Stewart J. College Algebra: Concepts and Contexts

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144 CHAPTER 2

■

Linear Functions and Models

Farming has always been an important part of the U.S. economy. In the 19th

century the United States was very much an agrarian society; more than 75% of the

labor force was engaged in some aspect of farming. Westward expansion of the U.S.

population was fueled to a great extent by the search for new land to homestead

and farm. As a result there was a dramatic increase in the number of farms in the

United States. Most farms were family owned and operated. In the mid-20th cen-

tury new automated farming methods and improved strains of crops led to an in-

crease in farm productivity. Large corporate farming enterprises arose, and many

smaller farmers found it impossible to remain profitable and sold their land to the

corporate enterprises. As a result the number of farms decreased as the size of in-

dividual farms increased. Because of the efficiencies of scale, fewer workers were

required to operate these larger farms, resulting in a population shift from rural to

urban areas.

example

Number of Farms in the United States

Year

Farms

(ⴛ 1000)

1850 1449

1860 2044

1870 2660

1880 4009

1890 4565

1900 5740

1910 6366

1920 6454

1930 6295

1940 6102

1950 5388

1960 3711

1970 2780

1980 2440

1990 2143

2000 2172

Farming in the 19th century

Farming in the 21st century

The table in the margin gives the number of farms in the United States from 1850 to

2000.

(a) Draw a scatter plot of the data.

(b) Find the average rate of change in the number of farms between the following

years: (i) 1860 and 1890; (ii) 1950 and 1970.

of decline?

Solution

(a) A scatter plot is shown in Figure 3.

2000

1000

3000

4000

5000

6000

7000

1860 1880 1900 1920 1940 1960 1980 2000

figure 3 Number of farms in the

United States (in thousands)

Stephen Mcsweeny/Shutterstock.com 2009

IN CONTEXT ➤

SECTION 2.1

■

Working with Functions: Average Rate of Change 145

(b) (i) The average rate of change between 1860 and 1890 is

/year

So the number of farms increased at an average rate of 84,000 farms per year.

(ii) The average rate of change between 1950 and 1970 is

/year

Since the average rate of change is negative, the number of farms decreased at

an average rate of 130,400 farms per year.

between 1950 and 1960.

■ NOW TRY EXERCISE 25 ■

average rate of change =

2780 - 5388

1970 - 1950

=-130.4 thousand farms

average rate of change =

4565 - 2044

1890 - 1860

= 84.0 thousand farms

■ Average Speed of a Moving Object

example

Average Speed in a Bicycle Race

James and Jodi compete in a bicycle race. The graphs in Figure 4 on the next page

show the distance each has traveled as a function of time. We plot the time (in hours)

on the t-axis and the distance (in miles) on the y-axis.

(a) Describe the bicycle race.

(b) Find James’s and Jodi’s average speeds for the entire race.

(d) Find James’s average speed between t ⫽ 2 hours and t ⫽ 4 hours.

(e) Find Jodi’s average speed in the final hour of the race.

For a moving object, let be the distance it has traveled at time t. Then

the average rate of change of the function s from time t

to time t

is called

the average speed:

Average speed =

net change in distance

change in time

s 1t

2- s 1t

- t

s 1t2

If you drive your car a distance of 60 miles in 2 hours, then your average speed is

/hour

In general, if a function represents the distance traveled, the average rate of change

of the function is the average speed of the moving object.

Average Speed of a Moving Object

60 miles

2 hours

= 30 miles

146 CHAPTER 2

■

Linear Functions and Models

Solution

(a) From the graphs in Figure 4 we see that the race had a fair start with both

James and Jodi starting the race at time zero. However, the differences in the

graphs show that they don’t always cycle at the same pace. James travels at a

steady speed throughout the race, but Jodi varies her speed, traveling slowly in

the beginning of the race and speeding up in the end. Even though the graphs

are very different, in the end the race is a tie, since both competitors finish the

race at the same time.

(b) The average speed for Jodi and James is the same, since they both travel 60

miles in 5 hours:

So they each cycle at an average speed of 12 miles per hour.

distance Jodi traveled in this time is 38 ⫺ 10 ⫽ 28 miles. So Jodi’s average

speed in this time interval is

(d) Similarly, James’s average speed between and is

(e) Jodi’s average speed in the final hour of the race is

■ NOW TRY EXERCISE 27 ■

average speed =

net change in distance

change in time

60 - 38

5 - 4

= 22 mi

average speed =

net change in distance

change in time

48 - 24

4 - 2

= 12 mi

t = 4 ht = 2 h

average speed =

net change in distance

change in time

38 - 10

4 - 2

= 14 mi

average speed =

net change in distance

change in time

60 mi - 0 mi

5 h - 0 h

= 12 mi

1234

James’s bicycle race

1234

Jodi’s bicycle race

figure 4

SECTION 2.1

■

Working with Functions: Average Rate of Change 147

■ Functions Defined by Algebraic Expressions

The concept of average rate of change applies to any function. In the next example

we find average rates of change for a function defined by an algebraic expression.

example

Average Rate of Change of a Function

Find the average rate of change of the function between the follow-

ing values of x.

(a) and

(b) and

Solution

(a) The average rate of change of f between and is

So on the interval the values of the function f decrease an average of

2 units for each unit change in x.

(b) The average rate of change of f between and is

So on the interval [2, 5] the values of the function f increase an average of

7 units for each unit change in x.

■ NOW TRY EXERCISES 13 AND 15 ■

From the graphs in Figure 5 we can see why the average rate of change of f is

positive between and but negative between and .x = 1x =-3x = 5x = 2

average rate of change =

f 15 2- f 122

5 - 2

+ 42- 12

+ 42

5 - 2

= 7

x = 5x = 2

3- 3, 14

average rate of change =

f 11 2- f 1- 32

1 - 1- 32

+ 42- 11- 32

+ 42

1 + 3

=-2

x = 1x =-3

x = 5x = 2

x = 1x =-3

f 1x2= x

+ 4

2345_3_2_5_4 _1 1

A net increase in the value of f between

x=2 and x=5

A net decrease in the value of f between

x=_3 and x=1

2345_3_2_5_4 _1 1

figure 5

148 CHAPTER 2

■

Linear Functions and Models

2.1 Exercises

CONCEPTS

Fundamentals

1. (a) The average rate of change of a function between and is

(b) If and , then the average rate of change of f between and

2. The graphs of functions f, g, and h are shown. Between and the function

________ has average rate of change of 0, the function ________ has positive average

rate of change, and the function

________ has negative average rate of change.

x = 3x = 0

- 

= ________.

x = 5

x = 2f 152= 10f 122= 3

change in ____

ⵧ

2- f 1

ⵧ

x = bx = ay = f 1x 2

It is known that when an object is dropped in a vacuum, the distance the object falls

in t seconds is modeled by the function

where s is measured in feet (ignoring the effects of wind resistance on the speed).

Use the function s to find the average speed of a bungee jumper during the follow-

ing time intervals:

(a) The first second of the jump (that is, between and )

(b) The third second of the jump (that is, between and )

Solution

(a) The average speed of the bungee jumper in the first second of the jump is

(b) Similarly, the average speed of the bungee jumper in the third second of the jump is

■ NOW TRY EXERCISE 29 ■

average speed =

s 132- s 122

3 - 2

1613

2- 1612

= 80 ft

average speed =

s 112- s 102

1 - 0

1611

2 ft - 1610

2 ft

1 s - 0 s

= 16 ft

t = 3t = 2

t = 1t = 0

s 1t2= 16t

example

Bungee Jumping

SECTION 2.1

■

Working with Functions: Average Rate of Change 149

Think About It

3. True or false?

(a) If a function has positive net change between and , then the function

has positive average rate of change between and .

(b) If a function has positive average rate of change between and , then the

function has positive net change between and .

4. The graphs of the functions f and are shown in the margin. The function

_______ ( f or )

has a greater average rate of change between and . The function

_______

( f or ) has a greater average rate of change between and . The functions f

and have the same average rate of change between and .

5. Graphs of the functions f, , and h are shown below. What can you say about the average

rate of change of each function on the successive intervals [0, 1], [1, 2], [2, 3], . . .?

x ⫽ ________x ⫽ ________g

x = 2x = 1g

x = 1x = 0

6. The graph of a function f is shown below. Find x-values a and b so that the average rate

of change of f between a and b is

(a) 0 (b) 2 (c) (d) - 2- 1

y y y

1222333

SKILLS

7–10 ■ The graph of a function is given. Determine the average rate of change of the

function between the indicated points.

7. 8.

150 CHAPTER 2

■

Linear Functions and Models

9. 10.

_1 5

11–12 ■ A function is given by a table.

(a) Determine the average rate of change of the function between the given values of x.

(b) Graph the function.

change is the largest. What is this rate of change?

11. (i) Between and

(ii) Between and x = 9x = 4

x = 4x = 2

x 0 1 2 3 4 5 6 7 8 9 10

F(x) 10 20 50 70 90 80 65 60 50 60 80

12. (i) Between and

(ii) Between and x = 90x = 20

x = 50x = 0

x 0 10 20 30 40 50 60 70 80 90 100

G(x) 3.3 3.0 2.5 1.7 1.7 0.8 2.2 4.5 5.0 5.5 6.0

13–20 ■ A function is given. Determine the average rate of change of the function between

the given values of x.

13. 14.

15. 16.

17. 18.

19. 20. k 1x 2= x

;x = 2, x = 4k 1x 2=

;x = 1, x = 3

h 1x 2= 2x - x

;x = 2, x = 4h 1x 2= x

+ 3x;x =-1, x = 1

g1x 2=

+ 4;x =-2, x = 0g1x 2= 1 - 2x

;x = 0, x = 1

f 1x 2= 5 - 7x;

x =-1, x = 3f 1x 2= 3x + 2;x = 2, x = 5

21. Population of Atlanta In the latter part of the 20th century the United States

experienced a large population shift from the cities to the suburbs. This is true of

Atlanta, for example, whose population grew steadily for its first hundred years, then

began to decline. Within the last two decades Atlanta’s population has started to rise

again, as seen in the table at the top of the next page.

(a) Draw a scatter plot of the data.

(b) Find the average rate of change of the population of Atlanta between the following

years: (i) 1850 and 1950 (ii) 1950 and 2000 (iii) 1950 and 1970

CONTEXTS

SECTION 2.1

■

Working with Functions: Average Rate of Change 151

22. Cooling Soup When a bowl of hot soup is left in a room, the soup eventually cools

down to the temperature of the room. The temperature of the soup is a function of

time t. The temperature changes more slowly as the soup gets closer to room

temperature. The table below shows the temperature (in degrees Fahrenheit) of the soup

t minutes after it was set down.

(a) What was the temperature of the soup when it was initially placed on the table?

(b) Find the average rate of change of the temperature of the soup over the first 20

minutes and over the next 20 minutes. On which interval did the temperature

decline more quickly?

T 1t 2

Yea r Population

1850 2572

1860 9554

1870 21,789

1880 37,409

1890 65,533

1900 89,872

1910 154,839

1920 200,616

Year Population

1930 270,688

1940 302,288

1950 331,000

1960 487,000

1970 497,000

1980 425,000

1990 394,017

2000 416,474

(min)

(ⴗF)

0 200

5 172

10 150

15 133

20 119

25 108

30 100

(min)

(ⴗF)

35 94

40 89

50 81

60 77

90 72

120 70

150 70

Age

(yr)

Height

(in.)

0 19.25

1 28.00

2 32.50

3 36.25

4 39.63

6 44.50

8 49.25

10 54.38

Age

(yr)

Height

(in.)

12 58.75

14 64.00

15 66.50

16 69.13

17 69.50

18 69.75

19 69.88

20 69.88

Population of Atlanta, Georgia

Year 1985 1986 1987 1990 1995 1997 2000 2001 2005 2006 2007

Number of books 420 460 1300

the greatest average rate of increase.

23. Growth Rate Jason’s height is a function of his age x (in years). At various

stages in his life he grows at different rates, as shown in the table in the margin, which

gives his height every year on his birthday.

(a) Find , and .

(b) Find the average rate of change in Jason’s height from birth to 4 years and from 4 years

to 8 years. Over which period did Jason have the faster average rate of growth?

greatest average rate of growth between his 14th and 20th birthdays.

24. Rare Book Collection Between 1985 and 2007 a rare book collector purchased

books for his collection at the rate of 40 books per year. Use this information to

complete the table below showing the number of books in his collection between 1985

and 2007. (Note that not every year is given in the table.)

H 18 2H 10 2, H 14 2

H 1x 2

100

500

d (m)

t (s)

152 CHAPTER 2

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Linear Functions and Models

25. Currency Exchange Rates The euro was introduced in 1990 as a common currency

for twelve member countries of the European Union. The table in the margin shows the

value of the euro in U.S. dollars on the first business day of each year from 1999 to

2008.

(a) Draw a scatter plot of the data.

(b) Find the average rate of change of the value of the euro in U.S. dollars between the

following years: (i) 1999 and 2008 (ii) 2002 and 2006 (iii) 2005 and 2008

largest average rate of increase in terms of the U.S. dollar.

26. Rate of Increase of an Investment Julia invested $500 in a mutual fund on June 30,

2000, using a generous high school graduation gift from her aunt. Every June 30th she

records the amount in the fund. In early 2001 the stock market crashed, and by June 30

that year Julia’s investment had lost half its value. Over the course of the next year the

fund again lost half its value, but in 2003 its value tripled. The value of the mutual fund

continued to increase, and by June 30, 2006, Julia’s investment was worth $600. By

June 2007 the mutual fund had a 30% increase, that is, it increased by 30% of its value

on June 30, 2006. The value of Julia’s mutual fund is a function where t is the year.

(a) Find A(2000), A(2001), A(2002), A(2003), A(2006), and A(2007).

(b) Find the annual average rate of change of the function A between 2000 and 2007.

(d) Which is greater: the annual rate of change of A between 2002 and 2003 or

between 2003 and 2006?

27. Speed Skating At the 2006 Winter Olympics in Turin, Italy, the United States won

three gold medals in men’s speed skating. The graph in the margin shows distance as a

function of time for two speed skaters racing in a 500-meter event.

(a) Who won the race?

(b) Find the average speed during the first 10 seconds for each skater.

28. 100-Meter Race A 100-meter race ends in a three-way tie for first place. The graph

shows distance as a function of time for each of the three winners.

(a) Find the average speed for each winner.

(b) Describe the differences in the way the three runners ran the race.

A 1t 2

t (s)

d (m)

100

29. Phoenix Mars Lander On August 4, 2007, NASA’s Phoenix Mars Lander was

launched into space to search for life in the icy northern region of the planet Mars; it

touched down on Mars on May 25, 2008. As the ship raced into space, its jet fuel tanks

dropped off when they were used up. The distance one of the tanks falls in t seconds is

modeled by the function

s 1t2= 16t

Year Value (U.S. $)

1999 0.86

2000 1.01

2001 0.94

2002 0.89

2003 1.05

2004 1.26

2005 1.35

2006 1.18

2007 1.32

2008 1.46

Value of the Euro in

U.S. dollars

SECTION 2.2

■

Linear Functions: Constant Rate of Change 153

where s is measured in feet per second and is the instant the tank left the ship. Use

the function s to find the average speed of a tank during the following time intervals.

(a) The first 10 seconds after separation from the ship

(b) The first 30 seconds after separation from the ship

30. A Falling Skydiver When a skydiver jumps out of an airplane from a height of 13,000

feet, her height h (in feet) above the ground after t seconds is given by the function

(a) Use the function h to find the average speed of the skydiver during the first 5 seconds.

(b) The skydiver opens her parachute after 24 seconds. What is her average speed

during her 24 seconds of free fall?

31. Falling Cannonballs Galileo Galilei is said to have dropped cannonballs of different

sizes from the Tower of Pisa to demonstrate that their speed is independent of their

mass. The function

models the height h (in feet) of the cannonball above the ground t seconds after it is

dropped. Use the function h to find the average speed of the cannonball during the

following time intervals.

(a) The first 2 seconds (b) The first 3 seconds (c) Between t ⫽ 2 and t ⫽ 3.25

32. Path of Bullet A bullet is shot straight upward with an initial speed of 800 ft/s. The

height of the bullet after t seconds is modeled by the function

where h is measured in feet. Use the function h to find the average speed of the bullet

during the given time intervals.

(a) The first 5 seconds (b) The first 40 seconds (c) Between t ⫽ 10 and t ⫽ 40

h 1t2=-16t

+ 800t

h 1t2= 183.27 - 16t

h 1t2= 13,000 - 16t

t = 0

2.2 Linear Functions: Constant Rate of Change

■

Linear Functions

■

Linear Functions and Rate of Change

■

Linear Functions and Slope

■

Using Slope and Rate of Change

IN THIS SECTION… we consider functions for which the average rate of change is

constant. Such functions have the form , and their graphs are straight lines. We’ll

see that the number m can be interpreted as the rate of change of f or the slope of the graph of f.

In Section 2.1 we studied the average rate of change of a function on an interval.

Most of the functions we considered had different average rates of change on differ-

ent intervals. But what if a function has constant average rate of change? That is,

what if the function has the same average rate of change on every interval?

The graph in Figure 1(b) on page 154 shows the number of chocolates pro-

duced by a chocolate-manufacturing machine in t minutes (the start of a work shift

is represented by t ⫽ 0). We can see that chocolates are produced by the machine at

a fixed rate of 100 per minute. The graph in Figure 1(a) on page 154 shows the num-

ber of chocolates produced by a malfunctioning machine (which sometimes de-

stroys chocolates it has produced). That machine’s production rate varies wildly

g1t 2

f 1t 2

f 1x 2= b + mx

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