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Thomas Calculus

George B. Thomas, Maurice D. Weir, Joel Hass, Frank R. Giordano

Chapter 14

Partial Derivatives

Educators


Problem 1

In Exercises $1-12,$ (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 2

In Exercises $1-12,$ (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 3

In Exercises $1-12,$ (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 4

In Exercises $1-12,$ (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 5

In Exercises $1-12,$ (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 6

In Exercises $1-12,$ (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 7

In Exercises $1-12,$ (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 8

In Exercises $1-12,$ (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 9

In Exercises $1-12,$ (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 10

In Exercises $1-12,$ (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 11

In Exercises $1-12,$ (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 12

In Exercises $1-12,$ (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 13

Exercises $13-18$ show level curves for the functions graphed in $(a)-(f) .$ Match each set of curves with the appropriate function.
(GRAPH NOT COPY)

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Problem 14

Exercises $13-18$ show level curves for the functions graphed in $(a)-(f) .$ Match each set of curves with the appropriate function.
(GRAPH NOT COPY)

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Problem 15

Exercises $13-18$ show level curves for the functions graphed in $(a)-(f) .$ Match each set of curves with the appropriate function.
(GRAPH NOT COPY)

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Problem 16

Exercises $13-18$ show level curves for the functions graphed in $(a)-(f) .$ Match each set of curves with the appropriate function.
(GRAPH NOT COPY)

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Problem 17

Exercises $13-18$ show level curves for the functions graphed in $(a)-(f) .$ Match each set of curves with the appropriate function.
(GRAPH NOT COPY)

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Problem 18

Exercises $13-18$ show level curves for the functions graphed in $(a)-(f) .$ Match each set of curves with the appropriate function.
(GRAPH NOT COPY)

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Problem 19

Display the values of the functions in Exercises $19-28$ 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 20

Display the values of the functions in Exercises $19-28$ 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.
$$
f(x, y)=4-y^{2}
$$

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Problem 21

Display the values of the functions in Exercises $19-28$ 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 22

Display the values of the functions in Exercises $19-28$ 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 23

Display the values of the functions in Exercises $19-28$ 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.
$$
f(x, y)=-\left(x^{2}+y^{2}\right)
$$

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Problem 24

Display the values of the functions in Exercises $19-28$ 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 25

Display the values of the functions in Exercises $19-28$ 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 26

Display the values of the functions in Exercises $19-28$ 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.
$$
f(x, y)=4 x^{2}+y^{2}+1
$$

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Problem 27

Display the values of the functions in Exercises $19-28$ 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 28

Display the values of the functions in Exercises $19-28$ 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 29

In Exercises $29-32$ , find an equation for 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 30

In Exercises $29-32$ , find an equation for 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 31

In Exercises $29-32$ , find an equation for the level curve of the function $f(x, y)$ that passes through the given point.
$$
f(x, y)=\int_{x}^{y} \frac{d t}{1+t^{2}}, \quad(-\sqrt{2}, \sqrt{2})
$$

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Problem 32

In Exercises $29-32$ , find an equation for the level curve of the function $f(x, y)$ that passes 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 33

In Exercises $33-40,$ sketch a typical level surface for the function.
$$
f(x, y, z)=x^{2}+y^{2}+z^{2}
$$

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Problem 34

In Exercises $33-40,$ 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 35

In Exercises $33-40,$ sketch a typical level surface for the function.
$$
f(x, y, z)=x+z
$$

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Problem 36

In Exercises $33-40,$ sketch a typical level surface for the function.
$$
f(x, y, z)=z
$$

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Problem 37

In Exercises $33-40,$ sketch a typical level surface for the function.
$$
f(x, y, z)=x^{2}+y^{2}
$$

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Problem 38

In Exercises $33-40,$ sketch a typical level surface for the function.
$$
f(x, y, z)=y^{2}+z^{2}
$$

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Problem 39

In Exercises $33-40,$ sketch a typical level surface for the function.
$$
f(x, y, z)=z-x^{2}-y^{2}
$$

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Problem 40

In Exercises $33-40,$ 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 41

In Exercises $41-44$ , 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 42

In Exercises $41-44$ , 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 43

In Exercises $41-44$ , find an equation for the level surface of the function through the given point.
$$
g(x, y, z)=\sum_{n=0}^{\infty} \frac{(x+y)^{n}}{n ! z^{n}}, \quad(\ln 2, \ln 4,3)
$$

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Problem 44

In Exercises $41-44$ , find an equation for the level surface of the function through the given point.
$$
g(x, y, z)=\int_{x}^{y} \frac{d \theta}{\sqrt{1-\theta^{2}}}+\int_{\sqrt{2}}^{z} \frac{d t}{t \sqrt{t^{2}-1}}, \quad(0,1 / 2,2)
$$

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Problem 45

The maximum value of a function on a line in space Does the function $f(x, y, z)=x y z$ have a maximum value on the line$x=20-t, y=t, z=20 ?$ If so, what is it? Give reasons for your answer. (Hint: Along the line, $w=f(x, y, z)$ is a differentiable function of $t . )$

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Problem 46

The minimum value of a function on a line in space Does the function $f(x, y, z)=x y-z$ have a minimum value on the line $x=t-1, y=t-2, z=t+7 ?$ If so, what is it? Give reasons for your answer. (Hint: Along the line, $w=f(x, y, z)$ is a differentiable function of $t . )$

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Problem 47

The Concorde's sonic booms Sound waves from the Concorde at bend as the temperature changes and below the altitude at which the plane flies. The sonic boom carpet is the region on the ground that receives shock waves directly from the plane, not reflected from the atmosphere or diffracted along the ground. The carpet is determined by the grazing rays striking the ground from the point directly under the plane. (See accompanying figure.)
The width $w$ of the region in which people on the ground hear the concorde's sonic boom directly, not reflected from a layer in the atmosphere, is a function of
$T=$ air temperature at ground level (in degrees Kelvin)
$h=$ the Concorde's allitude (in kilometers)
$d=$ the Vertical temperature gradient (temperature drop in degrees Kelvin per kilometer).
The formula for $w$ is
$$w=4\left(\frac{T h}{d}\right)^{1 / 2}$$
The Washington-bound Concorde approached the United States from Europe on a course that took it south of Nantucket Island at an altitude of $16.8 \mathrm{km} .$ If the surface temperature is 290 $\mathrm{K}$ and the vertical temperature gradient is 5 $\mathrm{K} / \mathrm{km}$ , how many kilometers south of Nantucket did the plane have to be flown to keep its sonic boom carpet away from the island? (From "Concorde Sonic Booms as an Atmospheric Probe" by N. K. Bal- achandra, W. L. Donn, and D. H. Rind, Science, Vol. 197 (July 1 , 1977), pp. $47-49 .$ )

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Problem 48

As you know, the graph of a real-valued function of a single real variable is a set in a two-coordinate space. The graph of a realvalued function of two independent real variables is a set in a three-coordinate space. The graph of a real-valued function of three independent real variables is a set in a four-coordinate space. How would you define the graph of a real-valued function $f\left(x_{1}, x_{2}, x_{3}, x_{4}\right)$ of four independent real variables? How would you define the graph of a real-valued function $f\left(x_{1}, x_{2}, x_{3}, \ldots, x_{n}\right)$ of $n$ independent real variables?

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Problem 49

Use a CAS to perform the following steps for each of the functions in Exercises $49-52 .$
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.
$$
\begin{array}{l}{f(x, y)=x \sin \frac{y}{2}+y \sin 2 x, \quad 0 \leq x \leq 5 \pi \quad 0 \leq y \leq 5 \pi} \\ {P(3 \pi, 3 \pi)}\end{array}
$$

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Problem 50

Use a CAS to perform the following steps for each of the functions in Exercises $49-52 .$
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.
$$
\begin{array}{l}{f(x, y)=(\sin x)(\cos y) e^{\sqrt{x^{2}+y^{2}} / 8}, \quad 0 \leq x \leq 5 \pi} \\ {0 \leq y \leq 5 \pi, \quad P(4 \pi, 4 \pi)}\end{array}
$$

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Problem 51

Use a CAS to perform the following steps for each of the functions in Exercises $49-52 .$
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.
$$
\begin{array}{l}{f(x, y)=\sin (x+2 \cos y), \quad-2 \pi \leq x \leq 2 \pi} \\ {-2 \pi \leq y \leq 2 \pi, \quad P(\pi, \pi)}\end{array}
$$

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Problem 52

Use a CAS to perform the following steps for each of the functions in Exercises $49-52 .$
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.
$$
\begin{array}{l}{f(x, y)=e^{\left(x^{01}-y\right)} \sin \left(x^{2}+y^{2}\right), \quad 0 \leq x \leq 2 \pi} \\ {-2 \pi \leq y \leq \pi, \quad P(\pi,-\pi)}\end{array}
$$

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Problem 53

Use a CAS to plot the level surfaces in Exercises $53-56$
$$
4 \ln \left(x^{2}+y^{2}+z^{2}\right)=1
$$

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Problem 54

Use a CAS to plot the level surfaces in Exercises $53-56$
$$
x^{2}+z^{2}=1
$$

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Problem 55

Use a CAS to plot the level surfaces in Exercises $53-56$
$$
x+y^{2}-3 z^{2}=1
$$

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Problem 56

Use a CAS to plot the level surfaces in Exercises $53-56$
$$
\sin \left(\frac{x}{2}\right)-(\cos y) \sqrt{x^{2}+z^{2}}=2
$$

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Problem 57

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 $16.6 .$ ) Use a CAS to plot the surfaces in Exercises $57-60 .$ 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 58

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 $16.6 .$ ) Use a CAS to plot the surfaces in Exercises $57-60 .$ 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 59

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 $16.6 .$ ) Use a CAS to plot the surfaces in Exercises $57-60 .$ 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 60

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 $16.6 .$ ) Use a CAS to plot the surfaces in Exercises $57-60 .$ 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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