Single Variable Calculus: Early Transcendentals

James Stewart, Daniel K. Clegg, Saleem Watson, Lothar Redlin

Chapter 14

Partial Derivatives - all with Video Answers

Educators

Section 1

Functions of Several Variables

Problem 1

If $f(x, y)=x^2 y /\left(2 x-y^2\right)$, find
(a) $f(1,3)$
(b) $f(-2,-1)$
(c) $f(x+h, y)$
(d) $f(x, x)$

James Kiss

James Kiss

Numerade Educator

Problem 2

If $g(x, y)=x \sin y+y \sin x$, find
(a) $g(\pi, 0)$
(b) $g(\pi / 2, \pi / 4)$
(c) $g(0, y)$
(d) $g(x, y+h)$

James Kiss

Numerade Educator

Problem 3

Let $g(x, y)=x^2 \ln (x+y)$.
(a) Evaluate $g(3,1)$.
(b) Find and sketch the domain of $g$.
(c) Find the range of $g$.

James Kiss

Numerade Educator

Problem 4

Let $h(x, y)=e^{\sqrt{y-x^2}}$.
(a) Evaluate $h(-2,5)$.
(b) Find and sketch the domain of $h$.
(c) Find the range of $h$.

James Kiss

Numerade Educator

Problem 5

Let $F(x, y, z)=\sqrt{y}-\sqrt{x-2 z}$.
(a) Evaluate $F(3,4,1)$.
(b) Find and describe the domain of $F$.

James Kiss

Numerade Educator

Problem 6

Let $f(x, y, z)=\ln \left(z-\sqrt{x^2+y^2}\right)$.
(a) Evaluate $f(4,-3,6)$.
(b) Find and describe the domain of $f$.

James Kiss

Numerade Educator

Problem 7

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

ag

Alan Ghazarians

Numerade Educator

Problem 8

Find and sketch the domain of the function.
$f(x, y)=\sqrt[4]{x-3 y}$

ag

Alan Ghazarians

Numerade Educator

Problem 9

Find and sketch the domain of the function.
$q(x, y)=\sqrt{x}+\sqrt{4-4 x^2-y^2}$

James Kiss

Numerade Educator

Problem 10

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

James Kiss

Numerade Educator

Problem 11

Find and sketch the domain of the function.
$g(x, y)=\frac{x-y}{x+y}$

Kian Manafi

Numerade Educator

Problem 12

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

Kian Manafi

Numerade Educator

Problem 13

Find and sketch the domain of the function.
$p(x, y)=\frac{\sqrt{x y}}{x+1}$

James Kiss

Numerade Educator

Problem 14

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

ag

Alan Ghazarians

Numerade Educator

Problem 15

Find and sketch the domain of the function.
$f(x, y, z)=\sqrt{4-x^2}+\sqrt{9-y^2}+\sqrt{1-z^2}$

ag

Alan Ghazarians

Numerade Educator

Problem 16

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

Problem 17

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

Problem 18

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

Problem 19

In Example 3 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 $f(T, 20)=-49$ ?" 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.

James Kiss

Numerade Educator

Problem 20

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.

Table 3 can't copy Apparent temperature as a function of temperature and humidity
(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 $h$.

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Bcrypt_Sha256$$2B$12$8Zlyrq/Mry7Neqzpqy/Nau80Tlrvo9Bu3Mbvzip.Gbblju6Hggsfy Bcrypt_Sha256$$2B$12$8Zlyrq/Mry7Neqzpqy/Nauq4.Oroa01Tmuxwldd7Bnvez4Kszcu..

Numerade Educator

Problem 21

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.
Table 4 can't copy Wave height as a function of wind speed and duration

Christian Otero

Christian Otero

Numerade Educator

Problem 22

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

Numerade Educator

Problem 23

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

Ashley Boni

Numerade Educator

Problem 24

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

Abigail Martyr

Numerade Educator

Problem 25

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

Linda Hand

Numerade Educator

Problem 26

Sketch the graph of the function.
$f(x, y)=\cos y$

Khushbu Rani

Numerade Educator

Problem 27

Sketch the graph of the function.
$f(x, y)=\sin x$

Khushbu Rani

Numerade Educator

Problem 28

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

Abigail Martyr

Numerade Educator

Problem 29

Sketch the graph of the function.
$f(x, y)=x^2+4 y^2+1$

ag

Alan Ghazarians

Numerade Educator

Problem 30

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

Vikash Ranjan

Numerade Educator

Problem 31

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

Vikash Ranjan

Numerade Educator

Problem 32

Match the function with its graph (labeled I-VI). Give reasons for your choices.
(a) $f(x, y)=\frac{1}{1+x^2+y^2}$
(b) $f(x, y)=\frac{1}{1+x^2 y^2}$
(c) $f(x, y)=\ln \left(x^2+y^2\right)$
(d) $f(x, y)=\cos \sqrt{x^2+y^2}$
(e) $f(x, y)=|x y|$
(f) $f(x, y)=\cos (x y)$
Graph I-VI can't copy

James Kiss

Numerade Educator

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?
Graph can't copy

Lucas Finney

Numerade Educator

Problem 34

Shown is a contour map of atmospheric pressure in North America on a particular day. On the level curves (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? (See the discussion preceding Example 9.)
Figure can't copy

James Kiss

Numerade Educator

Problem 35

Level curves (isothermals) are shown for the typical water temperature (in ${ }^{\circ} \mathrm{C}$ ) in Long Lake (Minnesota) 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 .
Figure can't copy

ag

Alan Ghazarians

Numerade Educator

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?
Graph I-II can't copy

Lucas Finney

Numerade Educator

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

Problem 38

Make a rough sketch of a contour map for the function whose graph is shown.
Graph can't copy

Lucas Finney

Numerade Educator

Problem 39

The body mass index (BMI) of a person is defined by

$$
B(m, h)=\frac{m}{h^2}
$$

where $m$ is the person's mass (in kilograms) and $h$ is the person's height (in meters). Draw the level curves $B(m, h)=18.5, B(m, h)=25, B(m, h)=30$, and $B(m, h)=40$. A rough guideline is that a person is underweight if the BMI is less than 18.5 ; optimal if the BMI lies between 18.5 and 25 ; overweight if the BMI lies between 25 and 30; and obese if the BMI exceeds 30 . Shade the region corresponding to optimal BMI. Does someone who weighs 62 kg and is 152 cm tall fall into the optimal category?

Bobby Barnes

University of North Texas

Problem 40

The body mass index is defined in Exercise 39. Draw the level curve of this function corresponding to someone who is 200 cm tall and weighs 80 kg . Find the weights and heights of two other people with that same level curve.

ag

Alan Ghazarians

Numerade Educator

Problem 41

A contour map of a function is shown. Use it to make a rough sketch of the graph of $f$.
Graph can't copy

Carson Merrill

Numerade Educator

Problem 42

A contour map of a function is shown. Use it to make a rough sketch of the graph of $f$.
Graph can't copy

Carson Merrill

Numerade Educator

Problem 43

A contour map of a function is shown. Use it to make a rough sketch of the graph of $f$.
Graph can't copy

Carson Merrill

Numerade Educator

Problem 44

A contour map of a function is shown. Use it to make a rough sketch of the graph of $f$.
Graph can't copy

Carson Merrill

Numerade Educator

Problem 45

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

Ashley Boni

Numerade Educator

Problem 46

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

ag

Alan Ghazarians

Numerade Educator

Problem 47

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

Check back soon!

Problem 48

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

ag

Alan Ghazarians

Numerade Educator

Problem 49

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

Amrita Bhasin

Numerade Educator

Problem 50

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

ag

Alan Ghazarians

Numerade Educator

Problem 51

Draw a contour map of the function showing several level curves.
$f(x, y)=\sqrt[3]{x^2+y^2}$

Check back soon!

Problem 52

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

ag

Alan Ghazarians

Numerade Educator

Problem 53

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

ag

Alan Ghazarians

Numerade Educator

Problem 54

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

ag

Alan Ghazarians

Numerade Educator

Problem 55

A thin metal plate, located in the $x y$-plane, has temperature $T(x, y)$ at the point $(x, y)$. Sketch some level curves (isothermals) if the temperature function is given by

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

James Kiss

Numerade Educator

Problem 56

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

Numerade Educator

Problem 57

Graph the function using various domains and viewpoints. If your software also produces level curves, then plot some contour lines of the same function and compare with the graph.
$f(x, y)=x y^2-x^3$ (monkey saddle)

James Kiss

Numerade Educator

Problem 58

Graph the function using various domains and viewpoints. If your software also produces level curves, then plot some contour lines of the same function and compare with the graph.
$f(x, y)=x y^3-y x^3$ (dog saddle)

James Kiss

Numerade Educator

Problem 59

Graph the function using various domains and viewpoints. If your software also produces level curves, then plot some contour lines of the same function and compare with the graph.
$f(x, y)=e^{-\left(x^2+y^2\right) / 3}\left(\sin \left(x^2\right)+\cos \left(y^2\right)\right)$

James Kiss

Numerade Educator

Problem 60

Graph the function using various domains and viewpoints. If your software also produces level curves, then plot some contour lines of the same function and compare with the graph.
$f(x, y)=\cos x \cos y$

James Kiss

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.
$z=\sin (x y)$

Graph A-F can't copy
Graph I-VI can't copy

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.
$z=e^x \cos y$

Graph A-F can't copy
Graph I-VI can't copy

Lucas Finney

Numerade Educator

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.
$z=\sin (x-y)$

Graph A-F can't copy
Graph I-VI can't copy

Check back soon!

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.
$z=\sin x-\sin y$

Graph A-F can't copy
Graph I-VI can't copy

Ashley Boni

Numerade Educator

Problem 65

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.
$z=\left(1-x^2\right)\left(1-y^2\right)$

Graph A-F can't copy
Graph I-VI can't copy

Check back soon!

Problem 66

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.
$z=\frac{x-y}{1+x^2+y^2}$

Graph A-F can't copy
Graph I-VI can't copy

ag

Alan Ghazarians

Numerade Educator

Problem 67

Describe the level surfaces of the function.
$f(x, y, z)=2 y-z+1$

James Kiss

Numerade Educator

Problem 68

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

James Kiss

Numerade Educator

Problem 69

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

James Kiss

Numerade Educator

Problem 70

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

James Kiss

Numerade Educator

Problem 71

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

Numerade Educator

Problem 72

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) $g(x, y)=f(x+3, y-4)$

ag

Alan Ghazarians

Numerade Educator

Problem 73

Graph the function using various domains and viewpoints that give good views 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"?
$f(x, y)=3 x-x^4-4 y^2-10 x y$

Nick Johnson

Numerade Educator

Problem 74

Graph the function using various domains and viewpoints that give good views 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"?
$f(x, y)=x y e^{-x^2-y^2}$

Nick Johnson

Numerade Educator

Problem 75

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

ag

Alan Ghazarians

Numerade Educator

Problem 76

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

ag

Alan Ghazarians

Numerade Educator

Problem 77

Investigate the family of functions $f(x, y)=e^{c x^2+y^2}$. How does the shape of the graph depend on $c$ ?

James Kiss

Numerade Educator

Problem 78

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 $b$ ?

Nick Johnson

Numerade Educator

Problem 79

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.

Nick Johnson

Numerade Educator

Problem 80

Graph the functions

$$
\begin{aligned}
& f(x, y)=\sqrt{x^2+y^2} \\
& f(x, y)=e^{\sqrt{x^2+y^2}} \\
& f(x, y)=\ln \sqrt{x^2+y^2} \\
& f(x, y)=\sin \left(\sqrt{x^2+y^2}\right) \\
& f(x, y)=\frac{1}{\sqrt{x^2+y^2}}
\end{aligned}
$$

and

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

Numerade Educator

Problem 81

(a) Show that, by taking logarithms, the general CobbDouglas function $P=b L^a K^{1-a}$ 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 4) to make a table of values of $\ln (L / K)$ and $\ln (P / K)$ for the years 1899-1922. Then 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

Numerade Educator