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C. The Average Rate of Change Formula

C. The Average Rate of Change Formula

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Average Rates of Change Applied to Projectile Velocity

A projectile is any object that is thrown, shot, or cast upward, with no continuing

source of propulsion. The object’s height (in feet) after t sec is modeled by the function

h1t2 ϭ Ϫ16t2 ϩ vt ϩ k, where v is the initial velocity of the projectile, and k is the

height of the object at t ϭ 0. For instance, if a soccer ball is kicked vertically upward

from ground level (k ϭ 0) with an initial speed of 64 ft/sec, the height of the ball t sec

later is h1t2 ϭ Ϫ16t2 ϩ 64t. From Section 3.3, we recognize the graph will be a

parabola and evaluating the function for t ϭ 0 to 4 produces Table 3.1 and the graph

shown in Figure 3.62. Experience tells us the ball is traveling at a faster rate immediately after being kicked, as compared to when it nears its maximum height where it

¢height

momentarily stops, then begins its descent. In other words, the rate of change

¢time

has a larger value at any time prior to reaching its maximum height. To quantify this

we’ll compute the average rate of change between (a) t ϭ 0.5 and t ϭ 1, and compare

it to the average rates of change between, (b) t ϭ 1 and t ϭ 1.5, and (c) t ϭ 1.5 and

t ϭ 2.

Table 3.1

Time in

seconds

WORTHY OF NOTE

Keep in mind the graph of h

represents the relationship between

the soccer ball’s height in feet and

the elapsed time t. It does not

model the actual path of the ball.



EXAMPLE 4







Figure 3.62



Height in

feet



0



0



1



48



2



64



3



48



4



0



h(t)

80



(2, 64)

60



(3, 48)



(1, 48)

40

20



0



1



2



3



4



t



5



Average Rates of Change Applied to Projectiles

For the projectile function h1t2 ϭ Ϫ16t2 ϩ 64t, find the average rate of change for

a. t ʦ 30.5, 1 4 .

b. t ʦ 31, 1.54 .

c. t ʦ 31.5, 2.04 .

Then graph the secant lines representing these average rates of change and comment.



Solution







h1t2 2 Ϫ h1t1 2

¢h

ϭ

yields

¢t

t2 Ϫ t 1

h11.52 Ϫ h112

h122 Ϫ h11.52

¢h

¢h

b.

c.

ϭ

ϭ

¢t

1.5 Ϫ 1

¢t

2 Ϫ 1.5

60 Ϫ 48

64 Ϫ 60

ϭ

ϭ

0.5

0.5

ϭ 24

ϭ8



Using the given intervals in the formula

a.



h112 Ϫ h10.52

¢h

ϭ

¢t

1 Ϫ 10.52

48 Ϫ 28

ϭ

0.5

ϭ 40



For t ʦ 3 0.5, 1 4 , the average rate of change is 40

1,

meaning the height of the ball is increasing at an

average rate of 40 ft/sec. For t ʦ 31, 1.5 4 , the average

rate of change has slowed to 24

1 , and the soccer ball’s

height is increasing at only 24 ft/sec. In the interval

[1.5, 2], the average rate of change has slowed to

8 ft/sec. The secant lines representing these rates of

change are shown in the figure, where we note the line

from the first interval (in red), has a much steeper

slope than the line from the third interval (in gold).



h(t)

80



(2, 64)



60



(1.5, 60)

(1, 48)



40



(0.5, 28)

20



(4, 0)



(0, 0)

0



1



2



3



4



5



Now try Exercises 27 through 34



t







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The calculation for average rates of change can be applied to any function y ϭ f 1x2

and will yield valuable information—particulary in an applied context. For practice

with other functions, see Exercises 35 to 42.

You may have had the experience of riding in the external elevator of a modern

building, with a superb view of the surrounding area as you rise from the bottom floor.

For the first few floors, you note you can see much farther than from ground level. As

you ride to the higher floors, you can see still farther, but not that much farther due to

the curvature of the Earth. This is another example of a nonconstant rate of change.

EXAMPLE 5







Average Rates of Change Applied to Viewing Distance

The distance a person can see depends their elevation

above level ground. On a clear day, this viewing

distance can be approximated by the function

d1h2 ϭ 1.2 1h, where d(h) represents the viewing

distance (in miles) at height h (in feet) above level

ground. Find the average rate of change to the

nearest 100th, for

a. h ʦ 3 9, 16 4

b. h ʦ 3 196, 225 4

c. Graph the function along with the lines

representing the average rate of change and comment on what you notice.



Solution







d1h2 2 Ϫ d1h1 2

¢d

ϭ

¢h

h2 Ϫ h1

d12252 Ϫ d11962

¢d

b.

ϭ

¢h

225 Ϫ 196

18 Ϫ 16.8

ϭ

29

Ϸ 0.04



Use the points given in the formula:



d1162 Ϫ d192

¢d

ϭ

¢h

16 Ϫ 9

4.8 Ϫ 3.6

ϭ

7

Ϸ 0.17

0.17

¢d

c. For h ʦ 39, 16 4,

Ϸ

, meaning the viewing distance is increasing at an

¢h

1

average rate of 0.17 miles (about 898 feet) for each 1 ft increase in elevation.

¢d

0.04

For h ʦ 3 196, 225 4,

and the viewing distance is increasing at a rate

Ϸ

¢h

1

of only 0.04 miles (about 211 feet) for each increase of 1 ft. We’ll sketch the

graph using the points (0, 0), (9, 3.6), (16, 4.8), (196, 16.8) and (225, 18),

along with (100, 12) and (169, 15.6) to help round out the graph (Figure 3.63).



a.



Figure 3.63



Viewing distance



20



(225, 18)

(169, 15.6)



16



(196, 16.8)



12



(100, 12)

8



(16, 4.8)

(9, 3.6)



4

0

0



20



40



60



80



100



120



140



160



180



200



220



240



Height



C. You’ve just seen how

we can calculate average rates

of change using the average

rate of change formula



Note the slope of the secant line through the points (9, 3.6) and (16, 4.8), has a

much steeper slope than the line through (196, 16.8) and (225, 18).

Now try Exercises 51 through 56







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Section 3.4 Quadratic Models; More on Rates of Change



3.4 EXERCISES





CONCEPTS AND VOCABULARY



Fill in each blank with the appropriate word or phrase. Carefully reread the section if needed.



1. Data that indicate a gradual increase to a maximum

value, matched by a like decrease, might best be

modeled using

regression. In the case

described, the coefficient of the x2-term will be

.



2. For linear functions, the rate of change is

. For

functions, the rate of

change is not constant.



3. The

rate of change of a function can be

found by calculating the

of the line that

passes through two points on the graph of the

function. For nonlinear functions, this is called

a

.



4. The average rate of change of a function f(x)

between x1 and x2 is given by the formula

. To avoid division by

, x1

cannot equal x2.



5. Discuss/Explain the differences between the three

types of regressions we have studied so far (linear,

power, and quadratic). Include examples that

highlight the different behavior of each of these

types of models.





6. Given f 1x2 ϭ 2x2 Ϫ 12x, compare the average rate

¢y

of change near the vertex, with

on either side

¢x

of the vertex. Include specific intervals and values

in your discussion.



DEVELOPING YOUR SKILLS



Use a graphing calculator and the tables shown to find

(a) a quadratic regression equation that models the

data, (b) the output of the function for x ‫ ؍‬3.5, and

(c) the positive value of x where f(x) ϭ 39. Round to

nearest thousandths as necessary.



7.



9.



8.



x



x



y



1



Ϫ5.69



1



3



0.73



2



Use a graphing calculator and the tables shown to find

(a) a quadratic regression equation y ϭ f(x) that models

the data, (b) the value of f(x) at the vertex, and (c) the

two values of x where f(x) ϭ 25. Round to nearest

thousandths.



11.



12.



x



y



x



y



4.29



1



37.8



2



Ϫ48.7



10.72



3



12.1



3



Ϫ66.2



15.3



5



Ϫ59.3



y



5



18.75



4



37.74



5



7



48.37



5



58.33



6



28.2



7



8.5



7



113.67



8



71.9



8



98.6



x



y



x



y



8



67.53



x



y



10.



13.



14.



x



y



0



67.6



0



Ϫ7.6



1



67.8



1



Ϫ1.0



2



63.4



2



Ϫ13.9



5



29.6



10



30.4



9.9



20



50.1



3



68.5



3



Ϫ14.2



10



4



67.7



4



Ϫ17.5



15



3.3



30



56.7



Ϫ8.2



40



76.5



5



55.1



5



Ϫ5.1



20



6



51.7



6



3.2



25



Ϫ17.1



50



71.9



Ϫ7.9



60



73.6



7



47.3



7



27.2



30



8



31.4



8



42.6



35



4.6



70



55.4



6.8



80



53.2



9



27.2



9



88.4



40



10



9.9



10



105.3



45



25.7



90



19.3



50



54.8



100



5.2



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Using the graphs shown, find the average rate of change

of f and g for the intervals specified.

y

7



y



f(x)

(4, 0.9)

Ϫ6



(Ϫ5, Ϫ3.6)

(Ϫ3, Ϫ4)



(5, 13.5)



15



(6, 4.1)



5



(0, 0)



7 x



(2, Ϫ1.5)

(Ϫ1, Ϫ3.6)



Ϫ5



(4, 3.2)



Ϫ6



6 x

Ϫ5



(Ϫ5, Ϫ8.5)



31. Exercise 27.



g(x)



10



(Ϫ3, 3.9)



Graph the function and secant lines representing the

average rates of change for the exercises given. Comment

on what you notice in terms of the projectile’s velocity.



(1, Ϫ2.5)



Ϫ10



15. f 1x2 for x ʦ 3 Ϫ5, Ϫ1 4

16. f 1x2 for x ʦ 3 Ϫ1, 4 4



32. Exercise 28.



17. f 1x2 for x ʦ 3 2, 64



18. f 1x2 for x ʦ 3 Ϫ3, 2 4



19. f 1x2 for x ʦ 3Ϫ5, Ϫ34



20. f 1x2 for x ʦ 3 Ϫ3, Ϫ1 4

21. g1x2 for x ʦ 3Ϫ3, 14

22. g1x2 for x ʦ 3 0, 14



33. Exercise 29.



500

450

400

350

300

250

200

150

100

50

0



20

18

16

14

12

10

8

6

4

2

0



70



1



1.5



2



25. g1x2 for x ʦ 3 Ϫ3, 54



40



60



24. g1x2 for x ʦ 3 0, 44



50



30



26. g1x2 for x ʦ 3 0, 5 4



20

10

0



34. Exercise 30.



1



3



2



5



4



6



7



8



2.5



3



3.5



16

14

12

10

8



27. h1t2 ϭ Ϫ16t ϩ 160t, h in feet

a. [2, 5]

b. [3, 5]

c. [4, 5]

d. [5, 7]

2



28. h1t2 ϭ Ϫ16t ϩ 32t, h in feet

a. [0.25, 1]

b. [0.5, 1]

c. [0.75, 1]

d. [1, 1.5]



0.5



80



23. g1x2 for x ʦ 3 Ϫ3, 4 4



The functions shown are projectile equations where h(t)

¢h

is the height of the projectile after t sec. Calculate

¢t

over the intervals indicated. Include units of measurement

in your answer.



1 2 3 4 5 6 7 8 9 10



6

4

2

0



0.5



1



1.5



2



2



29. h1t2 ϭ Ϫ4.9t2 ϩ 34.3t ϩ 2.6, h in meters

a. [1, 6]

b. [2, 5]

c. [2.5, 3.5]

d. [3.5, 4.5]

30. h1t2 ϭ Ϫ4.9t2 ϩ 14.7t ϩ 4.1, h in meters

a. [1, 2]

b. [0.2, 2.8]

c. [0.5, 1.5]

d. [1.5, 2.5]



Graph each function in an appropriate window, then

find the average rate of change for the interval specified.

Round to hundredths as needed.



35. y ϭ x3 Ϫ 8; 32, 5 4



3

36. y ϭ 2

x ϩ 5; 3 Ϫ5, 34



37. y ϭ 2Ϳx ϩ 3Ϳ; 3 Ϫ4, 04



38. y ϭ Ϳ3x ϩ 1ͿϪ2; 3Ϫ2, 2 4

39. F ϭ 9.8m; [70, 100]

40. ␳ ϭ 0.2m; [1.3, 1.5]

41. A ϭ ␲r2; [5, 7]

4

42. V ϭ ␲r3; [5, 7]

3



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WORKING WITH FORMULAS



43. Height of a falling object: h1t2 ‫ ؍‬؊16t 2 ؊ v0 t ؉ h0

Neglecting air resistance, the height of an object

that is thrown straight downward with velocity v0

from a height of h0 is given by the formula shown,

where h(t) represents the height at time t. The

Earth’s longest vertical drop (on land) is the Rupal

Face on Nanga Parbat (Pakistan), which rises

15,000 ft above its base. From the top of this rock

face, a climber’s piton hammer slips from her hand

and is projected downward with an initial velocity

of 6 ft/sec. Determine the hammer’s height after

(a) t ϭ 5 sec and (b) t ϭ 7 sec. (c) Use the results

to calculate the average rate of change over this







337



Section 3.4 Quadratic Models; More on Rates of Change



2-sec interval. (d) Repeat parts a, b, and c for

t ϭ 10 sec and t ϭ 12 sec and comment.

44. The Difference Quotient: D1x2 ‫؍‬



f 1x ؉ h2 ؊ f 1x2

h



As we’ll see in Section 3.6, the difference quotient

is closely related to the average rate of change.

a. Given f 1x2 ϭ x2 and h ϭ 0.1, evaluate D(3)

using the formula.

b. Calculate the average rate of change of f (x)

over the interval [3, 3.1] and comment on what

you notice.



APPLICATIONS



45. Registration for 5-km

Registration

race: A local community

Day

Total

hosts a popular 5-km race

1

791

to raise money for breast

2

1688

cancer research. Due to

3

2407

certain legal restrictions,

4

3067

only the first 5000

registrants will be allowed

5

3692

to compete. The table

shows the cumulative number of registered

participants at the end of the day, for the first

5 days. (a) Use a graphing calculator to find a

quadratic regression equation that models the data.

Use this equation to estimate (b) the number of

participants after 1 week of registration, (c) the

number of days it will take for the race to fill up,

and (d) the maximum number of participants that

would have signed up had there been no limit.

Round to the nearest hundredth when necessary.

46. Concert tickets: In San

Ticket Sales

Francisco, the Javier

Week

Total

Mendoza Band has

1

17,751

scheduled a concert at

2

31,266

Candlestick Park. Once

3

45,311

the tickets go on sale, the

band is sure to sell out

4

54,986

this 70,000 person venue.

The table shows the cumulative number of tickets

sold each week, for the first 4 weeks. (a) Use a

graphing calculator to find a quadratic regression

equation that models the data. Use this equation to

(b) estimate the number of tickets sold after 5 weeks,

(c) estimate the number of weeks it will take for

the concert to sell out, and (d) estimate the number

of fans that won’t get to attend the show.



47. Guided tours: A tour

No. of

Start-up

guide for Kalaniohana

Tourists

Time (sec)

Tours noticed that for

2

206

groups of two to seven

4

115

people, the average time

6

63

it took to organize them at

9

79

the beginning of a tour

actually decreased as the

11

154

group size increased. For

13

269

groups of eight or more,

however, the logistics (and questions asked)

actually caused a significant increase in the start

time required. Using the given table and a

graphing calculator, (a) find a quadratic regression

equation that models the data. Use this equation to

(b) estimate how long it would take to get a group

of five tourists ready, (c) estimate the tour capacity

if start-up time can be no longer than 10 min, and

(d) estimate the fastest start time that could be

expected. Round to the nearest hundredth as

necessary.

48. Gardening: The

Water

No. of

production of a garden

Total (gal)

Tomatoes

can be diminished not

77

11

only by lack of water, but

132

25

also by overwatering.

198

29

Shay has kept diligent

2

records of her 100-ft

256

20

tomato garden’s weekly

315

1

production, as well as

the amount of water it received through watering

and rain. Use the given table and a graphing

calculator to (a) find a quadratic regression

equation that models the data. Then use this

equation to (b) estimate how many tomatoes she



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can expect when the garden receives 156 gal of

water per week, (c) estimate how much water

the garden received if there were 15 tomatoes

produced per week, and (d) estimate the maximum

number of tomatoes she can expect from the garden

in a week. Round to the nearest ten-thousandth as

necessary.

49. Weight of a

fetus: The

growth rate of a

Full term

fetus in the

(40, 3200)

mother’s womb

(36, 2600)

(by weight in

grams) is

(32, 1600)

modeled by the

(29, 1100)

graph shown

(25, 900)

here, beginning

with the 25th

Age (weeks)

week of

gestation. (a) Calculate the average rate of change

(slope of the secant line) between the 25th week

and the 29th week. Is the slope of the secant line

positive or negative? Discuss what the slope means

in this context. (b) Is the fetus gaining weight faster

between the 25th and 29th week, or between the

32nd and 36th week? Compare the slopes of both

secant lines and discuss.

3800



3600

3400



Weight (g)



3200



3–58



For Exercises 51 to 56, use the formula for the average

f 1x2 2 ؊ f 1x1 2

rate of change

.

x2 ؊ x1



51. Average rate of change: For f 1x2 ϭ x3, (a) calculate

the average rate of change for the interval x ϭ Ϫ2

to x ϭ Ϫ1 and (b) calculate the average rate of

change for the interval x ϭ 1 to x ϭ 2. (c) What do

you notice about the answers from parts (a) and

(b)? (d) Sketch the graph of this function along

with the lines representing these average rates of

change and comment on what you notice.



2800

2400



2000

1600

1200

800



24



26



28



30



32



34



36



38



40



42



50. Fertility rates:

Over the years,

(60, 3.6)

fertility rates for

(10, 3.4)

women in the

(20, 3.2)

(50, 3.0)

United States

(70, 2.4)

(average number

of children per

(40, 2.2)

(90, 2.0)

(80, 1.8)

woman) have

varied a great

deal, though in

the twenty-first

Year (10 → 1910)

century they’ve

begun to level out. The graph shown models this

fertility rate for most of the twentieth century.

(a) Calculate the average rate of change from the

years 1920 to 1940. Is the slope of the secant line

positive or negative? Discuss what the slope means

in this context. (b) Calculate the average rate of

change from the year 1940 to 1950. Is the slope of

the secant line positive or negative? Discuss what

the slope means in this context. (c) Was the fertility

rate increasing faster from 1940 to 1950, or from

1980 to 1990? Compare the slope of both secant

lines and comment.

Rate (children per woman)



4.0



3.0



2.0



52. Average rate of change: Knowing the general

3

shape of the graph for f 1x2 ϭ 1x, (a) is the

average rate of change greater between x ϭ 0 and

x ϭ 1 or between x ϭ 7 and x ϭ 8? Why?

(b) Calculate the rate of change for these intervals

and verify your response. (c) Approximately how

many times greater is the rate of change?

53. Height of an arrow: If an

arrow is shot vertically from a

bow with an initial speed of

192 ft/sec, the height of the

arrow can be modeled by the

function h1t2 ϭ Ϫ16t2 ϩ 192t,

where h(t) represents the height of the arrow

after t sec (assume the arrow was shot from

ground level).

a. What is the arrow’s height at t ϭ 1 sec?

b. What is the arrow’s height at t ϭ 2 sec?

c. What is the average rate of change from t ϭ 1

to t ϭ 2?

d. What is the rate of change from t ϭ 10 to

t ϭ 11? Why is it the same as (c) except for

the sign?



1.0



10



20



30



40



50



60



70



80



90



100



110



Source: Statistical History of the United States from Colonial Times to Present



54. Height of a water rocket: Although they have

been around for decades, water rockets continue to

be a popular toy. A plastic rocket is filled with

water and then pressurized using a handheld pump.

The rocket is then released and off it goes! If the

rocket has an initial velocity of 96 ft/sec, the height

of the rocket can be modeled by the function

h1t2 ϭ Ϫ16t2 ϩ 96t, where h(t) represents the

height of the rocket after t sec (assume the rocket

was shot from ground level).

a. Find the rocket’s height at t ϭ 1 and

t ϭ 2 sec.

b. Find the rocket’s height at t ϭ 3 sec.

c. Would you expect the average rate of change

to be greater between t ϭ 1 and t ϭ 2, or

between t ϭ 2 and t ϭ 3? Why?

d. Calculate each rate of change and discuss your

answer.



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55. Velocity of a falling object: The impact velocity

of an object dropped from a height is modeled by

v ϭ 12gs, where v is the velocity in feet per

second (ignoring air resistance), g is the

acceleration due to gravity (32 ft/sec2 near the

Earth’s surface), and s is the height from which

the object is dropped.

a. Find the velocity at s ϭ 5 ft and s ϭ 10 ft.

b. Find the velocity at s ϭ 15 ft and s ϭ 20 ft.

c. Would you expect the average rate of change

to be greater between s ϭ 5 and s ϭ 10, or

between s ϭ 15 and s ϭ 20?

d. Calculate each rate of change and discuss your

answer.







339



56. Temperature drop: One day in November, the town

of Coldwater was hit by a sudden winter storm that

caused temperatures to plummet. During the storm,

the temperature T (in degrees Fahrenheit) could be

modeled by the function T1h2 ϭ 0.8h2 Ϫ 16h ϩ 60,

where h is the number of hours since the storm

began. Graph the function and use this information

to answer the following questions.

a. What was the temperature as the storm began?

b. How many hours until the temperature dropped

below zero degrees?

c. How many hours did the temperature remain

below zero?

d. What was the coldest temperature recorded

during this storm?



EXTENDING THE CONCEPT



57. The function A1t2 ϭ 200t gives the amount of air

(in cubic inches) that a compressor has pumped out

after t sec. The volume of a spherical balloon being

4

inflated by this compressor is given by V1r2 ϭ ␲r3,

3

with the radius of the balloon modeled by

r1t2 ϭ Ϫ0.02t2 ϩ 0.76t ϩ 2.26, where r(t) is the

radius after t sec.

a. If the balloon pops when the radius is at its

maximum, what is the maximum volume of

the balloon?

b. What amount of air (the volume) was needed

to pop the balloon?

¢r

c. Calculate the average rates of change

and

¢t

¢V

during the first second of inflation and the

¢r

last second of inflation. Compare the results.





Section 3.4 Quadratic Models; More on Rates of Change



58. In Exercise 44, you were provided with a formula

called the difference quotient. This formula can be

derived by finding the average rate of change of a

function f (x) over an interval 3x, x ϩ h4 .

a. Use the definition of average rate of change to

derive the formula for the difference quotient.

b. Find and simplify the difference quotient of

f 1x2 ϭ x2 ϩ 3x.

59. The floor function f 1x2 ϭ :x ; and the ceiling

function g1x2 ϭ
produce some interesting average rates of change.

The average rate of change of these two functions

over any interval 1 unit or longer must lie within

what range of values?



MAINTAINING YOUR SKILLS



60. (1.1) Complete the squares in x and y to find the

center and radius of the circle defined by

x2 ϩ y2 ϩ 6x Ϫ 8y ϭ 0. Then graph the circle on a

graphing calculator.

61. (1.5) Solve the following inequalities graphically

using the Intersection-of-Graphs method. Round to

nearest hundredths when necessary.

xϪ5

6 2

a.

xϪ2

2

b. x Ϫ 13 Ϫ x2 7 710.2x ϩ 0.52

5



62. (3.3) Find an equation of the quadratic function

with vertex 1Ϫ5, Ϫ32 and y-intercept (0, 7).

|x ϩ 3| x 6 Ϫ2

63. (2.2/2.5) Given f 1x2 ϭ • 1

, find

Ϫ x

x Ն Ϫ2

2

the equation of g(x), given its graph is the same as

f(x) but translated right 3 units and reflected across

the x-axis.



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The Algebra of Functions



LEARNING OBJECTIVES

In Section 3.5 you will see how we can:



A. Compute a sum or

difference of functions

and determine the

domain of the result

B. Compute a product or

quotient of functions and

determine the domain

C. Interpret operations on

functions graphically and

numerically

D. Apply the algebra of

functions in context



In Section 2.2, we created new functions graphically by applying transformations to

basic functions. In this section, we’ll use two (or more) functions to create new functions algebraically. Previous courses often contain material on the sum, difference,

product, and quotient of polynomials. Here we’ll combine functions with the basic

operations, noting the result is also a function that can be evaluated, graphed, and analyzed. We call these basic operations on functions the algebra of functions.



A. Sums and Differences of Functions

This section introduces the notation used for basic operations on functions. Here we’ll

note the result is also a function whose domain depends on the original functions. In

general, if f and g are functions with overlapping domains, f 1x2 ϩ g1x2 ϭ 1 f ϩ g21x2

and f 1x2 Ϫ g1x2 ϭ 1 f Ϫ g2 1x2 .

Sums and Differences of Functions

For functions f and g with domains P and Q respectively,

the sum and difference of f and g are defined by:

1 f ϩ g21x2 ϭ f 1x2 ϩ g1x2



Domain of result



1 f Ϫ g21x2 ϭ f 1x2 Ϫ g1x2



EXAMPLE 1A



Solution











PʝQ

PʝQ



Evaluating a Difference of Functions



Given f 1x2 ϭ x2 Ϫ 5x and g1x2 ϭ 2x Ϫ 9,

a. Determine the domain of h1x2 ϭ 1 f Ϫ g21x2 .



b. Find h(3) using the definition.



a. Since the domain of both f and g is ‫ޒ‬, their intersection is ‫ޒ‬, so the domain of

h is also ‫ޒ‬.

b. h1x2 ϭ 1 f Ϫ g21x2

given difference

by definition

ϭ f 1x2 Ϫ g1x2

substitute 3 for x

h132 ϭ f 132 Ϫ g132

2

ϭ 3 132 Ϫ 5132 4 Ϫ 32132 Ϫ 9 4 evaluate

multiply

ϭ 39 Ϫ 15 4 Ϫ 36 Ϫ 94

subtract

ϭ Ϫ6 Ϫ 3 Ϫ34

result

ϭ Ϫ3



If the function h is to be graphed or evaluated numerous times, it helps to compute

a new function rule for h, rather than repeatedly apply the definition.

EXAMPLE 1B







For the functions f, g, and h, as defined in Example 1A,

a. Find a new function rule for h.



Solution



340







a. h1x2 ϭ 1 f Ϫ g21x2

ϭ f 1x2 Ϫ g1x2

ϭ 1x2 Ϫ 5x2 Ϫ 12x Ϫ 92

ϭ x2 Ϫ 7x ϩ 9



b. Use the result to find h(3).

given difference

by definition

replace f (x ) with 1x 2 Ϫ 5x2 and g(x) with 12x Ϫ 92

distribute and combine like terms



3–60



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Section 3.5 The Algebra of Functions



b. h132 ϭ 132 2 Ϫ 7132 ϩ 9

ϭ 9 Ϫ 21 ϩ 9

ϭ Ϫ3



substitute 3 for x

multiply

result



Notice the result from Part (b) is identical to that in Example 1A.

Now try Exercises 7 through 10 ᮣ







CAUTION



EXAMPLE 2







Solution







WORTHY OF NOTE

If we did try to evaluate h1Ϫ12 , the

result would be 1 ϩ 1Ϫ3, which is

not a real number. While it’s true we

could write 1 ϩ 1Ϫ3 as 1 ϩ i13

and consider it an “answer,” our

study here focuses on real numbers

and the graphs of functions in a

coordinate system where x and y

are both real.



From Example 1A, note the importance of using grouping symbols with the algebra of

functions. Without them, we could easily confuse the signs of g when computing the difference. Also, note that any operation applied to the functions f and g simply results in an

expression representing a new function rule for h, and is not an equation that needs to be

factored or solved.



Evaluating a Sum of Functions



For f 1x2 ϭ x2 and g1x2 ϭ 1x Ϫ 2,

a. Determine the domain of h1x2 ϭ 1 f ϩ g21x2 .

b. Find a new function rule for h.

c. Evaluate h(3).

d. Evaluate h1Ϫ12 .



a. The domain of f is ‫ޒ‬, while the domain of g is x ʦ 32, q 2 . Since their

intersection is 3 2, q 2 , this is the domain of the new function h.

b. h1x2 ϭ 1 f ϩ g21x2

given sum

ϭ f 1x2 ϩ g1x2

by definition

ϭ x2 ϩ 1x Ϫ 2

substitute x2 for f (x ) and 1x Ϫ 2 for g (x ) (no other simplifications possible)

c. h132 ϭ 132 2 ϩ 13 Ϫ 2 substitute 3 for x

result

ϭ 10

d. x ϭ Ϫ1 is outside the domain of h.

Now try Exercises 11 through 14 ᮣ



This “intersection of domains” is illustrated in Figure 3.64.

Figure 3.64

Domain of f: x ⑀ R

Domain of g: x ⑀ [2, ϱ)



A. You’ve just seen how

we can compute a sum or

difference of functions and

determine the domain of the

result



Ϫ3 Ϫ2 Ϫ1



0



1



2



Ϫ3 Ϫ2 Ϫ1



0



1



2



3



4



5



6



7



8



9



10



11



3



4



5



6



7



8



9



10



11



9



10



11



[

Intersection



Domain of h ϭ f ϩ g: x ⑀ [2, ϱ)



Ϫ3 Ϫ2 Ϫ1



[

0



1



2



3



4



5



6



7



8



B. Products and Quotients of Functions

The product and quotient of two functions is defined in a manner similar to that for

sums and differences. For example, if f and g are functions with overlapping domains,

f 1x2

f

1 f # g21x2 ϭ f 1x2 # g1x2 and a b1x2 ϭ

. As you might expect, for quotients we must

g

g1x2

stipulate g1x2 0.



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C. The Average Rate of Change Formula

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