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Thomas Calculus

George B. Thomas, Jr.

Chapter 14

Partial Derivatives

Educators


Problem 1

In Exercises $1-4,$ find the specific function values.
$$
\begin{array}{ll}{f(x, y)=x^{2}+x y^{3}} \\ {\text { a. } f(0,0)} & {\text { b. } f(-1,1)} \\ {\text { c. } f(2,3)} & {\text { d. } f(-3,-2)}\end{array}
$$

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Problem 2

In Exercises $1-4,$ find the specific function values.
$$
f(x, y)=\sin (x y)
$$
$$
\begin{array}{ll}{\text { a. } f\left(2, \frac{\pi}{6}\right)} & {\text { b. } f\left(-3, \frac{\pi}{12}\right)} \\ {\text { c. } f\left(\pi, \frac{1}{4}\right)} & {\text { d. } f\left(-\frac{\pi}{2},-7\right)}\end{array}
$$

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Problem 3

In Exercises $1-4,$ find the specific function values.
$$
f(x, y, z)=\frac{x-y}{v^{2}+z^{2}}
$$
$$
\begin{array}{ll}{\text { a. } f(3,-1,2)} & {\text { b. } f\left(1, \frac{1}{2},-\frac{1}{4}\right)} \\ {\text { c. } f\left(0,-\frac{1}{3}, 0\right)} & {\text { d. } f(2,2,100)}\end{array}
$$

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Problem 4

In Exercises $1-4,$ find the specific function values.
$$
f(x, y, z)=\sqrt{49-x^{2}-y^{2}-z^{2}}
$$
$$
\begin{array}{ll}{\text { a. } f(0,0,0)} & {\text { b. } f(2,-3,6)} \\ {\text { c. } f(-1,2,3)} & {\text { d. } f\left(\frac{4}{\sqrt{2}}, \frac{5}{\sqrt{2}}, \frac{6}{\sqrt{2}}\right)}\end{array}
$$

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Problem 5

In Exercises $5-12,$ find and sketch the domain for each function.
$$
f(x, y)=\sqrt{y-x-2}
$$

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Problem 6

In Exercises $5-12,$ 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

In Exercises $5-12,$ 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

In Exercises $5-12,$ 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

In Exercises $5-12,$ find and sketch the domain for each function.
$$
f(x, y)=\cos ^{-1}\left(y-x^{2}\right)
$$

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Problem 10

In Exercises $5-12,$ find and sketch the domain for each function.
$$
f(x, y)=\ln (x y+x-y-1)
$$

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Problem 11

In Exercises $5-12,$ 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

In Exercises $5-12,$ 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

In Exercises $13-16,$ 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.
$$
f(x, y)=x+y-1, \quad c=-3,-2,-1,0,1,2,3
$$

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Problem 14

In Exercises $13-16,$ 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.
$$
f(x, y)=x^{2}+y^{2}, \quad c=0,1,4,9,16,25
$$

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Problem 15

In Exercises $13-16,$ 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.
$$
f(x, y)=x y, \quad c=-9,-4,-1,0,1,4,9
$$

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Problem 16

In Exercises $13-16,$ 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 Exercises $17-30,$ (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 if the domain is an
open region, a closed region, or neither, and (f) decide if the domain
is bounded or unbounded.
$$
f(x, y)=y-x
$$

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Problem 18

In Exercises $17-30,$ (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 if the domain is an
open region, a closed region, or neither, and (f) decide if the domain
is bounded or unbounded.
$$
f(x, y)=\sqrt{y-x}
$$

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Problem 19

In Exercises $17-30,$ (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 if the domain is an
open region, a closed region, or neither, and (f) decide if the domain
is bounded or unbounded.
$$
f(x, y)=4 x^{2}+9 y^{2}
$$

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Problem 20

In Exercises $17-30,$ (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 if the domain is an
open region, a closed region, or neither, and (f) decide if the domain
is bounded or unbounded.
$$
f(x, y)=x^{2}-y^{2}
$$

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Problem 21

In Exercises $17-30,$ (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 if the domain is an
open region, a closed region, or neither, and (f) decide if the domain
is bounded or unbounded.
$$
f(x, y)=x y
$$

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Problem 22

In Exercises $17-30,$ (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 if the domain is an
open region, a closed region, or neither, and (f) decide if the domain
is bounded or unbounded.
$$
f(x, y)=y / x^{2}
$$

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Problem 23

In Exercises $17-30,$ (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 if the domain is an
open region, a closed region, or neither, and (f) decide if the domain
is bounded or unbounded.
$$
f(x, y)=\frac{1}{\sqrt{16-x^{2}-y^{2}}}
$$

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Problem 24

In Exercises $17-30,$ (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 if the domain is an
open region, a closed region, or neither, and (f) decide if the domain
is bounded or unbounded.
$$
f(x, y)=\sqrt{9-x^{2}-y^{2}}
$$

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Problem 25

In Exercises $17-30,$ (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 if the domain is an
open region, a closed region, or neither, and (f) decide if the domain
is bounded or unbounded.
$$
f(x, y)=\ln \left(x^{2}+y^{2}\right)
$$

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Problem 26

In Exercises $17-30,$ (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 if the domain is an
open region, a closed region, or neither, and (f) decide if the domain
is bounded or unbounded.
$$
f(x, y)=e^{-\left(x^{2}+y^{2}\right)}
$$

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Problem 27

In Exercises $17-30,$ (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 if the domain is an
open region, a closed region, or neither, and (f) decide if the domain
is bounded or unbounded.
$$
f(x, y)=\sin ^{-1}(y-x)
$$

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Problem 28

In Exercises $17-30,$ (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 if the domain is an
open region, a closed region, or neither, and (f) decide if the domain
is bounded or unbounded.
$$
f(x, y)=\tan ^{-1}\left(\frac{y}{x}\right)
$$

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Problem 29

In Exercises $17-30,$ (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 if the domain is an
open region, a closed region, or neither, and (f) decide if the domain
is bounded or unbounded.
$$
f(x, y)=\ln \left(x^{2}+y^{2}-1\right)
$$

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Problem 30

In Exercises $17-30,$ (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 if the domain is an
open region, a closed region, or neither, and (f) decide if the domain
is bounded or unbounded.
$$
f(x, y)=\ln \left(9-x^{2}-y^{2}\right)
$$

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Problem 31

Exercises $31-36$ show level curves for the functions graphed in (a)-(f)
on the following page. Match each set of curves with the appropriate
function.

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Problem 32

Exercises $31-36$ show level curves for the functions graphed in (a)-(f)
on the following page. Match each set of curves with the appropriate
function.

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Problem 33

Exercises $31-36$ show level curves for the functions graphed in (a)-(f)
on the following page. Match each set of curves with the appropriate
function.

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Problem 34

Exercises $31-36$ show level curves for the functions graphed in (a)-(f)
on the following page. Match each set of curves with the appropriate
function.

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Problem 35

Exercises $31-36$ show level curves for the functions graphed in (a)-(f)
on the following page. Match each set of curves with the appropriate
function.

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Problem 36

Exercises $31-36$ show level curves for the functions graphed in (a)-(f)
on the following page. Match each set of curves with the appropriate
function.

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Problem 37

Display the values of the functions in Exercises $37-48$ 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 Exercises $37-48$ 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 Exercises $37-48$ 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 Exercises $37-48$ 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 Exercises $37-48$ 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 Exercises $37-48$ 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 Exercises $37-48$ 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 Exercises $37-48$ 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 Exercises $37-48$ 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 Exercises $37-48$ 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 Exercises $37-48$ 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 Exercises $37-48$ 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

In Exercises $49-52$ , find an equation for and sketch the graph of the
level curve of the function $f(x, y)$ that passes through the given point.
$$
f(x, y)=16-x^{2}-y^{2}, \quad(2 \sqrt{2}, \sqrt{2})
$$

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Problem 50

In Exercises $49-52$ , find an equation for and sketch the graph of the
level curve of the function $f(x, y)$ that passes through the given point.
$$
f(x, y)=\sqrt{x^{2}-1}, \quad(1,0)
$$

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Problem 51

In Exercises $49-52$ , find an equation for and sketch the graph of the
level curve of the function $f(x, y)$ that passes through the given point.
$$
f(x, y)=\sqrt{x+y^{2}-3}, \quad(3,-1)
$$

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Problem 52

In Exercises $49-52$ , 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

In Exercises $53-60,$ sketch a typical level surface for the function.
$$
f(x, y, z)=x^{2}+y^{2}+z^{2}
$$

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Problem 54

In Exercises $53-60,$ 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

In Exercises $53-60,$ sketch a typical level surface for the function.
$$
f(x, y, z)=x+z
$$

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Problem 56

In Exercises $53-60,$ sketch a typical level surface for the function.
$$
f(x, y, z)=z
$$

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Problem 57

In Exercises $53-60,$ sketch a typical level surface for the function.
$$
f(x, y, z)=x^{2}+y^{2}
$$

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Problem 58

In Exercises $53-60,$ sketch a typical level surface for the function.
$$
f(x, y, z)=y^{2}+z^{2}
$$

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Problem 59

In Exercises $53-60,$ sketch a typical level surface for the function.
$$
f(x, y, z)=z-x^{2}-y^{2}
$$

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Problem 60

In Exercises $53-60,$ 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

In Exercises $61-64,$ 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

In Exercises $61-64,$ 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

In Exercises $61-64,$ 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

In Exercises $61-64,$ 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

In Exercises $65-68,$ 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.
$$
f(x, y)=\sum_{n=0}^{\infty}\left(\frac{x}{y}\right)^{n}, \quad(1,2)
$$

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Problem 66

In Exercises $65-68,$ 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.
$$
g(x, y, z)=\sum_{n=0}^{\infty} \frac{(x+y)^{n}}{n ! z^{n}}, \quad(\ln 4, \ln 9,2)
$$

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Problem 67

In Exercises $65-68,$ 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.
$$
f(x, y)=\int_{x}^{y} \frac{d \theta}{\sqrt{1-\theta^{2}}}, \quad(0,1)
$$

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Problem 68

In Exercises $65-68,$ 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 in
Exercises $69-72 .$
$$
\begin{array}{l}{\text { a. Plot the surface over the given rectangle. }} \\ {\text { b. Plot several level curves in the rectangle. }} \\ {\text { c. Plot the level curve of } f \text { through the given point. }}\end{array}
$$
$$
\begin{array}{l}{f(x, y)=x \sin \frac{y}{2}+y \sin 2 x, \quad 0 \leq x \leq 5 \pi, \quad 0 \leq y \leq 5 \pi} \\ {P(3 \pi, 3 \pi)}\end{array}
$$

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Problem 70

Use a CAS to perform the following steps for each of the functions in
Exercises $69-72 .$
$$
\begin{array}{l}{\text { a. Plot the surface over the given rectangle. }} \\ {\text { b. Plot several level curves in the rectangle. }} \\ {\text { c. Plot the level curve of } f \text { through the given point. }}\end{array}
$$
$$
\begin{array}{l}{f(x, y)=(\sin x)(\cos y) e^{\sqrt{x^{2}+y^{2}} / 8}, \quad 0 \leq x \leq 5 \pi} \\ {0 \leq y \leq 5 \pi, \quad P(4 \pi, 4 \pi)}\end{array}
$$

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Problem 71

Use a CAS to perform the following steps for each of the functions in
Exercises $69-72 .$
$$
\begin{array}{l}{\text { a. Plot the surface over the given rectangle. }} \\ {\text { b. Plot several level curves in the rectangle. }} \\ {\text { c. Plot the level curve of } f \text { through the given point. }}\end{array}
$$
$$
\begin{array}{l}{f(x, y)=\sin (x+2 \cos y), \quad-2 \pi \leq x \leq 2 \pi} \\ {-2 \pi \leq y \leq 2 \pi, \quad P(\pi, \pi)}\end{array}
$$

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Problem 72

Use a CAS to perform the following steps for each of the functions in
Exercises $69-72 .$
$$
\begin{array}{l}{\text { a. Plot the surface over the given rectangle. }} \\ {\text { b. Plot several level curves in the rectangle. }} \\ {\text { c. Plot the level curve of } f \text { through the given point. }}\end{array}
$$
$$
\begin{array}{l}{f(x, y)=e^{\left(l^{a 1}-y\right)} \sin \left(x^{2}+y^{2}\right), \quad 0 \leq x \leq 2 \pi} \\ {-2 \pi \leq y \leq \pi, \quad P(\pi,-\pi)}\end{array}
$$

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Problem 73

Use a CAS to plot the implicitly defined level surfaces in Exercises
$73-76 .$
$$
4 \ln \left(x^{2}+y^{2}+z^{2}\right)=1
$$

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Problem 74

Use a CAS to plot the implicitly defined level surfaces in Exercises
$73-76 .$
$$
x^{2}+z^{2}=1
$$

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Problem 75

Use a CAS to plot the implicitly defined level surfaces in Exercises
$73-76 .$
$$
x+y^{2}-3 z^{2}=1
$$

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Problem 76

Use a CAS to plot the implicitly defined level surfaces in Exercises
$73-76 .$
$$
\sin \left(\frac{x}{2}\right)-(\cos y) \sqrt{x^{2}+z^{2}}=2
$$

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Problem 77

Parametrized Surfaces Just as you describe curves in the plane
parametrically 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
16.5.) Use a CAS to plot the surfaces in Exercises $77-80 .$ Also plot
several level curves in the $x y$ -plane.
$$
\begin{array}{l}{x=u \cos v, \quad y=u \sin v, \quad z=u, \quad 0 \leq u \leq 2} \\ {0 \leq v \leq 2 \pi}\end{array}
$$

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Problem 78

Parametrized Surfaces Just as you describe curves in the plane
parametrically 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
16.5.) Use a CAS to plot the surfaces in Exercises $77-80 .$ Also plot
several level curves in the $x y$ -plane.
$$
\begin{array}{l}{x=u \cos v, \quad y=u \sin v, \quad z=v, \quad 0 \leq u \leq 2} \\ {0 \leq v \leq 2 \pi}\end{array}
$$

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Problem 79

Parametrized Surfaces Just as you describe curves in the plane
parametrically 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
16.5.) Use a CAS to plot the surfaces in Exercises $77-80 .$ Also plot
several level curves in the $x y$ -plane.
$$
\begin{array}{l}{x=(2+\cos u) \cos v, \quad y=(2+\cos u) \sin v, \quad z=\sin u} \\ {0 \leq u \leq 2 \pi, \quad 0 \leq v \leq 2 \pi}\end{array}
$$

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Problem 80

Parametrized Surfaces Just as you describe curves in the plane
parametrically 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
16.5.) Use a CAS to plot the surfaces in Exercises $77-80 .$ Also plot
several level curves in the $x y$ -plane.
$$
\begin{array}{l}{x=2 \cos u \cos v, \quad y=2 \cos u \sin v, \quad z=2 \sin u} \\ {0 \leq u \leq 2 \pi, \quad 0 \leq v \leq \pi}\end{array}
$$

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