Applied Mathematics: For the Managerial, Life, and Social Sciences

Soo T. Tan

Chapter 10

Applications of the Derivative - all with Video Answers

Educators

Section 1

Applications of the First Derivative

Problem 1

You are given the graph of a function $f$ Determine the intervals where $f$ is increasing, constant, or decreasing.

Tyler Moulton

Tyler Moulton

Numerade Educator

Problem 2

You are given the graph of a function $f$ Determine the intervals where $f$ is increasing, constant, or decreasing.

Tyler Moulton

Numerade Educator

Problem 3

You are given the graph of a function $f$ Determine the intervals where $f$ is increasing, constant, or decreasing.

Tyler Moulton

Numerade Educator

Problem 4

You are given the graph of a function $f$ Determine the intervals where $f$ is increasing, constant, or decreasing.

Tyler Moulton

Numerade Educator

Problem 5

You are given the graph of a function $f$ Determine the intervals where $f$ is increasing, constant, or decreasing.

Tyler Moulton

Numerade Educator

Problem 6

You are given the graph of a function $f$ Determine the intervals where $f$ is increasing, constant, or decreasing.

Tyler Moulton

Numerade Educator

Problem 7

You are given the graph of a function $f$ Determine the intervals where $f$ is increasing, constant, or decreasing.

Tyler Moulton

Numerade Educator

Problem 8

You are given the graph of a function $f$ Determine the intervals where $f$ is increasing, constant, or decreasing.

Tyler Moulton

Numerade Educator

Problem 9

The graph of the function $f$ shown in the accompanying figure gives the elevation of that part of the Boston Marathon course that includes the notorious Heartbreak Hill. Determine the intervals (stretches of the course) where the function $f$ is increasing (the runner is laboring), where it is constant (the runner is taking a breather), and where it is decreasing (the runner is coasting).

Nicole Krahulik

Nicole Krahulik

Numerade Educator

Problem 10

Among the important factors in determining the structural integrity of an aircraft is its age. Advancing age makes planes more likely to crack. The graph of the function $f$, shown in the accompanying figure, is referred to as a "bathtub curve" in the airline industry. It gives the fleet damage rate (damage due to corrosion, accident, and metal fatigue) of a typical fleet of commercial aircraft as a function of the number of years of service.
a. Determine the interval where $f$ is decreasing. This corresponds to the time period when the fleet damage rate is dropping as problems are found and corrected during the initial "shakedown" period.
b. Determine the interval where $f$ is constant. After the initial shakedown period, planes have few structural problems, and this is reflected by the fact that the function is constant on this interval.
c. Determine the interval where $f$ is increasing. Beyond the time period mentioned in part (b), the function is increasing - reflecting an increase in structural defects due mainly to metal fatigue.

Nicole Krahulik

Nicole Krahulik

Numerade Educator

Problem 11

Refer to the following figure:
What is the sign of the following?
a. $f^{\prime}(2)$
b. $f^{\prime}(x)$ in the interval $(1,3)$
c. $f^{\prime}(4)$
d. $f^{\prime}(x)$ in the interval $(3,6)$
e. $f^{\prime}(7)$
f. $f^{\prime}(x)$ in the interval $(6,9)$
g. $f^{\prime}(x)$ in the interval $(9,12)$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 12

Refer to the following figure:
a. What are the critical numbers of $f$. Give reasons for your
answers.
b. Draw the sign diagram for $f^{\prime}$.
c. Find the relative extrema of $f$.

Nicole Krahulik

Nicole Krahulik

Numerade Educator

Problem 13

Find the interval(s) where the function is increasing and the interval(s) where it is decreasing.
$$
f(x)=3 x+5
$$

Nishant Tyagi

Numerade Educator

Problem 14

Find the interval(s) where the function is increasing and the interval(s) where it is decreasing.
$$
f(x)=4-5 x
$$

Nishant Tyagi

Numerade Educator

Problem 15

Find the interval(s) where the function is increasing and the interval(s) where it is decreasing.
$$
f(x)=x^{2}-3 x
$$

Nishant Tyagi

Numerade Educator

Problem 16

Find the interval(s) where the function is increasing and the interval(s) where it is decreasing.
$$
f(x)=2 x^{2}+x+1
$$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 17

Find the interval(s) where the function is increasing and the interval(s) where it is decreasing.
$$
g(x)=x-x^{3}
$$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 18

Find the interval(s) where the function is increasing and the interval(s) where it is decreasing.
$$
f(x)=x^{3}-3 x^{2}
$$

Jonathon Brumley

Jonathon Brumley

Numerade Educator

Problem 19

Find the interval(s) where the function is increasing and the interval(s) where it is decreasing.
$$
g(x)=x^{3}+3 x^{2}+1
$$

Nishant Tyagi

Numerade Educator

Problem 20

Find the interval(s) where the function is increasing and the interval(s) where it is decreasing.
$$
f(x)=x^{3}-3 x+4
$$

Tyler Moulton

Numerade Educator

Problem 21

Find the interval(s) where the function is increasing and the interval(s) where it is decreasing.
$$
f(x)=\frac{1}{3} x^{3}-3 x^{2}+9 x+20
$$

Nishant Tyagi

Numerade Educator

Problem 22

Find the interval(s) where the function is increasing and the interval(s) where it is decreasing.
$$
f(x)=\frac{2}{3} x^{3}-2 x^{2}-6 x-2
$$

Nishant Tyagi

Numerade Educator

Problem 23

Find the interval(s) where the function is increasing and the interval(s) where it is decreasing.
$$
h(x)=x^{4}-4 x^{3}+10
$$

Nishant Tyagi

Numerade Educator

Problem 24

Find the interval(s) where the function is increasing and the interval(s) where it is decreasing.
$$
g(x)=x^{4}-2 x^{2}+4
$$

Nishant Tyagi

Numerade Educator

Problem 25

Find the interval(s) where the function is increasing and the interval(s) where it is decreasing.
$$
f(x)=\frac{1}{x-2}
$$

Tyler Moulton

Numerade Educator

Problem 26

Find the interval(s) where the function is increasing and the interval(s) where it is decreasing.
$$
h(x)=\frac{1}{2 x+3}
$$

Tyler Moulton

Numerade Educator

Problem 27

Find the interval(s) where the function is increasing and the interval(s) where it is decreasing.
$$
h(t)=\frac{t}{t-1}
$$

Donna Densmore

Numerade Educator

Problem 28

Find the interval(s) where the function is increasing and the interval(s) where it is decreasing.
$$
g(t)=\frac{2 t}{t^{2}+1}
$$

Tyler Moulton

Numerade Educator

Problem 29

Find the interval(s) where the function is increasing and the interval(s) where it is decreasing.
$$
f(x)=x^{3 / 5}
$$

Tyler Moulton

Numerade Educator

Problem 30

Find the interval(s) where the function is increasing and the interval(s) where it is decreasing.
$$
f(x)=x^{2 / 3}+5
$$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 31

Find the interval(s) where the function is increasing and the interval(s) where it is decreasing.
$$
f(x)=\sqrt{x+1}
$$

Jonathon Brumley

Jonathon Brumley

Numerade Educator

Problem 32

Find the interval(s) where the function is increasing and the interval(s) where it is decreasing.
$$
f(x)=(x-5)^{2 / 3}
$$

Tyler Moulton

Numerade Educator

Problem 33

Find the interval(s) where the function is increasing and the interval(s) where it is decreasing.
$$
f(x)=\sqrt{16-x^{2}}
$$

Tyler Moulton

Numerade Educator

Problem 34

Find the interval(s) where the function is increasing and the interval(s) where it is decreasing.
$$
g(x)=x \sqrt{x+1}
$$

Tyler Moulton

Numerade Educator

Problem 35

Find the interval(s) where the function is increasing and the interval(s) where it is decreasing.
$$
f(x)=x^{2} e^{-x}
$$

Tyler Moulton

Numerade Educator

Problem 36

Find the interval(s) where the function is increasing and the interval(s) where it is decreasing.
$$
f(x)=e^{-x^{2} / 2}
$$

Tyler Moulton

Numerade Educator

Problem 37

Find the interval(s) where the function is increasing and the interval(s) where it is decreasing.
$$
f(x)=\frac{\ln x}{x}
$$

Tyler Moulton

Numerade Educator

Problem 38

Find the interval(s) where the function is increasing and the interval(s) where it is decreasing.
$$
f(x)=\ln x^{2}
$$

Tyler Moulton

Numerade Educator

Problem 39

Find the interval(s) where the function is increasing and the interval(s) where it is decreasing.
$$
f(x)=\frac{x^{2}-1}{x}
$$

Tyler Moulton

Numerade Educator

Problem 40

Find the interval(s) where the function is increasing and the interval(s) where it is decreasing.
$$
h(x)=\frac{x^{2}}{x-1}
$$

Tyler Moulton

Numerade Educator

Problem 41

You are given the graph of a function $f$. Determine the relative maxima and relative minima, if any.

Nicole Krahulik

Nicole Krahulik

Numerade Educator

Problem 42

You are given the graph of a function $f$. Determine the relative maxima and relative minima, if any.

Nicole Krahulik

Nicole Krahulik

Numerade Educator

Problem 43

You are given the graph of a function $f$. Determine the relative maxima and relative minima, if any.

Nicole Krahulik

Nicole Krahulik

Numerade Educator

Problem 44

You are given the graph of a function $f$. Determine the relative maxima and relative minima, if any.

Nicole Krahulik

Nicole Krahulik

Numerade Educator

Problem 45

You are given the graph of a function $f$. Determine the relative maxima and relative minima, if any.

Nicole Krahulik

Nicole Krahulik

Numerade Educator

Problem 46

You are given the graph of a function $f$. Determine the relative maxima and relative minima, if any.

Nicole Krahulik

Nicole Krahulik

Numerade Educator

Problem 47

You are given the graph of a function $f$. Determine the relative maxima and relative minima, if any.

Nicole Krahulik

Nicole Krahulik

Numerade Educator

Problem 48

You are given the graph of a function $f$. Determine the relative maxima and relative minima, if any.

Nicole Krahulik

Nicole Krahulik

Numerade Educator

Problem 49

Match the graph of the function with the graph of its derivative in (a)-(d).

Nicole Krahulik

Nicole Krahulik

Numerade Educator

Problem 50

Match the graph of the function with the graph of its derivative in (a)-(d).

Nicole Krahulik

Nicole Krahulik

Numerade Educator

Problem 51

Match the graph of the function with the graph of its derivative in (a)-(d).

Nicole Krahulik

Nicole Krahulik

Numerade Educator

Problem 52

Match the graph of the function with the graph of its derivative in (a)-(d).

Nicole Krahulik

Nicole Krahulik

Numerade Educator

Problem 53

Find the relative maxima and relative minima, if any, of each function.
$$
f(x)=x^{2}-4 x
$$

Nishant Tyagi

Numerade Educator

Problem 54

Find the relative maxima and relative minima, if any, of each function.
$$
g(x)=x^{2}+3 x+8
$$

Jonathon Brumley

Jonathon Brumley

Numerade Educator

Problem 55

Find the relative maxima and relative minima, if any, of each function.
$$
h(t)=-t^{2}+6 t+6
$$

Nishant Tyagi

Numerade Educator

Problem 56

Find the relative maxima and relative minima, if any, of each function.
$$
f(x)=\frac{1}{2} x^{2}-2 x+4
$$

Nishant Tyagi

Numerade Educator

Problem 57

Find the relative maxima and relative minima, if any, of each function.
$$
f(x)=x^{5 / 3}
$$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 58

Find the relative maxima and relative minima, if any, of each function.
$$
f(x)=x^{2 / 3}+2
$$

Tyler Moulton

Numerade Educator

Problem 59

Find the relative maxima and relative minima, if any, of each function.
$$
g(x)=x^{3}-3 x^{2}+4
$$

Nishant Tyagi

Numerade Educator

Problem 60

Find the relative maxima and relative minima, if any, of each function.
$$
f(x)=x^{3}-3 x+6
$$

Nishant Tyagi

Numerade Educator

Problem 61

Find the relative maxima and relative minima, if any, of each function.
$$
f(x)=\frac{1}{2} x^{4}-x^{2}
$$

Nishant Tyagi

Numerade Educator

Problem 62

Find the relative maxima and relative minima, if any, of each function.
$$
h(x)=\frac{1}{2} x^{4}-3 x^{2}+4 x-8
$$

Tyler Moulton

Numerade Educator

Problem 63

Find the relative maxima and relative minima, if any, of each function.
$$
F(x)=\frac{1}{3} x^{3}-x^{2}-3 x+4
$$

Nishant Tyagi

Numerade Educator

Problem 64

Find the relative maxima and relative minima, if any, of each function.
$$
F(t)=3 t^{5}-20 t^{3}+20
$$

Nishant Tyagi

Numerade Educator

Problem 65

Find the relative maxima and relative minima, if any, of each function.
$$
g(x)=x^{4}-4 x^{3}+8
$$

Tyler Moulton

Numerade Educator

Problem 66

Find the relative maxima and relative minima, if any, of each function.
$$
f(x)=3 x^{4}-2 x^{3}+4
$$

Jonathon Brumley

Jonathon Brumley

Numerade Educator

Problem 67

Find the relative maxima and relative minima, if any, of each function.
$$
g(x)=\frac{x+1}{x}
$$

Jonathon Brumley

Jonathon Brumley

Numerade Educator

Problem 68

Find the relative maxima and relative minima, if any, of each function.
$$
h(x)=\frac{x}{x+1}
$$

Tyler Moulton

Numerade Educator

Problem 69

Find the relative maxima and relative minima, if any, of each function.
$$
f(x)=x+\frac{9}{x}+2
$$

Tyler Moulton

Numerade Educator

Problem 70

Find the relative maxima and relative minima, if any, of each function.
$$
g(x)=2 x^{2}+\frac{4000}{x}+10
$$

Tyler Moulton

Numerade Educator

Problem 71

Find the relative maxima and relative minima, if any, of each function.
$$
f(x)=\frac{x}{1+x^{2}}
$$

Tyler Moulton

Numerade Educator

Problem 72

Find the relative maxima and relative minima, if any, of each function.
$$
g(x)=\frac{x}{x^{2}-1}
$$

Tyler Moulton

Numerade Educator

Problem 73

Find the relative maxima and relative minima, if any, of each function.
$$
f(x)=x e^{-x}
$$

Tyler Moulton

Numerade Educator

Problem 74

Find the relative maxima and relative minima, if any, of each function.
$$
f(x)=x^{2} e^{-x}
$$

Tyler Moulton

Numerade Educator

Problem 75

Find the relative maxima and relative minima, if any, of each function.
$$
f(x)=x-\ln x
$$

Tyler Moulton

Numerade Educator

Problem 76

Find the relative maxima and relative minima, if any, of each function.
$$
f(x)=x^{2} \ln x
$$

Tyler Moulton

Numerade Educator

Problem 77

A stone is thrown straight up from the roof of an $80-\mathrm{ft}$ building. The distance (in feet) of the stone from the ground at any time $t$ (in seconds) is given by
$$
h(t)=-16 t^{2}+64 t+80
$$
When is the stone rising, and when is it falling? If the stone were to miss the building, when would it hit the ground? Sketch the graph of $h$. Hint: The stone is on the ground when $h(t)=0$.

Ekaveera Kumar

Numerade Educator

Problem 78

The Mexican subsidiary of ThermoMaster manufactures an indoor-outdoor thermometer. Management estimates that the profit (in dollars) realizable by the company for the manufacture and sale of $x$ units of thermometers each week is
$$
P(x)=-0.001 x^{2}+8 x-5000
$$
Find the intervals where the profit function $P$ is increasing and the intervals where $P$ is decreasing.

Jonathon Brumley

Jonathon Brumley

Numerade Educator

Problem 79

Based on a study conducted in 1997 , the percent of the U.S. population by age afflicted with Alzheimer's disease is given by the function
$P(x)=0.0726 x^{2}+0.7902 x+4.9623 \quad(0 \leq x \leq 25)$
where $x$ is measured in years, with $x=0$ corresponding to age 65 yr. Show that $P$ is an increasing function of $x$ on the interval $(0,25)$. What does your result tell you about the relationship between Alzheimer's disease and age for the population that is age $65 \mathrm{yr}$ and older?

Carolyn Behr-Jerome

Carolyn Behr-Jerome

Numerade Educator

Problem 80

Almost half of companies let other firms manage some of their Web operations-a practice called Web hosting. Managed services -monitoring a customer's technology services-is the fastest growing part of Web hosting. Managed services sales are expected to grow in accordance with the function
$$
f(t)=0.469 t^{2}+0.758 t+0.44 \quad(0 \leq t \leq 6)
$$
where $f(t)$ is measured in billions of dollars and $t$ is measured in years, with $t=0$ corresponding to 1999 .
a. Find the interval where $f$ is increasing and the interval where $f$ is decreasing.
b. What does your result tell you about sales in managed services from 1999 through 2005 ?

Mutahar Mehkri

Numerade Educator

Problem 81

The height (in feet) attained by a rocket $t$ sec into flight is given by the function
$$
h(t)=-\frac{1}{3} t^{3}+16 t^{2}+33 t+10 \quad(t \geq 0)
$$
When is the rocket rising, and when is it descending?

Nishant Tyagi

Numerade Educator

Problem 82

Following the lead of the National Wildlife Federation, the Department of the Interior of a South American country began to record an index of environmental quality that measured progress and decline in the environmental quality of its forests. The index for the years 1998 through 2008 is approximated by the function
$$
I(t)=\frac{1}{3} t^{3}-\frac{5}{2} t^{2}+80 \quad(0 \leq t \leq 10)
$$
where $t=0$ corresponds to 1998 . Find the intervals where the function $I$ is increasing and the intervals where it is decreasing. Interpret your results.

Nicole Krahulik

Nicole Krahulik

Numerade Educator

Problem 83

The average speed of a vehicle on a stretch of Route 134 between 6 a.m. and 10 a.m. on a typical weekday is approximated by the function
$$
f(t)=20 t-40 \sqrt{t}+50 \quad(0 \leq t \leq 4)
$$
where $f(t)$ is measured in miles per hour and $t$ is measured in hours, with $t=0$ corresponding to 6 a.m. Find the interval where $f$ is increasing and the interval where $f$ is decreasing and interpret your results.

Jonathon Brumley

Jonathon Brumley

Numerade Educator

Problem 84

The average cost (in dollars) incurred by Lincoln Records each week in pressing $x$ compact discs is given by
$$
\bar{C}(x)=-0.0001 x+2+\frac{2000}{x} \quad(0<x \leq 6000)
$$
Show that $\bar{C}(x)$ is always decreasing over the interval $(0,6000)$

Melissa Munoz

Numerade Educator

Problem 85

Refer to Exercise 80 . Sales in the Webhosting industry are projected to grow in accordance with the function
$f(t)=-0.05 t^{3}+0.56 t^{2}+5.47 t+7.5 \quad(0 \leq t \leq 6)$
where $f(t)$ is measured in billions of dollars and $t$ is measured in years, with $t=0$ corresponding to 1999 .
a. Find the interval where $f$ is increasing and the interval where $f$ is decreasing. Hint: Use the quadratic formula.
b. What does your result tell you about sales in the Webhosting industry from 1999 through 2005 ?

Carson Merrill

Numerade Educator

Problem 86

According to a study from the American Medical Association, the number of medical school applicants from academic year $1997-1998(t=0)$ through the academic year 2002-2003 is approximated by the function
$N(t)=-0.0333 t^{3}+0.47 t^{2}-3.8 t+47 \quad(0 \leq t \leq 5)$
where $N(t)$ measured in thousands.
a. Show that the number of medical school applicants had been declining over the period in question. Hint: Use the quadratic formula.
b. What was the largest number of medical school applicants in any one academic year for the period in question? In what academic year did that occur?

Karl Schaefer

University of Chicago

Problem 87

The sales of functional food products-those that promise benefits beyond basic nutrition-have risen sharply in recent years. The sales (in billions of dollars) of foods and beverages with herbal and other additives is approximated by the function
$S(t)=0.46 t^{3}-2.22 t^{2}+6.21 t+17.25 \quad(0 \leq t \leq 4)$
where $t$ is measured in years, with $t=0$ corresponding to the beginning of 1997 . Show that $S$ is increasing on the interval $[0,4]$. Hint: Use the quadratic formula.

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 88

Based on data from the Central Provident Fund of a certain country (a government agency similar to the Social Security Administration), the estimated cash in the fund in 2003 is given by
$$
\begin{aligned}
A(t)=&-96.6 t^{4}+403.6 t^{3} \\
&+660.9 t^{2}+250 \quad(0 \leq t \leq 5)
\end{aligned}
$$
where $A(t)$ is measured in billions of dollars and $t$ is measured in decades, with $t=0$ corresponding to $2003 .$ Find the interval where $A$ is increasing and the interval where $A$ is decreasing and interpret your results. Hint: Use the quadratic formula.

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 89

U.S. telephone company spending on fiber-optic links to homes and businesses from 2001 to 2006 is projected to be
$S(t)=-2.315 t^{3}+34.325 t^{2}+1.32 t+23 \quad(0 \leq t \leq 5)$ billion dollars in year $t$, where $t$ is measured in years with $t=0$ corresponding to 2001 . Show that $S^{\prime}(t)>0$ for all $t$ in the interval $[0,5]$. What conclusion can you draw from this result? Hint: Use the quadratic formula.

Carson Merrill

Numerade Educator

Problem 90

According to the South Coast Air Quality Management District, the level of nitrogen dioxide, a brown gas that impairs breathing, present in the atmosphere on a certain May day in downtown Los Angeles is approximated by
$$
A(t)=0.03 t^{3}(t-7)^{4}+60.2 \quad(0 \leq t \leq 7)
$$
where $A(t)$ is measured in pollutant standard index (PSI) and $t$ is measured in hours, with $t=0$ corresponding to 7 a.m. At what time of day is the air pollution increasing, and at what time is it decreasing?

Nicole Krahulik

Nicole Krahulik

Numerade Educator

Problem 91

The concentration (in milligrams/cubic centimeter) of a certain drug in a patient's body $t$ hr after injection is given by
$$
C(t)=\frac{t^{2}}{2 t^{3}+1} \quad(0 \leq t \leq 4)
$$
When is the concentration of the drug increasing, and when is it decreasing?

Nicole Krahulik

Nicole Krahulik

Numerade Educator

Problem 92

The number of crash fatalities per 100,000 vehicle miles of travel (based on 1994 data) is approximated by the model
$$
f(x)=\frac{15}{0.08333 x^{2}+1.91667 x+1} \quad(0 \leq x \leq 11)
$$
where $x$ is the age of the driver in years, with $x=0$ corresponding to age 16 . Show that $f$ is decreasing on $(0,11)$ and interpret your result.

Ashley Boni

Numerade Educator

Problem 93

The amount of nitrogen dioxide, a brown gas that impairs breathing, present in the atmosphere on a certain May day in the city of Long Beach is approximated by
$$
A(t)=\frac{136}{1+0.25(t-4.5)^{2}}+28 \quad(0 \leq t \leq 11)
$$
where $A(t)$ is measured in pollutant standard index (PSI) and $t$ is measured in hours, with $t=0$ corresponding to 7 a.m. Find the intervals where $A$ is increasing and where $A$ is decreasing and interpret your results.

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 94

The 1980s saw a trend toward oldfashioned punitive deterrence as opposed to the more liberal penal policies and community-based corrections popular in the 1960 s and early $1970 \mathrm{~s}$. As a result, prisons became more crowded, and the gap between the number of people in prison and the prison capacity widened. The number of prisoners (in thousands) in federal and state prisons is approximated by the function
$$
N(t)=3.5 t^{2}+26.7 t+436.2 \quad(0 \leq t \leq 10)
$$
where $t$ is measured in years, with $t=0$ corresponding to
1984. The number of inmates for which prisons were designed is given by
$$
C(t)=24.3 t+365 \quad(0 \leq t \leq 10)
$$
where $C(t)$ is measured in thousands and $t$ has the same meaning as before. Show that the gap between the number of prisoners and the number for which the prisons were designed has been widening at any time $t$. Hint: First, write a function $G$ that gives the gap between the number of prisoners and the number for which the prisons were designed at any time $t$. Then show that $G^{\prime}(t)>0$ for all values of t in the interval $(0,10)$.

James Kiss

Numerade Educator

Problem 95

U.S. NURSING SHORTAGE The demand for nurses between 2000 and 2015 is estimated to be
$$
D(t)=0.0007 t^{2}+0.0265 t+2 \quad(0 \leq t \leq 15)
$$
where $D(t)$ is measured in millions and $t=0$ corresponds to the year 2000 . The supply of nurses over the same time period is estimated to be
$$
S(t)=-0.0014 t^{2}+0.0326 t+1.9 \quad(0 \leq t \leq 15)
$$
where $S(t)$ is also measured in millions.
a. Find an expression $G(t)$ giving the gap between the demand and supply of nurses over the period in question.
b. Find the interval where $G$ is decreasing and where it is increasing. Interpret your result.
c. Find the relative extrema of $G$. Interpret your result.

Carson Merrill

Numerade Educator

Problem 96

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false.
If $f$ is decreasing on $(a, b)$, then $f^{\prime}(x)<0$ for each $x$ in $(a, b) .$

Nicole Krahulik

Nicole Krahulik

Numerade Educator

Problem 97

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false.
If $f$ and $g$ are hoth increasing on $(a, b)$, then $f+g$ is increasing on $(a, b)$.

Nicole Krahulik

Nicole Krahulik

Numerade Educator

Problem 98

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false.
If $f$ and $g$ are both decreasing on $(a, b)$, then $f-g$ is decreasing on $(a, b)$.

Carson Merrill

Numerade Educator

Problem 99

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false.
If $f(x)$ and $g(x)$ are positive on $(a, b)$ and both $f$ and $g$ are increasing on $(a, b)$, then $f g$ is increasing on $(a, b)$.

Nicole Krahulik

Nicole Krahulik

Numerade Educator

Problem 100

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false.
If $f^{\prime}(c)=0$, then $f$ has a relative maximum or a relative minimum at $x=c$.

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 101

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false.
If $f$ has a relative minimum at $x=c$, then $f^{\prime}(c)=0$.

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 102

Using Theorem 1 , verify that the linear function $f(x)=m x+$ $b$ is (a) increasing everywhere if $m>0$, (b) decreasing everywhere if $m<0$, and $(\mathrm{c})$ constant if $m=0$.

Jonathon Brumley

Jonathon Brumley

Numerade Educator

Problem 103

Show that the function $f(x)=x^{3}+x+1$ has no relative extrema on $(-\infty, \infty)$.

Matt Just

Numerade Educator

Problem 104

Let $f(x)=x^{2}+a x+b$. Determine the constants $a$ and $b$ so that $f$ has a relative minimum at $x=2$ and the relative minimum value is 7 .

Nicole Krahulik

Nicole Krahulik

Numerade Educator

Problem 105

Let $f(x)=a x^{3}+6 x^{2}+b x+4$. Determine the constants $a$ and $b$ so that $f$ has a relative minimum at $x=-1$ and a relative maximum at $x=2$.

Nicole Krahulik

Nicole Krahulik

Numerade Educator

Problem 106

Let
$$
f(x)=\left\{\begin{array}{ll}
-3 x & \text { if } x<0 \\
2 x+4 & \text { if } x \geq 0
\end{array}\right.
$$
a. Compute $f^{\prime}(x)$ and show that it changes sign from negative to positive as we move across $x=0$.
b. Show that $f$ does not have a relative minimum at $x=0 .$ Does this contradict the first derivative test? Explain your answer.

Nicole Krahulik

Nicole Krahulik

Numerade Educator

Problem 107

Let
$$
f(x)=\left\{\begin{array}{ll}
-x^{2}+3 & \text { if } x \neq 0 \\
2 & \text { if } x=0
\end{array}\right.
$$
a. Compute $f^{\prime}(x)$ and show that it changes sign from positive to negative as we move across $x=0$.
b. Show that $f$ does not have a relative maximum at $x=0$. Does this contradict the first derivative test? Explain your answer.

Nicole Krahulik

Nicole Krahulik

Numerade Educator

Problem 108

Let
$$
f(x)=\left\{\begin{array}{ll}
\frac{1}{x^{2}} & \text { if } x>0 \\
x^{2} & \text { if } x \leq 0
\end{array}\right.
$$
a. Compute $f^{\prime}(x)$ and show that it does not change sign as we move across $x=0$.
b. Show that $f$ has a relative minimum at $x=0$. Does this contradict the first derivative test? Explain your
answer.

Carson Merrill

Numerade Educator

Problem 109

Show that the quadratic function
$$
f(x)=a x^{2}+b x+c \quad(a \neq 0)
$$
has a relative extremum when $x=-b / 2 a$. Also, show that the relative extremum is a relative maximum if $a<0$ and a relative minimum if $a>0$.

Suzanne W.

Numerade Educator

Problem 110

Show that the cubic function
$$
f(x)=a x^{3}+b x^{2}+c x+d \quad(a \neq 0)
$$
has no relative extremum if and only if $b^{2}-3 a c \leq 0$.

Matt Just

Numerade Educator

Problem 111

Refer to Example 6, page 561 .
a. Show that $f$ is increasing on the interval $(0,1)$.
b. Show that $f(0)=-1$ and $f(1)=1$ and use the result of part (a) together with the intermediate value theorem to conclude that there is exactly one root of $f(x)=$ 0 in $(0,1)$

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 112

Show that the function
$$
f(x)=\frac{a x+b}{c x+d}
$$
does not have a relative extremum if $a d-b c \neq 0 .$ What can you say about $f$ if $a d-b c=0$ ?

Oswaldo Jiménez

Oswaldo Jiménez

Numerade Educator