238 Chapter 3 Applications of Differentiation

Review Exercises See CalcChat.com for tutorial help and worked-out solutions to odd-numbered exercises.

Finding Extrema on a Closed Interval In Exercises 1–8,find the absolute extrema of the function on the closed interval.

1. 2.

3. 4.

5. 6.

7.

8.

Using Rolle’s Theorem In Exercises 9–12, determinewhether Rolle’s Theorem can be applied to on the closedinterval . If Rolle’s Theorem can be applied, find all values of in the open interval such that IfRolle’s Theorem cannot be applied, explain why not.

9.

10.

11.

12.

Using the Mean Value Theorem In Exercises 13–18, deter-mine whether the Mean Value Theorem can be applied to onthe closed interval If the Mean Value Theorem can beapplied, find all values of in the open interval such that

If the Mean Value Theorem cannot be applied, explain why not.

13.

14.

15.

16.

17.

18.

19. Mean Value Theorem Can the Mean Value Theorem beapplied to the function

on the interval Explain.

20. Using the Mean Value Theorem

(a) For the function determine thevalue of guaranteed by the Mean Value Theorem on theinterval

(b) Demonstrate the result of part (a) for on the interval

Intervals on Which Is Increasing or Decreasing InExercises 21–26, identify the open intervals on which the function is increasing or decreasing.

21.

22.

23.

24.

25.

26.

Applying the First Derivative Test In Exercises 27–34,(a) find the critical numbers of (if any), (b) find the open interval(s) on which the function is increasing or decreasing,(c) apply the First Derivative Test to identify all relativeextrema, and (d) use a graphing utility to confirm your results.

27.

28.

29.

30.

31.

32.

33.

34.

Finding Points of Inflection In Exercises 35–40, find thepoints of inflection and discuss the concavity of the graph of thefunction.

35.

36.

37.

38.

39.

40.

Using the Second Derivative Test In Exercises 41–46,find all relative extrema. Use the Second Derivative Test whereapplicable.

41.

42.

43.

44. h�t� � t � 4�t � 1

g�x� � 2x2�1 � x2� f �x� � 2x3 � 11x2 � 8x � 12

f �x� � �x � 9�2

f �x� � tan x4

, �0, 2��

f �x� � x � cos x, �0, 2�� f �x� � 3x � 5x3

g�x� � x�x � 5

f �x� � 6x4 � x2

f �x� � x3 � 9x2

�0, 4�g�x� �32

sin��x2

� 1� ,

f �x� � cos x � sin x, �0, 2��

f �x� �x2 � 3x � 4

x � 2

f �x� �x � 4

x2

g�x� �x3 � 8x

4

h�t� �14

t 4 � 8t

f �x� � 4x3 � 5x

f �x� � x2 � 6x � 5

f

�0, 2�� f �x� � sin x � cos x,

x > 0h�x� � �x�x � 3�,g�x� � �x � 1�3

f �x� � �x � 1�2�x � 3�h�x� � �x � 2�1�3 � 8

f �x� � x2 � 3x � 12

f

�0, 4�.f �x� � 2x2 � 3x � 1

�x1, x2�.c

f �x� � Ax2 � Bx � C,

��2, 1�?

f �x� �1x2

f �x� � �x � 2x, �0, 4�

f �x� � x � cos x, ���

2,

�

2�f �x� � 2x � 3�x, ��1, 1�f �x� � �5 � x�, �2, 6�

f �x� �1x, �1, 4�

f �x� � x2�3, �1, 8�

f��c� �f �b� � f �a�

b � a.

�a, b�c[a, b].

f

���, ��f �x� � sin 2x,

��2, 2�f �x� �x2

1 � x2,

��3, 2�f �x� � �x � 2��x � 3�2,

�0, 4�f �x� � 2x2 � 7,

f��c� � 0.�a, b�c[a, b�

f

�0, 2��f �x� � sin 2x,

�0, 2��g�x� � 2x � 5 cos x,

�0, 2�f �x� �x

�x2 � 1,��4, 4�f �x� �

4xx2 � 9

,

�0, 9�h�x� � 3�x � x,�0, 4�f �x� � �x � 2,

��6, 1�f �x� � x3 � 6x2,��4, 0�f �x� � x2 � 5x,

Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).

Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

Review Exercises 239

45.

46.

Think About It In Exercises 47 and 48, sketch the graph ofa function having the given characteristics.

47. 48.

for or

for does not exist.

for

for for

for or for

for

49. Writing A newspaper headline states that “The rate ofgrowth of the national deficit is decreasing.” What does thismean? What does it imply about the graph of the deficit as afunction of time?

50. Inventory Cost The cost of inventory depends on theordering and storage costs according to the inventory model

Determine the order size that will minimize the cost, assumingthat sales occur at a constant rate, is the number of units soldper year, is the cost of storing one unit for one year, is thecost of placing an order, and is the number of units per order.

51. Modeling Data Outlays for national defense (in billions of dollars) for selected years from 1970 through 2010are shown in the table, where is time in years, with corresponding to 1970. (Source: U.S. Office of Managementand Budget)

(a) Use the regression capabilities of a graphing utility to finda model of the form

for the data.

(b) Use a graphing utility to plot the data and graph the model.

(c) For the years shown in the table, when does the modelindicate that the outlay for national defense was at a maximum? When was it at a minimum?

(d) For the years shown in the table, when does the modelindicate that the outlay for national defense was increasingat the greatest rate?

52. Modeling Data The manager of a store recorded the annual sales (in thousands of dollars) of a product over aperiod of 7 years, as shown in the table, where is the time inyears, with corresponding to 2006.

(a) Use the regression capabilities of a graphing utility to finda model of the form

for the data.

(b) Use a graphing utility to plot the data and graph the model.

(c) Use calculus and the model to find the time when saleswere increasing at the greatest rate.

(d) Do you think the model would be accurate for predictingfuture sales? Explain.

Finding a Limit In Exercises 53–62, find the limit.

53. 54.

55. 56.

57. 58.

59. 60.

61. 62.

Horizontal Asymptotes In Exercises 63–66, use a graphing utility to graph the function and identify any horizontal asymptotes.

63. 64.

65. 66.

Analyzing the Graph of a Function In Exercises 67–76,analyze and sketch a graph of the function. Label any intercepts,relative extrema, points of inflection, and asymptotes. Use agraphing utility to verify your results.

67. 68.

69. 70.

71.

72.

73.

74. f �x� �2x

1 � x2

f �x� �5 � 3xx � 2

f �x� � �x � 3��x � 2�3

f �x� � x1�3�x � 3�2�3

f �x� � �x2 � 4�2f �x� � x�16 � x2

f �x� � 4x3 � x4f �x� � 4x � x2

f �x� �3x

�x2 � 2h�x� �

2x � 3x � 4

g�x� �5x2

x2 � 2f �x� �

3x

� 2

limx→��

x

2 sin xlim

x→��

6xx � cos x

limx→�

x3

�x2 � 2lim

x→� 5 cos x

x

limx→��

�x2 � x

�2xlim

x→��

3x2

x � 5

limx→�

4x3

x4 � 3lim

x→�

2x2

3x2 � 5

limx→��

1 � 4xx � 1

limx→�

�8 �1x�

t

S � at3 � bt2 � ct � d

t 6 7 8 9 10 11 12

S 5.4 6.9 11.5 15.5 19.0 22.0 23.6

t � 6t

S

D � at4 � bt3 � ct2 � dt � e

t 25 30 35 40

D 272.1 294.4 495.3 693.6

t 0 5 10 15 20

D 81.7 86.5 134.0 252.7 299.3

t � 0t

D

xsr

Q

C � �Qx �s � �x

2� r.

C

3 < x < 4f � �x� > 0

x � 2f � �x� < 0x > 4x < 3f� �x� < 0

2 < x < 4f��x� > 0x > 5f��x� < 0

f��4� � 03 < x < 5f��x� > 0

f��2�x < 3f��x� > 0

x > 4x < 2f��x� < 0f��3� � f��5� � 0

f �6� � 0f �0� � 4,f �0� � f �6� � 0

f

�0, 4��h�x� � x � 2 cos x,

f �x� � 2x �18x

Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).

Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

240 Chapter 3 Applications of Differentiation

75.

76.

77. Maximum Area A rancher has 400 feet of fencing withwhich to enclose two adjacent rectangular corrals (see figure).What dimensions should be used so that the enclosed area willbe a maximum?

78. Maximum Area Find the dimensions of the rectangle ofmaximum area, with sides parallel to the coordinate axes, thatcan be inscribed in the ellipse given by

79. Minimum Length A right triangle in the first quadranthas the coordinate axes as sides, and the hypotenuse passesthrough the point Find the vertices of the triangle suchthat the length of the hypotenuse is minimum.

80. Minimum Length The wall of a building is to be bracedby a beam that must pass over a parallel fence 5 feet high and4 feet from the building. Find the length of the shortest beamthat can be used.

81. Maximum Length Find the length of the longest pipethat can be carried level around a right-angle corner at theintersection of two corridors of widths 4 feet and 6 feet.

82. Maximum Length A hallway of width 6 feet meets ahallway of width 9 feet at right angles. Find the length of thelongest pipe that can be carried level around this corner. [Hint:If is the length of the pipe, show that

where is the angle between the pipe and the wall of thenarrower hallway.]

83. Maximum Volume Find the volume of the largest rightcircular cone that can be inscribed in a sphere of radius

84. Maximum Volume Find the volume of the largest rightcircular cylinder that can be inscribed in a sphere of radius

Using Newton’s Method In Exercises 85–88, approxi-mate the zero(s) of the function. Use Newton’s Method andcontinue the process until two successive approximations differby less than 0.001. Then find the zero(s) using a graphing utility and compare the results.

85.

86.

87.

88.

Finding Point(s) of Intersection In Exercises 89 and 90,apply Newton’s Method to approximate the -value(s) of theindicated point(s) of intersection of the two graphs. Continuethe process until two successive approximations differ by lessthan 0.001. [Hint: Let

89. 90.

Comparing y and dy In Exercises 91 and 92, use the information to evaluate and compare and

Function x-Value Differential of x

91.

92.

Finding a Differential In Exercises 93 and 94, find the differential of the given function.

93. 94.

95. Volume and Surface Area The radius of a sphere ismeasured as 9 centimeters, with a possible error of 0.025 centimeter.

(a) Use differentials to approximate the possible propagatederror in computing the volume of the sphere.

(b) Use differentials to approximate the possible propagatederror in computing the surface area of the sphere.

(c) Approximate the percent errors in parts (a) and (b).

96. Demand Function A company finds that the demand forits commodity is

where is the price in dollars and is the number of units.Find and compare the values of and as changes from7 to 8.

xdp�pxp

p � 75 �14

x

y � �36 � x2y � x�1 � cos x�

dy

�x � dx � 0.1x � 2y � x3 � 6x

�x � dx � 0.01x � 3y � 0.5x2

dy.�y�

x

f

g

y

1 2 3

1

3

x

f g

y

1 2−2−1

1

3

g�x� � x2 � 2x � 1g�x� � x5 � 2

f �x� � sin xf �x� � 1 � x

h�x� � f �x� � g�x�.�

x

f �x� � 3�x � 1 � x

f �x� � x4 � x3 � 3x2 � 2

f �x� � x3 � 2x � 1

f �x� � x3 � 3x � 1

r.

r

r

r.

�

L � 6 csc � � 9 csc��

2� ��

L

�1, 8�.

x2

144�

y 2

16� 1.

x x

y

f �x� � x2 �1x

f �x� � x3 � x �4x

Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).

Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

A36 Answers to Odd-Numbered Exercises

21. (a) 0.9 (b) 1.04 23. (a) 8.035 (b) 7.95

25. (a) (b) 0.625%

27. (a) (b) about 1.19%29. (a) (b) (c) 0.6%; 0.4%31. 27.5 mi; About 7.3% 33. (a) (b)35. 6407 ft

37.

Calculator: 9.97

39.

Calculator: 4.99841.

43. The value of becomes closer to the value of as decreases.

45.

47. True 49. True

Review Exercises for Chapter 3 (page 238)

1. Maximum: 3. Maximum:Minimum: Minimum:

5. Maximum: 7. Maximum:Minimum: Minimum:

9. 11. Not continuous on

13. 15. is not differentiable at

17.19. No; The function has a discontinuity at which is in the

interval 21. Increasing on Decreasing on 23. Increasing on Decreasing on 25. Increasing on Decreasing on 27. (a) Critical number:

(b) Increasing on Decreasing on (c) Relative minimum:(d)

29. (a) Critical number:(b) Increasing on Decreasing on (c) Relative minimum:(d)

31. (a) Critical number: Discontinuity:(b) Increasing on

Decreasing on and (c) Relative minimum:(d)

33. (a) Critical numbers:

(b) Increasing on

Decreasing on and

(c) Relative minimum:

Relative maximum:

(d)

35. Concave upward:Concave downward:

37. No point of inflection; Concave upward:39. Concave upward:

Concave downward:41. Relative minimum:43. Relative maxima:

Relative minimum:45. Relative maximum: Relative minimum:47. 49. Increasing and concave down

51. (a)(b)

(c) Maximum in 2010; Minimum in 1970 (d) 2010

00

800

40

D � 0.00188t 4 � 0.1273t2 � 2.672t2 � 7.81t � 77.1

x2 3 4 5

5

2

6

(0, 0)

(6, 0)

3

7

−1

4

1

y

7

(3, f (3))

(5, f (5))

�3, 12���3, �12�;�0, 0�

���2�2, 1�2�;��2�2, 1�2�,��9, 0�

�3��2, 2���0, ��2�,���2, 3��2�;�3��2, 3��2�;���2, ��2�,

��5, �����, 3)

�3, ��;�3, �54);

0

−2

2

2p

�7�

4, �2�

�3�

4, ��2�;

�7�

4, 2���0,

3�

4 ��3�

4,

7�

4 �;

x �3�

4,

7�

4

−10

−2

8

5

��8, � 116�

�0, �����, �8���8, 0�;

x � 0x � �8;

−2

−15

10

6

�2, �12����, 2��2, ��;

t � 2

−3

−5

3

9

�3, �4����, 3��3, ��;

x � 3�0, 1��1, ��;

�1, 73����, 1�, �73, ��;

���, �32���3

2, ��;��2, 1�.

x � 0,f� �0� � 1

x � 5.ff��2744729 � �

37

��2, 2�f �0� � f �4��2.73, 0.88���3, �2

3 ��2�, 17.57�;�3, 23�;�0, �2���5

2, �254 �

�4, 0�;�0, 0�;

f �4.02� � �4 �1

2�4�0.02� � 2 �

14

�0.02�

dy �1

2�x dxf �x� � �x;

�x�ydy

y � 2 � x�4 y � 2 �

14 x

6

−2

−6

(0, 2)yf

6 y � f �0� � f��0��x � 0�

f �624� � 4�625 �1

4�625�3�4 ��1� � 4.998

f �x� � 4�x, dy �1

4x3�4 dx

f �99.4� � �100 �1

2�100 ��0.6� � 9.97

f �x� � �x, dy �1

2�x dx

216 sec � 3.6 min14%

±5.4 in.2±20.25 in.3±10.75 cm2

±58

in.2

Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).

Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

53. 8 55. 57. 59. 0 61. 663.

65. 67.

69. 71.

73. 75.

77. and 79.81. 14.05 ft 83. 85.87. 89.91.93. 95. (a)

(b) (c) About 0.83%; About 0.56%

P.S. Problem Solving (page 241)

1. Choices of may vary.

(a) One relative minimum at for

(b) One relative maximum atfor

(c) Two relative minima forwhen

(d) If then there are threecritical points; if thenthere is only one critical point.

3. All where is a real number 5. Proof7. The bug should head towards the midpoint of the opposite side.

Without calculus, imagine opening up the cube. The shortestdistance is the line passing through the midpoint as shown.

9. 11. Proof

13. Greatest slope: Least slope:

15. Proof 17. Proof; Point of inflection:19. (a)

(b)

Chapter 4Section 4.1 (page 251)

1. Proof 3. 5.OriginalIntegral Rewrite Integrate Simplify

7.

9.

11. 13.

15. 17.

19. 21.

23. 25.27. 29. 31.33. Answers will vary. Sample answer:

35. 37.39. 41.

43. (a) Answers will vary. (b)

Sample answer:

45. (a) (b)(c)

15

−8

−15

12

y � x2 � 6

3

−9

−3

9

−4 4

−5

( 1, 3)−

5

x− 4

−5

5

4

y

y �x3

3� x �

73

f �x� � �4�x � 3xf �x� � x2 � x � 4h�t� � 2t 4 � 5t � 11f �x� � 3x2 � 8

y

x−1−2−3 1 2 3

2

3

5

f(x) = 4x + 2

f ′

f(x) = 4x

tan y � Ctan � � cos � � Ct � csc t � C5 sin x � 4 cos x � Cx3 �

12 x2 � 2x � C

23 x3�2 � 12x1�2 � C�1��4x4� � C

35 x 5�3 � C2

5 x 5�2 � x2 � x � C

16 x6 � x � C1

2 x2 � 7x � C

�2�x

� Cx�1�2

�1�2� C�x�3�2 dx� 1

x�x dx

34

x 4�3 � Cx 4�3

4�3� C�x1�3 dx� 3�x dx

y �25 x 5�2 � Cy � 3t3 � C

5

3

−3

−3(0, 0)

P(x)

f (x)

P�x� � x � x2

�1, 0���3

3,

34���

�33

, 34�;

c � 2b � 1,a � 6,

Q

Px

PQ,

cc,

a � 0,a < 0,

x � ±��a�2a < 0

a < 0�0, 1�

a � 0�0, 1�

x

54

y

678

32

−1−2

2−2

a = −3

a = −2

a = −1

a = 1 a = 3 a = 2 a = 0

a

±1.8� cm2

±8.1� cm3dy � �1 � cos x � x sin x� dxdy � 0.03�y � 0.03005;

�0.755�0.795�2.182,�1.532, �0.347, 1.87932�r3�81

�0, 0�, �5, 0�, �0, 10�y �2003 ftx � 50 ft

x21

x = 0

5

10

−1

−5

−2(−1, −6)

(1, 6)

yy

x

x = 2

y = −3−1−2 1 3 4 5 6

−2

−4

−5

−6

1

2

2

53 , 0( (

52

0, −( (

(−3, 0)

(−1, −1.59)

(0, 0)x

21−1−2

1

2

3

4

−4−5

−3

y

8x

4 62

6

4

−2−6

2

−8

8

−8

y

(−4, 0) (4, 0)

(0, 0)

2, −8−2 ))

2, 82 ))

x

(2, 4)

(4, 0)(0, 0)

5

4

2

3

5321

1

y

−6

−4

8

12

y = 2

−5

−7

3

5

y = −2

��23

Answers to Odd-Numbered Exercises A37

Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).

Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.