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1 Working with Functions: Average Rate of Change

1 Working with Functions: Average Rate of Change

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SECTION 2.1

Working with Functions: Average Rate of Change

143

In general, we find the average rate of change of a function by calculating the

net change in the function values and dividing by the net change in the x-values.

y

f(b)

f(b)-f(a)

Net change

in y

Average Rate of Change of a Function

The average rate of change of the function y = f 1x 2 between x = a and

x = b is

f(a)

a

b

x

average rate of change =

b-a

Change in x

f i g u r e 1 Average rate of change

of a function f

example 1

x

Hours

f(x)

Tiles

0

1

2

3

4

5

6

7

8

0

21

69

126

189

216

245

347

403

net change in y f 1b2 - f 1a2

=

change in x

b - a

The graph in Figure 1 shows that f 1b2 - f 1a2 is the net change in the value of

f and b - a is the change in the value of x.

Average Rate of Installation

Sima is installing new Italian ceramic flooring in her house. The table in the margin

gives the total number f 1x2 of tiles she has installed after working for x hours.

(a) Find the average rate of installation in the first hour.

(b) Find the average rate of installation for the first 4 hours.

(c) Find the average rate of installation from hour 6 to hour 8.

(d) Draw a graph of f and use the graph to find the hour in which Sima had the

fastest average rate of installation.

Solution

(a) We find the average rate of change of the function f between x = 0 and x = 1:

average rate of change =

f 112 - f 102

1 - 0

=

21 tiles - 0 tiles

= 21 tiles/hour

1 hour

So the average rate of installation for the first hour is 21 tiles per hour.

(b) We find the average rate of change of the function f between x = 0 and x = 4:

average rate of change =

4 - 0

=

189 - 0

= 47.25 tiles/hour

4

So the average rate of installation for the first 4 hours is about 47 tiles per hour.

(c) We find the average rate of change of the function f between x = 6 and x = 8:

y

400

300

average rate of change =

200

100

0

f 142 - f 102

1 2 3 4 5 6 7 8 x

figure 2

f 182 - f 162

8 - 6

=

403 - 245

= 79 tiles/hour

2

So the average rate of installation for this time period is about 79 tiles per hour.

(d) A scatter plot is shown in Figure 2. We can see from the graph that the

steepest rise in the graph is during the seventh hour (from hour 6 to hour 7), so

the fastest average rate of installation occurred in the seventh hour.

NOW TRY EXERCISE 21

CHAPTER 2

Linear Functions and Models

IN CONTEXT ➤

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 century new automated farming methods and improved strains of crops led to an increase 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 individual 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.

Stephen Mcsweeny/Shutterstock.com 2009

144

Farming in the 19th century

Farming in the 21st century

e x a m p l e 2 Number of Farms in the United States

Year

Farms

(؋ 1000)

1850

1860

1870

1880

1890

1900

1910

1920

1930

1940

1950

1960

1970

1980

1990

2000

1449

2044

2660

4009

4565

5740

6366

6454

6295

6102

5388

3711

2780

2440

2143

2172

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.

(c) In which decade did the number of farms experience the greatest average rate

of decline?

Solution

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

y

7000

6000

5000

4000

3000

2000

1000

f i g u r e 3 Number of farms in the

United States (in thousands)

0

1860 1880 1900 1920 1940 1960 1980 2000 x

SECTION 2.1

Working with Functions: Average Rate of Change

145

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

average rate of change =

4565 - 2044

= 84.0 thousand farms/year

1890 - 1860

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

average rate of change =

2780 - 5388

= - 130.4 thousand farms/year

1970 - 1950

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

an average rate of 130,400 farms per year.

(c) From the graph we see that the steepest drop in a single decade occurred

between 1950 and 1960.

2

NOW TRY EXERCISE 25

■ Average Speed of a Moving Object

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

60 miles

= 30 miles/hour

2 hours

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

For a moving object, let s 1t 2 be the distance it has traveled at time t. Then

the average rate of change of the function s from time t1 to time t2 is called

the average speed:

Average speed =

net change in distance s 1t 2 2 - s 1t1 2

=

change in time

t 2 - t1

e x a m p l e 3 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.

(c) Find Jodi’s average speed between t ϭ 2 hours and t ϭ 4 hours.

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

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

146

CHAPTER 2

Linear Functions and Models

y

y

60

60

50

50

40

40

30

30

20

20

10

10

0

1 2 3 4 5 t

James’s bicycle race

0

1 2 3 4 5 t

Jodi’s bicycle race

figure 4

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:

average speed =

net change in distance 60 mi - 0 mi

=

= 12 mi/h

change in time

5h - 0h

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

(c) The total time elapsed is 4 Ϫ 2 ϭ 2 hours. From the graph we see that the

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

speed in this time interval is

average speed =

net change in distance 38 - 10

=

= 14 mi/h

change in time

4 -2

(d) Similarly, James’s average speed between t = 2 h and t = 4 h is

average speed =

net change in distance 48 - 24

=

= 12 mi/h

change in time

4 -2

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

average speed =

net change in distance 60 - 38

=

= 22 mi/h

change in time

5 -4

NOW TRY EXERCISE 27

SECTION 2.1

2

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.

e x a m p l e 4 Average Rate of Change of a Function

Find the average rate of change of the function f 1x2 = x 2 + 4 between the following values of x.

(a) x = - 3 and x = 1

(b) x = 2 and x = 5

Solution

(a) The average rate of change of f between x = - 3 and x = 1 is

average rate of change =

f 112 - f 1- 3 2

1 - 1- 32

=

11 2 + 42 - 11- 32 2 + 42

1 +3

= -2

So on the interval 3- 3, 14 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 x = 2 and x = 5 is

average rate of change =

f 152 - f 122

5 -2

=

15 2 + 42 - 12 2 + 42

5 -2

=7

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

y

y

30

30

20

20

10

10

_5 _4 _3_2 _1 0

1 2 3 4 5 x

A net increase in the value of f between

x=2 and x=5

figure 5

_5 _4 _3_2 _1 0

1 2 3 4 5 x

A net decrease in the value of f between

x=_3 and x=1

148

CHAPTER 2

Linear Functions and Models

e x a m p l e 5 Bungee Jumping

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

s 1t2 = 16t 2

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 following time intervals:

(a) The first second of the jump (that is, between t = 0 and t = 1)

(b) The third second of the jump (that is, between t = 2 and t = 3)

Solution

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

average speed =

s 112 - s 102

1 -0

=

1611 2 2 ft - 1610 2 2 ft

1s - 0s

= 16 ft/s

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

average speed =

s 132 - s 122

3 -2

1613 2 2 - 1612 2 2

=

1

= 80 ft/s

NOW TRY EXERCISE 29

2.1 Exercises

CONCEPTS

Fundamentals

1. (a) The average rate of change of a function y = f 1x 2 between x = a and x = b is

change in ____ f 1

=

change in ____

ٗ 2 - f 1ٗ 2 .

ٗ-ٗ

(b) If f 122 = 3 and f 152 = 10, then the average rate of change of f between x = 2 and

x = 5 is

-

= ________.

2. The graphs of functions f, g, and h are shown. Between x = 0 and x = 3 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.

y

y

y

f

h

g

0

1

x

0

1

x

0

1

x

SECTION 2.1

149

Working with Functions: Average Rate of Change

y

6

5

4

3

2

1

3. True or false?

(a) If a function has positive net change between x = 0 and x = 1, then the function

has positive average rate of change between x = 0 and x = 1.

(b) If a function has positive average rate of change between x = 0 and x = 1, then the

function has positive net change between x = 0 and x = 1.

g

4. The graphs of the functions f and g are shown in the margin. The function _______ ( f or g)

has a greater average rate of change between x = 0 and x = 1. The function _______

( f or g) has a greater average rate of change between x = 1 and x = 2. The functions f

f

0

and g have the same average rate of change between x ϭ ________ and x ϭ ________.

1

2

x

5. Graphs of the functions f, g, 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], . . .?

y

y

y

6

5

4

3

2

1

6

5

4

3

2

1

6

5

4

3

2

1

f

0

1

2

x

3

g

0

1

2

0

x

3

h

1

2

3

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) - 1

(d) - 2

y

1

0

SKILLS

7–10

7.

x

1

The graph of a function is given. Determine the average rate of change of the

function between the indicated points.

8.

y

y

5

4

3

2

0

1

4

x

0

1

5

x

x

150

CHAPTER 2

Linear Functions and Models

9.

10.

y

y

6

4

2

0

_1

0

1

5

x

x

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.

(c) From your graph, find the two successive points between which the average rate of

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

11. (i) Between x = 2 and x = 4

(ii) Between x = 4 and x = 9

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 x = 0 and x = 50

(ii) Between x = 20 and x = 90

x

G(x)

13–20

0

10

20

30

40

50

60

70

80

90

100

3.3

3.0

2.5

1.7

1.7

0.8

2.2

4.5

5.0

5.5

6.0

A function is given. Determine the average rate of change of the function between

the given values of x.

13. f 1x2 = 3x + 2; x = 2, x = 5

2

15. g1x2 = 1 - 2x ;

16. g1x2 =

1 2

2x

+ 4; x = - 2, x = 0

17. h 1x2 = x + 3x; x = - 1, x = 1

18. h 1x2 = 2x - x 2;

6

19. k 1x 2 = ;

x

20. k 1x 2 = x 3;

2

CONTEXTS

x = 0, x = 1

14. f 1x2 = 5 - 7x; x = - 1, x = 3

x = 1, x = 3

x = 2, x = 4

x = 2, x = 4

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

SECTION 2.1

Working with Functions: Average Rate of Change

151

(c) Use the scatter plot to find the decade in which Atlanta’s population experienced

the greatest average rate of increase.

Population of Atlanta, Georgia

Year

Population

Year

Population

1850

1860

1870

1880

1890

1900

1910

1920

2572

9554

21,789

37,409

65,533

89,872

154,839

200,616

1930

1940

1950

1960

1970

1980

1990

2000

270,688

302,288

331,000

487,000

497,000

425,000

394,017

416,474

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 T 1t 2 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?

Age

(yr)

Height

(in.)

Age

(yr)

Height

(in.)

0

1

2

3

4

6

8

10

19.25

28.00

32.50

36.25

39.63

44.50

49.25

54.38

12

14

15

16

17

18

19

20

58.75

64.00

66.50

69.13

69.50

69.75

69.88

69.88

t

(min)

T

(؇F)

t

(min)

T

(؇F)

0

5

10

15

20

25

30

200

172

150

133

119

108

100

35

40

50

60

90

120

150

94

89

81

77

72

70

70

23. Growth Rate Jason’s height H 1x2 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 H 102, H 14 2 , and H 18 2 .

(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?

(c) Draw a graph of H, and use the graph to find the year in which Jason had the

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.)

Year

1985

1986

Number of books

420

460

1987

1990

1995

1997

2000

2001

2005

2006

2007

1300

152

CHAPTER 2

Linear Functions and Models

Value of the Euro in

U.S. dollars

Year

Value (U.S. \$)

1999

2000

2001

2002

2003

2004

2005

2006

2007

2008

0.86

1.01

0.94

0.89

1.05

1.26

1.35

1.18

1.32

1.46

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

(c) Use the scatter plot to find the year in which the value of the Euro experienced the

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 A 1t2 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.

(c) Draw a scatter plot of A using your data from part (a).

(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.

(c) Find the average speed during the last 15 seconds for each skater.

d (m)

500

A

B

100

0

10

t (s)

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.

d (m)

100

A

B

50

0

C

5

10 t (s)

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 2

SECTION 2.2

Linear Functions: Constant Rate of Change

153

where s is measured in feet per second and t = 0 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

h 1t2 = 13,000 - 16t 2

(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

h 1t2 = 183.27 - 16t 2

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 ϭ 2 and t ϭ 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

h 1t 2 = - 16t 2 + 800t

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 ϭ 10 and t ϭ 40

2

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

Ralph Hagen/www.CartoonStock.com

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

constant. Such functions have the form f 1x2 = b + mx, 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 different 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 f 1t 2 produced by a chocolate-manufacturing machine in t minutes (the start of a work shift

is represented by t ϭ 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 number of chocolates g1t2 produced by a malfunctioning machine (which sometimes destroys chocolates it has produced). That machine’s production rate varies wildly

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