Calculus Concepts

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

Chapter 5

Accumulating Change: Limits of Sums and the Definite Integral - all with Video Answers

Educators

Section 1

An Introduction to Results of Change

Problem 1

Bacteria Growth The growth rate of bacteria (in thousand organisms per hour) in milk at room temperature is $b(t),$ where $t$ is the number of hours that the milk has been at room temperature.
a. What does the area of the region between the graph of $b$ lying above the $t$ -axis and the $t$ -axis represent?
b. What are the units of measure of
i. The height and width of region in part $a$ ?
ii. The area of the region between the graph of $b$ and the $t$ -axis?

Nick Johnson

Nick Johnson

Numerade Educator

Problem 2

Learning Curve The rate at which an assembly-line worker is learning a new skill can be represented by $s(t)$ percentage points per hour, where $t$ is the time the worker has spent on task.
a. What does the area of the region between the graph of $s$ and the $t$ -axis represent?
b. What are the units of measure of
i. The height and width of region in part $a$ ?
ii. The area of the region between the graph of $s$ and the $t$ -axis?

Nick Johnson

Numerade Educator

Problem 3

Braking Distance The distance required for a car to stop is a function of the speed of the car when the brakes are applied. The rate of change of the stopping distance can be expressed in feet per $\mathrm{mph}$, where the input variable is the speed of the car (in mph) when the brakes are first applied.
a. What does the area of the region between the rate-ofchange graph and the input axis from $40 \mathrm{mph}$ to 60 mph represent?
b. What are the units of measure of
i. The height and width of the region in part $a$ ?
ii. The area of the region in part $a$ ?

Nick Johnson

Numerade Educator

Problem 4

Car Acceleration The acceleration of a car (in feet per second per second) during a test conducted by a car manufacturer is given by $A(t),$ where $t$ is the number of seconds since the beginning of the test.
a. What does the area of the region between the portion of the graph of $A$ lying above the $t$ -axis and the $t$ -axis tell about the car?
b. What are the units or measure of
i. The height and width of the region in part $a$ ?
ii. The area of the region between the graph of $A$ and the $t$ -axis?

Nick Johnson

Numerade Educator

Problem 5

Atmospheric $\mathrm{CO}_{2}$ The rate of change in atmospheric carbon dioxide (measured in parts per million per year) is shown in the figure. The data was collected at Law Dome East Antarctica.
$c^{\prime}(t)$ ppm
a. What are the units of measure of height, width, and area of the region between the rate-of-change graph and the horizontal axis from 1958 to 2008 ?
b. How much did atmospheric carbon dioxide increase berween 1958 and 2008

Nick Johnson

Numerade Educator

Problem 6

Girl Scout Cookies The marginal revenue (rate of change of revenue with respect to quantity sold) from a council's sale of Girl Scout cookies is shown in the figure.a. What are the units of measure of height, width, and area of the region between the marginal revenue graph and the horizontal axis from 0 to $3.95 ?$
b. Calculate the area of the region between the marginal revenue graph and the horizontal axis between 0 and 3.95. Write a sentence of interpretation for this result.
c. Calculate the area of the region between the marginal revenue graph and the horizontal axis between 3 and 3.95. Write a sentence of interpretation for this result.

Nick Johnson

Numerade Educator

Problem 7

Commuting Time A student estimates that his daily commute to college consists of 10 minutes driving at a speed of 30 mph to a divided highway, followed by 5 minutes in which he accelerates to 70 miles per hour, and 15 minutes driving at 70 mph before slowing to exit and enter the parking lot. The figure shows his velocity in terms of time.
$v(t)$ miles per hour
a. What are the units of measure of height, width, and area of the region between the speed graph and the horizontal axis?
b. How far does the student drive on his commute from home before exiting to the parking lot?

Nick Johnson

Numerade Educator

Problem 8

Robot Speed A prototype robot takes 1 minute to accelerate to $10 \mathrm{mph}(880$ feet per minute). The robot maintains that speed for 2 minutes and then takes half a minute to come to a complete stop. The robot's acceleration and deceleration are constant. The figure shows the robot's speed.
$v(t)$ feet
per
a. Calculate the area of the shaded region between the graph and the horizontal axis.
b. What is the practical interpretation of the area found in part $a$ ?

Nick Johnson

Numerade Educator

Problem 9

North Dakota Population (Historic) The rate of change of the population of North Dakota from 1970 through 1990 can be modeled as
$p^{\prime}(t)=\left\{\begin{array}{cl}3.87 & \text { when } 0 \leq t<15 \\ -7.39 & \text { when } 15 \leq t \leq 30\end{array}\right.$
where $p^{\prime}$ is measured in thousand people per year and $t$ represents the number of years since $1970 .$ The figure shows a graph of $p$ ? (Source: Based on data from Statistical Abstract, 1994$)$
$p^{\prime}(t)$ thousand people per year
a. Calculate the area of the region between the graph of $p$ and the horizontal axis from 0 to $15 .$ Interpret the result.
b. Calculate the area of the region between the graph of $p$ ' and the horizontal axis from 15 to $30 .$ Interpret the result.
c. Was the population of North Dakota in 1990 greater or less than the population in $1970 ?$ By how much did the population change between 1970 and $2000 ?$

Nick Johnson

Numerade Educator

Problem 10

Cottage Cheese Consumption The rate of change of the per capita consumption of cottage cheese in the United States between 1980 and 1996 can be modeled as
where $c^{\prime}$ is measured in pounds per year and $t$ is the number of years since $1980 .$ The figure shows a graph of $c^{\prime}$. (Source: Based on data from Statistical Abstract, 2001
$c^{\prime}(t)$ pounds
a. Calculate the area of the region between the graph and the horizontal axis between $t=0$ and $t=19$
b. Interpret the area in part $a$ in the context of cottage cheese consumption.

Nick Johnson

Numerade Educator

Problem 11

Hospital Stay The rate of change of the length of the average hospital stay between 1993 and 2000 can be modeled as
$$
s^{\prime}(t)=\left\{\begin{array}{ll}
0.082 t-0.39 & \text { when } 0 \leq t<5 \\
-0.1 t & \text { when } 5 \leq t \leq 7
\end{array}\right.
$$
where $s^{\prime}$ is measured in days per year and $t$ is the number of years since $1993 .$ The figure shows a graph of $s^{\prime}$. (Source: Based on data from Statistical Abstract, 2001$)$
a. Calculate the area of the region lying above the axis between the graph and the $t$ -axis.
b. Calculate the total area of the regions lying below the axis between the graph and the $t$ -axis.
c. Use the answers to parts $a$ and $b$ to estimate by how much the average hospital stay changed between 1993 and 2000 .

Nick Johnson

Numerade Educator

Problem 12

Gap Earnings The rate of change in the net earnings of Gap, Inc., from 1992 through 2001 can be modeled as
$g^{\prime}(t)=\left\{\begin{array}{ll}45.86 t-133.31 & \text { when } 0<t<9 \\ -567 & \text { when } 9<t<11\end{array}\right.$
where $g$ ' is measured in million dollars per year and $t$ is the number of years since $1990 .$ The figure shows a graph of $g^{\prime}$ (Source: Based on data from the Gap, Inc., Annual Report, 2001$)$
$g^{\prime}(t)$ million
per year
a. Calculate the area of the region lying above the axis between the graph and the $t$ -axis.
b. Calculate the total area of the regions lying below the axis between the graph and the $t$ -axis.
c. Use the answers to parts $a$ and $b$ to estimate by how much Gap, Inc., earnings changed berween 1990 and 2000 .

Nick Johnson

Numerade Educator

Problem 13

Cell Phone Bill The figure shows the rate of change in the average monthly cell phone bill in the United States between 1990 and 2008 .
$c^{\prime}(x)$ dollars
(Source: Based on information from CTIA Wireless Industry Survey, 2008 )
a. Calculate the area of the region lying above the axis between the graph and the $x$ -axis.
b. Calculate the total area of the regions lying below the axis between the graph and the $x$ -axis.
c. Use the answers to parts $a$ and $b$ to estimate by how much the average monthly cell phone bill in the United States changed between 1990 and 2008.

Nick Johnson

Numerade Educator

Problem 14

Median Family Income The figure shows the rate of change of the median family income in the United States during 2008 (in 2008 inflation-adjusted dollars) as a function of the size of the family.
$f^{\prime}(n)$ dollars
per person
(Source: Based on data from U.S. Burcau of the Census, 2008 American Community Survey)
a. Calculate the total area of the regions lying above the $t$ -axis between the graph and the $t$ -axis.
b. Calculate the area of the region lying below the axis between the graph and the $t$ -axis.
c. Use the answers to parts $a$ and $b$ to estimate by how much the median family income changed when the family size increased from 2 to 7 people. Interpret the answer.

Nick Johnson

Numerade Educator

Problem 15

Pencil Production The figure shows the rate of change of profit at various production levels for a pencil manufacturer.
$$
p^{\prime}(n)
$$
dollars per box
Use the graph of $p^{\prime}$ to complete the statements about $p$. Write NA if the statement cannot be completed using the information provided.
a. Profit is increasing when daily production is between and between
b. The profit when 500 boxes of pencils are produced is dollars.
c. Profit is higher than nearby profits at a production level of $\quad$ boxes, and it is lower than nearby profits at a production level of $\quad$ boxes of pencils.
d. The profit is decreasing most rapidly when boxes are produced.
e. The units of measure of the area of the region between a graph of the rate of change of profit and the production-level axis between production levels of 100 and 200 boxes is

Nick Johnson

Numerade Educator

Problem 16

Orchard Cost The figure shows the rate of change of cost for an orchard in Florida at various production levels during grapefruit season. Use the graph of $c^{\prime}$ to complete the statements about $c$. Write NA if the statement cannot be completed using the information provided.
a. Cost is increasing when grapefruit production is between $\quad$ and $\quad$ cartons
b. The cost to produce 100 cartons of grapefruit is dollars.
c. The cost is lower than nearby costs at a production level of $\quad$ cartons.
d. It is higher than nearby costs at a production level of cartons of grapefruit.
e. The cost is increasing most rapidly when tons are produced.
f. The units of measure of the area of the region between a graph of the rate of change of cost and the productionlevel axis between production levels of 50 and 150 cartons is

Nick Johnson

Numerade Educator

Problem 17

Wild Turkey Restoration In $1951,$ wild turkeys werc harvested in only two counties in Tennessee. Through intensive restoration efforts, wildlife officials were able to restore turkeys to all 95 counties by $1995 .$ The figure shows the rate-of-change graph of Tennessee counties with wild turkeys.
a. What are the units on the area of the region between the graph of $c^{\prime}$ and the $x$ -axis?
b. Write a sentence interpreting the behavior of $c^{\prime}$ at $x=27.4$
c. What does the area of the region between the graph of $c^{\prime}$ and the $x$ -axis from $c=0$ to $c=27.4$ represent?

Nick Johnson

Numerade Educator

Problem 18

Girl Scout Cookies The marginal cost (rate of change of cost) from a council's sale of Girl Scout cookies is shown in the figure.
$C^{\prime}(q)$ dollars
a. What are the units of measure for the area of the region between the marginal-cost graph and the horizontal axis?
b. Write a sentence of interpretation for $C^{\prime}(3.395)$
c. Write a sentence of interpretation for the area of the region between the marginal-cost graph and the horizontal axis between $q=0$ and $q=3.395$

Nick Johnson

Numerade Educator

Problem 19

Explain how area and accumulated change are related and how they differ.

Nick Johnson

Numerade Educator

Problem 20

Describe how to locate a relative maximum, a relative minimum, and an inflection point on a function graph by using its rate-of-change graph.

Nick Johnson

Numerade Educator