• Home
  • Textbooks
  • Calculus
  • Partial Derivatives

Calculus

James Stewart

Chapter 14

Partial Derivatives - all with Video Answers

Educators

Section 1

14.1 Exercises

01:57

Problem 1

In Example 2 we considered the function $W=f(T, v)$, where $W$ is the wind-chill index, $T$ is the actual temperature, and $v$ is the wind speed. A numerical representation is given in Table 1.
(a) What is the value of $f(-15,40)$ ? What is its meaning?
(b) Describe in words the meaning of the question "For what value of $v$ is $f(-20, v)=-30$ ?" Then answer the question.
(c) Describe in words the meaning of the question "For what value of $T$ is $f(T, 20)=-49 ?^{\prime \prime}$ Then answer the question.
(d) What is the meaning of the function $W=f(-5, v)$ ? Describe the behavior of this function.
(e) What is the meaning of the function $W=f(T, 50)$ ? Describe the behavior of this function.

Farnood Ensan
Farnood Ensan
Numerade Educator
02:44

Problem 2

The temperature-humidity index $I$ (or humidex, for short) is the perceived air temperature when the actual temperature is $T$ and the relative humidity is $h$, so we can write $I=f(T, h)$. The following table of values of $I$ is an excerpt from a table compiled by the National Oceanic \& Atmospheric Administration.
(a) What is the value of $f(95,70)$ ? What is its meaning?
(b) For what value of $h$ is $f(90, h)=100$ ?
(c) For what value of $T$ is $f(T, 50)=88$ ?
(d) What are the meanings of the functions $I=f(80, h)$ and $I=f(100, h)$ ? Compare the behavior of these two functions of $h$.

Bcrypt_Sha256$$2B$12$8Zlyrq/Mry7Neqzpqy/Nau80Tlrvo9Bu3Mbvzip.Gbblju6Hggsfy Bcrypt_Sha256$$2B$12$8Zlyrq/Mry7Neqzpqy/Nauq4.Oroa01Tmuxwldd7Bnvez4Kszcu..
Bcrypt_Sha256$$2B$12$8Zlyrq/Mry7Neqzpqy/Nau80Tlrvo9Bu3Mbvzip.Gbblju6Hggsfy Bcrypt_Sha256$$2B$12$8Zlyrq/Mry7Neqzpqy/Nauq4.Oroa01Tmuxwldd7Bnvez4Kszcu..
Numerade Educator
00:53

Problem 3

A manufacturer has modeled its yearly production function $P$ (the monetary value of its entire production in millions of dollars) as a Cobb-Douglas function

$$
P(L, K)=1.47 L^{0.65} K^{0.35}
$$

where $L$ is the number of labor hours (in thousands) and $K$ is the invested capital (in millions of dollars). Find $P(120,20)$ and interpret it.

IL
Isaac Linn
Numerade Educator
02:30

Problem 4

Verify for the Cobb-Douglas production function

$$
P(L, K)=1.01 L^{0.75} K^{0.25}
$$
discussed in Example 3 that the production will be doubled if both the amount of labor and the amount of capital are doubled. Determine whether this is also true for the general production function

$$
P(L, K)=b L^a K^{1-a}
$$

Farnood Ensan
Farnood Ensan
Numerade Educator
01:37

Problem 5

A model for the surface area of a human body is given by the function

$$
S=f(w, h)=0.1091 w^{0.425} h^{0.725}
$$

where $w$ is the weight (in pounds), $h$ is the height (in inches), and $S$ is measured in square feet.
(a) Find $f(160,70)$ and interpret it.
(b) What is your own surface area?

IL
Isaac Linn
Numerade Educator
04:48

Problem 6

The wind-chill index $W$ discussed in Example 2 has been modeled by the following function:

$$
W(T, v)=13.12+0.6215 T-11.37 v^{0.16}+0.3965 T v^{0.16}
$$

Check to see how closely this model agrees with the values in Table 1 for a few values of $T$ and $v$.

Bcrypt_Sha256$$2B$12$8Zlyrq/Mry7Neqzpqy/Nau80Tlrvo9Bu3Mbvzip.Gbblju6Hggsfy Bcrypt_Sha256$$2B$12$8Zlyrq/Mry7Neqzpqy/Nauq4.Oroa01Tmuxwldd7Bnvez4Kszcu..
Bcrypt_Sha256$$2B$12$8Zlyrq/Mry7Neqzpqy/Nau80Tlrvo9Bu3Mbvzip.Gbblju6Hggsfy Bcrypt_Sha256$$2B$12$8Zlyrq/Mry7Neqzpqy/Nauq4.Oroa01Tmuxwldd7Bnvez4Kszcu..
Numerade Educator
02:49

Problem 7

The wave heights $h$ in the open sea depend on the speed $v$ of the wind and the length of time $t$ that the wind has been blowing at that speed. Values of the function $h=f(v, t)$ are recorded in feet in Table 4.
(a) What is the value of $f(40,15)$ ? What is its meaning?
(b) What is the meaning of the function $h=f(30, t)$ ? Describe the behavior of this function.
(c) What is the meaning of the function $h=f(v, 30)$ ? Describe the behavior of this function.

Christian Otero
Christian Otero
Numerade Educator
01:41

Problem 8

A company makes three sizes of cardboard boxes: small, medium, and large. It costs $$\$ 2.50$$ to make a small box, $$\$ 4.00$$ for a medium box, and $$\$ 4.50$$ for a large box. Fixed costs are $$\$ 8000$$.
(a) Express the cost of making $x$ small boxes, $y$ medium boxes, and $z$ large boxes as a function of three variables: $C=f(x, y, z)$.
(b) Find $f(3000,5000,4000)$ and interpret it.
(c) What is the domain of $f$ ?

James Kiss
James Kiss
Numerade Educator
02:11

Problem 9

Let $g(x, y)=\cos (x+2 y)$.
(a) Evaluate $g(2,-1)$.
(b) Find the domain of $g$.
(c) Find the range of $g$.

Adithya Ramanujam
Adithya Ramanujam
Numerade Educator
02:28

Problem 10

Let $F(x, y)=1+\sqrt{4-y^2}$.
(a) Evaluate $F(3,1)$.
(b) Find and sketch the domain of $F$.
(c) Find the range of $F$.

Adithya Ramanujam
Adithya Ramanujam
Numerade Educator
02:40

Problem 11

Let $f(x, y, z)=\sqrt{x}+\sqrt{y}+\sqrt{z}+\ln \left(4-x^2-y^2-z^2\right)$.
(a) Evaluate $f(1,1,1)$.
(b) Find and describe the domain of $f$.

Adithya Ramanujam
Adithya Ramanujam
Numerade Educator
02:31

Problem 12

Let $g(x, y, z)=x^3 y^2 z \sqrt{10-x-y-z}$.
(a) Evaluate $g(1,2,3)$.
(b) Find and describe the domain of $g$.

Adithya Ramanujam
Adithya Ramanujam
Numerade Educator
01:06

Problem 13

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

Amrita Bhasin
Amrita Bhasin
Numerade Educator
01:12

Problem 14

Find and sketch the domain of the function.
$f(x, y)=\sqrt{x y}$

Ashley Boni
Ashley Boni
Numerade Educator
02:16

Problem 15

Find and sketch the domain of the function.
$f(x, y)=\ln \left(9-x^2-9 y^2\right)$

Sarah Gift
Sarah Gift
Numerade Educator
01:02

Problem 16

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

Sarah Gift
Sarah Gift
Numerade Educator
01:08

Problem 17

Find and sketch the domain of the function.
$f(x, y)=\sqrt{1-x^2}-\sqrt{1-y^2}$

Amrita Bhasin
Amrita Bhasin
Numerade Educator
02:04

Problem 18

Find and sketch the domain of the function.
$f(x, y)=\sqrt{y}+\sqrt{25-x^2-y^2}$

Sarah Gift
Sarah Gift
Numerade Educator
01:00

Problem 19

Find and sketch the domain of the function.
$f(x, y)=\frac{\sqrt{y-x^2}}{1-x^2}$

Amrita Bhasin
Amrita Bhasin
Numerade Educator

Problem 20

Find and sketch the domain of the function.
$f(x, y)=\arcsin \left(x^2+y^2-2\right)$

Check back soon!
04:02

Problem 21

Find and sketch the domain of the function.
$f(x, y, z)=\sqrt{1-x^2-y^2-z^2}$

WT
Worawit Tepsan
University of Houston
01:07

Problem 22

Find and sketch the domain of the function.
$f(x, y, z)=\ln \left(16-4 x^2-4 y^2-z^2\right)$

ag
Alan Ghazarians
Numerade Educator
00:34

Problem 23

Sketch the graph of the function.
$f(x, y)=1+y$

Abigail Martyr
Abigail Martyr
Numerade Educator
01:01

Problem 24

Sketch the graph of the function.
$f(x, y)=2-x$

Lauren Shelton
Lauren Shelton
Numerade Educator
03:22

Problem 25

Sketch the graph of the function.
$f(x, y)=10-4 x-5 y$

Linda Hand
Linda Hand
Numerade Educator
00:58

Problem 26

Sketch the graph of the function.
$f(x, y)=e^{-y}$

Lucas Finney
Lucas Finney
Numerade Educator
00:34

Problem 27

Sketch the graph of the function.
$f(x, y)=y^2+1$

Abigail Martyr
Abigail Martyr
Numerade Educator

Problem 28

Sketch the graph of the function.
$f(x, y)=1+2 x^2+2 y^2$

Check back soon!
00:24

Problem 29

Sketch the graph of the function.
$f(x, y)=9-x^2-9 y^2$

Abigail Martyr
Abigail Martyr
Numerade Educator
00:24

Problem 30

Sketch the graph of the function.
$f(x, y)=\sqrt{4 x^2+y^2}$

Vikash Ranjan
Vikash Ranjan
Numerade Educator
00:24

Problem 31

Sketch the graph of the function.
$f(x, y)=\sqrt{4-4 x^2-y^2}$

Vikash Ranjan
Vikash Ranjan
Numerade Educator
06:20

Problem 32

Match the function with its graph (labeled I-VI). Give reasons for your choices.
(a) $f(x, y)=|x|+|y|$
(b) $f(x, y)=|x y|$
(c) $f(x, y)=\frac{1}{1+x^2+y^2}$
(d) $f(x, y)=\left(x^2-y^2\right)^2$
(e) $f(x, y)=(x-y)^2$
(f) $f(x, y)=\sin (|x|+|y|)$

Ashley Boni
Ashley Boni
Numerade Educator
01:40

Problem 33

A contour map for a function $f$ is shown. Use it to estimate the values of $f(-3,3)$ and $f(3,-2)$. What can you say about the shape of the graph?

Stephen Hobbs
Stephen Hobbs
Numerade Educator
01:18

Problem 34

Shown is a contour map of atmospheric pressure in North America on August 12, 2008. On the level curves (called isobars) the pressure is indicated in millibars (mb).
(a) Estimate the pressure at $C$ (Chicago), $N$ (Nashville), $S$ (San Francisco), and $V$ (Vancouver).
(b) At which of these locations were the winds strongest?

ag
Alan Ghazarians
Numerade Educator
02:25

Problem 35

Level curves (isothermals) are shown for the water temperature (in ${ }^{\circ} \mathrm{C}$ ) in Long Lake (Minnesota) in 1998 as a function of depth and time of year. Estimate the temperature in the lake on June 9 (day 160) at a depth of 10 m and on June 29 (day 180) at a depth of 5 m .

Michael Mccarthy
Michael Mccarthy
Numerade Educator
01:21

Problem 36

Two contour maps are shown. One is for a function $f$ whose graph is a cone. The other is for a function $g$ whose graph is a paraboloid. Which is which, and why?

Ashley Boni
Ashley Boni
Numerade Educator
View

Problem 37

Locate the points $A$ and $B$ on the map of Lonesome Mountain (Figure 12). How would you describe the terrain near A? Near $B$ ?

ag
Alan Ghazarians
Numerade Educator
00:36

Problem 38

Make a rough sketch of a contour map for the function whose graph is shown.

ag
Alan Ghazarians
Numerade Educator
00:21

Problem 39

39-42 A contour map of a function is shown. Use it to make a rough sketch of the graph of $f$.

ag
Alan Ghazarians
Numerade Educator
00:21

Problem 40

39-42 A contour map of a function is shown. Use it to make a rough sketch of the graph of $f$.

ag
Alan Ghazarians
Numerade Educator
00:21

Problem 41

39-42 A contour map of a function is shown. Use it to make a rough sketch of the graph of $f$.

ag
Alan Ghazarians
Numerade Educator
00:21

Problem 42

39-42 A contour map of a function is shown. Use it to make a rough sketch of the graph of $f$.

ag
Alan Ghazarians
Numerade Educator
01:21

Problem 43

Draw a contour map of the function showing several level curves.
$f(x, y)=(y-2 x)^2$

Amrita Bhasin
Amrita Bhasin
Numerade Educator

Problem 44

Draw a contour map of the function showing several level curves.
$f(x, y)=x^3-y$

Check back soon!

Problem 45

Draw a contour map of the function showing several level curves.
$f(x, y)=\sqrt{x}+y$

Check back soon!
00:53

Problem 46

Draw a contour map of the function showing several level curves.
$f(x, y)=\ln \left(x^2+4 y^2\right)$

Vikash Ranjan
Vikash Ranjan
Numerade Educator
04:42

Problem 47

Draw a contour map of the function showing several level curves.
$f(x, y)=y e^x$

Aparna Shakti
Aparna Shakti
Numerade Educator
01:53

Problem 48

Draw a contour map of the function showing several level curves.
$f(x, y)=y \sec x$

Farnood Ensan
Farnood Ensan
Numerade Educator
00:36

Problem 49

Draw a contour map of the function showing several level curves.
$f(x, y)=\sqrt{y^2-x^2}$

ag
Alan Ghazarians
Numerade Educator
02:23

Problem 50

Draw a contour map of the function showing several level curves.
$f(x, y)=y /\left(x^2+y^2\right)$

Ashley Boni
Ashley Boni
Numerade Educator
00:54

Problem 51

Sketch both a contour map and a graph of the function and compare them.
$f(x, y)=x^2+9 y^2$

ag
Alan Ghazarians
Numerade Educator
01:27

Problem 52

Sketch both a contour map and a graph of the function and compare them.
$f(x, y)=\sqrt{36-9 x^2-4 y^2}$

ag
Alan Ghazarians
Numerade Educator
09:07

Problem 53

A thin metal plate, located in the $x y$-plane, has temperature $T(x, y)$ at the point $(x, y)$. The level curves of $T$ are called isothermals because at all points on such a curve the temperature is the same. Sketch some isothermals if the temperature function is given by

$$
T(x, y)=\frac{100}{1+x^2+2 y^2}
$$

Deven Gill
Deven Gill
Numerade Educator
02:50

Problem 54

If $V(x, y)$ is the electric potential at a point $(x, y)$ in the $x y$-plane, then the level curves of $V$ are called equipotential curves because at all points on such a curve the electric potential is the same. Sketch some equipotential curves if $V(x, y)=c / \sqrt{r^2-x^2-y^2}$, where $c$ is a positive constant.

James Kiss
James Kiss
Numerade Educator
00:41

Problem 55

Use a computer to graph the function using various domains and viewpoints. Get a printout of one that, in your opinion, gives a good view. If your software also produces level curves, then plot some contour lines of the same function and compare with the graph.
$f(x, y)=x y^2-x^3 \quad$ (monkey saddle)

Adithya Ramanujam
Adithya Ramanujam
Numerade Educator
02:29

Problem 56

Use a computer to graph the function using various domains and viewpoints. Get a printout of one that, in your opinion, gives a good view. If your software also produces level curves, then plot some contour lines of the same function and compare with the graph.
$f(x, y)=x y^3-y x^3 \quad$ (dog saddle)

Deven Gill
Deven Gill
Numerade Educator
00:31

Problem 57

Use a computer to graph the function using various domains and viewpoints. Get a printout of one that, in your opinion, gives a good view. If your software also produces level curves, then plot some contour lines of the same function and compare with the graph.
$f(x, y)=e^{-\left(x^2+y^2\right) / 3}\left(\sin \left(x^2\right)+\cos \left(y^2\right)\right)$

ag
Alan Ghazarians
Numerade Educator
00:20

Problem 58

Use a computer to graph the function using various domains and viewpoints. Get a printout of one that, in your opinion, gives a good view. If your software also produces level curves, then plot some contour lines of the same function and compare with the graph.
$f(x, y)=\cos x \cos y$

ag
Alan Ghazarians
Numerade Educator

Problem 59

Match the function (a) with its graph (labeled A-F below) and (b) with its contour map (labeled I-VI). Give reasons for your choices.
$z=\sin (x y)$

Check back soon!
00:39

Problem 60

Match the function (a) with its graph (labeled A-F below) and (b) with its contour map (labeled I-VI). Give reasons for your choices.
$z=e^x \cos y$

ag
Alan Ghazarians
Numerade Educator

Problem 61

Match the function (a) with its graph (labeled A-F below) and (b) with its contour map (labeled I-VI). Give reasons for your choices.
$z=\sin (x-y)$

Check back soon!

Problem 62

Match the function (a) with its graph (labeled A-F below) and (b) with its contour map (labeled I-VI). Give reasons for your choices.
$z=\sin x-\sin y$

Check back soon!
00:26

Problem 63

Match the function (a) with its graph (labeled A-F below) and (b) with its contour map (labeled I-VI). Give reasons for your choices.
$z=\left(1-x^2\right)\left(1-y^2\right)$

ag
Alan Ghazarians
Numerade Educator
01:12

Problem 64

Match the function (a) with its graph (labeled A-F below) and (b) with its contour map (labeled I-VI). Give reasons for your choices.
$z=\frac{x-y}{1+x^2+y^2}$

ag
Alan Ghazarians
Numerade Educator
00:50

Problem 65

Describe the level surfaces of the function.
$f(x, y, z)=x+3 y+5 z$

Farnood Ensan
Farnood Ensan
Numerade Educator
02:32

Problem 66

Describe the level surfaces of the function.
$f(x, y, z)=x^2+3 y^2+5 z^2$

Jacquelyn Trost
Jacquelyn Trost
Numerade Educator

Problem 67

Describe the level surfaces of the function.
$f(x, y, z)=y^2+z^2$

Check back soon!
01:05

Problem 68

Describe the level surfaces of the function.
$f(x, y, z)=x^2-y^2-z^2$

Farnood Ensan
Farnood Ensan
Numerade Educator
01:38

Problem 69

Describe how the graph of $g$ is obtained from the graph of $f$.
(a) $g(x, y)=f(x, y)+2$
(b) $g(x, y)=2 f(x, y)$
(c) $g(x, y)=-f(x, y)$
(d) $g(x, y)=2-f(x, y)$

Farnood Ensan
Farnood Ensan
Numerade Educator
01:32

Problem 70

Describe how the graph of $g$ is obtained from the graph of $f$.
(a) $g(x, y)=f(x-2, y)$
(b) $g(x, y)=f(x, y+2)$
(c) $g(x, y)=f(x+3, y-4)$

Lauren Shelton
Lauren Shelton
Numerade Educator
00:45

Problem 71

Use a computer to graph the function using various domains and viewpoints. Get a printout that gives a good view of the "peaks and valleys." Would you say the function has a maximum value? Can you identify any points on the graph that you might consider to be "local maximum points"? What about "local minimum points"?
$f(x, y)=3 x-x^4-4 y^2-10 x y$

ag
Alan Ghazarians
Numerade Educator
01:09

Problem 72

Use a computer to graph the function using various domains and viewpoints. Get a printout that gives a good view of the "peaks and valleys." Would you say the function has a maximum value? Can you identify any points on the graph that you might consider to be "local maximum points"? What about "local minimum points"?
$f(x, y)=x y e^{-x^2-y^2}$

Christian Otero
Christian Otero
Numerade Educator
02:19

Problem 73

Use a computer to graph the function using various domains and viewpoints. Comment on the limiting behavior of the function. What happens as both $x$ and $y$ become large? What happens as $(x, y)$ approaches the origin?
$f(x, y)=\frac{x+y}{x^2+y^2}$

Christian Otero
Christian Otero
Numerade Educator
01:55

Problem 74

Use a computer to graph the function using various domains and viewpoints. Comment on the limiting behavior of the function. What happens as both $x$ and $y$ become large? What happens as $(x, y)$ approaches the origin?
$f(x, y)=\frac{x y}{x^2+y^2}$

Nick Johnson
Nick Johnson
Numerade Educator
02:36

Problem 75

Use a computer to investigate the family of functions $f(x, y)=e^{-x^2+y^2}$. How does the shape of the graph depend on $c$ ?

James Kiss
James Kiss
Numerade Educator
02:19

Problem 76

Use a computer to investigate the family of surfaces

$$
z=\left(a x^2+b y^2\right) e^{-x^2-y^2}
$$

How does the shape of the graph depend on the numbers $a$ and $b$ ?

Adithya Ramanujam
Adithya Ramanujam
Numerade Educator
01:34

Problem 77

Use a computer to investigate the family of surfaces $z=x^2+y^2+c x y$. In particular, you should determine the transitional values of $c$ for which the surface changes from one type of quadric surface to another.

ag
Alan Ghazarians
Numerade Educator
01:08

Problem 78

Graph the functions

$$
\begin{aligned}
& f(x, y)=\sqrt{x^2+y^2} \\
& f(x, y)=e^{\sqrt{x^2+y^2}}
\end{aligned}
$$

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

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

and

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

In general, if $g$ is a function of one variable, how is the graph of

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

obtained from the graph of $g$ ?

Carson Merrill
Carson Merrill
Numerade Educator
01:08

Problem 79

(a) Show that, by taking logarithms, the general CobbDouglas function $P=b L^a K^{1-\alpha}$ can be expressed as

$$
\ln \frac{P}{K}=\ln b+\alpha \ln \frac{L}{K}
$$

(b) If we let $x=\ln (L / K)$ and $y=\ln (P / K)$, the equation in part (a) becomes the linear equation $y=\alpha x+\ln b$. Use Table 2 (in Example 3) to make a table of values of $\ln (L / K)$ and $\ln (P / K)$ for the years 1899-1922. Then use a graphing calculator or computer to find the least squares regression line through the points $(\ln (L / K), \ln (P / K))$.
(c) Deduce that the Cobb-Douglas production function is $P=1.01 L^{0.75} K^{0.25}$.

Carson Merrill
Carson Merrill
Numerade Educator