EXERCISES 3.1
In Exercises 1–34,find the derivative of the function f by using the rules of differentiation.
1. f x( ) 3 2. f x( ) 365 3. f x( )x6 4. f x( )x9 5. f x( )x2.1 6. f x( )x0.8 7. f x( ) 4 x2 8. f x( ) 4x3
9. f x( )r2 10. ( ) 4 3 f r 3r 11. f x( ) 9 x1 3 12. 5 4 5
( ) 4 f x x
13. f x( ) 5 x 14. f u( ) 2
u 15. f x( ) 7 x12 16. f x( ) 0.3 x1.2 17. f x( ) 5 x23x7 18. f x( )x33x21 19. f x( ) 2x32x26 20. f x( )x42x25 21. f x( ) 0.03 x20.4x10
22. f x( ) 0.002 x30.05x20.1x20
23.
3 2
2 4 3
( ) x x
f x x
24.
3 2 2 1
( ) x x x
f x x
25. f x( ) 5 x43x5 22
26. ( ) 6 4 3 2 3 2 2 3 1 f x x 3x x x
27. f x( ) 2 x14x2 28. ( ) 1( 3 6) f x 3 x x 29. f t( ) 44 33 2
t t t
30. f x( ) 53 22 1 200 x
x x
31. f x( ) 3 x5 x 32. f t( ) 4 t2 t3
33. f x( ) 22 31 3
x x
34. f x( ) 23 4 1
x x
35. Let f x( ) 2 x34x. Find:
a. f ( 2) b. f (0) c. f (2) 36. Let f x( ) 4 x5 42x3 2x. Find
a. f (0) b. f (16)
In Exercises 37– 40, find each limit by evaluating the derivative of a suitable function at an appropriate point.
Hint: Look at the definition of the derivative.
37.
3 0
(1 ) 1
limh
h h
38.
5 1
lim 1 1
x
x x
Hint: Let h x 1 39.
2 0
3(2 ) (2 ) 10
limh
h h
h
40.
2 0 2
1 (1 ) limt (1 )
t
t t
In Exercises 41– 44, find the slope and an equation of the tangent line to the graph of the function f at the specified point.
41. f x( ) 2 x23x4; (1,3)
42. ( ) 5 2 2 2
f x 3x x ; ( 1, 5)
3 43. f x( )x43x32x2 x 1; (1, 0)
44. 1
( )
f x x
x; (4, )5 2 45. Let f x( )x3.
a. Find the point on the graph of f where the tangent line is horizontal.
b. Sketch the graph of f and draw the horizontal tang- ent line.
46. Let f x( )x34x2. Find the points(s) on the graph of f where the tangent line is horizontal.
47. Letf x( )x31.
a. Find the point(s) on the graph of f where the slope of the tangent line is equal to 12.
b. Find the equation(s) of the tangent line(s) of part (a).
c. Sketch the graph of f showing the tangent line(s).
48. Let ( ) 2 3 2 12 6
f x 3x x x . Find the values of x for which:
a. f x( ) 12 b. f x( ) 0 c. f x( ) 12
49. Let ( ) 1 4 1 3 2
4 3
f x x x x . Find the point(s) on the graph of f where the slope of the tangent line is equal to :
a. 2x b. 0 c. 10x
50. A straight line perpendicular to and passing through the point of tangency of the tangent line is called the normal to the curve. Find an equation of the tangent line and the normal to the curve yx33x1 at the point (2,3).
51. GROWTH OF A CANCEROUS TUMOR The volume of a spherical cancerous tumor is given by the function
4 3
( ) 3 V r r
where r is the radius of the tumor in centimeters. Find the rate of change in the volume of the tumor when
a. 2
r 3cm b. 5
r 4cm
52. VELOCITY OF BLOOD IN AN ARTERY The velocity (in centi- meters/second) of blood r cm from the central axis of an artery is given by
2 2
υ( )r k R( r )
where k is a constant and R is the radius of the artery (see the accompanying figure). Suppose k = 1000 and R = 0.2cm. Find υ(0.1) and υ(0.1) and interpret your results.
53. SALES OF DIGITAL CAMERAS According to projections made in 2004, the worldwide shipments of digital point-and-shoot cameras are expected to grow in accordance with the rule
0.8766
( ) 16.3
N t t (1 t 6)
where N t( ) is measured in millions and t is measured in years, with t = 1 corresponding to 2001.
a. How many digital cameras were sold in 2001 (t = 1)?
b. How fast were sales increasing in 2001?
c. What were the projected sales in 2005?
d. How fast were the sales projected to grow in 2005?
Source: International Data Corp
54. ONLINE BUYERS As use of the Internet grows, so does the number of consumers who shop online. The number of online buyers, as a percent of net users, is expected to be
( ) 530.12
P t t (1 t 7)
where t is measured in years, with t = 1 corresponding to the beginning of 2002.
a. How many online buyers, as a percent of net users, are there expected to be at the beginning of 2007?
b. How fast is the number of online buyers, as a percent of net users, expected to be changing at the beginning of 2007?
Source: Strategy Analytics
55. MARRIED HOUSEHOLDS WITH CHILDREN The percent of families that were married households with children between 1970 and 2000 is approximately
0.27
( ) 49.6
p t t (1 t 4)
where t is measured in decades, with t = 1 correspo- nding to 1970.
a. what percent of families were married households with children in 1970? In 1980? In 1990? In 2000?
b. How fast was the percent of families that were mar- ried households with children changing in 1980? In 1990?
Source: U.S. Census Bureau
56. EFFECT OF STOPPING ON AVERAGE SPEED According to data from a study, the average speed of your trip A (in mph) is related to the number of stops/mile you make on the trip x by the equation
0.45
A 26.5
x
Compute dA dx/ for x = 0.25 and x = 2. How is the rate of change of the average speed of your trip affected by the number of stops/mile?
Source: General Motors
57. PORTABLE PHONES The percent of the U.S. population with portable phones is projected to be
( ) 24.40.34
P t t (1 t 10)
where t is measured in years, with t = 1 corresponding to the beginning of 1998.
a. What percent of the U.S. population is expected to have portable phones by the beginning of 2006?
b. How fast is the percent of the U.S. population with portable phones expected to be changing at the beginning of 2006?
Source: Banc America Robertson Stephens
58. DEMAND FUNCTIONS The demand function for the Lumi- nar desk lamp is given by
( ) 0.1 2 0.4 35 p f x x x
where x is the quantity demanded (measured in thousa- nds) and P is the unit price in dollars.
a. Find f x( ).
b. What is the rate of change of the unit price when the quantity demanded is 10,000 units ( x = 10)? What is the unit price at that level of demand?
59. STOPPING DISTANCE OF A RACING CAR During a test by the editors of an auto magazine, the stopping distance s (in feet) of the MacPherson X-2 racing car conformed to the rule
( ) 120 15 2
s f t t t (t0)
where t was the time (in seconds) after the brakes were applied.
a. Find an expression for the car’s velocity υ at any time t.
b. What was the car’s velocity when the brakes were first applied?
c. What was the car’s stopping distance for that partic- ular test?
Hint: The stopping time is found by setting υ0. 60. SALES OF DSPS The sales of digital signal processors
(DSPs) in the billions of dollars is projected to be ( ) 0.142 0.68 3.1
s t t t (0 t 6)
where t is measured in years, with t = 0 corresponding to the beginning of 1997.
a. What were the sales of DSPs at the beginning of 1997?
What were the sales at the beginning of 2002?
b. How fast was the level of sales increasing at the beg- inning of 1997? How fast were sales increasing at the beginning of 2002?
Source: World Semiconductor Trade Statistics
61. CHILD OBESITY The percent of obese children, ages 12–
19, in the United States has grown dramatically in rec- ent years. The percent of obese children from 1980 thr- ough the year 2000 is approximated by the function
( ) 0.01052 0.735 5
p t t t (0 t 20) where t is measured in years, with t = 0 corresponding to the beginning of 1980.
a. What percent of children were obese at the begin- ning of 1980? At the beginning of 1990? At the beg- inning of the year 2000?
b. How fast was the percent of obese children changing at the beginning of 1985? At the beginning of 1990?
Source: Centers for Disease Control and Prevention 62. SPENDING ON MEDICARE Based on the current eligibility
requirement, a study conducted in 2004 showed that federal spending on entitlement programs, particularly Medicare, would grow enormously in the future. The study predicted that spending on Medicare, as a percent of the gross domestic product (GDP), will be
( ) 0.272 1.4 2.2
p t t t (0 t 5)
percent in year t , where t is measured in decades, with t = 0 corresponding to 2000.
a. How fast will the spending on Medicare, as a percent of the GDP, be growing in 2010? In 2020?
b. What will the predicted spending on Medicare be in 2010? In 2020?
Source: Congressional Budget Office
63. FISHERIES The total groundfish population on Georges Bank in New England between 1989 and 1999 is appr- oximated by the function
( ) 5.3032 53.977 253.8
f t t t (0 t 10) where f (t) is measured in thousands of metric tons and t is measured in years, with t = 0 corresponding to the beginning of 1989.
a. What was the rate of change of the groundfish popu- lation at the beginning of 1994? At the beginning of 1996?
b. Fishing restrictions were imposed on Dec.7, 1994.
Were the conservation measures effective?
Source: New England Fishery Management Council 64. WORKER EFFICIENCY An efficiency study conducted for
Elektra Electronics showed that the number of Space Commander walkie–talkies assembled by the average worker t hr after starting work at 8 a.m. is given by
3 2
( ) 6 15
N t t t t
a. Find the rate at which the average worker will be assembling walkie–talkies t hr after starting work.
b. At what rate will the average worker be assembling walkie–talkies at 10 a. m. ? At 11 a.m.?
c. How many walkie-talkies will the average worker assemble between 10 a.m. and 11 a. m.?
65. CONSUMER PRICE INDEX An economy’s consumer price index (CPI) is described by the function
3 2
( ) 0.2 3 100
I t t t (0 t 10) where t = 0 corresponds to 1997.
a. At what rate was the CPI changing in 2002? In 2004? In 2007?
b. What was the average rate of increase in the CPI over the period from 2002 to 2007?
66. EFFECT OF ADVERTISING ON SALES The relationship betw- een the amount of money x that Cannon Precision Inst- ruments spends on advertising and the company’s total sales S(x) is given by the function
3 2
( ) 0.002 0.6 500
S x x x x (0 x 200) where x is measured in thousands of dollars. Find the rate of change of the sales with respect to the amount of money spent on advertising. Are Cannon’s total sales increasing at a faster rate when the amount of money spent on advertising is (a) $100,000 or (b)
$150,000?
67. SUPPLY FUNCTIONS The supply function for a certain make of transistor radio is given by
54
( ) 0.0001 10 p f x x
where x is the quantity supplied and p is the unit price in dollars.
a. Find f x( ).
b. What is the rate of change of the unit price if the quantity supplied is 10,000 transistor radios?
68. POPULATION GROWTH A study prepared for a Sunbelt town’s chamber of commerce projected that the town’s population in the next 3 yr will grow according to the rule
( ) 50, 000 303 2 20
P t t t
where P t( ) denotes the population t mo from now.
How fast will the population be increasing 9 mo and 16 mo from now?
69. AVERAGE SPEED OF A VEHICLE ON A HIGHWAY The average speed of a vehicle on a stretch of Route 134 between 6 a.m. and 10 a.m. on a typical weekday is approximated by the function
( ) 20 40 50 (0 4) f t t t t
where f t( ) is measured in mph and t is measured in hours, with t0 corresponding to 6 a.m.
a. Computef t( ).
b. What is the average speed of a vehicle on that stret- ch of Route 134 at 6 a.m.? At 7 a.m.? At 8 a.m.?
c. How fast is the average speed of a vehicle on that stretch of Route 134 changing at 6:30 a.m.? At 7 a.m.? At 8 a.m.?
70. CURBING POPULATION GROWTH Five years ago, the gove- rnment of a Pacific Island state launched an extensive propaganda campaign toward curbing the country’s population growth. According to the Census Departm- ent, the population (measured in thousands of people) for the following 4 yr was
1 3
( ) 64 3000
P t 3t t
where t is measured in years and t0 corresponds to the start of the campaign. Find the rate of change of the population at the end of years 1, 2, 3, and 4. Was the plan working?
71. CONSERVATION OF SPECIES A certain species of turtle faces extinction because dealers collect truckloads of turtle eggs to be sold as aphrodisiacs. After severe conservation measures are implemented, it is hoped that the turtle population will grow according to the rule
3 2
( ) 2 3 4 1000 (0 10) N t t t t t
where N (t) denotes the population at the end of year t.
Find the rate of growth of the turtle population when t
= 2 and t = 8. What will be the population 10 yr after the conservation measures are implemented?
72. FLIGHT OF A ROCKET The altitude (in feet) of a rocket t sec into flight is given by
3 2
( ) 2 114 480 1 ( 0) s f t t t t t
a. Find an expression for the rocket’s velocity at any time t.
b. Compute the rocket’s velocity when t0, 20, 40, and 60. Interpret your results.
c. Using the results from the solution to part (b), find the maximum altitude attained by the rocket.
Hint: At its highest point, the velocity of the rocket is zero.
73. OBESITY IN AMERICA The body mass index (BMI) meas- ures body weight in relation to height. A BMI of 25 to 29.9 is considered overweight, a BMI of 30 or more is considered obese, and a BMI of 40 or more is morbidly
obese, The percent of the U.S. population that is obese is approximated by the function
3 2
( ) 0.0004 0.0036 0.8 12
P t t t t (0 t 13) where t is measured in years, with t = 0 corresponding to the beginning of 1991.
a. What percent of the U.S. population was deemed obese at the beginning of 1991? At the beginning of 2004?
b. How fast was the percent of the U.S. population that is deemed obese changing at the beginning of 1991?
At the beginning of 2004 ?
(Note: A formula for calculating the BMI of a person is given in Exercise 27,page 544.)
Source: Centers for Disease Control and Prevention 74. HEALTH-CARE SPENDING Despite efforts at cost containment,
the cost of the Medicare program is increasing. Two major reasons for this increase are an aging population and extensive use by physicians of new technologies. Based on data from the Health Care Financing Administration and the U.S. Census Bureau , health-care spending through the year 2000 may be approximated by the function
3 2
( ) 0.02836 0.05167 9.6088 41.9
S t t t t
(0 t 35)
where S(t) is the spending in billions of dollars and t is measured in years, with t = 0 corresponding to the beginning of 1965.
a. Find an expression for the rate of change of health–
care spending at any time t.
b. How fast was health–care spending changing at the beginning of 1980?At the beginning of 2000?
c. What was the amount of health–care spending at the beginning of 1980?At the beginning of 2000?
Source: Health Care Financing Administration and U.S. Census Bureau
In Exercises 75 and 76, determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false.
75. If f and g are differentiable, then
[2 ( ) 5 ( )] 2 ( ) 5 ( )
d f x g x f x g x
dx
76. If f x( )x, then f x( )xx1.
77. Prove the power rule (Rule 2) for the special case n = 3.
Hint: Compute
3 3
0
( )
limh
x h x
h
.
0 3321 0 4 S f t . t
. t
0 t 2
where S is measured in millions of dollars and t = 0 corresponds to the date Security Products began opera- tions. How fast were the sales increasing at the beginn- ing of the company’s second year of operation?
Solutions to Self-Check Exercises 3.2 can be found on page 182.
Exercises 3.2
In Exercises 1–30, find the derivative of each function.
1. f x
2x x
21
2. f x
3x x2
1
3. f t
t 1 3 1
t
4. f x
2x3 3
x4
5. f x
2x1
x22
6. f x
x1 2
x23x1
7. f x
x31 2
x1
8. f x
x312x
3x22x
9. f w
w3w2 w 1
w22
10. f x
15x5
x21
x2 x 1
2811. f x
5x21 2 x1
12. f t
1 t 2t23
13. f x
x25x2
x2x14.
3
22 1 2 1
f x x x
x
15. f x
x22 16. g x
2x5417. f x
32xx11 18. f t
1 21 3 tt19.
21 f x 1
x
20.
2 1f u u
u
21.
2 41 f s s
s
22.
32 21 f x x
x
23.
21 f x x
x
24. f x
x2 1x
25.
22 21 f x x
x x
26.
21
2 2 3
f x x
x x
27.
1
2 1
2
x x
f x x
28. f x
3x21
x21x29.
2 2 14 4
x x
f x x x
30. f x
x3x31xIn Exercises 31–34, suppose f and g are functions that are differentiable at x = 1 and that f
1 2, f '
1 1,
1 2g and g'
1 3.Find the value of h(1). 31. h x
f x g x
32. h x
x21
g x
33.
h x xf x
x g x
34.
f x g x h x f x g x
In Exercises 35–38, find the derivative of each function and evaluate f x( ) at the given value of x.
35. f x
2x1
x23
; x136. f x
22xx11; x237.
4 22 1
2 1
f x x ; x
x x
38. f x
x2x x 3 2x ; x
4
In Exercises 39–42,find the slope and an equation of the tangent line to the graph of the function f at the specified point.
39. f x
x31
x22
;
2 18,
40.f x
xx21; 2 , 4341.
2 11 f x x
x
;
1 1,42.
1 2 3 2
1 2 5
( ) ; 4,
9 1
f x x
x
43. Find an equation of the tangent line to the graph of the
function f x( )
x31 3
x2 4x 2
at the point (1, 2). 44. Find an equation of the tangent line to the graph of thefunction 23
( ) 2
f x x
x
at the point (2, 3).
45. Let f x( ) ( x21)(2x). Find the point(s) on graph of f where the tangent line is horizontal.
46. Let 22
( ) 1
f x x
x
. Find the point(s) on the graph of f where the tangent line is horizontal.
47. Find the point(s) on the graph of the function f x( ) (x26)(x5) where the slope of the tangent line is equal to 2.
48. Find the point(s) on the graph of the function ( ) 1 1 f x x
x
where the slope of the tangent line is equal to 1
.2 49. A straight line perpendicular to and passing through
the point of tangency of the tangent line is called the normal to the curve. Find the equation of the tangent line and the normal to the curve 12
y 1
x
at the point
1
(1, )2 .
50. CONCENTRATION OF A DRUG IN THE BLOODSTREAM The concentration of a certain drug in a patient’s bloods- tream t hr after injection is given by
2
( ) 0.2 1 C t t
t
a. Find the rate at which the concentration of the drug is changing with respect to time.
b. How fast is the concentration changing 1
2 hr, 1 hr, and 2 hr after the injection?
51. COST OF REMDVING TOXICWASTE A city’s main well was recently found to be contaminated with trichloroethyl- ene, a cancer-causing chemical, as a result of an aband- oned chemical dump leaching chemicals into the water.
A proposal submitted to the city’s council members indicates that the cost, measured in millions of dollars, of removing x% of the toxic pollutant is given by
( ) 0.5 100 C x x
x
Find C(80), C(90), C(95), and C(99). What does your result tell you about the cost of removing all of the pollutant?
52. DRUG DOSAGES Thomas Young has suggested the follo- wing rule for calculating the dosage of medicine for children 1 to 12 yr old. If a denotes the adult dosage (in milligrams) and if t is the child’s age (in years), then the child’s dosage is given by
( ) 12
D t at
t
Suppose the adult dosage of a substance is 500 mg.
Find an expression that gives the rate of change of a child’s dosage with respect to the child’s age. What is the rate of change of a child’s dosage with respect to his or her age for a 6-yr-old child? A 10-yr-old child?
53. EFFECT OF BACTERICIDE The number of bacteria N(t) in a certain culture t min after an experimental bactericide is introduced obeys the rule
2
10,000
( ) 2000
N t 1
t
Find the rate of change of the number of bacteria in the culture 1 min and 2 min after the bactericide is introd- uced. What is the population of the bacteria in the cult- ure 1 min and 2 min after the bactericide is introduced?
54. DEMAND FUNCTIONS The demand function for the Sicard wristwatch is given by
2
( ) 50 (0 20)
0.01 1
d x x
x
where x (measured in units of a thousand) is the quan- tity demanded per week and d(x) is the unit price in dollars.
a. Find d x( ).
b. Find d(5), d(10), and d(15) and interpret your results.
55. LEARNING CURVES From experience, Emory Secretarial School knows that the average student taking Advanc- ed Typing will progress according to the rule
60 180
( ) 6
N t t t
（ t 0)
where N(t) measures the number of words / minute the student can type after t wk in the course.
a. Find an expression forN t( )
b. Compute N t( ) for t1, 3, 4, and 7 and interpr-et your results.
c. Sketch the graph of the function N. Does it confirm the results obtained in part (b)?
d. What will be the average student’s typing speed at the end of the 12-wk course?
56. BOX-OFFICE RECEIPTS The total worldwide box-office receipts for a long-running movie are approximated by the function
2 2
( ) 120 4 T x x
x
where T(x) is measured in millions of dollars and x is the number of years since the movie’s release. How
fast are the total receipts changing 1 yr, 3 yr, and 5 yr after its release?
57. FORMALDEHYDE LEVELS A study on formaldehyde level in 900 homes indicates that emissions of various chem- icals can decrease over time. The formaldehyde level (parts per million) in an average home in the study is given by
0.055 0.26
( ) (0 12)
2
f t t t
t
where t is the age of the house in years. How fast is the formaldehyde level of the average house dropping when it is new? At the beginning of its fourth year?
Source: Bonneville Power Administration
58. POPULATION GROWTH A major corporation is building a 4325-acre complex of homes, offices, schools, and churches in the rural community of Glen Cove. As a result of this development, the planners have estimated that Glen Cove’s population (in thousands) t yr from now will be given by
2 2
25 125 200
( ) 5 40
t t
P t t t
a. Find the rate at which Glen Cove’s population is ch- anging with respect to time.
b. What will be the population after 10 yr? At what rate will the population be increasing when t = 10?
In Exercises 59–62, determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false.
59. If f and g are differentiable, then
( ) ( )
( ) ( )d f x g x f x g x
dx
60. If f is differentiable, then
( )
( ) ( )d xf x f x xf x
dx
61. If f is differentiable, then
2
( ) ( )
2
d f x f x
dx x x
62. If f, g, and h are differentiable, then
2( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( )
d f x g x f x g x h x f x g x h x f x g x h x
dx h x h x
63. Extend the product rule for differentiation to the follo- wing case involving the product of three differentiable functions: Let h x( )u x( )υ( ) ( )x w x and show that
( ) ( )υ( ) ( )
h x u x x w x u x( )υ( ) ( ) x w x u x( )υ( ) ( ) x w x . Hint: Let f x( )u x( )υ( ), ( )x g x w x( ), and h x( )
( ) ( )
f x g x and apply the product rule to the function h.
64. Prove the quotient rule for differentiation (Rule 6).
Hint: Let k x( ) f x g x( ) ( ) and verify the following steps:
a. ( ) ( ) ( ) ( ) ( ) ( )
( ) ( )
k x h k x f x h g x f x g x h
h hg x h g x
b. By adding
f x g x( ) ( ) f x g x( ) ( )
to the nume- rator and simplifying, show that( ) ( ) 1
( ) ( )
k x h k x
h g x h g x
( ) ( )
f x h f x ( ) h g x
( ) ( )
g x h g x ( ) h f x
c. 0
( ) ( )
( ) lim
h
k x h k x
k x h
2( ) ( ) ( ) ( ) ( )
g x f x f x g x g x
Exercises 3.3
In Exercises 1–48, find the derivative of each function.
1. f x( ) (2 x1)5 2. f x( ) (1 x)3 3. f x( ) ( x23)5 4. f t( ) 2( t31)5 5. f x( ) (2 x x 2 3) 6. f x( ) 3( x3x)4
7. f x( ) (2 x5)2 8. ( ) 1(2 2 )3 f t 2 t t 9. f x( ) ( x24)3 2 10. f t( ) (3 t2 2t 1)3 2
11. f x( ) 3x2 12. f t( ) 3t2t
13. f x( )34x2 14. f x( ) 2x22x5
15. ( ) 1 3
(2 5) f x x
16. ( ) 22 4
( 2)
f x x
17. 1
( ) 2 5
f t t
18. ( ) 12
2 3
f x x
19. 41 3 2
(4 )
y x x
20. ( ) 3 42
f t 2
t t
21. f x( ) (3 x22x1)2 22. f t( ) (5 t32t2 t 4)3 23. f x( ) ( x21)3(x31)2 24. f t( ) (2 t1)4(2t1)4
25. f t( ) ( t1t2 3) 26. f( ) ( 342 3) 27. f x( ) x 1 x1
28. f u( ) (2 u1)3 2(u21)3 2
29. f x( ) 2 (3 4 ) x2 x 4 30. h t( )t2(3t4)3 31. f x( ) ( x1) (22 x1)4
32. g u( ) (1 u2 5) (1 2 ) u2 8
33.
3 4
( ) 2
f x x x
34.
1 4
( ) 1
f x x x
35.
5 2
( ) 2 1
s t t t
36.
3 2
2 1
( )
g s s
s
37. ( ) 1
3 2
g u u u
38. ( ) 2 1
2 1
g x x x
39.
2
2 4
( ) ( 1) f x x
x
40.
2
2 3
( ) 2
( )
g u u
u u
41.
2 3
2 4
(3 1)
( ) ( 1)
h x x x
42.
2 4
(2 1) ( ) (3 2) g t t
t
43. ( ) 22 2
1 f x x
x
44.
2 2
( ) 4
2 2 1
f t t
t t
45. 2
( ) 1
1 g t t
t
46.
2 2
( ) 1
1 f x x
x
47. f x( ) (3 x1) (4 x2 x 1)3
48. g t( ) (2 t3) (32 t21)3
In Exercises 49–54, find dy du du dx and dy
dx . 49. y u 4 3 and u3x21
50. y u and u7x2x2 51. y u 2 3 and u2x3 x 1
52. y2u21 and ux21
53. y u 1
u and ux3x 54. y 1
and u u x1
55. Suppose F x( )g f x
( )
and f(2) 3, f (2) 3, (3) 5g , and g(3) 4 , Find F (2)
56. Suppose h f g. Find h(0) given that f(0) 6 , (5) 2
f , g(0) 5 , and g(0) 3 .
57. Suppose F x( ) f x( 21). Find F (1) if f (2) 3 . 58. Let F x( ) f f x( ( )). Does it follow that F x( )
f x( )
2?Hint: Let f x( )x2.
59. Suppose h g f . Does it follow that hgf? Hint: Let f x( )x and g x( )x2.
60. Suppose h f g. Show that h
fg g
. In Exercises 61-64, find an equation of the tangent line to the graph of the function at the given point.61. f x( ) (1 x x)( 21); (2, 9)
62.
1 2
( ) ; (3, 4) 1
f x x x
63. f x( )x 2x27; (3, 15)
64. 28
( ) ; (2, 2) f x 6
x x
65. TELEVISION VIEWING The number of viewers of a telev- ision series introduced several years ago is approxima- ted by the function
( ) (60 2 ) ; 2 3
N t t (1 t 26)
where N(t) (measured in millions) denotes the number of weekly viewers of the series in the t th week. Find the rate of increase of the weekly audience at end of week 2 and at the end of week 2 and at the end of week 12. How many viewers were there in week 2? In week 24 ?
66. OUTSOURCING OF JOBS According to a study conducted in 2003,the total number of U.S jobs that are projected to leave the country by year t, where t = 0 corresponds to 2000,is
( ) 0.018425( 5)2.5
N t t (0 t 15)
where N(t) is measured in millions. How fast will the number of U.S. jobs that are outsourced be changing in 2005? In 2010 (t = 10)
Source: Forrester Research
67. WORKING MONHERS The percent of mothers who work outside the home and have children younger than age 6 yr is approximated by the function
0.205
( ) 33.55( 5) (0 21) P t t t
where t is measured in years, with t = 0 corresponding to the beginning of 1980. Compute P t( ). At what rate was the percent of these mothers changing at the beginning of 2000? What was the percent of these mothers at the beginning of 2000?
Source: U.S. Bureau of Labor Statistics
68. SELLING PRICE OF DVD RECORDERS The rise of digital music and the improvement to the DVD format are some of the reasons why the average selling price of standalone DVD recorders will drop in the coming years. The function
0.94
( ) 699 (0 5)
( 1)
A t t
t
gives the projected average selling price (in dollars) of standalone DVD recorders in year t, where t = 0 corresp- onds to the beginning of 2002. How fast was the average selling price of standalone DVD recorders falling at the beginning of 2002? How fast was it falling at the beginn- ing of 2006?
Source: Consumer Electronics Association
69. SOCIALLY RESPONSIBLE FUNDS Since its inception in 1971, socially responsible investments, or SRIs, have yielded returns to investors on par with investments in general.
The assets of socially responsible funds (in billions of dollars) from 1991 through 2001 is given by
( ) 23.7(0.2 1)1.32 (0 11)
f t t t
where t = 0 corresponds to the beginning of 1991.
a. Find the rate at which the assets of SRIs were chang- ing at the beginning of 2000.
b. What were the assets of SRIs at the beginning of 2000?
Source: Thomson Financial Wiesenberger
70. AGING POPULATION The population of Americans age 55 and over as a percent of the total population is approxi- mated by the function
( ) 10.72(0.9 10) (00.3 20)
f t t t
where t is measured in years, with t = 0 corresponding to the year 2000. At what rate was the percent of Ame- ricans age 55 and over changing at the beginning of 2000? At what rate will the percent of Americans age 55 and over be changing in 2010? What will be the percent of the population of Americans age 55 and over in 2010?
Source: U.S. Census Bureau
71. CONCENTRATION OF CARBON MONOXIDE (CO) IN THE AIR According to a joint study conducted by Oxnard’s Environmental Management Department and a state government agency, the concentration of CO in the air due to automobile exhaust t yr from now is given by
2 2 3
( ) 0.01(0.2 4 64) C t t t parts per million.
a. Find the rate at which the level of CO is changing with respect to time.
b. Find the rate at which the level of CO will be changing 5 yr from now.
72. CONTINUING EDUCATION ENROLLMENT The registrar of Kellogg University estimates that the tot student enro- llment in the Continuing Education division will be given by
20,000
( ) 21,000
1 0.2
N t t
where N t( ) denotes the number of students enrolled in the division t yr from now . Find an expression for
( )
N t . How fast is the student enrollment increasing currently? How fast will it be increasing 5yr from now?
73. AIR POLLUTION According to the South Coast Air Qual- ity Management District, the level of nitrogen dioxide, a brown gas that impairs breathing, present in the atmosphere on a certain May day in downtown Los Angeles is approximated by
3 4
( ) 0.03 ( 7) 60.2 (0 t 7)
A t t t
where A t( ) is measured in pollutant standard index and t is measured in hours, with t0 corresponding to 7 a.m.
a. Find A t( )
b. Find A(1), (3)A , and A(4) and interpret your results.
74. EFFECT OF LUXURY TAX ON CONSUMPTION Government economists of a developing country determined that the purchase of imported perfume is related to a proposed
“luxury tax” by the formula
( ) 10, 000 40 0.02 2 (0 200)
N x x x x
where N x( ) measures the percentage of normal cons- umption of perfume when a “luxury tax” of x% is imp- osed on it . Find the rate of change of N(x) for taxes of 10%, 100%, and 150%.
75. PULSE RATEOF AN ATHLETE The pulse rate (the number of heartbeats/minute) of a long-distance rzunner t sec after leaving the starting line is given by
1 2
300 2 25
( ) 2 ( 0)
25
t t
P t t
t
Compute P t( ). How fast is the athlete’s pulse rate increasing 10 sec, 60 sec, and 2 min into the run? What is her pulse rate 2 min into the run?
76. THURSTONE LEARNING MODEL Psychologist L. L. Thur- stone suggested the following relationship between learning time T and the length of a list n:
( )
T f n An n b
where A and b are constants that depend on the person and the task.
a. Compute dT/dn and interpret your result.
b. For a certain person and a certain task, suppose A = 4 and b = 4, Compute f (13) and f (29) and interp- ret your results.
77. OIL SPILLS In calm waters, the oil spilling from the ruptured hull of a grounded tanker spreads in all directions. Assuming that the area polluted is a circle and that its radius is increasing at a rate of 2 ft/sec, determine how fast the area is increasing when the radius of the circle is 40 ft.
78. ARTERIOSCLEROSIS Refer to Example 8, PAGE190.
Suppose the radius of an individual’s artery is 1cm and the thickness of the plaque (in centimeters) t yr from now is given by
2 2
( ) 0.5 (0 10) 10
h g t t t
t
How fast will the arterial opening be decreasing 5 yr from now?
79. TRAFFIC FLOW Opened in the late 1950s, the Central Artery in downtown Boston was designed to move 75,000 vehicles a day. The number of vehicles moved per day is approximated by the function
( ) 6.252 19.75 74.75 (0 5)
x f t t t t
where x is measured in thousands and t in decades, with t0 corresponding to the beginning of 1959.
Suppose the average speed of traffic flow in mph is given by
( ) 0.00075 2 67.5 (75 350)
Sg x x x
where x has the same meaning as before. What was the rate of change of the average speed of traffic flow at the beginning of 1999? What was the average speed of traffic flow at that time?
Hint: Sg f t
( )
.80. HOTEL OCCUPANCY RATES The occupancy rate of the all-suite Wonderland Hotel, located near an amusement park, is given by the function
3 2
10 10 200
( ) 60 (0 12)
81 3 9
r t t t t t
where t is measured in months, with t = 0 correspo- nding to the beginning of January. Management has estimated that the monthly revenue (in thousands of dollars/month) is approximated by the function
3 2
3 9
( ) (0 100)
5000 50
R r r r r where r is the occupancy rate.
a. Find an expression that gives the rate of change of Wonderland’s occupancy rate with respect to time.
b. Find an expression that gives the rate of change of Wonderland’s monthly revenue with respect to the occupancy rate.
c. What is the rate of change of wonderland’s monthly revenue with respect to time at the beginning of Jan- uary? At the beginning of July?
Hint: Use the chain rule to find R r
0
r
0 and
6
6R r r .
81. EFFECT OF HOUSING STARTS ON JOBS The president of a major housing construction firm claims that the num- ber of construction jobs created is given by
N(x) = 1.42x.
where x denotes the number of housing starts. Suppose the number of housing start in the next t mo is expected to be
2 2
7 140 700
( ) 3 80 550
t t
x t t t
million units/year. Find an expression that gives the rate at which the number of construction jobs will be created t mo from now. At what rate will construction jobs be created 1 yr from now?
82. DEMAND FOR PCS The quantity demanded per month, x, of a certain make of personal computer (PC) is related to the average unit price, p(in dollars), of PCs by the equation
100 2
( ) 810,000
x f p 9 p
It is estimated that t mo from now, the average price of a PC will be given by
( ) 400 200 (0 60)
1 1 8
p t t
t
dollars. Find the rate at which the quantity demanded per month of the PCs will be changing 16 mo from now.
83. DEMAND FOR WATCHES The demand equation for the Sicard wristwatch is given by
( ) 1050 p (0 50)
x f p p
p
where x (measured in units of a thousand) is the quan- tity demanded each week and p is the unit price in dol- lars. Find the rate of change of the quantity demanded of the wristwatches with respect to the unit price when the unit price is $25.
84. CRUISE SHIP BOOKINGS The management of Cruise World, operators of Caribbean luxury cruises, expects that the percent of young adults booking passage on their cruises in the years ahead will rise dramatically.
They have constructed the following model, which give the percent of young adult passengers in year t:
2 2
2 4
( ) 50 (0 5)
4 8
t t
p f t t
t t
Young adults normally pick shorter cruises and gener- ally spend less on their passage. The following model gives an approximation of the average amount of mon- ey R (in dollars) spent per passenger on a cruise when the percent of young adults is p:
( ) 1000 4 2 R p p
p
Find the rate at which the price of the average passage will be changing 2 yr form now.
In Exercises 85–88, determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false.
85. If f and g are differentiable and h f g then ( ) [ ( )] ( ).
h x f g x g x
86. If f is differentiable and c is a constant, then
[ ( )] ( )
d f cx cf cx
dx
87. If f is differentiable, then ( ) ( )
2 ( )
d f x
dx f x f x
88. If f is differentiable, then
1 1
d f f
dx x x
89. In Section 3.1, we proved that ( )n n1
d x nx
dx
for the special case when n = 2. Use the chain rule to show that
1/ 1 1/ 1
( n) n
d x x
dx n
for any nonzero integer n, assuming thatf x( )x1/nis differentiable
Hint: Letf x( )x1/nso that[ ( )]f x nx. Differentiable both sides with respect to x.
90. With the aid of Exercise 89, prove that ( )r r1
d x rx
dx
for any rational number r
Hint: Let r = m/n, where m and n are integers, with n ≠ 0, and writex (xm)1n
Exercises 3.4
1. PRODUTION COSTS The graph of a typical total cost fun- ction C(x) associated with the manufacture of x units of a certain commodity is shown in the following figure.
a. Explain why the function C is always increasing.
b. As the level of production x increases, the cost/unit drops so that C (x) increases but at a slower pace. However, a level of production is soon reached at which the cost/unit begins to increase dramatically (due to a shortage of raw material, overtime break-down of machinery due excessive stress and strain) so that C(x) continues to
