Blank B.E., Krantz S.G. Calculus: Single Variable

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95.

20

y  f(c)(x  c)  f(c)

y  20x  1

c  1.3124

y  x

 x

 1

97.

311

y  6x  25

c  1.826

y  x

 2x

 x

 2x  13

y  6(x  c)  f(c)

Section 4.3

1. Decreasing: ð2N; 21=2Þ; increasing:

ð21=2; NÞ; c 521/2: (absolute) minimum

3. Decreasing: (25, 3); increasing:

ð2N; 25Þ,ð3; NÞ; c 525: local maximum; c 5 3: local

minimum

5. Decreasing: ð2N; 23Þ,ð22; 0Þ; increasing:

ð23; 22Þ,ð0; NÞ; c 523: local minimum; c 522: local

maximum; c 5 0: local minimum

7. Decreasing: (25, 5); increasing:

ð2N; 25Þ,ð5; NÞ; c 525:

local maximum; c 5 5: local

minimum

9. Decreasing: ð2N; 1Þ,ð1; NÞ

11. Decreasing: (22, 24/3); increasing:

ð2N; 22Þ,ð24=3; NÞ; c 522: local maximum;

c 524/3: local minimum

13. Decreasing: ð23; 22Þ,ð22; 21Þ; increasing: ð2N; 23Þ

,ð21; NÞ; c 523 : local maximum; c 521 : local

minimum

15. Decreasing : ð23=2; 0Þ,ð0; 3=2Þ; increasing:

ð2N; 23=2Þ,ð3=2; NÞ; c 523=2:

local

maximum; c 5 3=2 : local minimum

17. Increasing: (2N, N)

19. Decreasing: (2N, 0); increasing: (0, N); c 5 0: local

minimum

21. Decreasing: ð2N; 0Þ,ð2; NÞ;

increasing: ð0; 2Þ; c

5 0 : local

minimum;

c 5 2 : local maximum

23. Decreasing:



2N; 2ln



lnð2Þ



=lnð2Þ



; increasing:



lnð2Þ



=lnð2Þ; N



; c 52ln



lnð2Þ



=lnð2Þ: local

minimum

25. Decreasing: (0, 1/e); increasing: (1/e,N); c 5 1/e: local

minimum

27. Decreasing: ð0; 1Þ,ð1; eÞ; increasing : ðe; NÞ; c 5 e :

local

minimum

29. Local minimum at 22/5

31. No extremum at 0; local minimum at 2

33. Local maximum at 21/27; no extremum at 0; local

minimum at 1/27

35. No extremum at 0; local minimum at 8

37. Local maximum at 25; local minimum at 21; no extre-

mum at 1

39. Local maximum at 21/3; local minimum at 3; no extre-

mum at 4;

41. Decreasing: (21, 0) and (3/2, 2); increasing: (23/2, 21),

(0, 3/2), and (2, 5/2)

43. Local minima at 2

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ



2=lnð2Þ



lnð2Þ

and

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ



2=lnð2Þ



lnð2Þ

;

local maximum at 0

45. No local extremum at 8/3

47. Local maximum at

ﬃﬃﬃ

49. Local minimum at

ﬃﬃﬃ

2 2

51. Local minimum at 2

lnð3Þ

53. Local minimum at 1=

ﬃﬃﬃ

77. Decreasing on (2N, 2.33344); increasing on (2.33344, N)

79. Decreasing on (2N, 21.3363) and (0.3449, 2.8134);

increasing on (21.3363, 0.3449) and (2.8134, N)

81. Decreasing on (2N, 24.0471) and (20.46916, 0.83853);

increasing on (24.0471, 20.46916) and (0.83853,N)

83. Decreasing on (0.36835, 215.9861); increasing on ( 2N,

0.36835) and (215.9861, N)

85. Local maximum at 20.724068; local minimum at

0.4628463

87. Local minimum at 2.04289; local maxima at 0.50638 and

2.95695

Section 4.3 A-31

Section 4.4

1. 25 m (each side)

3. (2

21/3

1/6

)

5. 10/3 in (each side)

7. 22 9.

ﬃﬃﬃ

11. Height 5 width 5 2 1 4

ﬃﬃﬃ

13. Base: 4

ﬃﬃﬃ

=3; height: 8/3

15. Side length of base: 2  9

1/3

17. 6

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

33ð6 1

ﬃﬃﬃ

19. 5π/3 in

21. 100 23. 25

25. 21/4, 2

27. 21/2, 31/2

29. 23, 9

31. 12, 13

33. 1, e 2 1

35. The positive integer in (0, α/β) nearest α/(2β)

41. ðb 1

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1 b

Þ=a

43. Circumference of circle: πL/(π 1 4)

45. Minimum area: πL

/(6π

1 8π 1 8); maximum area:

/(4π)

47. (b/a)

1/4

49. (2b)

1/6

53. (r/R)

55. 5000 m

57. 8/π 59. ‘/6

63. 20.2228589, 1.059793

65. 10(2 2 2

2/3

1 2

1/3

)/3 cm from the stronger source

67.

arccos



gh=ðv

1 ghÞ



69. 1.029

71. 0.64

Section 4.5

1. Concave up: (0,N); concave down: no interval; inﬂection

points: none; critical points: 4; local minimum value at

x 5 4; local maximum value: none

3. Concave up: (0, 9); concave down: (9,N); inﬂection point:

(9, 4); critical points: 3; local minimum value at x 5 3;

local maximum value: none

5. Concave up: (2N, 23  2

1/3

) and (0, N); concave down:

(23  2

1/3

, 0); inﬂection point: (23  2

1/3

, 0); critical

points: 3; local minimum value at x 5 3; local maximum

value: none

7. Concave up: (23,N); concave down: (2N, 23); inﬂection

point: (23, 132); critical points: 27, 1; local minimum

value at x 5 1; local maximum value at x 527

9. Concave up: (1/2,N); concave down: (2N, 1/2); inﬂection

point: (1/2, 211/2); critical points: 21, 2; local minimum

value at x 5 2; local maximum value at x 521

11. Concave up: ð2N; ð1 2

ﬃﬃﬃ

Þ=2Þand ðð1 1

ﬃﬃﬃ

Þ=2; NÞ;

concave down : ðð1 2

ﬃﬃﬃ

Þ=2; ð1 1

ﬃﬃﬃ

Þ=2Þ; inflection

points: ðð1 2

ﬃﬃ

ﬃ

Þ=2; 271=4 1 5

ﬃﬃﬃ

Þ and ðð1 1

ﬃﬃﬃ

Þ=2;

271=4 2 5

ﬃﬃﬃ

Þ; critical points: 23/2, 0, 3; local

minimum values at x 523/2 and x 5 3; local

maximum value at x 5 0

13. Concave up: (21, 0) and (1,N); concave down: (2N, 21)

and (0, 1); inﬂection points: (21, 216), (0, 28), and (1, 0);

critical points: 21, 1; local minimum values: none; local

maximum values: none

15. Concave up: (23, 0) and (3,N); concave down: (2N, 23)

and (0, 3); inﬂection points: (23, 21/4), (0, 0), and (3, 1/4);

critical points : 2

ﬃﬃﬃ

and

ﬃﬃ

ﬃ

; local minimum value at

x 52

ﬃﬃﬃ

; local maximum value at x 5

ﬃﬃﬃ

17. Concave up: ð2

ﬃﬃﬃ

=3;

ﬃﬃﬃ

=3Þ; concave down:

ð2N; 2

ﬃﬃﬃ

=3Þ and ð

ﬃﬃﬃ

=3; NÞ; inﬂection points:

ð2

ﬃﬃﬃ

=3; 21=2Þ and ð

ﬃﬃﬃ

=3; 21=2Þ; critical point: 0; local

minimum value at x 5 0; local maximum values: none

19. Concave up: (π/2, 3π/2); concave down: (0, π/2) and

(3π/2,

2π); inﬂection points: (π/2, π/2) and (3π/2, 3π/2);

critical points: π/2; local minimum values: none; local

maximum values: none

21. Concave up: (0, π); concave down: (2π, 0); inﬂection

point: (0, 0); critical points: none; local minimum values:

none; local maximum values: none

23. Concave up: (0, π/4) and (3π/4, π); concave down: (π/4,

3π/4); inﬂection points: (π/4, 1/2) and (3π/4, 1/2); critical

points: π/2; local minimum value: none; local maximum

value at x 5 π /2

25. Concave up: (2N,N); concave down: nowhere; inﬂection

points: none; critical points: 0; local minimum value at

x 5 0; local maximum values: none

27. Concave up: (0,N); concave down: nowhere; inﬂection

points: none; critical points: 1; local minimum value at

x 5 1; local maximum values: none

29. Concave up: (0,N); concave down: nowhere; inﬂection

points: none; critical points: 2; local minimum value at

x 5 2; local maximum values: none

31. Concave up: (2N, 25/2) and (0,N); concave down:

(25/2, 0); inﬂection points: (25/2, f (25/2)) and (0, 0); cri-

tical points: 5/4, 0; local minimum value at x 5 5/4; local

maximum values: none

33. Concave

up: (2N,N); concave down: nowhere; inﬂection

points: none; critical points: ln(ln(2))/ln(2); local mini-

mum value at x 5 ln(ln(2))/ln(2); local maximum values:

none

A-32 Answers to Selected Exercises

35. Concave up: ð

ﬃﬃﬃ

=2; 1Þ; concave down: ð21;

ﬃﬃﬃ

=2Þ;

inﬂection point: ð

ﬃﬃﬃ

=2; π=4 2 1=2Þ; critical point:

ﬃﬃﬃ

=2;

local minimum values: none; local maximum value: none

37. Local maximum at 0; local minimum at 1

39. Local maximum at 21; local minimum at 1

41. Local maximum at 22; local minimum at 2

43. 21, 0 45. none 47. 25, 5

49. Inﬂection points: ð22a=

ﬃﬃﬃ

; 3a=2Þ and ð2a=

ﬃﬃﬃ

; 3a=2Þ;

concave up: ð2N; 22a=

ﬃﬃ

ﬃ

Þ and ð2a=

ﬃﬃﬃ

; NÞ; concave

down: ð22a=

ﬃﬃﬃ

; 2a=

ﬃﬃﬃ

51. Concave up for all x

53. Inﬂection point: ðm

ðcÞ; cÞ; concave up on

ð0; m

ðcÞÞand concave down on ðm

ðcÞ;NÞ

57. Concave down on (0, 2π); local maximum at π

59. Concave up on ð2N; 2

ﬃﬃﬃ

=3Þ and ð

ﬃﬃﬃ

=3; NÞ; concave

down on ð2

ﬃﬃﬃ

=3;

ﬃﬃﬃ

=3Þ

x  tan (t),

y  cos

(t)

(1/

冑3, 3/4)

(1/冑3, 3/4)

61. jcosðxÞj=



1 1 sin

ðxÞ



3=2

63. 2jxj

=ðx

1 1Þ

3=2

65. 1=

ﬃﬃﬃ

67. 2

lnð2Þ

73. Critical point: 1.27846 local maximum; inﬂection points:

(22.39936,22.19968) and (2.39936, 0.19968)

75. Critical points: 20.80778 local minimum and 6.14167

local maximum; inﬂection points: (20.34378, 22.43554),

(0.53289,21.72839), and (5.01770, 346.83)

77. Critical points: 21.44403 local maximum and 0.56043 local

minimum; inﬂection points: (21.82524, 1.07320),

(20.75879, 0.84663), and (1.22627, 0.59430)

79. Critical point: 0.70871 local maximum; inﬂection points:

(20.40297, 20.52219) and (1.44141, 0.21890)

Section 4.6

1. Local maximum: P 5 (21, 12); local minimum: R 5

(3, 220); point of inﬂection: Q 5 (1, 24); increasing:

(2N, 21),(3,N); decreasing: (1, 3); concave up: (1,N);

concave down: (2N,1)

60

40

20

4

(

)

 x

 3x

 9x  7

3. Local maximum: P 5 (0, 4); local minimum: R 5 (2, 0);

point of inﬂection: Q 5 (1, 2); increasing: (2N,0),

(2,N); decreasing: (0, 2); concave up: (1,N); concave

down: (2N,1)

2

10

4312

5. Local maximum: S 5 (2, 1/4); local minimum:

Q 5 (22, 21/4); points of inﬂection:

P 5 ð22

ﬃﬃﬃ

; 2

ﬃﬃﬃ

=8Þ, R 5 (0, 0), and T 5 ð2

ﬃﬃﬃ

;

ﬃﬃﬃ

=8Þ;

increasing: (22, 2); decreasing: (2N, 22),(2, N);

concave up: ð2 2

ﬃﬃﬃ

; 0Þ,ð2

ﬃﬃﬃ

; NÞ; concave down:

ð2N; 22

ﬃﬃﬃ

Þ,ð0; 2

ﬃﬃﬃ

Þ; horizontal asymptote: y 5 0

10 5105

7. Local maximum: P 5 (0, 21); increasing: (2N, 22),

(22, 0); decreasing: (0, 2),(2, N); concave up: (2N,2),

(2,N); concave down: (22, 2); vertical asymptotes x 522

and x 5 2; horizontal asymptote: y 5 1

Section 4.6 A-33

x  2 x  2

55

9. Local minimum: P 5 (2

24/3

, f (2

24/3

)); increasing: (2

24/3

,N);

decreasing: (0, 2

24/3

); concave up: (0, N)

11. Points of inﬂection: (nπ, 2nπ) for every integer n; con-

cave down: (2kπ,(2k 1 1)π) for every integer k; concave

up: ((2k 2 1)π,2kπ) for every integer k

10 5

5

10

105

13. Local maximum: P 5 (227/8, 4/27); point of inﬂection:

(2(9/5)

, 100/729); increasing: (2N, 227/8); decreasing:

(227/8, 0),(0,N); concave up: (2N, 2(9/5)

),(0, N);

concave down: (2(9/5)

, 0); vertical asymptote: x 5 0;

horizontal asymptote: y 5 0

2

8

15. Local maximum: Q 5 (3/2, f (3/2)); points of inﬂection:

P 5



3=2 2 9

ﬃﬃﬃ

=10; f ð3=2 2 9

ﬃﬃﬃ

=10Þ



; O 5 ð0; 0Þ; and

R 5



3=2 1 9

ﬃﬃﬃ

=10; f ð3=2 1 9

ﬃﬃﬃ

=10Þ



; increasing:

(2N23),(23, 3/2); decreasing: (3/2,N); concave up:

ð2N; 23Þ,ð3=2 2 9

ﬃﬃﬃ

=10; 0Þ, ð3=2 1 9

ﬃﬃ

ﬃ

=10; NÞ;

concave down: ð23; 3=2 2 9

ﬃﬃﬃ

Þ ,ð0; 3=2 1 9

ﬃﬃﬃ

Þ; vertical

asymptote x 523; horizontal asymptote: y 5 0

x  3

10 5

5

105

17. Local minimum: P 5 (25/6, f (25/6)); points of inﬂection:

Q 5 (25/3, f (25/3)) and R 5 (21 , 0); increasing: (25/3,N);

decreasing: (2N, 25/3); concave up: (2N, 25/3),(21,N);

concave down: (25/3, 21)

5

19. Local minimum: P 5 (1/2, 23 2

24/3

); points of inﬂection:

Q 5 (0, 0) and R 5 (21, 3); increasing: (1/2,N); decreas-

ing: (2N, 1/2); concave up: (2N, 21),(0,N); concave

down: (21, 0)

A-34 Answers to Selected Exercises

4 2

642

y  x

13

(x  2)

21. Local maximum: P 5 (1, 1/2); point of inﬂection:

Q 5



1 1 2=

ﬃﬃﬃ

; f ð1 1 2=

ﬃﬃﬃ



; increasing: (0, 1); decreas-

ing: (1,N); concave up: ð1 1 2=

ﬃﬃﬃ

; NÞ; concave down:

ð0; 1 1 2=

ﬃﬃﬃ

0.5

0.3

0.1

246810

y  

x/(x  1)

23. Local minimum: P 5 (2, 12); point of inﬂection: Q 5

(22

4/3

, 0); increasing: (2,N); decreasing: (2N, 0) and

(0, 2); concave up: (2N, 22

4/3

),(0,N); concave down:

(22

4/3

,0)

y  x

 16/x

4

10

2466

25. Local maximum: Q 5 (22, f (22)); points of inﬂection:

P 5



22 2 6

ﬃﬃﬃ

=5; f ð22 2 6

ﬃﬃﬃ

=5Þ



; R 5 ð0; 0Þ;

S 5



22 1 6

ﬃﬃﬃ

=5; f ð22 1 6

ﬃﬃﬃ

=5Þ



; increasing:

(2N, 22); decreasing: (22, 4),(4,N); concave up:

ð2N; 22 2 6

ﬃﬃﬃ

=5Þ,ð0; 22 1 6

ﬃﬃ

ﬃ

=5Þ,ð4; NÞ;

concave down: ð22 2 6

ﬃﬃﬃ

=5; 0Þ , ð22 1 6

ﬃﬃﬃ

=5; 4Þ;

vertical asymptote x 5 4; horizontal asymptote: y 5 0

x  4

10 5

4

2

105

27. Local maxima: P 5 ð2

ﬃﬃﬃ

=3; 2

ﬃﬃﬃ

=9Þand

T5 ð

ﬃﬃﬃ

=3; 2

ﬃﬃﬃ

=9Þ; local minimum: R 5 (0, 0); points

of inﬂection: Q 5



ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

27 2 3

ﬃﬃﬃﬃﬃ

=6;

f ð2

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

27 2 3

ﬃﬃﬃﬃﬃ



and S5



ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

27 2 3

ﬃﬃﬃﬃﬃ

=6;

f ð

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

27 2 3

ﬃﬃﬃﬃﬃ



;increasing:½21; 2

ﬃﬃ

ﬃ

=3Þ,ð0;

ﬃﬃﬃ

=3Þ;

decreasing: ð2

ﬃﬃﬃ

=3; 0Þ,ð

ﬃﬃﬃ

=3; 1; concave up:

ð2

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

27 2 3

ﬃﬃﬃﬃﬃ

=6;

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

27 2 3

ﬃﬃﬃﬃﬃ

=6Þ; concave down:

½21; 2

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

27 2 3

ﬃﬃﬃﬃﬃ

=6Þ,ð

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

2723

ﬃﬃﬃﬃﬃ

=6; 1

1.0

0.5

0.4

0.3

0.2

0.1

1.0

0.5

29. Local maximum: P 5 ð29=2;

ﬃﬃﬃﬃﬃ

=3Þ; point of inﬂection:

Q 5 ð26; 1=

ﬃﬃﬃ

Þ; increasing: (2N, 29/2); decreasing:

(29/2, 0) and (0,N); concave up: (2N, 26) and (0,N);

concave down: (26, 0); vertical asymptote x 5 0; hor-

izontal asymptotes: y 521 and y 5 1

5

510



y  1

y  1

 8x  18



31. Local minimum: Q 5 (22, 0); local maximum: R 5 (1/3,

f (1/3)); point of inﬂection: P 5 (213/3, f (213/3));

Section 4.6 A-35

increasing: (2N, 22),(1/3,N); decreasing: (22, 1/3);

concave up: (213/3, 22),(22,N); concave down:

(2N, 213/3)

8

30

4

20

10

33. Local minimum: Q 5 (2

21/3

, f (2

21/3

)); point of inﬂection:

P 5 (21, 1); increasing: (2

21/3

,N); decreasing: (2N,0),

(0, 2

21/3

); concave up: (2N, 21),(0,N); concave down:

(21, 0); vertical asymptote: x 5 0

2

10

35. No local extrema; no points of inﬂection (concavity

changes at points 2π/2 1 2kπ for integral k, but these

points are not in the domain of f); increasing: on its

entire domain; concave up: intervals of the form (2π/2 1

2kπ, π/2 1 2kπ) for integral k; concave down: intervals

of the form: (π/2 1 2kπ,3π/2 1 2kπ) for integral k; ver-

tical asymptotes: lines of the form x 5 π/2 1 2kπ for

integral k

5

10

5

37. Local minima: P 5 (21, 0) and R 5 (0, 0); local maximum:

Q 5 (21/3, 4/27); points of inﬂection: s 5 (22/3, 2/27) and

R 5 (0, 0); increasing: (21, 21/3),(0,N); decreasing:

(2N, 21),(21/3, 0); concave up: (2N, 22/3),(0,N);

concave down: (22/3, 0)

1.5 0.5

0.5

39. Local minimum: Q 5



214=13 2 3

ﬃﬃﬃ

=13;

f ð214=13 2 3

ﬃﬃﬃ

=13Þ



; local maximum:

R 5



214=13 1 3

ﬃﬃﬃ

=13; f ð214=13 1 3

ﬃﬃﬃ

=13Þ



;

point of inﬂection: P 5 (22, 24); concave up:

(22, 21),(1,N); concave down: (2N, 22) ,(21, 1);

vertical asymptotes: x 521 and x 5 1

20

41. Increasing: (2N,0),(0, N); concave up: (2N, 0);

concave down: (0, N); vertical asymptote: x 5 0;

skew-asymptote: y 5 x/2

10

y  (x

 4)(2x)

y  x2

45. Global minimum: P 5 (24.39926, 2125.33442); points of

inﬂection: Q 5 (22.94338, 276.22328) and

A-36 Answers to Selected Exercises

R 5 (20.05662, 24.05450); increasing: (24.39926,N);

decreasing: (2N, 24.39926); concave up: (2N,N)and

(20.05662,N); concave down: (22.94338, 20.05662)

50

100

246 2

y  x

 6x

 x

 x  4

47. Local maxima: P 5 (21.6012, f (21.6012)) and

T 5 (0.39915, f (0.39915)); local minima:

R 5 (21.1839, f (21.1839)) and

V 5 (1.5859, f (1.5859)); points of inﬂection:

Q 5 (21.4117, f (21.4117)), S 5 (20.31451, f (20.31451))

and U 5 (1.1262, f (1.1262)); increasing:

(2N, 21.6012),( 21.1839, 0.39915),(1.5859,N);

decreasing: (21.6012, 21.1839),(0.39915, 1.5859);

concave up: (21.4117, 20.31451),(1.1262,N):

concave down: (2N, 21.4117),(20.31451, 1.1262)

2

49. Local maximum: P 5 (0.92388, 1.20711); local minimum:

R 5 (20.38268, 20.20711); point of inﬂection:

Q 5(0.51201, 0.70196); increasing: (20.38268, 0.92388);

decreasing: (21, 20.38268) and (0.92388, 1); concave

up: (21, 0.51201): concave down: (0.51201, 1)

0.5

0.5 0.5 11

 x

 x 

1  x

51. Global minima: P 5 (22, 0) and V 5 (2 , 0); global

maximum: S 5 (0, 2); points of inﬂection

Q 5 (21.7321, f (21.7321)) , R 5 (20.58288, f (20.58288)),

T 5 (0.58288, f (0.58288)), and U 5 (1.7321, f (1.7321));

increasing: (22, 0); decreasing: (0, 2); concave up:

(21.7321, 20.58288),(0.58288, 1.7321): concave down:

(22, 21.7321),(20.58288, 0.58288),(1.7321, 2)

2

0.5

1.0

1.5

R T

53. Local maximum: P 5 (21.49714, 9.06986); loca l minimum:

Q 5 (1.97537, 23.00630); points of inﬂection:

R 5 (22.58587, 6.34340), S 5 (0, 15/4), and

T 5 (3.39777, 21.12363); increasing: (2N, 21.49714) and

(1.97537,N); decreasing: (21.49714, 1.97537); concave up:

(2N, 22.58587) and (0, 3.39777): concave down:

(22.58587, 0) and (3.39777,N); skew asymptote: y 5 x

10

105

y  x

y 

 20x  15

16  x



Section 4.7

1. 21 3. 1/5 5. 21/8

7. 1/4 9. Does not exist. 11. 2

13. 1/3 15. 1 17. 0

19. 1 21. 4/π

23. 24/3

25. 0 27. 0 29. 1

31. 24/π 33. 1 35. 1/e

37. 1 39. 1 41. 21

43. 0 45. Does not exist. 47. 0

49. 21 51. 21/2 53. 1

55. 1 57. 1/2 59. 21/2

61. 1/e

63. e

65. e

Section 4.7 A-37

67. a(a

2 1)/(2b(b

2 1))

69. 0 71. 1/3 73. 2

75. 1/e

77.

0.4

0.4

0.2

Horizontal asymptote: y  0

468

f(x)  x/e

79.

24

2

6468

Horizontal

asymptotes: y  0, y  1

f(x)  e

/(e

 x)

81.

Horizontal asymptote: y  0

f(x)  x ln (x)/(1  x

)

0.1

0.2

0.3

10008006004002005

lim

x→0



f(x)  0

f(x)  x ln (x)/(1  x

)

0.2

0.2

21 345

89. The limit is 1/2.

1

0.45

0.55

0.60

y 

1  cos(x)

sin(x)

91. The limit is 1.

1.4

1.0

0.6

1.0

0.2 0.4 0.6 0.8

y  x

(1  sin(x)x)

93.

1

0.45

0.55

0.60

(

)

 1  cos

(

)

, g

(

)

 x sin

(

)

y 

f (x)

g(x)

y 

f(x)

g(x)

95.

0.4

0.5

0.6

(

)

 1  cos

(

)

, g

(

)



(

x  p

)

sin

(

)

y 

f (x)

g(x)

y 

f(x)

g(x)

Section 4.8

1. x

5 2.5, x

5 2.45

3. x

5 4.6667, x

5 4.5641

5. x

5 1.9, x

5 1.8883

7. x

5 1.6667, x

5 1.5911

9. x

5 2.45, x

5 2.4495

11. x

5 3.5, x

5 3.1786

13. 1 whatever value of x

is chosen

29. 0.995598772 31. 0.62902

33. 12 35. 11

37. M 5 4, 4, 3, and 4 for Exercises 33, 34, 35, and 36,

respectively

39. 1.080

41. a. 7.75%; b. 5.82%

A-38 Answers to Selected Exercises

43. The second: It has an effective yield of 6.50%, whereas

the ﬁrst has an effective yield of 6.25%.

45. The second: It has an effective yield of 7.37%, whereas

the ﬁrst has an effective yield of 7.00%.

49. 1.02876, occuring at 0.51282

Section 4.9

1 C

3. e

1 C

ðx 1 2Þ

3=2

1 C

7. 21=x 2 1=ð6x

Þ1 C

ðx 1 1Þ

1 C

11.

exp ðexÞ

13. 2

24=3

2 12x

1=3

1 C

15.

sinð4xÞ1 x

1 C

17.

tanð8xÞ1 C

19.

ð3x 2 2Þ

1 C

21. 1 23. 23/2 25. 3e 2 4

27. 4

ﬃﬃﬃﬃﬃ

m=s

29. 155:5ft; 291:61 ft=s

31. 121=54 ft=s

33. 4:738 s

35.

sin

ðxÞ1 C; or 2

cos

ðxÞ1 C; or 2

cosð2xÞ1 C

37. tanðxÞ2 x 1 C

39.

x 1

sinð2xÞ1 C

41. sec(x) 1 C

43.

ð2Þ

xln2

1 C

45. Local maximum at 23; local minimum at 2

47. G( f (x)) 1 C

49. sin(x

1 7) 1C

51.

expðx

Þ1 C

55. expð2xÞ=

ﬃﬃﬃﬃﬃﬃ

πx

57. 420.9 ft; 95 ft/s

59. Impact at 17 mi/hr

61. 484 ft

63. 3825=4ft; 260

ﬃﬃﬃﬃﬃ

ft=s

65. 120 s

67. H 2 kgt 1 k

gð1 2 e

2t=k

69. 1839.6 yr

71. TN 5 40



C; τ

5 25:54 s; τ

5 20:27s; τ

5 14:39 s

73. vðtÞ5 8:855ð1 1 expð4:427tÞÞ

2 4:427; v

524:427;

vð1:717Þ5 0:999 v

1

4

3

2

1234

v(t)

REVIEW EXERCIS ES FOR CHAPTER 4

1. 26/5 3. 6

5. 3/(32π) cm/min

7. 4/15 m/min

9. 2105

11. 1/2 ft/s

13. c 5

ﬃﬃﬃ

; f uðcÞ5 21

15. c 529; f uðcÞ521=81

17. 21 19. e 2 1 21. 8 2 ln(2) 23. 11

25. Local maximum at 22; local minimum at 2

27. Local minimum at 16

1/3

29. Local maximum at 22; local minimum at 2

31. Local minimum at 1; local maximum at e

33. Local maximum at 5

35. Local maximum at 1; local minimum at 2

37. No extremum at 0; local minimum at 1/2; no extremum at 1

39. 214, 67 41. 2e,1/e

43. 10/(4 1 π) 47.

1=3

49. 4

ﬃﬃﬃ

51. a. (2N, 21) and (1,N) b. (21, 1) c. 21, 1

53. a. (2N,3) b. (3,N) c. 3

55. a. (0, 6) b. (6,N) c. 6

57. a. (e

5/6

,N) b. (0, e

5/6

) c. e

5/6

59. a. (23, 0) and (0, N) b. (2N, 23) c. 23

61. 2 63. 23 65. 22 67. 0

69. 4 71. e

73. 1/2

75. 3.181725815, 3.146284845

Review Exercises for Chapter 4 A-39

77. 6.823500280, 6.854079461

79. sec(x) 1 C

81. (3x 1 1)

/9 1 C

83. x

/4 2 x

/2 1 C

85. 3x

7=3

=7 2 6x

2=3

1 C

87.

tanð3xÞ1 C

89.

9=2

1 C

Section 5.1

1. 30 3. 2108

5. 73/6 7. 118

9. 19/4 11.

j51

ðj 1 1Þ

13.

j51

ð5 1 4jÞ 15.

j54

1=j

17. 650 19. 806

21. 676 23. (e

2 1)/(e 2 1)

25. ln(720) 27. 23

29. (2 2 log

(3))/4 31. 3π

33. 197/60 35. π

37. 14 39. π ð2 1

ﬃﬃﬃ

Þ=24

41. 17/6 43. 581/1200

45. 16/3 47. p 5 5050; q 5 100!

49. N(2N 21) (2N 1 1)/3 65. ab 2 A

69. 0.16 71. 1.57

73. 1.94 75. 0.30

Section 5.2

1. a. 11/6 b. ln(7)

3. a. 72 b. 224/3

5. a. 378 b. 696

7. a. 42 b. 37

9. a. 2(e

1 1 1 e

) b. e

2 e

11. a. π/3 b. 2

ﬃﬃﬃ

13. lower: 45/64; upper: 365/64

15. lower: 2π; upper: π/3

17. 2

ﬃﬃﬃ

19. 5 21. 9/2 23. 13/2

25. 11 27. 14/3 29. 1 31. 1

33. 195/4 35. e 2 1 37. 21 39. 7/ln(2)

41. 1 2

ﬃﬃﬃ

=3 43. 21/2

45. Rðf ; L

Þ5 1; Rðf ; SÞ5

ﬃﬃﬃ

; Rðf ; U

Þ5 e

47. Rðf ; L

Þ5 7=12; Rðf ; SÞ5 24=35; Rðf ; U

Þ5 5=6

49. Rðf ; L

Þ5 lnð6Þ; Rðf ; SÞ5 lnð105=8Þ; Rðf ; U

Þ 5 lnð24Þ

51. 2/3 53. 1/2

55. 1 2 1/e

57. Rðf ; L

Þ5 11 , 33=2 5

f ðxÞdx , 22 5 Rðf ; U

59. s

5 10

1/3

, s

5 68

1/3

61. s

5 1/(2 ln(3/2)), s

5 1/(2 ln(4/3))

71. 0.375 73. 3.14195155

75. Rðf ; L

Þ5 8:035809924; Rðf ; U

Þ5 8:244835710

77. Rðf ; L

Þ5 1:893551751; Rðf ; U

Þ5 1:908235239

Section 5.3

1. 4 3. 8 5. 10 7. 75

9. 264 11. 218 13. 25

15. f

avg

5 4; c 5 3

17. f

avg

5 14/9; c 5 196/81

19. f

avg

5 41; c 5

ﬃﬃﬃ

21. 18 23. 8/9 25. 2/π 27. 4/π

29. A 5 9; B 5 18

31. A 5 4/5; B 5 4/3

33.

f ðxÞdx 5 0;

gðxÞdx 521

35.

f ðxÞdx 525;

gðxÞdx 521

37. 6 2 5ln(3)

39. 3 2 1/e

41. 4/15 43. 6e 2 9 45. No

49. A 5 3/5; B 5 3

51. A 5 2=

ﬃﬃﬃ

; B 5 4=e

53. F(x) 5 x

1 x 22

55. F(x) 5 1/x 2 x

57. F(x) 5 2x 2 2

59. j

ðπx 2 3Þdxj5 3 2 π=2 5 1:429:::;

jπx 2 3 jdx 5 9=π 2 3 1 π=2 5 1:435:::

61. j

ð1 2 9x

Þdxj5 24;

j1 2 9x

jdx 5 224=9 5 24:888:::

65. A 5 6/17; B 5 2.8406 . . .

67. A 5 π/2; B 5 5.2413 . . .

69. f

avg

5 π=4 1 2=π; c 5 0:74445:::

71. f

avg

1 lnð3Þ; c 5 4:18032:::

Section 5.4

1. (e

2 1)/3 3. 6 2 3

ﬃﬃﬃ

5. 1 7. 6 2 2

ﬃﬃﬃ

A-40 Answers to Selected Exercises