Answers

Chapter 4

Section 4.1

1. 10 m³/min

2. 0.5 m/min

4. −1

5. 5π²

7. 40

9. 900 cm³/s

11. −0.000625 cm/h

13. ≈ −0.178 cm/min

15. ≈ 0.242 cm²/min

17. −0.75 m²/min

19. rad/s

21. 0.2 m/min

23. ≈ 1.273 m/min

25. (a) ≈ 1.698 m/min

(b) ≈ 2.116 m³/min

27. ≈ 4.366 units/s

29.

31. 1.75 kg/cm²/min

33. 100.8π ≈ 316.673 ft²/min

35. (a) ≈ 0.202 m/s

(b) ≈ 0.289 m/s

(c) ≈ 0.567 m/s

37. ≈ 15.08 km/s

39. 4 m/min

41. He is rising at ≈ 68.017 ft/min. He is moving horizontally at − ≈ − 32.270 ft/min.

43. ≈−1.671 m/s

45. (a) $150/day

(b) $200/day

(c) $50/day

47. ≈−4.864 lb/s

49. −100π ≈ −314.159 ft/s

51. −316.8 rad/h

53. (a) Answers will vary.

(b) Answers will vary.

55. ≈ 32.071 ft/s

57. ≈ 0.165 in²/min

Section 4.2

1. False

2. (b)

3. False

4. False

5. False

6. False

7. x₁: neither, x₂: local maximum, x₃: local minimum and absolute minimum, x₄: neither, x₅: local maximum, x₆: neither, x₇: local minimum, x₈: absolute maximum

11.

13. 4

15. 0, 2

17. −1, 0, 1

19. 0

21. 0

23. π

25.

27. 0, 2

29. −3, 0, 1

31. 3, 4

33.

35. 1

37. Absolute maximum 20 at x = 10, absolute minimum −16 at x = 4

39. Absolute maximum 16 at x = 4, absolute minimum −4 at x = 2

41. Absolute maximum 9 at x = 2, absolute minimum 0 at x = 1

43. Absolute maximum 1 at x = −1, 1, absolute minimum 0 at x = 0

45. Absolute maximum 4 at x = 4, absolute minimum 2 at x = 1

47. Absolute maximum π at x = π, absolute minimum 0 at x = 0

49.

51. Absolute maximum 0 at x = 0, absolute minimum

53. Absolute maximum at x = 5, absolute minimum 0 at x = −3, 1

55. Absolute maximum at x = 4, absolute minimum −1 at x = 2

57. Absolute maximum , absolute minimum 0 at x = 3

59. Absolute maximum 1 at x = 0, absolute minimum e − 3 at x = 1

61. Absolute maximum 9 at x = 3, absolute minimum 1 at x = 0

63. Absolute maximum 8 at x = 2, absolute minimum 0 at x = 0

65. (a) f′(x) = 12x³ − 6x² − 42x + 36

(b) −2, 1,

(c) Absolute minimum at x = −2, local maximum at x = 1, local minimum at x =

67. (a) f′(x) =

(b) −5, ≈−2.364, ≈1.977

(c) Local maximum at x ≈ −2.364,
local minimum at x ≈ 1.977,
neither a local minimum nor a local maximum at x = −5.

69. (a) f′(x) = 4x³ − 37.2x² + 98.48x − 68.64

(b) Absolute maximum 0 at x = 0, absolute minimum −37.2 at x = 5

(c)

71. (a) 50 mi/h

(b)

73. (a) 67.5°, 0.0183v₀² ft

(b)

(c) Answers will vary.

75. (a) See Student Solutions Manual.

(b) ≈846.858 m

77.

79. 5 m

81. (a) 0.04, 0.07

(b) Absolute maximum −3.719 at x = 1, absolute minimum −4.391 at x = −1

(c)

83. Absolute maximum at x = 3, absolute minimum at x = 2

85. (a) [−4,−3] [3, 4]

(b)

87. (c) (e)

89. See Student Solutions Manual.

91. See Student Solutions Manual.

93. (a) See Student Solutions Manual.

(b) Answers will vary.

Section 4.3

1. False

2. Answers will vary.

3. True

4. False

7. 1

11.

13. ±1, 0

15.

17. f (−2) ≠ f (1)

19. f is not differentiable at x = 0.

21. 1

23. e – 1

25.

27.

29.

31. f is increasing on (−∞,∞).

33. f is increasing on [−1, 0] [1,∞) and decreasing on (−∞,−1] [0, 1].

35. f is increasing on [ and decreasing on ].

37. f is increasing on and decreasing on .

39. f is increasing on [−1,∞) and decreasing on (−∞,−1].

41. f is increasing on and decreasing on

43. See Student Solutions Manual.

45. See Student Solutions Manual.

47. Answers will vary.

49. See Student Solutions Manual.

51. See Student Solutions Manual.

53. (a) See Student Solutions Manual.

(b) d(0) = d(5) = 0

(c) ≈2.892 ft; d(2.892) ≈ −0.423 ft.

(d)

55. See Student Solutions Manual.

57. See Student Solutions Manual.

59. Answers will vary.

61.

63. (a) is increasing on the interval (0, 0).

(b) Yes.

65. (b)

67. See Student Solutions Manual.

69. See Student Solutions Manual.

71. See Student Solutions Manual.

73. See Student Solutions Manual.

75. For b² −3ac ≤ 0, f is increasing on (−∞,∞) for a = 0 and decreasing on (−∞,∞) for a < 0. For b² −3ac = 0 and a = 0, f is increasing on (−∞, x₁] [x₂,∞) and decreasing on [x₁, x₂]. For b² −3ac > 0 and a < 0, f is increasing on [x₁, x₂] and decreasing on (−∞, x₁] [x₂,∞), .

77. See Student Solutions Manual.

79. See Student Solutions Manual.

81. 0 is the only critical number. If n is odd and ad − bc > 0, then f is increasing on the domain. If n is odd and ad −bc < 0, then f is decreasing on the domain. If n is even and ad −bc > 0, then f is increasing on the domain where x = 0 and decreasing on the domain where x < 0. If n is even and ad − bc < 0, then f is increasing on the domain where x < 0 and decreasing on the domain where x > 0.

83. See Student Solutions Manual.

Section 4.4

1. False

2. False

3. (a)

4. (b)

5. (a)

6. (c)

7. True

8. False

9. (a) Local maximum at (−1, 0), local minimum at (−2.5,−4) and (0.5,−4), inflection points (−1.8,−2) and (−0.2,−2)

(b) Increasing on [−2.5,−1] [0.5,∞), decreasing on (−∞,−2.5] [−1, 0.5], concave up on (−∞,−1.8) (−0.2,∞), concave down on (−1.8,−0.2)

11. (a) Local maximum at (−2, 3) and (12, 10), local minimum at (0, 0), no inflection point

(b) Increasing on (−∞,−2] [0, 12], decreasing on [−2, 0] [12,∞), never concave up, concave down on (−∞, 0) (0,∞)

13. (a) 0, 4

(b) Local maximum 2 at 0, local minimum −30 at 4

15. (a) 0, 1

(b) Local minimum −1 at 1

17. (a)

(b) Local maximum

19. (a) 1

(b) Local minimum e at 1

21. (a)

(b) Local minimum

23. (a) −1, 0, 1

(b) Local maximum 0 at 0, local minimum −3 at −1 and 1

25. (a)

(b) Local maximum

27. (a) −1, 0, 1

(b) Local maximum 1 at 0, local minimum 0 at −1 and 1

29. (a) kπ – tan⁻¹ , k an integer

(b) Local maximum local minimum

31. (a) The object moves right on (1,∞), left on (0, 1)

(b) t = 1

(c) The velocity is increasing on (0,∞)

(d)

(e)

33. (a) The object moves right on (1,∞), left on (0, 1)

(b) t = 1

(c) The velocity is increasing on (0,∞)

(d)

(e)

35. (a) The object moves right on (0,∞)

(b) No direction reverse

(c) The velocity is decreasing on (0,∞)

(d)

(e)

37. (a) The object moves right on

(b) t =

(c) The velocity is

(d)

(e)

39. (a) Concave up on (−∞,∞)

(b) No inflection point

41. (a) Concave up on (3,∞), concave down on (−∞, 3)

(b) (3,−52)

43. (a) Concave up on (−∞, 0) (2,∞), concave down on (0, 2)

(b) (0, 10), (2,−6)

45. (a) Concave up on Concave down on

(b) (−1.483, 0.295), (0.150, 0.328)

47. (a) Concave up on (e^7/12,∞), concave down on (0, e^7/12)

(b)

49. (a) Concave up on (0,∞), concave down on (−∞, 0)

(b) No inflection point

51. (a) Concave up on (−∞, 0), concave down on (0,∞)

(b) (0, 0)

53. (a) Concave up on (−∞, 0) (0, 3), concave down on (3,∞)

(b)

55. (a) No local extrema

(b) Concave up on (1,∞), concave down on (−∞, 1)

(c) (1,−1)

57. (a) Local minimum −3 at x = 1

(b) Concave up on (−∞, 0) (0,∞)

(c) No inflection point

59. (a) Local maximum 256 at x = 4, local minimum 0 at x = 0

(b) Concave up on (−∞, 0) (0, 3), concave down on (3,∞)

(c) (3, 162)

61. (a) Local maximum 64 at x = −2, local minimum −64 at x = 2

(b) Concave up on concave down on

(c) (0, 0),

63. (a) Local maximum 4e⁻² at x = −2, local minimum 0 at x = 0

(b) Concave up on , concave down on

(c)

65. (a) Local minimum 1 at x = 0

(b) Concave up on (−∞,∞)

(c) No inflection point

67. (a) Local minimum

(b) Concave up on concave down on

(c) (0, 0)

69. (a)

(b) Concave up on (−∞, 0) (0,∞)

(c) No inflection point

71. (a) Local minimum

(b) Concave up on (0,∞)

(c) No inflection point

73. (a) Local maximum local minimum

(b) Concave up on concave down on

75. (a) Local maximum local minimum 0 at x = 0

(b) Concave up on concave down on

(c)

77. (a) Local maximum 1 at $x = k \pi + \dfrac{\pi}{2}$, local minimum 0 at x = kπ, k an integer.

(b) Concave up on $\left(k \pi - \dfrac{\pi}{4}, k \pi + \dfrac{\pi}{4}\right)$, concave down on $\left(k \pi + \dfrac{\pi}{4}, k \pi + \dfrac{3\pi}{4}\right)$, k an integer.

(c) $\left(\dfrac{(2k+1)\pi}{4}, \dfrac{1}{2}\right)$, k an integer.

79. (a) Local maximum local minimum

(b) Concave up on (0, π), concave down on (π, 2π)

(c) (π, π)

81. (a) Local maximum −20 at x = 3, local minimum −21 at x = 2

(b) Same as (a)

(c) Answers will vary

83. (a) Local maximum 4e at x = 1, local minimum 0 at x = 3

(b) Same as (a)

(c) Answers will vary.

85. Answers will vary.

87. Answers will vary.

89. Answers will vary.

91. Answers will vary.

93. Answers will vary.

95. Answers will vary.

97. (a) Concave up on concave down on

(b)

(c)

99. (a) Concave up on (−∞, 0.135) (0.721, 5.144), concave down on (0.135, 0.721) (5.144,∞)

(b) (0.135, 2.433), (0.721, 2.139), (5.144, -0.072)

(c)

101. See Student Solutions Manual.

103. a = −3, b = 9

105. (a) N′(t) =

(b) N′(t) is increasing on the interval (0, ln 9999) and decreasing on the interval (ln 9999,∞).

(c) ln 9999

(d) (ln 9999, 5000)

(e) Answers will vary.

107. (a)

(b) Answers will vary.

(c)

(d) P′(t) is increasing on the interval (0, 133.843) and decreasing on the interval (133.843,∞).

(e) The rate of change in population is maximum when t = 133.843.

(f) (133.843, 381088358.9)

(g) Answers will vary.

109. (a) Local minimum B(2.80) ≈ −328.53 billion dollars, local maximum B(5.71) ≈ −170.80 billion dollars.

(b) Both represent a budget deficit.

(c) Concave up on (0, 4.26), concave down on (4.26, 9), (4.26,−249.27) is a point of inflection.

(d) To the left of the inflection point, the budget is increasing at an increasing rate. To the immediate right of the inflection point, the budget is increasing at a decreasing rate.

(e)
Answers will vary.

111.

113. Local maximum 2 at local minimum −2 at Inflection points

115. f″(x) does not exist at the critical number x = 0.

117. (e)

119. See Student Solutions Manual.

121. See Student Solutions Manual.

123. See Student Solutions Manual.

125. See Student Solutions Manual.

127. See Student Solutions Manual.

129. See Student Solutions Manual.

131. See Student Solutions Manual.

133. See Student Solutions Manual.

135. See Student Solutions Manual.

Section 4.5

1. False

2. False

3. False

4. False

5. Answers will vary.

6. Answers will vary.

7. (a) Yes, (b)

9. (a) No, (b) Answers will vary.

11.(a) Yes, (b)

13. (a) No, (b) Answers will vary.

15. (a) Yes, (b)

17. (a) Yes, (b)

19. (a) Yes, (b) 0 · ∞

21. (a) Yes, (b) ∞−∞

23. (a) Yes, (b) ∞⁰

25. (a) No, (b) Answers will vary.

27. , 5

29.

31. , 2

33. , −π

35.

37.

39.

41.

43. 0 · ∞, 0

45. 0 · ∞, 1

47. ∞−∞, 0

49. ∞−∞, −1

51. 0⁰, 1

53. ∞⁰, 1

55. ∞⁰, 1

57. 1^∞, 1

59. 2

61. 0

63. 0

65. 2

67. 1

69.

71. 0

73. 0

75. 1

77.

79.

81. −1

83. 1

85. 1

87. 1

89. 1

91. (a) 366

(b) Answers will vary.

(c)

93. (a)

(b) Answers will vary.

95. See Student Solutions Manual.

97. See Student Solutions Manual.

99. 0

101. See Student Solutions Manual.

103. See Student Solutions Manual.

105. (a) See Student Solutions Manual. f′(0) = 0

(b)

107. See Student Solutions Manual.

109. (a) 0

(b) a

(c) ∞

(d) Does not exist.

(e) 0

(f) a

(g) 1

(h) a

(i) Does not exist.

(j) e⁻¹

111. (a) f″(x)

(b) f‴(x)

(c)

113. See Student Solutions Manual.

Section 4.6

11.

13.

15.

17.

19.

21.

23.

25.

27.

29.

31.

33.

35.

37.

39.

41.

43. (a)

(b) Vertical asymptote: x = 1, horizontal asymptote: y = 0

(c) Decreasing on approximately (−∞,−2) (0, 1) (1,∞) and increasing on approximately (−2, 0), concave down on approximately (−∞,−4) (0.1, 1) and concave up on approximately (−4, 0) (0, 0.1) (1,∞)

(d) Local minimum value at approximately −2 and local maximum value at approximately 0

(e) They are the same.

(f) Points of inflection at approximately

45. (a)

(b) No asymptotes

(c) Increasing on and decreasing on for k an integer, f is concave up on and concave down on for k an integer.

(d) Local maximum value at approximately and local maximum value at approximately for k an integer.

(e) They are the same.

(f) for k an integer.

47. (a)

(b) Vertical asymptote: x = 1

(c) Increasing and concave down on approximately (1,∞)

(d) No local extrema

(e) They are the same.

(f) No points of inflection

49. (a)

(b) No asymptotes

(c) Increasing approximately on and decreasing on concave up approximately on (−π + 2kπ, 2kπ) and concave down approximately on (2kπ, π + 2kπ) for k an integer.

(d) Local maximum value at approximately and local minimum value at approximately for k an integer.

(e) They are the same.

(f) Points of inflection at approximately (kπ, 0), k an integer.

51. (a)

(b) No asymtotes.

(c) Decreasing on (-∞, 0) (4, 6); Increasing on (0, 4); Concave up on (-∞, 0); Concave down on (0, 6)

(d) Local minimum value of 0 at x = 0, and local maximum value of at x = 4.

(e) They are the same.

(f) There is an inflection point at (0, 0).

53. Answers will vary.

55. Answers will vary.

57. Answers will vary.

59. (a) Absolute maximum value e − 1 at x = and absolute minimum value 1 at x = 1

(b) Concave up on and concave down on (2, e)

(c)

61. x = ±2 are two vertical asymptotes.

63. y = 1 is a horizontal asymptote.

Section 4.7

1. 1,125,000 m²

7. 8 cm × 8 cm × 2 cm

11. Radius

13. $21 per day

15. (1, 1)

17. (1.784, 0.817)

19. (a) 40 mi/h

(b) ≈46.6 mi/h

(c) ≈51.0 mi/h

21. (a)

(b)

23. ≈9.582 m, ≈36.383^o

25.

27. 594 cases, $456 per case

29.

31.

33. 2 m from the weak light source.

35. minimum length: 28.359 cm, maximum length: 42.758 cm

37. (a) Cut 19.603 cm for a square and 15.397 cm for a circle.

(b) Cut 0 cm for a square and 35 cm for a circle.

(c)

39. See Student Solutions Manual.

41.

43.

45. See Student Solutions Manual.

47. Observer should stand approximately 4.583 m from the wall.

49. See Student Solutions Manual.

51. 20 ft

53. (−3.758, 6.547)

Section 4.8

1. Antiderivative

2. True

3. ln |x| + C

4. True

5. False

6. True

7. True

8. True

9. 2x + C

11.

13.

15.

17.

19. x⁴−x³ +x +C

21. 3x³−6x² +4x +C

23. 3x −2 ln|x|+C

25. x² −3 sin x +C

27.

29. 7 tan^-1x + C

31. y = x³ − x² + x – 5

33.

35.

37.

39. y = e^x − x + 1

41. s(t) = −16t² + 128t

43. s(t) = t³ + 18t + 2

45.

47.

49. F(x) = x cos x + sin x + 1

51. v_o = 10 m/s ≈ 38.74 mi/hr

53. 220 ft

55. 15.00625 m

57. 13.133 m/s

59. 8 N

61. ≈16.759 m/s

63. (a) See Student Solutions Manual.

(b) Answers will vary.

(c) I (x) = I₀e^−kx

(d) $-\dfrac{1}{2}\ln0.1$

Review Exercises

3. 318.953 mph

5. (−8,−9) is an absolute minimum, (−5, 0) is neither, (−2, 9) is a local maximum and an absolute maximum, (1, 0) is a local minimum, (3, 4) is a local maximum, (5, 0) is neither