Calculus

George B. Thomas, Jr. Maurice D. Weir, Joel Hass

Chapter 14

Partial Derivatives - all with Video Answers

Educators

+ 4 more educators

Section 1

Functions of Several Variables

Problem 1

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

Vysakh M

Vysakh M

Numerade Educator

Problem 2

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

Eric Carlsen

Numerade Educator

Problem 3

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

Claire Rochford

Claire Rochford

Numerade Educator

Problem 4

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

Brittany Knowlton

Brittany Knowlton

Numerade Educator

Problem 5

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

Regina Hays

Numerade Educator

Problem 6

In Exercises $5-12,$ find and sketch the domain for each function.
$$f(x, y)=\ln \left(x^{2}+y^{2}-4\right)$$

Chris Smith

Numerade Educator

Problem 7

In Exercises $5-12,$ find and sketch the domain for each function.
$$f(x, y)=\frac{(x-1)(y+2)}{(y-x)\left(y-x^{3}\right)}$$

Adam Dehollander

Adam Dehollander

Numerade Educator

Problem 8

In Exercises $5-12,$ find and sketch the domain for each function.
$$f(x, y)=\frac{\sin (x y)}{x^{2}+y^{2}-25}$$

IZ

Ivy Zhang

Numerade Educator

Problem 9

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

BS

Benjamin Sekutera

Numerade Educator

Problem 10

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

Brittany Knowlton

Brittany Knowlton

Numerade Educator

Problem 11

In Exercises $5-12,$ find and sketch the domain for each function.
$$f(x, y)=\sqrt{\left(x^{2}-4\right)\left(y^{2}-9\right)}$$

Melissa Munoz

Numerade Educator

Problem 12

In Exercises $5-12,$ find and sketch the domain for each function.
$$f(x, y)=\frac{1}{\ln \left(4-x^{2}-y^{2}\right)}$$

Chris Smith

Numerade Educator

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

Sandra Herrman

Numerade Educator

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

Brittany Knowlton

Brittany Knowlton

Numerade Educator

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

Brittany Knowlton

Brittany Knowlton

Numerade Educator

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.
$$f(x, y)=\sqrt{25-x^{2}-y^{2}}, \quad c=0,1,2,3,4$$

Brittany Knowlton

Brittany Knowlton

Numerade Educator

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.
$f(x, y)=y-x$

Regina Hays

Numerade Educator

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.
$$f(x, y)=\sqrt{y-x}$$

Regina Hays

Numerade Educator

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.
$$f(x, y)=4 x^{2}+9 y^{2}$$

Brian Lin

Numerade Educator

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.
$$f(x, y)=x^{2}-y^{2}$$

Brittany Knowlton

Brittany Knowlton

Numerade Educator

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.
$$f(x, y)=x y$$

Regina Hays

Numerade Educator

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.
$$f(x, y)=y / x^{2}$$

Brittany Knowlton

Brittany Knowlton

Numerade Educator

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.
$f(x, y)=\frac{1}{\sqrt{16-x^{2}-y^{2}}}$

Brittany Knowlton

Brittany Knowlton

Numerade Educator

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.
$$f(x, y)=\sqrt{9-x^{2}-y^{2}}$$

Brian Lin

Numerade Educator

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.
$$f(x, y)=\ln \left(x^{2}+y^{2}\right)$$

Brittany Knowlton

Brittany Knowlton

Numerade Educator

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.
$$f(x, y)=e^{-\left(x^{2}+y^{2}\right)}$$

Brittany Knowlton

Brittany Knowlton

Numerade Educator

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.
$$f(x, y)=\sin ^{-1}(y-x)$$

Regina Hays

Numerade Educator

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.
$$f(x, y)=\tan ^{-1}\left(\frac{y}{x}\right)$$

Regina Hays

Numerade Educator

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.
$$f(x, y)=\ln \left(x^{2}+y^{2}-1\right)$$

Regina Hays

Numerade Educator

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.
$$f(x, y)=\ln \left(9-x^{2}-y^{2}\right)$$

Brian Lin

Numerade Educator

Problem 31

Matching Surfaces with Level Curves
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.

WM

William Mead

Numerade Educator

Problem 32

Matching Surfaces with Level Curves
Exercises $31-36$ show level curves for the functions graphed in
$(a)$ - (f) on the following page. Match each set of curves with the ap-
propriate function.

WM

William Mead

Numerade Educator

Problem 33

Matching Surfaces with Level Curves
Exercises $31-36$ show level curves for the functions graphed in
$(a)$ - (f) on the following page. Match each set of curves with the ap-
propriate function.

WM

William Mead

Numerade Educator

Problem 34

Matching Surfaces with Level Curves
Exercises $31-36$ show level curves for the functions graphed in
$(a)$ - (f) on the following page. Match each set of curves with the ap-
propriate function.

WM

William Mead

Numerade Educator

Problem 35

Matching Surfaces with Level Curves
Exercises $31-36$ show level curves for the functions graphed in
$(a)$ - (f) on the following page. Match each set of curves with the ap-
propriate function.

WM

William Mead

Numerade Educator

Problem 36

Matching Surfaces with Level Curves
Exercises $31-36$ show level curves for the functions graphed in
$(a)$ - (f) on the following page. Match each set of curves with the ap-
propriate function.

WM

William Mead

Numerade Educator

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.
$$f(x, y)=y^{2}$$

Kevin Harmer

Numerade Educator

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.
$$f(x, y)=\sqrt{x}$$

WM

William Mead

Numerade Educator

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.
$$f(x, y)=x^{2}+y^{2}$$

Kevin Harmer

Numerade Educator

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.
$$f(x, y)=\sqrt{x^{2}+y^{2}}$$

WM

William Mead

Numerade Educator

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.
$$f(x, y)=x^{2}-y$$

WM

William Mead

Numerade Educator

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.
$$f(x, y)=4-x^{2}-y^{2}$$

Kevin Harmer

Numerade Educator

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.
$$f(x, y)=4 x^{2}+y^{2}$$

Kevin Harmer

Numerade Educator

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.
$$f(x, y)=1-|x|-|y|$$

WM

William Mead

Numerade Educator

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..
$$f(x, y)=1-|y|$$

WM

William Mead

Numerade Educator

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.
$$f(x, y)=1-|x|-|y|$$

WM

William Mead

Numerade Educator

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.
$$f(x, y)=\sqrt{x^{2}+y^{2}+4}$$

WM

William Mead

Numerade Educator

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.
$$f(x, y)=\sqrt{x^{2}+y^{2}-4}$$

WM

William Mead

Numerade Educator

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.
$$f(x, y)=16-x^{2}-y^{2}, \quad(2 \sqrt{2}, \sqrt{2})$$

Regina Hays

Numerade Educator

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.
$$f(x, y)=\sqrt{x^{2}-1}, \quad(1,0)$$

Brittany Knowlton

Brittany Knowlton

Numerade Educator

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.
$$f(x, y)=\sqrt{x+y^{2}-3}, \quad(3,-1)$$

WM

William Mead

Numerade Educator

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.
$$f(x, y)=\frac{2 y-x}{x+y+1}, \quad(-1,1)$$

Brittany Knowlton

Brittany Knowlton

Numerade Educator

Problem 53

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

Brittany Knowlton

Brittany Knowlton

Numerade Educator

Problem 54

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

Regina Hays

Numerade Educator

Problem 55

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

WM

William Mead

Numerade Educator

Problem 56

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

Regina Hays

Numerade Educator

Problem 57

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

Regina Hays

Numerade Educator

Problem 58

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

Regina Hays

Numerade Educator

Problem 59

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

Brittany Knowlton

Brittany Knowlton

Numerade Educator

Problem 60

In Exercises $53-60,$ sketch a typical level surface for the function.
$$f(x, y, z)=\left(x^{2} / 25\right)+\left(y^{2} / 16\right)+\left(z^{2} / 9\right)$$

Regina Hays

Numerade Educator

Problem 61

In Exercises $61-64$ , find an equation for the level surface of the function through the given point.
$$f(x, y, z)=\sqrt{x-y}-\ln z, \quad(3,-1,1)$$

Brittany Knowlton

Brittany Knowlton

Numerade Educator

Problem 62

In Exercises $61-64$ , find an equation for the level surface of the function through the given point.
$$f(x, y, z)=\ln \left(x^{2}+y+z^{2}\right), \quad(-1,2,1)$$

Regina Hays

Numerade Educator

Problem 63

In Exercises $61-64$ , find an equation for the level surface of the function through the given point.
$$g(x, y, z)=\sqrt{x^{2}+y^{2}+z^{2}}, \quad(1,-1, \sqrt{2})$$

Brittany Knowlton

Brittany Knowlton

Numerade Educator

Problem 64

In Exercises $61-64$ , find an equation for the level surface of the function through the given point.
$$g(x, y, z)=\frac{x-y+z}{2 x+y-z}, \quad(1,0,-2)$$

Regina Hays

Numerade Educator

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

Brittany Knowlton

Brittany Knowlton

Numerade Educator

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

Regina Hays

Numerade Educator

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

Brittany Knowlton

Brittany Knowlton

Numerade Educator

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.
$$g(x, y, z)=\int_{x}^{y} \frac{d t}{1+t^{2}}+\int_{0}^{z} \frac{d \theta}{\sqrt{4-\theta^{2}}}, \quad(0,1, \sqrt{3})$$

Regina Hays

Numerade Educator

Problem 69

Use a CAS to perform the following steps for each of the functions in Exercises $69-72 .$
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.
$$f(x, y)=x \sin \frac{y}{2}+y \sin 2 x, \quad 0 \leq x \leq 5 \pi, \quad 0 \leq y \leq 5 \pi$ $P(3 \pi, 3 \pi)$$

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 70

Use a CAS to perform the following steps for each of the functions in Exercises $69-72 .$
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.
$$f(x, y)=(\sin x)(\cos y) e^{\sqrt{x^{2}+y^{2}} / 8}, \quad 0 \leq x \leq 5 \pi$$
$$0 \leq y \leq 5 \pi, \quad P(4 \pi, 4 \pi)$$

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 71

Use a CAS to perform the following steps for each of the functions in Exercises $69-72 .$
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.
$$f(x, y)=\sin (x+2 \cos y), \quad-2 \pi \leq x \leq 2 \pi$$
$$2 \pi \leq y \leq 2 \pi, \quad P(\pi, \pi)$$

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 72

Use a CAS to perform the following steps for each of the functions in Exercises $69-72 .$
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.
$$f(x, y)=e^{\left(x^{01}-y\right)} \sin \left(x^{2}+y^{2}\right), \quad 0 \leq x \leq 2 \pi$$
$$-2 \pi \leq y \leq \pi, \quad P(\pi,-\pi)$$

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 73

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

Regina Hays

Numerade Educator

Problem 74

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

Regina Hays

Numerade Educator

Problem 75

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

Regina Hays

Numerade Educator

Problem 76

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

Dushyant Barot

Numerade Educator

Problem 77

Parametrized Surfaces Just as you describe curves in the plane parametrized 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 xy-plane.
$$x=u \cos v, \quad y=u \sin v, \quad z=u, \quad 0 \leq u \leq 2$$
$$0 \leq v \leq 2 \pi$$

Regina Hays

Numerade Educator

Problem 78

Parametrized Surfaces Just as you describe curves in the plane parametrized 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 xy-plane.
$$x=u \cos v, \quad y=u \sin v, \quad z=v, \quad 0 \leq u \leq 2$$
$$0 \leq v \leq 2 \pi$$

Regina Hays

Numerade Educator

Problem 79

Parametrized Surfaces Just as you describe curves in the plane parametrized 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 xy-plane.
$$x=(2+\cos u) \cos v, \quad y=(2+\cos u) \sin v, \quad z=\sin u$$
$$0 \leq u \leq 2 \pi, \quad 0 \leq v \leq 2 \pi$$

Regina Hays

Numerade Educator

Problem 80

Parametrized Surfaces Just as you describe curves in the plane parametrized 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 xy-plane.
$$x=2 \cos u \cos v, \quad y=2 \cos u \sin v, \quad z=2 \sin u$$
$$0 \leq u \leq 2 \pi, \quad 0 \leq v \leq \pi$$

Regina Hays

Numerade Educator