Chapter 3

Section 3.1

1. Chain

2. True

3. False

4. tan u; 1 + cos x

5. 100(x3 + 4x + 1)99 (3x2 + 4)

6. 6xe3x2+5

7. True

8. 2x cos(x2)

9. y = (x3 + 1)5; = 15x2(x3 + 1)4

11.

13.

15. f′(x) = 6(3x + 5)

17. f′(x) =

19. g′(x) = 8x(x2 + 5)3

21. f′(u) =

23. g′(x) = 3(4x + ex)2(4 + ex)

25. f′(x) = 2 tan x sec2 x

27. f′(z) = 2(tan z +cos z)(sec2 z −sin z)

29. y′ = 4x(x2 +4)(2x3 −1)3 +18x2(x2 +4)2(2x3 −1)2 = 2x(x2 +4)(2x3 −1)2(13x3 +36x −2)

31. y′ =

33. y′ = 4 cos(4x)

35. y′ = 2 cos(x2 + 2x − 1) · (2x + 2)

37. y′ =

39. y′ = 4 sec(4x) tan(4x)

41. y′ = e1/x

43. y′ =

45. y′ =

47. y′ = (ln 2)2sin x cos x

49. y′ = (ln 6)6sec x sec x tan x

51. y′ = 5e3x + 15xe3x

53. y′ = 2x sin(4x) + 4x2 cos(4x)

55. y′ = −ae−ax sin(bx) + be−ax cos(bx)

57. y′ =

59.

61.

63. y′ = −2e−2x cos(3x) − 3e−2x sin(3x)

65. y′ = −2xex2 sin(ex2)

67. y′ = ecos(4x)(−4 sin(4x))

69. y′ = 24 sin(3x) cos(3x)

71. (a) y′ = 6x2(x3 + 1)

(b) y′ = 6x2(x3 + 1)

(c) y′ = 6x5 + 6x2

73. (a) y = 0

(b)

75. (a) y =

(b)

77. (a) y = 2x + 1

(b)

79. = −25x8 cos(x5) − 20x3 sin(x5)

81. h′(1) = −12

83. h′(0) = 0

85. 78

87. = f′(x2 + 1)(2x)

89.

91. = f′(sin x) cos x

93. = f″;(cos x) sin2 xf′(cos x) cos x

95. (a)

(b)

(c)

(d)

97. a(t) =

99. = −3.597 × 10−6 ohms/K

101. (a) 102

(b)

(c) = 18.9e−0.21t

(d) A′(5) = 6.614 %/yr; A′(10) = 2.314 %/yr

(e)

103. (a) a(t) =

(b) mg/k

(c) 0

105. See Student Solutions Manual.

107. (a)

(b)

109. F′(1) = 2

111. See Student Solutions Manual.

113. See Student Solutions Manual.

115. See Student Solutions Manual.

117. See Student Solutions Manual.

119. See Student Solutions Manual.

121. (a) y(n)(x) = aneax

(b) y(n)(x) = (−1)nane−ax

123. (a) = a11 sin(ax)

(b) = a12 cos(ax)

(c) f(n)(x) = −an sin(ax), n = 1 + 4k, k = 0, 1, 2, 3,…; f(n)(x) = −an cos(ax), n = 2 + 4k, k = 0, 1, 2, 3,…; f(n)(x) = an sin(ax), n = 3 + 4k, k = 0, 1, 2, 3,…; f(n)(x) = an cos(ax), n = 4 + 4k, k = 0, 1, 2, 3,…

125. See Student Solutions Manual.

127. See Student Solutions Manual.

129. See Student Solutions Manual.

131. See Student Solutions Manual.

133. (a) −72,000π2 cm/sec2

(b) 72,000π2 kg · cm/sec2

Section 3.2

1. True

2. True

3. True

4.

5. False

6.

7. y′ =

9. y′ =

11.

13. y′ =

15. y′ =

17. y′ =

19. y′ =

21. y′ =

23.

25. y′ =

27. y′ =

29. y′ =

31. y′ =

33. y′ =

35. y′ =

37. y′ =

39. y′ = (x2 − 1)1/2 +

41. y′ =

43. y′ = (x2 cos x)1/2(2x cos xx2 sin x)

45. y′ = 3x(x2 − 3)1/2(6x + 1)5/3 + 10(x2 − 3)3/2(6x + 1)2/3 = (6x + 1)2/3(x2 − 3)1/2(28x2 + 3x − 30)

47. y′ =

49. y′ =

51. y′ =

53. y′ = sin−1 x +

55. y′ =

57.

59. y′ =

61. y′ =

63. (a) −1/2

(b) y =

(c)

65. (a) 3

(b) y = 3x – 8

(c)

67. (a) 1/2

(b) y =

(c)

69. y′ = −4/9; y″; = −100/243

71. g′(4) = −1/2

73. f′(−2) = 2

75. g′(0) = 1/2, g′(3) = 1/5

77. y′ = 3x/|x|

79. y′ =

81.

83.

85. y = −2x + 10

87. (a)

(b)

89. (a) y′ =

(b) y =

(c) (6,−3)

(d)

91. (a) y =

(b)

93. See Student Solutions Manual.

95. See Student Solutions Manual.

97. (a) y′ =

(b) y = −x + 2

(c)

(d)

99.

101. See Student Solutions Manual.

103. See Student Solutions Manual.

105. See Student Solutions Manual.

107. See Student Solutions Manual.

109. See Student Solutions Manual.

111. (a)

(b)

113. See Student Solutions Manual.

115. See Student Solutions Manual.

Section 3.3

1.

2. True

3. False

4. False

5.

6. e

7. y′ = 5/x

9. y′ =

11. y′ =

13. y′ = 1/x

15. y′ =

17. y′ = ln(x2 + 4) +

19. y′ =

21. y′ =

23. y′ =

25. y′ =

27. y′ =

29. y′ = cot θ

31. y′ =

33. y′ =

35. y′ =

37. y′ =

39. y′ =

41. y′ = cos(ln θ) ·

43. y′ =

45.

47.

49.

51. y′ =

53. y′ =

55. y′ =

57. y′ = (3x x (ln(3x) + 1)

59. y′ =

61. y′ = xx2 (2x ln x + x)

63. y′ =

65. y′ =

67. y′ = (sin x)x(ln(sin x) + x cot x)

69. y′ = (sin x)cos x (−sin x ln(sin x) + cos x cot x)

71. y′ =

73. y = 5x − 1

75. y = −5x + 18

77. e2

79. e1/3

81.

83.

85. y′ = xx(ln x + 1)

87. y′ = tan−1(x/a)

89. (a) See Student Solutions Manual.

(b) $6107.01

(c) 28.881years

(d) See Student Solutions Manual.

91. (a) dB/m

(b) Answers will vary.

93. See Student Solutions Manual.

95. See Student Solutions Manual.

97. See Student Solutions Manual.

Section 3.4

1. (d)

2. f (x0) + f ′(x0)(xx0)

3. True

4.

5. True

6. True

7. dy = (3x2 − 2) dx

9. dy = 12x(x2 + 1)1/2 dx

11. dy = (6 cos(2x) + 1) dx

13. dy = −e−x dx

15. dy = (ex + xex)dx

17. (a) dy = exdx

(b) (i) (ii) (iii)

(c) (i) 0.4043 (ii) 0.0141 (iii) 0.000136

19. (a)

(b) (i) dy ≈ 0.2646, Δy ≈ 0.2546 (ii) dy ≈ 0.0529, Δy ≈ 0.0525 (iii) dy ≈ 0.0053, Δy ≈ 0.0053;

(c) (i) 0.009952, (ii) 0.000431, (iii) 4 × 10−6

21. (a) dy = −sin x dx

(b) (i) dy = 0, Δy ≈ 0.1224 (ii) dy = 0, Δy ≈ 0.005 (iii) dy = 0, Δy ≈ 0.00005;

(c) (i) 0.1224, (ii) 0.005 (iii) 0.00005

23. (a) L(x) = 405x – 567

(b)

25. (a) L(x) =

(b)

27. (a) L(x) = x – 1

(b)

29. (a) L(x) =

(b)

31. (a) 0.006

(b) 0.00125

33. (a) See Student Solutions Manual.

(b) c3 = 1.155

35. (a) See Student Solutions Manual.

(b) c3 = 0.214

37. (a) See Student Solutions Manual.

(b) c3 = 3.136

39. (a) See Student Solutions Manual.

(b) Answers will vary.

41. (a) See Student Solutions Manual.

(b) c5 ≈ −0.567

43. (a) See Student Solutions Manual.

(b) c5 ≈ −0.791

45. (a) See Student Solutions Manual.

(b) c5 ≈ 4.796

47. 2π

49. 3.6π m3

51. (a) 1.6π m3

(b) Underestimate

(c) Answers will vary.

53. 30 ± 0.27m; 0.9%

55. 72 min

57. 9.998 kg

59. 0.333 %

61. 6%

63. 1.309

65. -0.703

67. 2.718

69. Answers will vary.

71. See Student Solutions Manual.

73. See Student Solutions Manual.

75. See Student Solutions Manual.

77. 9.871 in

Section 3.5

1. False

2. True

3. P5(x) = 3(x − 1)3 + 11(x − 1)2 + 7(x − 1) + 4

5. P5(x) = 2(x + 1)4 − 14(x + 1)3 + 30(x + 1)2 − 25(x + 1) + 7

7. P5(x) = (x−2)5+10(x−2)4+40(x−2)3+80(x−2)2+80(x−2)+32

9. P5(x) =

11. P5(x) = −(x −1)5+(x −1)4−(x −1)3+(x −1)2−(x −1)+1

13. P5(x) =

15. P5(x) =

17. P5(x) = x5 + x4 + x3 + x2 + x + 1

19. P5(x) = −6x5 + 5x4 − 4x3 + 3x2 − 2x + 1

21. P5(x) =

23. P5(x) =

25. P5(x) =

27. P5(x) =

29. f (x) = 3(x − 1)2 + 1

31. f (x) = (x − 1)3 + 4(x − 1)2 + 5(x − 1) − 6

33. (a) P4(x) =

(b)

35. (a) P5(x) =

(b)

37. (a) P4(x)

(b)

39. (a) P3(x) = 3.205 + 0.620x + 0.058x2 + 0.0034x3

(b)

41. See Student Solutions Manual.

43. P2 (x) = − λπ − (1 + λ)(x − π) See Student Solutions Manual.

Section 3.6

1. False

2. sinh x/ cosh x

3. False

4. (a)

5. False

6. False

7. True

8. False

9.

11. 1

13. 0

15. See Student Solutions Manual.

17. See Student Solutions Manual.

19. See Student Solutions Manual.

21. See Student Solutions Manual.

23. See Student Solutions Manual.

25. y′ = 3 cosh(3x)

27. y′ = 2x sinh(x2 + 1)

29. y′ =

31. y′ = 4 sinh x sinh(4x) + cosh x cosh(4x)

33. y′ = 2 cosh x sinh x

35. y′ = ex sinh x + ex cosh x

37. y′ = 2x sech xx2 tanh x sech x

39. y′ =

41. y′ =

43.

45.

47. y′ =

49. P6(x) =

51. Δ ≈ 0.031 radians ≈ 1.776°

53. (a) 625.092 ft

(b) 580.744 ft

(c) ±0.384

(d) ±2.766

(e) 82.026°

(f)

55. (a) See Student Solutions Manual.

(b) { x | x > 1 }

57. See Student Solutions Manual.

59. See Student Solutions Manual.

61. See Student Solutions Manual.

63. See Student Solutions Manual.

65. 0

67. See Student Solutions Manual.

Review Exercises

1. an(ax + b)n−1

3.

5. 3x(x2 + 4) 1/2

7.

9.

11.

13. sec(2x) + 2x sec(2x) tan(2x)

15.

17. cos v − sin2 v cos v = cos3 v

19. 1.05x ln 1.05

21.

23. 2 cot(2x)

25.

27.

29.

31.

33.

35.

37. tan−1(x)

39.

41.

43.

45.

47.

49. y′ =

51. y′ =

53.

55. e2/5

57. 0

59. See Student Solutions Manual.

61.

63. (a)

(b)

(c)

(d)

65. dy =

67. L(x) = 2x − 1

69. (a) dy = dx; Δy = tan(Δx)

(b) (i) dy = 0.5, Δy ≈ 0.546; (ii) dy = 0.1, Δy ≈ 0.100; (iii) dy = 0.01, Δy ≈ 0.010

71.

73. −2

75. P4(x) =

77. P6(x) =

79. (a) See Student Solutions Manual.

(b) 2.568

81. y = 6x − 3