Calculus and Its Applications

Marvin L. Bittinger, David J. Ellenbogen, Scott A. Surgent

Chapter 2

Applications of Differentiation - all with Video Answers

Educators

Section 1

Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs

Problem 1

Find the relative extrema of each function, if they exist. List each extremum along with the $x$ -value at which it occurs. Then sketch a graph of the function.
$$f(x)=x^{2}+4 x+5$$

Joseph Liao

Joseph Liao

Numerade Educator

Problem 2

Find the relative extrema of each function, if they exist. List each extremum along with the $x$ -value at which it occurs. Then sketch a graph of the function.
$$f(x)=x^{2}+6 x-3$$

Joseph Liao

Numerade Educator

Problem 3

Find the relative extrema of each function, if they exist. List each extremum along with the $x$ -value at which it occurs. Then sketch a graph of the function.
$$f(x)=5-x-x^{2}$$

Joseph Liao

Numerade Educator

Problem 4

Find the relative extrema of each function, if they exist. List each extremum along with the $x$ -value at which it occurs. Then sketch a graph of the function.
$$f(x)=2-3 x-2 x^{2}$$

Joseph Liao

Numerade Educator

Problem 5

Find the relative extrema of each function, if they exist. List each extremum along with the $x$ -value at which it occurs. Then sketch a graph of the function.
$$g(x)=1+6 x+3 x^{2}$$

Joseph Liao

Numerade Educator

Problem 6

Find the relative extrema of each function, if they exist. List each extremum along with the $x$ -value at which it occurs. Then sketch a graph of the function.
$$F(x)=0.5 x^{2}+2 x-11$$

Joseph Liao

Numerade Educator

Problem 7

Find the relative extrema of each function, if they exist. List each extremum along with the $x$ -value at which it occurs. Then sketch a graph of the function.
$$G(x)=x^{3}-x^{2}-x+2$$

Joseph Liao

Numerade Educator

Problem 8

Find the relative extrema of each function, if they exist. List each extremum along with the $x$ -value at which it occurs. Then sketch a graph of the function.
$$g(x)=x^{3}+\frac{1}{2} x^{2}-2 x+5$$

Joseph Liao

Numerade Educator

Problem 9

Find the relative extrema of each function, if they exist. List each extremum along with the $x$ -value at which it occurs. Then sketch a graph of the function.
$$f(x)=x^{3}-3 x+6$$

Joseph Liao

Numerade Educator

Problem 10

Find the relative extrema of each function, if they exist. List each extremum along with the $x$ -value at which it occurs. Then sketch a graph of the function.
$$f(x)=x^{3}-3 x^{2}$$

Joseph Liao

Numerade Educator

Problem 11

Find the relative extrema of each function, if they exist. List each extremum along with the $x$ -value at which it occurs. Then sketch a graph of the function.
$$f(x)=3 x^{2}+2 x^{3}$$

Joseph Liao

Numerade Educator

Problem 12

Find the relative extrema of each function, if they exist. List each extremum along with the $x$ -value at which it occurs. Then sketch a graph of the function.
$$f(x)=x^{3}+3 x$$

Joseph Liao

Numerade Educator

Problem 13

Find the relative extrema of each function, if they exist. List each extremum along with the $x$ -value at which it occurs. Then sketch a graph of the function.
$$g(x)=2 x^{3}-16$$

Joseph Liao

Numerade Educator

Problem 14

Find the relative extrema of each function, if they exist. List each extremum along with the $x$ -value at which it occurs. Then sketch a graph of the function.
$$F(x)=1-x^{3}$$

Joseph Liao

Numerade Educator

Problem 15

Find the relative extrema of each function, if they exist. List each extremum along with the $x$ -value at which it occurs. Then sketch a graph of the function.
$$G(x)=x^{3}-6 x^{2}+10$$

Joseph Liao

Numerade Educator

Problem 16

Find the relative extrema of each function, if they exist. List each extremum along with the $x$ -value at which it occurs. Then sketch a graph of the function.
$$f(x)=12+9 x-3 x^{2}-x^{3}$$

Joseph Liao

Numerade Educator

Problem 17

Find the relative extrema of each function, if they exist. List each extremum along with the $x$ -value at which it occurs. Then sketch a graph of the function.
$$g(x)=x^{3}-x^{4}$$

Joseph Liao

Numerade Educator

Problem 18

Find the relative extrema of each function, if they exist. List each extremum along with the $x$ -value at which it occurs. Then sketch a graph of the function.
$$f(x)=x^{4}-2 x^{3}$$

Joseph Liao

Numerade Educator

Problem 19

Find the relative extrema of each function, if they exist. List each extremum along with the $x$ -value at which it occurs. Then sketch a graph of the function.
$$f(x)=\frac{1}{3} x^{3}-2 x^{2}+4 x-1$$

Joseph Liao

Numerade Educator

Problem 20

Find the relative extrema of each function, if they exist. List each extremum along with the $x$ -value at which it occurs. Then sketch a graph of the function.
$$F(x)=-\frac{1}{5} x^{3}+3 x^{2}-9 x+2$$

Joseph Liao

Numerade Educator

Problem 21

Find the relative extrema of each function, if they exist. List each extremum along with the $x$ -value at which it occurs. Then sketch a graph of the function.
$$g(x)=2 x^{4}-20 x^{2}+18$$

Joseph Liao

Numerade Educator

Problem 22

Find the relative extrema of each function, if they exist. List each extremum along with the $x$ -value at which it occurs. Then sketch a graph of the function.
$$f(x)=3 x^{4}-15 x^{2}+12$$

Joseph Liao

Numerade Educator

Problem 23

Find the relative extrema of each function, if they exist. List each extremum along with the $x$ -value at which it occurs. Then sketch a graph of the function.
$$F(x)=\sqrt[3]{x-1}$$

Joseph Liao

Numerade Educator

Problem 24

Find the relative extrema of each function, if they exist. List each extremum along with the $x$ -value at which it occurs. Then sketch a graph of the function.
$$G(x)=\sqrt[3]{x+2}$$

Joseph Liao

Numerade Educator

Problem 25

Find the relative extrema of each function, if they exist. List each extremum along with the $x$ -value at which it occurs. Then sketch a graph of the function.
$$f(x)=1-x^{2 / 3}$$

Joseph Liao

Numerade Educator

Problem 26

Find the relative extrema of each function, if they exist. List each extremum along with the $x$ -value at which it occurs. Then sketch a graph of the function.
$$f(x)=(x+3)^{2 / 3}-5$$

Joseph Liao

Numerade Educator

Problem 27

Find the relative extrema of each function, if they exist. List each extremum along with the $x$ -value at which it occurs. Then sketch a graph of the function.
$$G(x)=\frac{-8}{x^{2}+1}$$

Joseph Liao

Numerade Educator

Problem 28

Find the relative extrema of each function, if they exist. List each extremum along with the $x$ -value at which it occurs. Then sketch a graph of the function.
$$F(x)=\frac{5}{x^{2}+1}$$

Joseph Liao

Numerade Educator

Problem 29

Find the relative extrema of each function, if they exist. List each extremum along with the $x$ -value at which it occurs. Then sketch a graph of the function.
$$g(x)=\frac{4 x}{x^{2}+1}$$

Joseph Liao

Numerade Educator

Problem 30

Find the relative extrema of each function, if they exist. List each extremum along with the $x$ -value at which it occurs. Then sketch a graph of the function.
$$g(x)=\frac{x^{2}}{x^{2}+1}$$

Joseph Liao

Numerade Educator

Problem 31

Find the relative extrema of each function, if they exist. List each extremum along with the $x$ -value at which it occurs. Then sketch a graph of the function.
$$f(x)=\sqrt[3]{x}$$

Joseph Liao

Numerade Educator

Problem 32

Find the relative extrema of each function, if they exist. List each extremum along with the $x$ -value at which it occurs. Then sketch a graph of the function.
$$f(x)=(x+1)^{1 / 3}$$

Joseph Liao

Numerade Educator

Problem 33

Find the relative extrema of each function, if they exist. List each extremum along with the $x$ -value at which it occurs. Then sketch a graph of the function.
$$g(x)=\sqrt{x^{2}+2 x+5}$$

Joseph Liao

Numerade Educator

Problem 34

Find the relative extrema of each function, if they exist. List each extremum along with the $x$ -value at which it occurs. Then sketch a graph of the function.
$$F(x)=\frac{1}{\sqrt{x^{2}+1}}$$

Joseph Liao

Numerade Educator

Problem 35

Check the results of Exercises $1-34$ using a calculator.

Joseph Liao

Numerade Educator

Problem 36

Check the results of Exercises $1-34$ using a calculator.

Joseph Liao

Numerade Educator

Problem 37

Check the results of Exercises $1-34$ using a calculator.

Joseph Liao

Numerade Educator

Problem 38

Check the results of Exercises $1-34$ using a calculator.

Joseph Liao

Numerade Educator

Problem 39

Check the results of Exercises $1-34$ using a calculator.

Joseph Liao

Numerade Educator

Problem 40

Check the results of Exercises $1-34$ using a calculator.

Joseph Liao

Numerade Educator

Problem 41

Check the results of Exercises $1-34$ using a calculator.

Joseph Liao

Numerade Educator

Problem 42

Check the results of Exercises $1-34$ using a calculator.

Joseph Liao

Numerade Educator

Problem 43

Check the results of Exercises $1-34$ using a calculator.

Joseph Liao

Numerade Educator

Problem 44

Check the results of Exercises $1-34$ using a calculator.

Joseph Liao

Numerade Educator

Problem 45

Check the results of Exercises $1-34$ using a calculator.

Joseph Liao

Numerade Educator

Problem 46

Check the results of Exercises $1-34$ using a calculator.

Joseph Liao

Numerade Educator

Problem 47

Check the results of Exercises $1-34$ using a calculator.

Joseph Liao

Numerade Educator

Problem 48

Check the results of Exercises $1-34$ using a calculator.

Joseph Liao

Numerade Educator

Problem 49

Check the results of Exercises $1-34$ using a calculator.

Joseph Liao

Numerade Educator

Problem 50

Check the results of Exercises $1-34$ using a calculator.

Joseph Liao

Numerade Educator

Problem 51

Check the results of Exercises $1-34$ using a calculator.

Joseph Liao

Numerade Educator

Problem 52

Check the results of Exercises $1-34$ using a calculator.

Joseph Liao

Numerade Educator

Problem 53

Check the results of Exercises $1-34$ using a calculator.

Joseph Liao

Numerade Educator

Problem 54

Check the results of Exercises $1-34$ using a calculator.

Joseph Liao

Numerade Educator

Problem 55

Check the results of Exercises $1-34$ using a calculator.

Joseph Liao

Numerade Educator

Problem 56

Check the results of Exercises $1-34$ using a calculator.

Joseph Liao

Numerade Educator

Problem 57

Check the results of Exercises $1-34$ using a calculator.

Joseph Liao

Numerade Educator

Problem 58

Check the results of Exercises $1-34$ using a calculator.

Joseph Liao

Numerade Educator

Problem 59

Check the results of Exercises $1-34$ using a calculator.

Joseph Liao

Numerade Educator

Problem 60

Check the results of Exercises $1-34$ using a calculator.

Joseph Liao

Numerade Educator

Problem 61

Check the results of Exercises $1-34$ using a calculator.

Joseph Liao

Numerade Educator

Problem 62

Check the results of Exercises $1-34$ using a calculator.

Joseph Liao

Numerade Educator

Problem 63

Check the results of Exercises $1-34$ using a calculator.

Joseph Liao

Numerade Educator

Problem 64

Check the results of Exercises $1-34$ using a calculator.

Joseph Liao

Numerade Educator

Problem 65

Check the results of Exercises $1-34$ using a calculator.

Joseph Liao

Numerade Educator

Problem 66

Check the results of Exercises $1-34$ using a calculator.

Joseph Liao

Numerade Educator

Problem 67

Check the results of Exercises $1-34$ using a calculator.

Joseph Liao

Numerade Educator

Problem 68

Check the results of Exercises $1-34$ using a calculator.

Joseph Liao

Numerade Educator

Problem 69

Draw a graph to match the description given. Answers will vary.
$f(x)$ is increasing over $(-\infty, 2)$ and decreasing over $(2, \infty)$

Joseph Liao

Numerade Educator

Problem 70

Draw a graph to match the description given. Answers will vary.
$g(x)$ is decreasing over $(-\infty,-3)$ and increasing over $(-3, \infty)$

Joseph Liao

Numerade Educator

Problem 71

Draw a graph to match the description given. Answers will vary.
$G(x)$ is decreasing over $(-\infty, 4)$ and $(9, \infty)$ and increasing over (4,9)

Joseph Liao

Numerade Educator

Problem 72

Draw a graph to match the description given. Answers will vary.

Joseph Liao

Numerade Educator

Problem 73

Draw a graph to match the description given. Answers will vary.
$g(x)$ has a positive derivative over $(-\infty,-3)$ and a negative derivative over $(-3, \infty)$

Joseph Liao

Numerade Educator

Problem 74

Draw a graph to match the description given. Answers will vary.
$f(x)$ has a negative derivative over $(-\infty, 1)$ and a positive derivative over $(1, \infty)$

Joseph Liao

Numerade Educator

Problem 75

Draw a graph to match the description given. Answers will vary.
$F(x)$ has a negative derivative over $(-\infty, 2)$ and (5,9) and a positive derivative over (2,5) and $(9, \infty)$

Joseph Liao

Numerade Educator

Problem 76

Draw a graph to match the description given. Answers will vary.
$G(x)$ has a positive derivative over $(-\infty,-2)$ and
(4,7) and a negative derivative over (-2,4) and
$(7, \infty)$

Joseph Liao

Numerade Educator

Problem 77

Draw a graph to match the description given. Answers will vary.
$f(x)$ has a positive derivative over $(-\infty, 3)$ and (3,9) a negative derivative over $(9, \infty),$ and a derivative equal to 0 at $x=3$

Joseph Liao

Numerade Educator

Problem 78

Draw a graph to match the description given. Answers will vary.
$g(x)$ has a negative derivative over $(-\infty, 5)$ and (5,8) a positive derivative over $(8, \infty),$ and a derivative equal to 0 at $x=5$

Joseph Liao

Numerade Educator

Problem 79

Draw a graph to match the description given. Answers will vary.
$F(x)$ has a negative derivative over $(-\infty,-1)$ and a positive derivative over $(-1, \infty),$ and $F^{\prime}(-1)$ does not exist.

Joseph Liao

Numerade Educator

Problem 80

Draw a graph to match the description given. Answers will vary.
$G(x)$ has a positive derivative over $(-\infty, 0)$ and $(3, \infty)$ and a negative derivative over $(0,3),$ but neither $G^{\prime}(0)$ nor $G^{\prime}(3)$ exists.

Joseph Liao

Numerade Educator

Problem 81

Draw a graph to match the description given. Answers will vary.
$f(x)$ has a negative derivative over $(-\infty,-2)$ and
$(1, \infty)$ and a positive derivative over $(-2,1),$ and $f^{\prime}(-2)=0,$ but $f^{\prime}(1)$ does not exist.

Joseph Liao

Numerade Educator

Problem 82

Draw a graph to match the description given. Answers will vary.
$g(x)$ has a positive derivative over $(-\infty,-3)$ and
$(0,3),$ a negative derivative over (-3,0) and $(3, \infty)$ and a derivative equal to 0 at $x=-3$ and $x=3,$ but $g^{\prime}(0)$ does not exist.

Joseph Liao

Numerade Educator

Problem 83

Draw a graph to match the description given. Answers will vary.
$H(x)$ is increasing over $(-\infty, \infty),$ but the derivative does not exist at $x=1$

Joseph Liao

Numerade Educator

Problem 84

Draw a graph to match the description given. Answers will vary.
$K(x)$ is decreasing over $(-\infty, \infty),$ but the derivative does not exist at $x=0$ and $x=2$

Joseph Liao

Numerade Educator

Problem 85

Consider this graph.
Explain the idea of a critical value. Then determine which $x$ -values are critical values, and state why.
(GRAPH CAN'T COPY)

Joseph Liao

Numerade Educator

Problem 86

Consider this graph.
Using the graph and the intervals noted, explain how to relate the concept of the function being increasing or decreasing to the first derivative.
(GRAPH CAN'T COPY)

Joseph Liao

Numerade Educator

Problem 87

Employment. According to the U.S. Bureau of Labor Statistics, the number of professional services employees Fluctuated during the period $2000-2009,$ as modeled by $E(t)=-28.31 t^{3}+381.86 t^{2}-1162.07 t+16,905.87$
where $t$ is the number of years since $2000(t=0$ corresponds to 2000 ) and $E$ is thousands of employees. Find the relative extrema of this function, and sketch the graph. Interpret the meaning of the relative extrema.

Joseph Liao

Numerade Educator

Problem 88

Advertising. Brody Electronics estimates that it will sell N units of a new toy after spending $a$ thousands of dollars on advertising, where $N(a)=-a^{2}+300 a+6, \quad 0 \leq a \leq 300$
Find the relative extrema and sketch a graph of the function.

Joseph Liao

Numerade Educator

Problem 89

Temperature during an illness. The temperature of a person during an intestinal illness is given by $T(t)=-0.1 t^{2}+1.2 t+98.6, \quad 0 \leq t \leq 12$
where $T$ is the temperature $\left(^{\circ} \mathrm{F}\right)$ at time $t,$ in days. Find the relative extrema and sketch a graph of the function.

Joseph Liao

Numerade Educator

Problem 90

Solar eclipse. On January $15,2010,$ the longest annular solar eclipse until 3040 occurred over Africa and the Indian Ocean (in an annular eclipse, the sun is partially obscured by the moon and looks like a ring). The path of the full eclipse on the earth's surface is modeled by $f(x)=0.0125 x^{2}-1.157 x+22.864, \quad 15<x<90$
where $x$ is the number of degrees of longitude east of the prime meridian and $f(x)$ is the number of degrees of latitude north (positive) or south (negative) of the equator. (Source: NASA.) Find the longitude and latitude of the southernmost point at which the full eclipse could be viewed.
(IMAGE CAN'T COPY)

Joseph Liao

Numerade Educator

Problem 91

The graph of a derivative $f^{\prime}$ is shown. Use the information in each graph to determine where f is increasing or decreasing and the $x$ -values of any extrema. Then sketch a possible graph of $f$
(GRAPH CAN'T COPY)

Joseph Liao

Numerade Educator

Problem 92

The graph of a derivative $f^{\prime}$ is shown. Use the information in each graph to determine where f is increasing or decreasing and the $x$ -values of any extrema. Then sketch a possible graph of $f$
(GRAPH CAN'T COPY)

Joseph Liao

Numerade Educator

Problem 93

The graph of a derivative $f^{\prime}$ is shown. Use the information in each graph to determine where f is increasing or decreasing and the $x$ -values of any extrema. Then sketch a possible graph of $f$
(GRAPH CAN'T COPY)

Joseph Liao

Numerade Educator

Problem 94

The graph of a derivative $f^{\prime}$ is shown. Use the information in each graph to determine where f is increasing or decreasing and the $x$ -values of any extrema. Then sketch a possible graph of $f$
(GRAPH CAN'T COPY)

Joseph Liao

Numerade Educator

Problem 95

The graph of a derivative $f^{\prime}$ is shown. Use the information in each graph to determine where f is increasing or decreasing and the $x$ -values of any extrema. Then sketch a possible graph of $f$
(GRAPH CAN'T COPY)

Joseph Liao

Numerade Educator

Problem 96

The graph of a derivative $f^{\prime}$ is shown. Use the information in each graph to determine where f is increasing or decreasing and the $x$ -values of any extrema. Then sketch a possible graph of $f$
(GRAPH CAN'T COPY)

Joseph Liao

Numerade Educator

Problem 97

$$\text {Graph each function. Then estimate any relative extrema.}$$
$$\begin{aligned}
f(x)=&-x^{6}-4 x^{5}+54 x^{4}+160 x^{3}-641 x^{2} \\
&-828 x+1200
\end{aligned}$$

Joseph Liao

Numerade Educator

Problem 98

$$\text {Graph each function. Then estimate any relative extrema.}$$
$$f(x)=x^{4}+4 x^{3}-36 x^{2}-160 x+400$$

Joseph Liao

Numerade Educator

Problem 99

$$\text {Graph each function. Then estimate any relative extrema.}$$
$$f(x)=\sqrt[3]{\left|4-x^{2}\right|}+1 \quad \text { 100. } f(x)=x \sqrt{9-x^{2}}$$

Joseph Liao

Numerade Educator

Problem 100

$$\text {Graph each function. Then estimate any relative extrema.}$$
$$f(x)=x \sqrt{9-x^{2}}$$

Joseph Liao

Numerade Educator

Problem 101

Use your calculator's absolute-value feature to graph the follow. ing functions and determine relative extrema and intervals over which the function is increasing or decreasing. State the $x$ -values at which the derivative does not exist.
$$f(x)=|x-2|$$

Joseph Liao

Numerade Educator

Problem 102

Use your calculator's absolute-value feature to graph the follow. ing functions and determine relative extrema and intervals over which the function is increasing or decreasing. State the $x$ -values at which the derivative does not exist.
$$f(x)=|2 x-5|$$

Joseph Liao

Numerade Educator

Problem 103

Use your calculator's absolute-value feature to graph the follow. ing functions and determine relative extrema and intervals over which the function is increasing or decreasing. State the $x$ -values at which the derivative does not exist.
$$f(x)=\left|x^{2}-1\right|$$

Joseph Liao

Numerade Educator

Problem 104

Use your calculator's absolute-value feature to graph the follow. ing functions and determine relative extrema and intervals over which the function is increasing or decreasing. State the $x$ -values at which the derivative does not exist.
$$f(x)=\left|x^{2}-3 x+2\right|$$

Joseph Liao

Numerade Educator

Problem 105

Use your calculator's absolute-value feature to graph the follow. ing functions and determine relative extrema and intervals over which the function is increasing or decreasing. State the $x$ -values at which the derivative does not exist.
$$f(x)=\left|9-x^{2}\right|$$

Joseph Liao

Numerade Educator

Problem 106

Use your calculator's absolute-value feature to graph the follow. ing functions and determine relative extrema and intervals over which the function is increasing or decreasing. State the $x$ -values at which the derivative does not exist.
$$f(x)=\left|-x^{2}+4 x-4\right|$$

Joseph Liao

Numerade Educator

Problem 107

Use your calculator's absolute-value feature to graph the follow. ing functions and determine relative extrema and intervals over which the function is increasing or decreasing. State the $x$ -values at which the derivative does not exist.
$$f(x)=\left|x^{3}-1\right|$$

Joseph Liao

Numerade Educator

Problem 108

Use your calculator's absolute-value feature to graph the follow. ing functions and determine relative extrema and intervals over which the function is increasing or decreasing. State the $x$ -values at which the derivative does not exist.
$$f(x)=\left|x^{4}-2 x^{2}\right|$$

Joseph Liao

Numerade Educator

Problem 109

Life expectancy and daily caloric intake.
a) Use the regression procedures of Section R.6 to fit a cubic function $y=f(x)$ to the data in the table, where $x$ is daily caloric intake and $y$ is life expectancy. Then fit a quartic function and decide which fits best. Explain.
b) What is the domain of the function?
c) Does the function have any relative extrema? Explain.
(TABLE CAN'T COPY)

Carson Merrill

Numerade Educator

Problem 110

Infant mortality and daily caloric intake.
a) Use the regression procedures of Section R.6 to fit a cubic function $y=f(x)$ to the data in the table, where $x$ is daily caloric intake and $y$ is infant mortality. Then fit a quartic function and decide which fits best. Explain.
b) What is the domain of the function?
c) Does the function have any relative extrema? Explain.
(TABLE CAN'T COPY)

Carson Merrill

Numerade Educator

Problem 111

Describe a procedure that can be used to select an appropriate viewing window for the functions given in
(a) Exercises $1-16$ and (b) Exercises $97-100$

Carson Merrill

Numerade Educator