Coburn J.W., Herdlick J.D. College Algebra: Graphs and Models

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EXAMPLE 7

䊳

Computing a Difference Quotient and Average Rates of Change

For ,

a. Compute the difference quotient.

b. Find the average rates of change in the intervals [1.9, 2.0] and [3.6, 3.7].

c. Sketch the graph of f along with the secant lines and comment on what you notice.

Solution

䊳

a. For

Using this result in the difference quotient yields,

eliminate parentheses

combine like terms

factor out

result

b. For the interval [1.9, 2.0], and

. The slope of the secant line is

. For the

interval [3.6, 3.7], and .

The slope of this secant line is

c. After sketching the graph of f and the secant

lines from each interval (see the ﬁgure), we

note the slope of the ﬁrst line (in red) is

negative and very near zero, while the slope of

the second (in blue) is positive and very steep.

Now try Exercises 39 through 50

䊳

It is important that you see these calculations

as much more than just an algebraic exercise. If

the parabola were modeling the distance a rocket

has traveled after the retro-rockets ﬁred, the differ-

ence quotient could provide us with valuable in-

formation regarding the decreasing velocity of the

rocket, and even help pinpoint the moment the

velocity was zero.

¢y

¢x

⫽ 213.62⫺ 4 ⫹ 0.1 ⫽ 3.3

h ⫽ 0.1x ⫽ 3.6

¢y

¢x

⫽ 211.92⫺ 4 ⫹ 0.1 ⫽⫺0.1

h ⫽ 0.1

x ⫽ 1.9

⫽ 2x ⫺ 4 ⫹ h

⫽

12x ⫹ h ⫺ 42

⫽

2xh ⫹ h

⫺ 4h

⫽

⫹ 2xh ⫹ h

⫺ 4x ⫺ 4h ⫺ x

⫹ 4x

1x ⫹ h2⫺ f 1x2

⫽

⫹ 2xh ⫹ h

⫺ 4x ⫺ 4h2⫺ 1x

⫺ 4x2

⫽ x

⫹ 2xh ⫹ h

⫺ 4x ⫺ 4h

1x2⫽ x

⫺ 4x, f 1x ⫹ h2⫽ 1x ⫹ h2

⫺ 41x ⫹ h2

1x2⫽ x

⫺ 4x

360 CHAPTER 3 Quadratic Functions and Operations on Functions 3–80

College Algebra G&M—

6⫺4

⫺5

substitute into the

difference quotient

cob19545_ch03_353-370.qxd 8/10/10 7:51 PM Page 360

EXAMPLE 8

䊳

Use the difference quotient to ﬁnd an average rate of change formula for the

reciprocal function Then use the formula to ﬁnd the average rate of

change on the intervals [0.5, 0.51] and [3, 3.01]. Note for both intervals.

Solution

䊳

For we have and we compute as follows:

difference quotient

substitute for and for

common denominator

simplify numerator

invert and multiply

This is the formula for computing rates of change given

To ﬁnd on the interval [0.5, 0.51] we have:

substitute 0.5 for

, 0.01 for

result (approximate)

On this interval y is decreasing by about 4 units

for every 1 unit x is increasing. The slope of the

secant line (in blue) through these points is

For the interval [3, 3.01] we have

substitute 3 for

, 0.01 for

result (approximate)

On this interval, y is decreasing 1 unit for every

9 units x is increasing. The slope of the secant line

(in red) through these points is .

Now try Exercises 51 through 54

䊳

In context, suppose the graph of f were modeling the declining area of the rainforest

over time. The difference quotient would give us information on just how fast the de-

struction was taking place (and give us plenty of cause for alarm).

m ⬇ ⫺

⬇ ⫺

¢y

¢x

⫽

⫺1

13 ⫹ 0.012132

m ⬇ ⫺4.

⬇

⫺4

¢y

¢x

⫽

⫺1

10.5 ⫹ 0.01210.52

¢y

¢x

1x2⫽

⫽

⫺1

1x ⫹ h2x

⫽

⫺h

1x ⫹ h2x

⫽

x ⫺ 1x ⫹ h2

1x ⫹ h2x

f 1x2

f 1x ⫹ h2

x ⫹ h

⫽

1x ⫹ h2

⫺

¢y

¢x

⫽

1x ⫹ h2⫺ f 1x2

1x ⫹ h2⫽

x ⫹ h

f 1x2⫽

h ⫽ 0.01

1x2⫽

3–81 Section 3.6 The Composition of Functions and the Difference Quotient 361

College Algebra G&M—

5⫺2

⫺2

C. You’ve just seen how

we can apply the difference

quotient to ﬁnd average rates

of change for nonlinear

functions

cob19545_ch03_353-370.qxd 8/10/10 7:51 PM Page 361

D. Applications of Composition and the Difference Quotient

Suppose that due to a collision, an oil tanker is spewing oil into the open ocean. The oil

is spreading outward in a shape that is roughly circular, with the radius of the circle mod-

eled by the function where t is the time in minutes and ris measured in feet.

How could we determine the area of the oil slick in terms of t? As you can see, the radius

depends on the time and the area depends on the radius. In diagram form we have:

It is possible to create a direct relationship between the elapsed time and the area

of the circular spill using a composition of functions.

EXAMPLE 9

䊳

Composition and the Area of an Oil Spill

Given and

a. Write A directly as a function of t by computing .

b. Find the area of the oil spill after 30 min.

Solution

䊳

a. The function A squares inputs, then multiplies by .

(

) is the input for

square input, multiply by

substitute 2 for

(

)

result

Since the result contains no variable r, we can now compute the area of the

spill directly, given the elapsed time t (in minutes):

b. To ﬁnd the area after 30 min, use

composite function

substitute 30 for

simplify

result (rounded to the nearest unit)

After 30 min, the area of the spill is approximately 377 ft

Now try Exercises 57 through 62

䊳

EXAMPLE 10

䊳

Composition and Related Populations

Using a statistical study, environmentalists ﬁnd

a current population of 600 wolves in the

county, and believe the wolf population w will

decrease as the human population x grows,

according to the formula .

Further, the population of rodents and small

animals r depends on the number of wolves w,

with the rodent population estimated by

a. Evaluate r(w) for to ﬁnd the current rodent population, then use a

composition to ﬁnd a function modeling how the human population relates

directly to the number of rodents.

b. Use the function to estimate the number of rodents in the county, if the human

population grows by 35,000.

w ⫽ 600

r1w2⫽ 10,000 ⫺ 9.5w

w1x2⫽ 600 ⫺ 0.02x

⬇ 377

⫽ 120␲

A1302⫽ 4␲1302

A1t2⫽ 4␲t

t ⫽ 30.

A1t2⫽ 4␲t.

⫽ 4␲t

1t ⫽ 321t4

␲

␲ ⫽ 3r1t24

␲

1A ⴰ r21t2⫽ A3r1t24

␲

1A ⴰ r21t2

A1r2⫽ ␲r

,r1t2⫽ 21t

Elapsed time t Radius depends on time: r(t) Area depends on radius: A(r)

r1t2⫽ 21t,

362 CHAPTER 3 Quadratic Functions and Operations on Functions 3–82

College Algebra G&M—

cob19545_ch03_353-370.qxd 11/25/10 3:48 PM Page 362

Solution

䊳

a. Since , there are currently 4300 rodents in the county. The number

of rodents and small animals is a function of w: , and the

number of wolves w is a function of human population .

To ﬁnd a function for r in terms of x, we use the composition .

depends on

compose

with

(

)

distribute

simplify,

depends on

b. Evaluating r(35,000) gives 10,950 rodents, showing a decline in the wolf

population will eventually cause the rodent population to ﬂourish, with an adverse

effect on humans. In cases like these, a careful balance should be the goal.

Now try Exercises 63 and 64

䊳

You might be familiar with Galileo Galilei and his studies of gravity. According to

popular history, he demonstrated that unequal weights will fall equal distances in equal

time periods, by dropping cannonballs from the upper ﬂoors of the Leaning Tower of

Pisa. Neglecting air resistance, the distance an object falls is modeled by the function

, where d(t) represents the distance fallen after t sec. Due to the effects of

gravity, the velocity of the object increases as it falls. In other words, the velocity or the

average rate of change is a nonconstant (increasing) quantity. We can analyze

this rate of change using the difference quotient.

EXAMPLE 11

䊳

Applying the Difference Quotient in Context

A construction worker drops a heavy wrench from atop a

girder of a new skyscraper. Use the function to

a. Compute the distance the wrench has fallen after

2 sec and after 7 sec.

b. Find a formula for the velocity of the wrench (average

rate of change in distance per unit time).

c. Use the formula to ﬁnd the rate of change in the

intervals [2, 2.01] and [7, 7.01].

d. Graph the function and the secant lines representing

the average rate of change. Comment on what you

notice.

Solution

䊳

a. Substituting and in the given function

yields

evaluate

square input

multiply

After 2 sec, the wrench has fallen 64 ft; after 7 sec, the wrench has fallen 784 ft.

b. For , which we compute separately.

substitute for

square binomial

distribute 16 ⫽ 16t

⫹ 32th ⫹ 16h

⫽ 161t

⫹ 2th ⫹ h

t ⫹ h d1t ⫹ h2⫽ 161t ⫹ h2

d1t2⫽ 16t

, d1t ⫹ h2⫽ 161t ⫹ h2

⫽ 784 ⫽ 64

⫽ 161492 ⫽ 16142

d1t2⫽ 16t

d172⫽ 16172

d122⫽ 16122

t ⫽ 7t ⫽ 2

d1t2⫽ 16t

¢distance

¢time

d1t2⫽ 16t

r1x2⫽ 4300 ⫹ 0.19x

⫽ 10,000 ⫺ 5700 ⫹ 0.19x

r3w1x24⫽ 10,000 ⫺ 9.53600 ⫺ 0.02x4

r1w2⫽ 10,000 ⫺ 9.5w

1r ⴰ w21x2⫽ r3w1x24

x: w1x2⫽ 600 ⫺ 0.02x

r1w2⫽ 10,000 ⫺ 9.5w

r16002⫽ 4300

3–83 Section 3.6 The Composition of Functions and the Difference Quotient 363

College Algebra G&M—

cob19545_ch03_353-370.qxd 8/12/10 6:02 PM Page 363

Using this result in the difference quotient yields

substitute into the difference quotient

eliminate parentheses

combine like terms

factor out

and simplify

result

For any number of seconds t and h a small increment of time thereafter, the

velocity of the wrench is modeled by .

c. For the interval and :

substitute 2 for

and 0.01 for

Two seconds after being dropped, the velocity of the wrench is close to

64.16 ft/sec (44 mph). For the interval and :

substitute 7 for

and 0.01 for

Seven seconds after being dropped, the velocity of the wrench is

approximately 224.16 ft/sec (about 153 mph).

The velocity increases with time, as indicated by the steepness of each

secant line.

Now try Exercises 65 through 68

䊳

200

400

600

800

1000

23456789

Time in seconds

Distance fallen (ft)

⫽ 224 ⫹ 0.16 ⫽ 224.16

¢distance

¢time

⫽

32172⫹ 1610.012

h ⫽ 0.013t, t ⫹ h4⫽ 37, 7.014, t ⫽ 7

⫽ 64 ⫹ 0.16 ⫽ 64.16

¢distance

¢time

⫽

32122⫹ 1610.012

h ⫽ 0.013t, t ⫹ h4⫽ 32, 2.014, t ⫽ 2

¢distance

¢time

ⴝ

32t ⴙ 16h

⫽ 32t ⫹ 16h

⫽

h132t ⫹ 16h2

⫽

32th ⫹ 16h

⫽

16t

⫹ 32th ⫹ 16h

⫺ 16t

d1t ⫹ h2⫺ d1t2

⫽

116t

⫹ 32th ⫹ 16h

2⫺ 16t

364 CHAPTER 3 Quadratic Functions and Operations on Functions 3–84

College Algebra G&M—

D. You’ve just seen how

we can apply the composition

of functions and the difference

quotient in context

cob19545_ch03_353-370.qxd 8/10/10 7:52 PM Page 364

3.6 EXERCISES

2. The notation indicates that g(x) is the input

value for f(x), which is written .

1f ⴰ g21x2

3. For functions f and g, the domain of is the

set of all x in the of g, such that

is in the domain of f.

1f ⴰ g21x2

4. The average rate of change formula becomes the

quotient by substituting for

and for x

5. Discuss/Explain how and why using the the

difference quotient differs from using the average

rate of change formula.

6. Discuss/Explain how the domain of

is determined, given and

.g1x2⫽

x ⫺ 1

1x2⫽ 12x ⫹ 7

1f ⴰ g21x2

䊳

CONCEPTS AND VOCABULARY

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

1. When evaluating functions, if the input value is a

function itself, the process is called a of

functions.

3–85 Section 3.6 The Composition of Functions and the Difference Quotient 365

College Algebra G&M—

䊳

DEVELOPING YOUR SKILLS

7. Given ﬁnd f(7),

f(2a), and .

8. Given ﬁnd g(3t), and

For each pair of functions below, ﬁnd (a) h(x) ⫽ ( f ⴰ g)(x)

and (b) H(x) ⫽ (g ⴰ f)(x), and (c) determine the domain

of each result.

9. and

10. and

11. and

12. and

13. and

14. and

15. and

16. and

17. and

18. and

For the functions f and g given, h(x) ⫽ ( f ⴰ g)(x). Use a

calculator to evaluate h(⫺3), h( ), , and h(5). If

an error message is received, explain why.

19.

20. , g1x2⫽ x ⫺ 2f

1x2⫽⫺x

⫺ 15x

g1x2⫽ x ⫹ 1f

1x2⫽ x

⫹ 3x ⫺ 4,

b22

g1x2⫽ 3x ⫺ 5f 1x2⫽

冟

x ⫺ 2

冟

g1x2⫽⫺3x ⫹ 1f

1x2⫽

冟

⫺ 5

g1x2⫽ x ⫺ 2f

1x2⫽ x

⫺ 4x ⫹ 2

g1x2⫽ x ⫹ 3f

1x2⫽ x

⫹ x ⫺ 4

g1x2⫽ 3x ⫹ 2f

1x2⫽ 2x

⫺ 1

g1x2⫽ x ⫹ 2f

1x2⫽ x

⫺ 3x

g1x2⫽ 4x ⫺ 1f

1x2⫽ 1x ⫹ 5

g1x2⫽ 3x ⫹ 4f 1x2⫽ 1x ⫺ 3

g1x2⫽ 29 ⫺ xf 1x2⫽ x ⫹ 3

g1x2⫽ 2x ⫺ 5f

1x2⫽ 1x ⫹ 3

g1t ⫹ 12

g1⫺32, g122,g1x2⫽ x

⫺ 9x,

1a ⫺ 22

1⫺22,f 1x2⫽ x

⫺ 5x ⫺ 14,

21. ,

22. ,

23. ,

24. ,

For the functions f(x) and g(x) given, analyze the

domain of (a) ( )(x) and (b) ( )(x), then (c) ﬁnd

the actual compositions and comment.

25. and

26. and

27. and

28. and

29. For and

ﬁnd h(5) in two ways:

a. b. f[g(5)]

30. For , and

ﬁnd in two ways:

a. b. p3q1⫺2241p ⴰ q21⫺22

H1⫺22H1x2⫽ 1p ⴰ q21x2,

p1x2⫽ x

⫺ 8, q1x2⫽ x ⫹ 2

1f ⴰ g2152

h1x2⫽ 1f ⴰ g21x2,

1x2⫽ x

⫺ 8, g1x2⫽ x ⫹ 2,

g1x2⫽

x ⫺ 2

f 1x2⫽

g1x2⫽

x ⫺ 5

f 1x2⫽

g1x2⫽

x ⫺ 2

1x2⫽

⫺3

g1x2⫽

1x2⫽

x ⫹ 3

g ⴰ ff ⴰ g

g1x2⫽ x

⫺ 11f 1x2⫽

x ⫹ 2

g1x2⫽ x

⫺ 9f 1x2⫽ 24 ⫺ 3x

g1x2⫽

2x ⫺ 1

1x2⫽

g1x2⫽

x ⫺ 5

1x2⫽ 1x ⫹ 82

cob19545_ch03_353-370.qxd 11/29/10 10:37 AM Page 365

31. For , ﬁnd two

functions f and g such that .

32. For , ﬁnd two functions

p and q such that .

33. Given , and

, ﬁnd and

34. Given and , ﬁnd

(a) , (b) , (c) , and

(d) .

35. Reading a graph: Use

the given graph to ﬁnd the

result of the operations

indicated.

Note

, and

so on.

c. d.

e. f.

g. h.

36. Reading a graph: Use

the given graph to ﬁnd the

result of the operations

indicated.

Note

, and so on.

c. d.

e. f.

g. h.

37. Given , , (a) state

the domain of f and g, then (b) use a graphing

calculator to study the graph of .

Finally, (c) algebraically determine the domain of

h, and reconcile it with the graph.

h1x2⫽ 1f ⴰ g21x2

g1x2⫽

⫺3

x ⫹ 2

f 1x2⫽ 32x ⫹ 1

1p ⴰ p21721q ⴰ q21⫺12

1q ⴰ p21221p ⴰ q21⫺22

1p ⴰ q21021p ⴰ q2142

1p ⴰ q2112

1p ⴰ q21⫺42

q152⫽ 6

p1⫺12⫽⫺2,

1g ⴰ f 21421f ⴰ g2162

1g ⴰ f 21221f ⴰ g21⫺22

1f ⴰ g21021f ⴰ g2142

1f ⴰ g2112

1f ⴰ g21⫺42

g1⫺42⫽⫺1

1⫺42⫽ 5,

1g ⴰ f 21x2

1f ⴰ g21x21g ⴰ g21x21f ⴰ f 21x2

g1x2⫽

x ⫺ 3

f 1x2⫽ 2x ⫹ 3

q1x2⫽ g3f

13h1x2424

p1x2⫽ f 3g13h1x2424h1x2⫽ x ⫹ 4

1x2⫽ 2x ⫺ 1, g1x2⫽ x

⫺ 1

1p ⴰ q21x2⫽ H1x2

H1x2⫽ 2

⫺ 5 ⫹ 2

1f ⴰ g21x2⫽ h1x2

h1x2⫽ 11x ⫺ 2

⫹ 12

⫺ 5

38. Given , (a) state

the domain of f and g, then (b) use a graphing

calculator to study the graph of .

Finally, (c) algebraically determine the domain of

h, and reconcile it with the graph.

Compute and simplify the difference quotient

for each function given.

39. 40.

41. 42.

43. 44.

45. 46.

Use the difference quotient to ﬁnd: (a) a rate of change

formula for the functions given and (b)/(c) calculate the

rate of change in the intervals shown. Then (d) sketch

the graph of each function along with the secant lines

and comment on what you notice.

47. 48.

[ ], [0.50, 0.51] [1.9, 2.0], [5.0, 5.01]

49.

[ ], [0.40, 0.41]

50. (Hint: Rationalize the numerator.)

[1, 1.1], [4, 4.1]

Use the difference quotient to ﬁnd a rate of change

formula for the functions given, then calculate the rate

of change for the intervals indicated. Comment on how

the rate of change in each interval corresponds to the

graph of the function.

51. 52.

53. 54.

31.00, 1.014, 34.00, 4.0143⫺2.01, ⫺2.004, 30.40, 0.414

r1x2⫽ 1xg1x2⫽ x

⫹ 1

30.00, 0.014, 33.00, 3.014

30.50, 0.514, 31.50, 1.514

f 1x2⫽ x

⫺ 4xj1x2⫽

v1x2⫽ 2x

⫺2.1, ⫺2

g1x2⫽ x

⫹ 1

⫺3.0, ⫺2.9

j1x2⫽ x

⫺ 6xg1x2⫽ x

⫹ 2x

g1x2⫽

⫺3

1x2⫽

r1x2⫽ x

⫺ 5x ⫹ 2q1x2⫽ x

⫹ 2x ⫺ 3

p1x2⫽ x

⫺ 2j1x2⫽ x

⫹ 3

g1x2⫽ 4x ⫹ 1f

1x2⫽ 2x ⫺ 3

f 1x ⴙ h2ⴚ f 1x2

h1x2⫽ 1f ⴰ g21x2

1x2⫽ 1x, g1x2⫽

⫺ 2x ⫺ 3

366 CHAPTER 3 Quadratic Functions and Operations on Functions 3–86

College Algebra G&M—

⫺4

f(x)

g(x)

⫺4

Exercise 35

8⫺4

⫺4

p(x)

q(x)

Exercise 36

䊳

WORKING WITH FORMULAS

55. Transformations via composition: For

and , (a) show that

, then (b) verify the graph

of h is the same as that of f, shifted 2 units to the right.

h1x2⫽ 1 f ⴰ g21x2⫽ x

⫺ 1

g1x2⫽ x ⫺ 2f 1x2⫽ x

⫹ 4x ⫹ 3

56. Compound annual growth:

The amount of money A in a savings account t yr

after an initial investment of P dollars depends on

the interest rate r. If $1000 is invested for 5 yr, ﬁnd

f(r) and g(r) such that A1r2⫽ 1f ⴰ g21r2.

A1r2ⴝ P11 ⴙ r2

cob19545_ch03_353-370.qxd 8/10/10 7:54 PM Page 366

3–87 Section 3.6 The Composition of Functions and the Difference Quotient 367

College Algebra G&M—

䊳

APPLICATIONS

57. International shoe sizes: Peering inside her

athletic shoes, Morgan notes the following shoe

sizes: US 8.5, UK 6, EUR 40. The function that

relates the U.S. sizes to the European (EUR) sizes

is , where x represents the U.S.

size and g(x) represents the EUR size. The function

that relates European sizes to sizes in the United

Kingdom (UK) is , where x

represents the EUR size and f(x) represents the UK

size. Find the function h(x) that relates the U.S.

measurement directly to the UK measurement by

ﬁnding Find the UK size for a

shoe that has a U.S. size of 13.

58. Currency conversion: On a trip to Europe, Megan

had to convert American dollars to euros using the

function where x represents the

number of dollars and E(x) is the equivalent

number of euros. Later, she converts her euros to

Japanese yen using the function

where x represents the number of euros and Y(x)

represents the equivalent number of yen.

(a) Convert 100 U.S. dollars to euros. (b) Convert

the answer from part (a) into Japanese yen.

then use M(x) to convert $100

directly to yen. Do parts (b) and (c) agree?

Source:

2005

World Almanac,

p. 231

59. Currency conversion: While traveling in the Far

East, Timi must convert U.S. dollars to Thai baht

using the function where x

represents the number of dollars and T(x) is the

equivalent number of baht. Later she needs to

convert her baht to Malaysian ringgit using the

function (a) Convert $100 to baht.

(b) Convert the result from part (a) to ringgit.

then use M(x) to convert $100

to ringgit directly. Do parts (b) and (c) agree?

Source:

2005

World Almanac,

p. 231

60. Spread of a ﬁre: Due to a lightning strike, a forest

ﬁre begins to burn and is spreading outward in a

shape that is roughly circular. The radius of the

circle is modeled by the function where t

is the time in minutes and r is measured in meters.

(a) Write a function for the area burned by the ﬁre

directly as a function of t by computing .

(b) Find the area of the circular burn after 60 min.

1A ⴰ r21t2

r1t2⫽ 2t,

M1x2⫽ 1R ⴰ T21x2,

R1x2⫽ 10.9x.

T1x2⫽ 41.6x,

M1x2⫽ 1Y ⴰ E21x2,

Y1x2⫽ 1061x,

E1x2⫽ 1.12x,

h1x2⫽ 1f ⴰ g21x2.

1x2⫽ 0.5x ⫺ 14

g1x2⫽ 2x ⫹ 23

61. Radius of a ripple: As Mark drops ﬁrecrackers

into a lake one 4th of July, each “pop” caused a

circular ripple that expanded with time. The

radius of the circle is a function of time t.

Suppose the function is where t is in

seconds and r is in feet. (a) Find the radius of the

circle after 2 sec. (b) Find the area of the circle

after 2 sec. (c) Express the area as a function of

time by ﬁnding and use A(t) to

ﬁnd the area of the circle after 2 sec. Do the

answers agree?

A1t2⫽ 1A ⴰ r21t2

r1t2⫽ 3t,

62. Expanding supernova: The surface area of a star

goes through an expansion phase prior to going

supernova. As the star begins expanding, the

radius becomes a function of time. Suppose this

function is , where t is in days and r(t)

is in gigameters (Gm). (a) Find the radius of the

star two days after the expansion phase begins.

(b) Find the surface area after two days. (c)

Express the surface area as a function of time by

ﬁnding then use h(t) to compute

the surface area after two days directly. Do the

answers agree?

h1t2⫽ 1S ⴰ r21t2,

r1t2⫽ 1.05t

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63. Composition and dependent relationships: In

the wild, the balance of nature is often very fragile,

with any sudden changes causing dramatic and

unforeseen changes. With a huge increase in

population and tourism near an African wildlife

preserve, the number of lions is decreasing due to

loss of habitat and a disruption in normal daily

movements. This is causing a related increase in

the hyena population, as the lion is one of the

hyena’s only natural predators. If this increase

remains unchecked, animals lower in the food

chain will suffer. If the lion population L depends

on the increase in human population x according

to the formula , and the

hyena population depends on the lion population

as modeled by the formula ,

(a) what is the current lion population

and hyena population? (b) Use a composition to

ﬁnd a function modeling how the hyena population

relates directly to the number of humans, and use

the function to estimate the number of hyenas in

the area if the human population grows by 16,000.

population of 625 hyenas as “extremely detrimental,”

at what point should the human population be

capped?

1x ⫽ 02

H1L2⫽ 650 ⫺ 0.5L

L1x2⫽ 500 ⫺ 0.015x

64. Composition and dependent relationships: The

recent opening of a landfill in the area has caused

the raccoon population to flourish, with an

adverse effect on the number of purple martins.

Wildlife specialists believe the population of

martins p will decrease as the raccoon population

r grows. Further, since mosquitoes are the primary

diet of purple martins, the mosquito population

m is likewise affected. If the first relationship is

modeled by the function and

the second by , (a) what is

the current number of purple martins and

mosquitoes? (b) Use a composition to find a

function modeling how the raccoon population

relates directly to the number of mosquitoes,

and use the function to estimate the number of

mosquitoes in the area if the raccoon population

1r ⫽ 02

m1p2⫽ 50,000 ⫺ 45p

p1r2⫽ 750 ⫺ 3.75r

grows by 50. (c) If the health department considers

36,500 mosquitoes to be a “dangerous level,”

what increase in the raccoon population will

bring this about?

65. Distance to the horizon: The distance that a

person can see depends on how high they’re

standing above level ground. On a clear day,

the distance is approximated by the function

, where d(h) represents the viewing

distance (in miles) at height h (in feet). Use

the difference quotient to find the average rate

of change in the intervals (a) [9, 9.01] and

(b) [225, 225.01]. Then (c) graph the function

along with the lines representing the average

rates of change and comment on what you

notice.

66. Projector lenses: A special magnifying lens is

crafted and installed in an overhead projector.

When the projector is x ft from the screen, the

size P(x) of the projected image is x

. Use the

difference quotient to find the average rate of

change for in the intervals (a) [1, 1.01]

and (b) [4, 4.01]. Then (c) graph the function

along with the lines representing the average

rates of change and comment on what you

notice.

67. Fortune and fame: Over the years there have

been a large number of what we know as “one

hit wonders,” persons or groups that published a

memorable or timeless song, but who were

unable to repeat the feat. In some cases, their

fame might be modeled by a quadratic function

as their popularity rose to a maximum, then

faded with time. Suppose the song She’s on Her

Way by Helyn Wheels rode to the top of the

charts in January of 1988, with demand for the

song modeled by . Here, d(t)

represents the demand in 1000s for month

. (a) How many times faster was

the demand growing in March (shortly after

the release) than in June? Use the difference

quotient and the intervals [3, 3.01] for March

1t ⫽ 1 S Jan2

d1t2⫽⫺2t

⫹ 27t

P1x2⫽ x

d1h2⫽ 1.21h

368 CHAPTER 3 Quadratic Functions and Operations on Functions 3–88

College Algebra G&M—

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3–89 Section 3.6 The Composition of Functions and the Difference Quotient 369

College Algebra G&M—

䊳

EXTENDING THE CONCEPT

69. Given and graph

each function on the same axes by plotting the

points that correspond to integer inputs for

Do you notice anything? Next, ﬁnd

and What

happened? Look closely at the functions f and g to

see how they are related. Can you come up with two

additional functions where the same thing occurs?

70. Given , and

, (a) ﬁnd the new function rule for

h and (b) determine the implied domain of h. Does

this implied domain include , and

as valid inputs? (c) Determine the actual

domain for and discuss the result.

h1x2⫽ 1

f ⴰ g21x2

x ⫽⫺3

x ⫽ 2, x ⫽⫺2

h1x2⫽ 1

f ⴰ g21x2

1x2⫽

⫺ 4

, g1x2⫽ 1x ⫹ 1

H1x2⫽ 1g ⴰ f21x2.h1x2⫽ 1f ⴰ g21x2

x 僆 3⫺3, 34.

g1x2⫽ 2

x ⫺ 2,f 1x2⫽ x

⫹ 2

and [6, 6.01] for June. (b) Determine the month

that demand reached its peak using a graphing

calculator. (c) Was the demand increasing or

decreasing in the month of August? At what

rate?

68. Velocity and fuel economy: It has long been

known that cars and trucks are more fuel efﬁcient

at certain speeds, which is why President Richard

Nixon lowered the speed limit on all federal

highways to 55 mph during the oil embargo of

1974. For heavier and less fuel-efﬁcient vehicles,

the miles per gallon for certain speeds can be

modeled by the function ,

where m(s) represents the mileage (in miles per

gallon) at speed s . (a) Use the

difference quotient to ﬁnd how many times

10 6 s ⱕ 802

m1s2⫽⫺0.01s

⫹ s

faster fuel efﬁciency is growing near mph

than near mph. Use the intervals [30, 30.1]

and [45, 45.1]. (b) Use a graphing calculator

to determine the speed(s) that maximizes fuel

efficiency for this vehicle. How many miles

per gallon are achieved? (c) Is fuel efficiency

increasing or decreasing at 70 mph? At what

rate?

s ⫽ 45

s ⫽ 30

71. Consider the functions and .

Both graphs appear similar in Quadrant I and both

may “ﬁt” a scatterplot fairly well, but there is a big

difference between them—they decrease as x gets

larger, but they decrease at very different rates.

(a) Assume and use the ideas from this

section to compute the rates of change for f and g

for the interval from to . Were you

surprised? (b) In the interval to ,

will the rate of decrease for each function be

greater or less than in the interval to

? Why?x ⫽ 0.51

x ⫽ 0.5

x ⫽ 0.81x ⫽ 0.8

x ⫽ 0.51x ⫽ 0.5

k ⫽ 1

g1x2⫽

f 1x2⫽

䊳

MAINTAINING YOUR SKILLS

72. (3.1) Find the sum and product of the complex

numbers and

73. (2.2) Draw a sketch of the functions from memory.

(a) (b) , and

冟

g1x2⫽ 2

xf 1x2⫽ 1x,

2 ⫺ 3i.2 ⫹ 3i

74. (3.2) Use the quadratic formula to solve

75. (1.4) Find an equation of the line perpendicular to

, that also goes through the origin.⫺2x ⫹ 3y ⫽ 9

⫺ 3x ⫹ 4 ⫽ 0.

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