Calculus Concepts

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

Chapter 2

Describing Change: Rates - all with Video Answers

Educators

Section 1

Measures of Change over an Interval

Problem 1

Rewrite the sentences to express how rapidly, on average, the quantity changed over the given interval.
Apple Stock Prices During a media event at which CEO Steve Jobs spoke, Apple shares opened at $\$ 156.86$ and dropped to $\$ 151.80$ fifty minutes into Jobs's keynote address.

Gregory Higby

Gregory Higby

Numerade Educator

Problem 2

Rewrite the sentences to express how rapidly, on average, the quantity changed over the given interval.
China Internet Users The number of Internet users in China grew from 12 million in 2000 to 103 million in $2005 .$ (Source: BDA [China], The Strategis Group and China Daily, July $22,2005 .$

Gregory Higby

Numerade Educator

Problem 3

Rewrite the sentences to express how rapidly, on average, the quantity changed over the given interval.
Iowa Community College Tuition The average community college tuition in Iowa during $2000-2001$ was $\$ 1,856 .$ In $2009-2010,$ the average community college tuition in Iowa was $\$ 3,660 .$

Gregory Higby

Numerade Educator

Problem 4

Rewrite the sentences to express how rapidly, on average, the quantity changed over the given interval.
Unemployment Rate The unemployment rate was $9.4 \%$ in July 2009, up from $5.4 \%$ in January $2003 .$

Gregory Higby

Numerade Educator

Problem 5

Calculate and write a sentence interpreting each of the following descriptions of change:
a. change
b. percentage change
c. average rate of change
Airline Profit AirTran posted a profit of $\$ 17.6$ million at the end of 2009 compared with a loss of $\$ 121.6$ million in 2008 .

Gregory Higby

Numerade Educator

Problem 6

Calculate and write a sentence interpreting each of the following descriptions of change:
a. change
b. percentage change
c. average rate of change
Airline Revenue For the second quarter of 2009 , AirTran posted revenue of $\$ 603.7$ million compared with revenue of $\$ 693.4$ million during the second quarter of 2008 .

Willis James

Numerade Educator

Problem 7

Calculate and write a sentence interpreting each of the following descriptions of change:
a. change
b. percentage change
c. average rate of change
ACT Scores The percentage of students meeting national mathematics benchmarks on the ACT increased from $40 \%$ in 2004 to $43 \%$ in 2008.

Willis James

Numerade Educator

Problem 8

Calculate and write a sentence interpreting each of the following descriptions of change:
a. change
b. percentage change
c. average rate of change
Native American Population The American Indian, Eskimo, and Aleut population in the United States was 362 thousand in 1930 and 4.5 million in $2005 .$

Willis James

Numerade Educator

Problem 9

October Madness The scatter plot shows the number of shares traded each day during October of $1987 .$ The behavior of the graph on October 19 and 20 has been referred to as "October Madness."
a. Calculate the percentage change and average rate of change in the number of shares traded per trading day between October 1 (when 193.2 million shares were traded) and October 30,1987 (when 303.4 shares were traded).
b. Draw a secant line whose slope is the average rate of change between October 1 and October $30,1987 .$
c. Write a sentence describing how the number of shares traded changed throughout the month. How well does the average rate of change calculated in part $a$ reflect what occurred throughout the month?

Carson Merrill

Numerade Educator

Problem 10

Lake Level (Historic) The figure shows the highest elevations above sea level attained by Lake Tahoe (located on the California-Nevada border) from 1982 through $1996 .$
a. Draw a secant line connecting the left and right endpoints of the graph. Calculate the slope of this line.
b. Write a sentence interpreting the slope in the context of Lake Tahoe levels.
c. Write a sentence summarizing how the level of the lake changed from 1982 through $1996 .$ How well does the answer to part $b$ describe the change in the lake level as shown in the graph?

Carson Merrill

Numerade Educator

Problem 11

Kelly Services Sales A graph of a model for the sales of services between 2004 and 2008 by Kelly Services, Inc., a leading global provider of staffing services, is shown below.
a. Use the graph to calculate the average rate of change in Kelly's sales of services between 2004 and $2007 .$ Interpret the result.
b. Calculate the percentage change in Kelly's sales between 2007 and 2008 . Interpret the result.
c. Calculate the change in Kelly's sales between 2004 and 2008

Willis James

Numerade Educator

Problem 12

Marriage Age The figure shows the median age at first marriage for men in the United States between 1970 and $2007 .$
a. Calculate how much and how rapidly the median marriage age increased from 1970 through $2007 .$
b. Did the median age at first marriage for men grow at the same rate from 1970 through 2000 as it did from 2000 through $2007 ?$

Willis James

Numerade Educator

Problem 13

Airline Profit The table gives the price, in dollars, of a round-trip flight from Denver to Chicago on a certain airline and the corresponding monthly profit for that airline that route
a. Find a model for the data.
b. Calculate the average rate of change of profit when the ticket price rises from $\$ 200$ to $\$ 350 .$
c. Calculate the average rate of change of profit when the ticket price rises from $\$ 350$ to $\$ 450 .$

Willis James

Numerade Educator

Problem 14

Cruise Tickets A travel agent vigorously promotes cruises to Alaska for several months. The table shows the
total (cumulative) sales every 3 weeks since the beginning of the special promotion.
a. Find a logistic model for cumulative sales. Why is a logistic model appropriate to model these data?
b. Calculate the percentage increase in the number of tickets sold between weeks 1 and 15 and between weeks 1 and $25 .$

Carson Merrill

Numerade Educator

Problem 15

Life Expectancy The life expectancies of black males in the United States at various ages for 2006 are as shown below.
a. How rapidly, on average, does the life expectancy change between birth and the 70 th year of life for black males in the United States?
b. Compare the average rates of change of life expectancy for the 10 -year periods between ages 10 and 20 and ages 20 and $30 .$

Willis James

Numerade Educator

Problem 16

Loan Principal The table shows the loan principal (the amount of money that a certain bank will lend) on the basis of a monthly payment of $\$ 600$ per month with a 30-year term.
a. What are the units for the change, the average rate of change, and the percentage change in the loan amount when the interest rate changes?
b. Determine the change, the average rate of change, and the percentage change in the loan amount when interest rates increase from $5 \%$ to $6.25 \%$
c. Consider the inverse relationship represented by reversing the columns in the table. What are the input units and the output units of the inverse relationship? What are the units for the change, the average rate of change, and the percentage change of the inverse relationship?
d. For the inverse relationship, determine the change, the average rate of change, and the percentage change in the monthly APR when the principal increases from $\$ 97.4$ thousand to $\$ 111.77$ thousand.

Carson Merrill

Numerade Educator

Problem 17

Mexico Internet Users The number of Internet users in Mexico between 2004 and 2008 can be modeled as
$u(t)=8.02\left(1.17^{t}\right)$ million users
where $t$ is the number of years since 2004 (Source: Based on data at www.internetworldstats.com/am/mx.htm)
a. On average, what was the rate of change in the number of Internet users in Mexico between 2004 and $2008 ?$
b. What was the percentage change in the number of Internet users in Mexico between 2004 and $2008 ?$
c. The population of Mexico in 2008 was 109,955,400 . What percentage of the Mexican population used the Internet in $2008 ?$

Willis James

Numerade Educator

Problem 18

AIDS Cases The number of AIDS cases diagnosed from 2000 through 2007 can be modeled as
$f(x)=3.23\left(1.06^{x}\right)$ hundred thousand cases
where $x$ is the number of years since $2000 .$ (Source: Based on data from U.S. Centers for Disease Control and Prevention)
a. Calculate and write a sentence of interpretation for the average rate of change in the number of persons diagnosed with AIDS between 2000 and $2007 .$
b. Calculate the percentage change in the number of persons diagnosed with AIDS between 2000 and $2007 .$

Willis James

Numerade Educator

Problem 19

ATM Surcharges $99.2 \%$ of ATMs levy a surcharge on users who are not account holders. The amount of the surcharge for non-account holders can be modeled as
$$
s(t)=0.72\left(1.081^{t}\right) \text { dollars }
$$
a. Calculate the average rate of change in the amount of the surcharge for non-account holders between 1998 and
2008. Write the result in a sentence of interpretation.
b. Calculate the change and the percentage change in the amount of the surcharge for non-account holders between 1998 and 2008 .
where $t$ is the number of years since $1995,$ data from $3 \leq t \leq 13$

Willis James

Numerade Educator

Problem 20

Social Networking In $2007,$ Bebo was the largest social networking site in the UK, Ireland, and New Zealand, and was the third largest social networking site in the United States. Bebo allows users to share photos with music and blogs and draw on members' whiteboards. The percent of Bebo users in the world between ages 17 and 33 can be modeled as
$$
u(x)=18-6.74 \ln x \text { percent }
$$
at age $x+16$ (Source: Based on data at www.pipl.com/statistics)
a. Calculate the average rate of change of the percent of world Bebo users who are between ages 20 and $30 .$
b. Calculate the percentage change in the percent of world Bebo users who are between ages 20 and $30 .$

Willis James

Numerade Educator

Problem 21

For the linear function
$$
f(x)=3 x+4
$$
a. Calculate the average rate of change and the percentage change in $f$ for each of the following intervals:
$$
\begin{array}{l}
\text { i. From } x=1 \text { to } x=3 \\
\text { ii. From } x=3 \text { to } x=5
\end{array}
$$
iii. From $x=5$ to $x=7$
b. On the basis of the results in part $a$ and the characteristics of linear functions presented in Chapter 1 , what generalizations can be made about percentage change and average rate of change for a linear function?

Willis James

Numerade Educator

Problem 22

For the exponential function
$$
f(x)=3\left(0.4^{x}\right)
$$
a. Calculate the percentage change and average rate of change of $f$ for each of the following intervals:
$$
\begin{array}{l}
\text { i. From } x=1 \text { to } x=3 \\
\text { ii. From } x=3 \text { to } x=5
\end{array}
$$
iii. From $x=5$ to $x=7$
b. On the basis of the results in part $a$ and the characteristics of exponential functions presented in Chapter 1 , what generalizations can be made about percentage change and average rate of change for an exponential function?

Willis James

Numerade Educator

Problem 23

Future Value The future value of $\$ 1400$ invested at a fixed interest rate compounded continuously is given in the table on page 138 .
a. Using the data, is it possible to calculate the average rate of change in the balance from the middle of the fourth year through the end of the fourth year? Explain how this could be done or why it cannot be done.
b. Find a model for the data and use the model to calculate the percentage change and the average rate of change in the balance over the last half of the fourth year.

Carson Merrill

Numerade Educator

Problem 24

Cesarean Deliveries $\quad$ The cesarean delivery rate jumped $27.5 \%$ between 2001 and 2006 and $2.6 \%$ between 2005 and $2006 .$
a. If the cesarean delivery rate was 31.1 per 100 live births in $2006,$ calculate the cesarean delivery rates in 2001 and $2005 .$
b. Use the information presented in the table to find a model for the cesarean delivery rate between 2000 and 2008 .

Carson Merrill

Numerade Educator