University Calculus: Early Transcendentals

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

Chapter 13

Partial Derivatives - all with Video Answers

Educators

Section 1

Functions of Several Variables

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. $f(-3,-2)$

Thomas Emment

Thomas Emment

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. $f\left(-\frac{\pi}{2},-7\right)$

Thomas Emment

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. $f(2,2,100)$

Thomas Emment

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. $f\left(\frac{4}{\sqrt{2}}, \frac{5}{\sqrt{2}}, \frac{6}{\sqrt{2}}\right)$

Thomas Emment

Numerade Educator

Problem 5

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

Amrita Bhasin

Numerade Educator

Problem 6

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

Abigail Martyr

Numerade Educator

Problem 7

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

Abigail Martyr

Numerade Educator

Problem 8

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

Abigail Martyr

Numerade Educator

Problem 9

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

Abigail Martyr

Numerade Educator

Problem 10

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

Abigail Martyr

Numerade Educator

Problem 11

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

Abigail Martyr

Numerade Educator

Problem 12

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

Abigail Martyr

Numerade Educator

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

Abigail Martyr

Numerade Educator

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

Abigail Martyr

Numerade Educator

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

Abigail Martyr

Numerade Educator

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

Abigail Martyr

Numerade Educator

Problem 17

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

Lucas Finney

Numerade Educator

Problem 18

(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 ( $\mathbf{f}$ ) decide if the domain is bounded or unbounded.
$$f(x, y)=\sqrt{y-x}$$

Lucas Finney

Numerade Educator

Problem 19

(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 ( $\mathbf{f}$ ) decide if the domain is bounded or unbounded.
$$f(x, y)=4 x^{2}+9 y^{2}$$

Lucas Finney

Numerade Educator

Problem 20

(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 ( $\mathbf{f}$ ) decide if the domain is bounded or unbounded.
$$f(x, y)=x^{2}-y^{2}$$

Lucas Finney

Numerade Educator

Problem 21

(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 ( $\mathbf{f}$ ) decide if the domain is bounded or unbounded.
$$f(x, y)=x y$$

Nick Johnson

Numerade Educator

Problem 22

(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 ( $\mathbf{f}$ ) decide if the domain is bounded or unbounded.
$$f(x, y)=y / x^{2}$$

Lucas Finney

Numerade Educator

Problem 23

(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 ( $\mathbf{f}$ ) decide if the domain is bounded or unbounded.
$$f(x, y)=\frac{1}{\sqrt{16-x^{2}-y^{2}}}$$

Carson Merrill

Numerade Educator

Problem 24

(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 ( $\mathbf{f}$ ) decide if the domain is bounded or unbounded.
$$f(x, y)=\sqrt{9-x^{2}-y^{2}}$$

Lucas Finney

Numerade Educator

Problem 25

(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 ( $\mathbf{f}$ ) decide if the domain is bounded or unbounded.
$$f(x, y)=\ln \left(x^{2}+y^{2}\right)$$

Lucas Finney

Numerade Educator

Problem 26

(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 ( $\mathbf{f}$ ) decide if the domain is bounded or unbounded.
$$f(x, y)=e^{-\left(x^{2}+y^{2}\right)}$$

Lucas Finney

Numerade Educator

Problem 27

(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 ( $\mathbf{f}$ ) decide if the domain is bounded or unbounded.
$$f(x, y)=\sin ^{-1}(y-x)$$

Lucas Finney

Numerade Educator

Problem 28

(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 ( $\mathbf{f}$ ) decide if the domain is bounded or unbounded.
$$f(x, y)=\tan ^{-1}\left(\frac{y}{x}\right)$$

Lucas Finney

Numerade Educator

Problem 29

(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 ( $\mathbf{f}$ ) decide if the domain is bounded or unbounded.
$$f(x, y)=\ln \left(x^{2}+y^{2}-1\right)$$

Lucas Finney

Numerade Educator

Problem 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 ( $\mathbf{f}$ ) decide if the domain is bounded or unbounded.
$$f(x, y)=\ln \left(9-x^{2}-y^{2}\right)$$

Lucas Finney

Numerade Educator

Problem 31

Show level curves for the functions graphed in (a)-(1) on the following page. Match each set of curves with the appropriate function.
(FIGURE CAN'T COPY)

Lucas Finney

Numerade Educator

Problem 32

Show level curves for the functions graphed in (a)-(1) on the following page. Match each set of curves with the appropriate function.
(FIGURE CAN'T COPY)

Lucas Finney

Numerade Educator

Problem 33

Show level curves for the functions graphed in (a)-(1) on the following page. Match each set of curves with the appropriate function.
(FIGURE CAN'T COPY)

Lucas Finney

Numerade Educator

Problem 34

Show level curves for the functions graphed in (a)-(1) on the following page. Match each set of curves with the appropriate function.
(FIGURE CAN'T COPY)

Lucas Finney

Numerade Educator

Problem 35

Show level curves for the functions graphed in (a)-(1) on the following page. Match each set of curves with the appropriate function.
(FIGURE CAN'T COPY)

Lucas Finney

Numerade Educator

Problem 36

Show level curves for the functions graphed in (a)-(1) on the following page. Match each set of curves with the appropriate function.
(FIGURE CAN'T COPY)

Lucas Finney

Numerade Educator

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

Abigail Martyr

Numerade Educator

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

Abigail Martyr

Numerade Educator

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

Abigail Martyr

Numerade Educator

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

Abigail Martyr

Numerade Educator

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

Abigail Martyr

Numerade Educator

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

Abigail Martyr

Numerade Educator

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

Abigail Martyr

Numerade Educator

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.
$$f(x, y)=6-2 x-3 y$$

Abigail Martyr

Numerade Educator

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

Abigail Martyr

Numerade Educator

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

Bobby Barnes

University of North Texas

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

Abigail Martyr

Numerade Educator

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

Abigail Martyr

Numerade Educator

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

Abigail Martyr

Numerade Educator

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

Lucas Finney

Numerade Educator

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

Lucas Finney

Numerade Educator

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

Abigail Martyr

Numerade Educator

Problem 53

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

Abigail Martyr

Numerade Educator

Problem 54

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

Abigail Martyr

Numerade Educator

Problem 55

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

Abigail Martyr

Numerade Educator

Problem 56

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

Abigail Martyr

Numerade Educator

Problem 57

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

Abigail Martyr

Numerade Educator

Problem 58

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

Abigail Martyr

Numerade Educator

Problem 59

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

Abigail Martyr

Numerade Educator

Problem 60

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

Abigail Martyr

Numerade Educator

Problem 61

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

Lucas Finney

Numerade Educator

Problem 62

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

Abigail Martyr

Numerade Educator

Problem 63

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

Abigail Martyr

Numerade Educator

Problem 64

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

Abigail Martyr

Numerade Educator

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

Lucas Finney

Numerade Educator

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

Abigail Martyr

Numerade Educator

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

Abigail Martyr

Numerade Educator

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

Abigail Martyr

Numerade Educator

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.
$$\begin{aligned}
&f(x, y)=x \sin \frac{y}{2}+y \sin 2 x, \quad 0 \leq x \leq 5 \pi, \quad 0 \leq y \leq 5 \pi\\
&P(3 \pi, 3 \pi)
\end{aligned}$$

Abigail Martyr

Numerade Educator

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.
$$\begin{array}{ll}
f(x, y)=(\sin x)(\cos y) e^{\sqrt{x^{2}+y^{3}} / 8}, & 0 \leq x \leq 5 \pi \\
0 \leq y \leq 5 \pi, & P(4 \pi, 4 \pi)
\end{array}$$

Abigail Martyr

Numerade Educator

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.
$$\begin{array}{l}
f(x, y)=\sin (x+2 \cos y), \quad-2 \pi \leq x \leq 2 \pi \\
-2 \pi \leq y \leq 2 \pi, \quad P(\pi, \pi)
\end{array}$$

Abigail Martyr

Numerade Educator

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.
$$\begin{array}{l}
f(x, y)=e^{\left(a^{\prime \prime}-y\right)} \sin \left(x^{2}+y^{2}\right), \quad 0 \leq x \leq 2 \pi \\
-2 \pi \leq y \leq \pi, \quad P(\pi,-\pi)
\end{array}$$

Lucas Finney

Numerade Educator

Problem 73

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

Abigail Martyr

Numerade Educator

Problem 74

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

Abigail Martyr

Numerade Educator

Problem 75

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

Abigail Martyr

Numerade Educator

Problem 76

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

Abigail Martyr

Numerade Educator

Problem 77

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.
$$\begin{array}{l}
x=u \cos v, \quad y=u \sin v, \quad z=u, \quad 0 \leq u \leq 2 \\
0 \leq v \leq 2 \pi
\end{array}$$

Lucas Finney

Numerade Educator

Problem 78

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.
$$\begin{aligned}
&x=u \cos v, \quad y=u \sin v, \quad z=v, \quad 0 \leq u \leq 2\\
&0 \leq v \leq 2 \pi
\end{aligned}$$

Lucas Finney

Numerade Educator

Problem 79

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.
$$\begin{array}{l}
x=(2+\cos u) \cos v, \quad y=(2+\cos u) \sin v, \quad z=\sin u \\
0 \leq u \leq 2 \pi, \quad 0 \leq v \leq 2 \pi
\end{array}$$

Lucas Finney

Numerade Educator

Problem 80

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.
$$\begin{array}{l}
x=2 \cos u \cos v, \quad y=2 \cos u \sin v, \quad z=2 \sin u \\
0 \leq u \leq 2 \pi, \quad 0 \leq v \leq \pi
\end{array}$$

Lucas Finney

Numerade Educator