Calculus Concepts

Donald R. LaTorre, John W. Kenelly, Sherry S. Biggers

Chapter 4

Analyzing Change: Applications of Derivatives - all with Video Answers

Educators

Section 1

Linearization and Estimates

Problem 1

The humidity is currently $32 \%$ and falling at a rate of 4 percentage points per hour.
a. Estimate the change in humidity over the next 20 minutes.
b. Estimate the humidity 20 minutes from now.

Gregory Higby

Gregory Higby

Numerade Educator

Problem 2

An airplane is flying at a speed of $300 \mathrm{mph}$ and accelerating at a rate of 200 mph per hour.
a. Estimate the change in the airplane's velocity over the next 5 minutes.
b. Estimate the airplane's speed in 5 minutes.

Gregory Higby

Numerade Educator

Problem 3

The daily production level is currently 100 filters and increasing by rate of 3 filters per day.
a. Estimate the change in daily production over the next week.
b. Estimate the daily production in a week.

Gregory Higby

Numerade Educator

Problem 4

Monthly sales are currently $\$ 500,000$ and decreasing by $\$ 2,000$ per month.
a. Estimate the change in monthly sales over the next two months.
b. Estimate sales in two months.

Gregory Higby

Numerade Educator

Problem 5

a. Write a linearization for $f$ with respect to $x$
b. Use the linearization to estimate $f$ at the given input.
$$
f(3)=17, f^{\prime}(3)=4.6 ; x=3.5
$$

Gregory Higby

Numerade Educator

Problem 6

a. Write a linearization for $f$ with respect to $x$
b. Use the linearization to estimate $f$ at the given input.
$$
g(7)=4, g^{\prime}(7)=-12.9 ; x=7.25
$$

Willis James

Numerade Educator

Problem 7

a. Write a linearization for $f$ with respect to $x$
b. Use the linearization to estimate $f$ at the given input.
$$
f(10)=5, f^{\prime}(10)=-0.3 ; x=10.4
$$

Gregory Higby

Numerade Educator

Problem 8

a. Write a linearization for $f$ with respect to $x$
b. Use the linearization to estimate $f$ at the given input.
$$
g(9)=12, g^{\prime}(9)=1.6 ; x=9.5
$$

Gregory Higby

Numerade Educator

Problem 9

New-Car Revenue The function $R(x)$ billion dollars models revenue from new-car sales for franchised new-car dealerships in the United States when $x$ million dollars are spent on associated advertising expenditures, data from $1.2 \leq x \leq 6.5$
(Source: Based on data from Statistical Abstract for data between 1980 and 2000$)$
a. Use $R(1.5)=141$ and $R^{\prime}(1.5)=38$ to write a linearization model with respect to $x$ at $c=1.5 .$
b. Estimate the revenue when $\$ 1.6$ million is spent on advertising.
c. Estimate the revenue when $\$ 2$ million is spent on advertising.

Gregory Higby

Numerade Educator

Problem 10

Study Time Suppose that a student's test grade out of 100 points is a function, $g$, of the time spent studying, $x$.
a. Write a linearization of $g$ with respect to $x$ given $g(5)=78$ and $g^{\prime}(5)=6$
b. Estimate the student's score after studying 6.57 hours.
c. If $g$ is concave down for $x>5,$ is the estimate $\operatorname{made}$ using a linearization an overestimate or an underestimate of the output of $g$. Explain.

Gregory Higby

Numerade Educator

Problem 11

CFC Emissions The estimated releases of CFC-11 between 1990 and 2009 are modeled as $g(x)$ thousand metric tons, where $x$ is the number of years since $1990 .$ See the graph.
a. Use the 2008 values $g(18)=38.3$ and $g^{\prime}(18)=-4.9$ to write a linearization model that could be used to estimate other values for CFC-11 releases. Use the linearization to estimate the amount of CFCs released into the atmosphere in $2009 .$
b. Use the 2007 values, $g(17)=42.2$ and $g^{\prime}(17)=-2.9$, to write a linearization of $g .$ Use the linearization to estimate the amount of CFCs released into the atmosphere in 2009
c. The actual amount of CFCs released into the atmosphere in 2009 was 37.7 thousand metric tons. Which of the estimates is closer?

Gregory Higby

Numerade Educator

Problem 12

South Carolina Population (Historic) The population of South Carolina between 1790 and 2000 can be modeled
as
$$
f(x)=268.79\left(1.013^{x}\right) \text { thousand people }
$$
where $x$ is the number of years since $1790 .$ (Source: Based on data from Statistical Abstract, 2001 )
a. Calculate the population and the rate of change of the population of South Carolina in 2000 .
b. Write the linearization of $f$ in 2000 .
c. Use the linearization to estimate the population in 2003 .

Gregory Higby

Numerade Educator

Problem 13

Future Value The future value of an investment after $t$ years is given by
$$
F(t)=120\left(1.126^{t}\right) \text { thousand dollars }
$$
a. Calculate the future value and the rate of change of the future value after 10 years.
b. Write the linearization of $F$ after 10 years.
c. Use the linearization to estimate the future value after 10.5 years.

Melissa Munoz

Numerade Educator

Problem 14

Mexico Population (Historic) The population of Mexico between 1921 and 2010 can be modeled as
$$
p(t)=8.028 e^{0.025 t} \text { million people }
$$
where $t$ is the number of years since 1900 . (Source: Based on data from Statistical Abstract, 2009 and www.inegi.gob.mx)
a. What was the population of Mexico and how rapidly was it growing in $2010 ?$
b. Write a linearization of $p$ in 2010 .
c. Use the linearization to estimate the population of Mexico in 2011 .

Gregory Higby

Numerade Educator

Problem 15

Life Insurance Costs The figure shows the annual cost for a one-million-dollar term life insurance policy as a function of the age of the insured person.
a. Estimate the slope of a tangent line drawn at $a=60$.
b. Write the linearization of $C$ at 60 .
c. Use the linearization function to estimate the annual cost for a one-million-dollar term life insurance policy for a 63 -year-old person.

Gregory Higby

Numerade Educator

Problem 16

New-Car Revenue The figure shows revenue from new-car sales as a function of advertising expenditures for franchised new-car dealerships in the United States.
a. Estimate the slope of a tangent line drawn at $a=2$.
b. Write the linearization of $R$ at 2 .
c. Use the linearization to estimate the revenue from newcar sales when $\$ 2.5$ billion is spent on advertising.

Gregory Higby

Numerade Educator

Problem 17

Crab Claw Weight For fiddler crabs, the weight of the claw can be modeled as
$$
f(w)=0.11 w^{1.54} \text { pounds }
$$
when the weight of the body is $w$ pounds. (Source: d'Arcy Thompson, $O n$ Growth and Form, Cambridge, UK: Cambridge University Press, $1961 .)$
a. Describe the direction and concavity of $f$ for positive values of $w$
b. For positive values of $w$, will a linearization of $f$ overestimate or underestimate the function values? Explain.

Gregory Higby

Numerade Educator

Problem 18

Dog Age A logarithmic model relating the age of a dog to the human age equivalent is
$$
b(d)=-17+28.4 \ln (d+2) \text { years }
$$
where $d$ is the chronological age of the dog, data from $0 \leq d \leq 14$
a. Describe the direction and concavity of $h$ for positive values of $d$
b. For positive values of $d$, will a linearization of $h$ overestimate or underestimate the function values? Explain why.

Gregory Higby

Numerade Educator

Problem 19

Sleeping Habits The percentage of people (aged 15 and above) in the United States who are asleep $t$ hours after 9: 00 P.M. can be modeled as
$$
\begin{aligned}
s(t) &=-2.63 t^{2}+29.52 t+13.52 \text { percent } \\
\text { data from } 0 & \leq t \leq 10 .
\end{aligned}
$$
(Source: Based on data from American Time Use Survey, March 2009, Bureau of Labor Statistics)
a. Describe the direction and concavity of $s$ over $0 \leq t \leq 10$
b. Will a linearization of $s$ always overestimate the function values? Explain.

Gregory Higby

Numerade Educator

Problem 20

Lizard Harvest The number of lizards harvested during March through October of a given year can be modeled as
$$
b(m)=15.3 \sin (0.805 \mathrm{~m}+2.95)+16.7 \text { lizards }
$$
where $m$ is the month of the year, data from $3 \leq m \leq 10$.
a. Describe the direction and concavity of $h$ over $3 \leq m \leq 10$
b. Will a linearization of $h$ always overestimate the function values? Explain.

Amy Jiang

Numerade Educator

Problem 21

Write a brief statement that explains why, when rates of change are used to approximate change in a function, approximations over shorter intervals are generally better answers than approximations over longer time intervals. Include graphical illustrations in the discussion.

Gregory Higby

Numerade Educator

Problem 22

Write a brief statement that explains why, when rates of change are used to approximate the change in a concave-up portion of a function, the approximation is an underestimate and, when rates of change are used to approximate change in a concave-down portion of a function, the approximation is an overestimate. Include graphical illustrations in the discussion.

Gregory Higby

Numerade Educator