$$

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Problem 59

Sketch a typical level surface for the function.
$$f(x, y, z)=z-x^{2}-y^{2}$$

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Problem 60

Sketch a typical level surface for the function.
$$f(x, y, z)=\left(x^{2} / 25\right)+\left(y^{2} / 16\right)+\left(z^{2} / 9\right)$$

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Problem 61

Find an equation for the level surface of the function through the given point.
$$f(x, y, z)=\sqrt{x-y}-\ln z, \quad(3,-1,1)$$

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Problem 62

Find an equation for the level surface of the function through the given point.
$$f(x, y, z)=\ln \left(x^{2}+y+z^{2}\right), \quad(-1,2,1)$$

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Problem 63

Find an equation for the level surface of the function through the given point.
$$g(x, y, z)=\sqrt{x^{2}+y^{2}+z^{2}}, \quad(1,-1, \sqrt{2})$$

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Problem 64

Find an equation for the level surface of the function through the given point.
$$g(x, y, z)=\frac{x-y+z}{2 x+y-z}, \quad(1,0,-2)$$

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Problem 65

Find and sketch the domain of $f .$ Then find an equation for the level curve or surface of the function passing through the given point.
$$f(x, y)=\sum_{n=0}^{\infty}\left(\frac{x}{y}\right)^{n}, \quad (1,2)$$

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Problem 66

Find and sketch the domain of $f .$ Then find an equation for the level curve or surface of the function passing through the given point.
$$g(x, y, z)=\sum_{n=0}^{\infty} \frac{(x+y)^{n}}{n ! z^{n}}, \quad(\ln 4, \ln 9,2)$$

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Problem 67

Find and sketch the domain of $f .$ Then find an equation for the level curve or surface of the function passing through the given point.
$$f(x, y)=\int_{x}^{y} \frac{d \theta}{\sqrt{1-\theta^{2}}}, \quad(0,1)$$

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Problem 68

Find and sketch the domain of $f .$ Then find an equation for the level curve or surface of the function passing through the given point.
$$g(x, y, z)=\int_{x}^{y} \frac{d t}{1+t^{2}}+\int_{0}^{z} \frac{d \theta}{\sqrt{4-\theta^{2}}}, \quad(0,1, \sqrt{3})$$

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Problem 69

Use a CAS to perform the following steps for each of the functions.
a. Plot the surface over the given rectangle.
b. Plot several level curves in the rectangle.
c. Plot the level curve of $f$ through the given point.
$$\begin{aligned}
&f(x, y)=x \sin \frac{y}{2}+y \sin 2 x, \quad 0 \leq x \leq 5 \pi, \quad 0 \leq y \leq 5 \pi, \quad P(3 \pi, 3 \pi)
\end{aligned}$$

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Problem 70

Use a CAS to perform the following steps for each of the functions.
a. Plot the surface over the given rectangle.
b. Plot several level curves in the rectangle.
c. Plot the level curve of $f$ through the given point.
$$\begin{array}{l}
f(x, y)=(\sin x)(\cos y) e^{\sqrt{x^{2}+y^{2}} / 8}, \quad 0 \leq x \leq 5 \pi, \quad
0 \leq y \leq 5 \pi, \quad P(4 \pi, 4 \pi)
\end{array}$$

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Problem 71

Use a CAS to perform the following steps for each of the functions.
a. Plot the surface over the given rectangle.
b. Plot several level curves in the rectangle.
c. Plot the level curve of $f$ through the given point.
$$\begin{array}{l} f(x, y)=\sin (x+2 \cos y), \quad-2 \pi \leq x \leq 2 \pi, \quad -2 \pi \leq y \leq 2 \pi, \quad P(\pi, \pi) \end{array}$$

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Problem 72

Use a CAS to perform the following steps for each of the functions.
a. Plot the surface over the given rectangle.
b. Plot several level curves in the rectangle.
c. Plot the level curve of $f$ through the given point.
$$\begin{array}{l} f(x, y)=e^{\left(x^{01}-y\right)} \sin \left(x^{2}+y^{2}\right), \quad 0 \leq x \leq 2 \pi, \quad -2 \pi \leq y \leq \pi, \quad P(\pi,-\pi)\end{array}$$

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Problem 73

Use a CAS to plot the implicitly defined level surfaces.
$$4 \ln \left(x^{2}+y^{2}+z^{2}\right)=1$$

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Problem 74

Use a CAS to plot the implicitly defined level surfaces.
$$x^{2}+z^{2}=1$$

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Problem 75

Use a CAS to plot the implicitly defined level surfaces.
$$x+y^{2}-3 z^{2}=1$$

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Problem 76

Use a CAS to plot the implicitly defined level surfaces.
$$\sin \left(\frac{x}{2}\right)-(\cos y) \sqrt{x^{2}+z^{2}}=2$$

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Problem 77

Just as you describe curves in the plane para metrically with a pair of equations $x=f(t), y=g(t)$ defined on some parameter interval $I,$ you can sometimes describe surfaces in space with a triple of equations $x=f(u, v), y=g(u, v), z=h(u, v)$ defined on some parameter rectangle $a \leq u \leq b, c \leq v \leq d .$ Many computer algebra systems permit you to plot such surfaces in parametric mode. (Parametrized surfaces are discussed in detail in Section 15.5.) Use a CAS to plot the surfaces. Also plot several level curves in the $x y$ -plane.
$$\begin{array}{l}
x=u \cos v, \quad y=u \sin v, \quad z=u, \quad 0 \leq u \leq 2, \quad 0 \leq v \leq 2 \pi
\end{array}$$

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Problem 78

Just as you describe curves in the plane para metrically with a pair of equations $x=f(t), y=g(t)$ defined on some parameter interval $I,$ you can sometimes describe surfaces in space with a triple of equations $x=f(u, v), y=g(u, v), z=h(u, v)$ defined on some parameter rectangle $a \leq u \leq b, c \leq v \leq d .$ Many computer algebra systems permit you to plot such surfaces in parametric mode. (Parametrized surfaces are discussed in detail in Section 15.5.) Use a CAS to plot the surfaces. Also plot several level curves in the $x y$ -plane.
$$\begin{array}{l}x=u \cos v, \quad y=u \sin v, \quad z=v, \quad 0 \leq u \leq 2, \quad
0 \leq v \leq 2 \pi
\end{array}$$

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Problem 79

Just as you describe curves in the plane para metrically with a pair of equations $x=f(t), y=g(t)$ defined on some parameter interval $I,$ you can sometimes describe surfaces in space with a triple of equations $x=f(u, v), y=g(u, v), z=h(u, v)$ defined on some parameter rectangle $a \leq u \leq b, c \leq v \leq d .$ Many computer algebra systems permit you to plot such surfaces in parametric mode. (Parametrized surfaces are discussed in detail in Section 15.5.) Use a CAS to plot the surfaces. Also plot several level curves in the $x y$ -plane.
$$\begin{array}{l}
x=(2+\cos u) \cos v, \quad y=(2+\cos u) \sin v, \quad z=\sin u, \quad 0 \leq u \leq 2 \pi, \quad 0 \leq v \leq 2 \pi
\end{array}$$

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Problem 80

Just as you describe curves in the plane para metrically with a pair of equations $x=f(t), y=g(t)$ defined on some parameter interval $I,$ you can sometimes describe surfaces in space with a triple of equations $x=f(u, v), y=g(u, v), z=h(u, v)$ defined on some parameter rectangle $a \leq u \leq b, c \leq v \leq d .$ Many computer algebra systems permit you to plot such surfaces in parametric mode. (Parametrized surfaces are discussed in detail in Section 15.5.) Use a CAS to plot the surfaces. Also plot several level curves in the $x y$ -plane.
$$\begin{array}{l}
x=2 \cos u \cos v, \quad y=2 \cos u \sin v, \quad z=2 \sin u, \quad 0 \leq u \leq 2 \pi, \quad 0 \leq v \leq \pi
\end{array}$$

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