Answers

Chapter 11

Section 11.1

1. {t | t ≤ 4}

2. False

3. True

4. True

5. False

6. u′ × v+u × v′

7. i − 2j

11. i−2k

13. 4i + j +8k

15. All real numbers

17. {t|t ≥ 0}

19. {t|t ≠ 0}

21. {t|t > 0}

23.

25.

27.

29.

31.

33.

35.

37. D

39. C

41.

43.

45.

47.

49. {t|t ≠ −1, 1}

51.

53. {t|t ≠ −1}

55.

57. r′ (t) = 8ti − 6t²j and r″(t) = 8i − 12tj

59. and

61. r′ (t) = j + 2tk and r″(t) = 2k

63. r′(t) = 2ti + 3t²j − k and r″(t) = 2i + 6tj

65. r′ (t) = sin(2t)i+sin(2t)j and r″(t) = 2 cos(2t)i+2 cos(2t)j

67. and

69. r′ (t) = (e^t cos t − e^t sin t)i + (e^t sin t + e^t cos t)j + k and r″(t) = −2e^t sin ti + 2e^t cos tj

71. r′ (t) = (1 − 3t²)i + (1 + 3t²)j − k and r″ (t) = −6ti + 6tj

73.

75.

77.

79. and

81. and

83.

85. See Student Solutions Manual.

87.

89.

91.

93.

95. (a) y = 1 − x

97. (a)

(b)

99. See Student Solutions Manual.

101. See Student Solutions Manual.

103. Answers will vary.

105. See Student Solutions Manual.

107. See Student Solutions Manual.

Section 11.2

1. False

2. False

3. False

4. False

5. True

6. True

7.

(a) r′(1) = i − 2j

(b) and

(c)

9.

(a) r′ (1) = 2i − j

(b) and

(c)

11. (a)

(b) and

(c)

13.

(a) r′ (0) = i − j

(b) and

(c)

15. r′ (0) = −3i + 2j − k

17.

19.

21. r′ (0) = i + j + k

23. Figure below not drawn to scale.

25. Figure below not drawn to scale.

27. and

29. and

31. and

33. and

35. and N(0) is undefined

37. and

39. and

41. and \({\rm {\bf N}}(0) = -\dfrac{\sqrt{2}}{2}{\rm {\bf i}} +\dfrac{\sqrt{2}}{2}{\rm {\bf j}}\)

43. s =

45. s =

47. s =

49. s =

51. s =

53. s =

55. (a)

(b) s ≈ 11.052

57. (a)

(b) s ≈ 33.637

59.

61. θ ≈ 37°

63.

65. θ = 90°

67. and

69. N (t) = −cos ti − sin tj, and

71.

73. s ≈ 10.516

75. (a) s =

(b) s ≈ 33.510

(c) See the Student Solutions Manual

Section 11.3

1. (d)

2. False

3. False

4. True

5. False

8. False

9. No

11. No

13. No

15. Yes

17. No

19. P R Q

21. κ =

23. κ =

25. κ =

27. κ = 0

29. κ =

31. κ =

33. κ =

35. κ =

37. κ =

39. κ =

41.

43.

45. ρ =

47. ρ = 1

49. ρ =

51. ρ =

53. ρ = 6

55. ρ =

57. ρ =

59. ρ =

61. ρ = 12a

63. See the Student Solutions Manual.

65.

67. \(\left(\pm\left(\dfrac{1}{5}\right)^{{1}/{4}},\;\pm\dfrac{1}{3}\left(\dfrac{1}{5}\right)^{{3}/{4}}\right)\)

69. α = 1

71. See the Student Solutions Manual.

73. κ =

75. κ = As t →∞, κ → ∞ as the graph curves more and more tightly around the origin.

77. See the Student Solutions Manual.

79.

81. See the Student Solutions Manual.

83. See the Student Solutions Manual.

85. See the Student Solutions Manual.

87. \(\kappa=\dfrac{\sqrt{2}}{2}\), \(\mathbf{T}(s)=\dfrac{\sqrt{2}}{2}\cos{s}\mathbf{i}-\dfrac{\sqrt{2}}{2}\sin{s}\mathbf{j}+\dfrac{\sqrt{2}}{2}\mathbf{k}\), \(\mathbf{N}(s)=-\sin{s}\mathbf{i}-\cos{s}\mathbf{j}\), and \(\mathbf{B}(s)=\dfrac{\sqrt{2}}{2}\cos{s}\mathbf{i}-\dfrac{\sqrt{2}}{2}\sin{s}\mathbf{j}-\dfrac{\sqrt{2}}{2}\mathbf{k}\)

89. κ =

91. κ =

93. κ =

95. See the Student Solutions Manual.

97.

99.

101. See the Student Solutions Manual.

Section 11.4

1. True

2. False

3. True

4. Tangential, normal

5. (a) v(t) = i + 2tj, a(t) = 2j, and

(b)

7. (a) v (t) = 2ti − j, a(t) = 2i, and

(b)

9. (a) v(t) = 4i + 3t²j, a(t) = 6tj, and

(b)

11. v (t) = 2ti + j − 9t²k, a(t) = 2i − 18tk, and

13. and

15. v (t) = −2 sin ti + cos tj + k, a(t) = −2 cos ti − sin tj, and

17. (a) v (t) = costi − sin tj + 2 cos(2t)k, a(t) = −sin ti − cos tj − 4 sin(2t)k,

(b) and

(c)

19. (a) F(t) = mr(t)

(b)

21. (a) F(t) = m e^t j

(b)

23. (a) F(t) = −4mr(t)

(b)

25. (a) F(t) = −mr(t)

(b)

27. (a) v(t) = 2i + j, a(t) = 0, and

(b) a_T = 0, a_N = 0

29. (a) v(t) = e^ti + 2e^2tj, a(t) = e^ti + 4e^2tj, and

(b)

31. (a) v(t) = 2 cos ti − sin tj, a(t) = −2 sin ti − cos tj, and

(b)

33. (a) v (t) = −3i + 2j − k, a (t) = 0, and

(b) a_T = 0, a_N = 0

35. (a) v (t) = i + 2tj + 3t²k, a(t) = 2j + 6tk, and

(b)

37. (a) v (t) = −sin tj + cos tk, a (t) = −cos tj − sin tk, and ν(t) = 1

(b) a_T = 0, a_N = 1

39. (a) and

(b)

41. (a) v (t) = (e^t cos t − e^t sin t)i + (e^t sin t + e^t cos t)j + e^tk, a(t) = −2e^t sin ti + 2e^t cos tj + e^tk, and

(b)

43. (a) v(t) = −a sin ti + b cos tj + ck, a(t) = −a cos ti − b sin tj, and

(b)

45. v(t) = (π − π cos(πt))i + π sin(πt)j, a(t) = π² sin(πt)i + π² cos(πt)j

47. and

49. See the Student Solutions Manual.

51. a_T = 0 and

53. (a) 350 N

(b) 35 km/h slower

55. 2.118 × 10¹³m N

57. \(4\sqrt{5}\;m\)

59. (a)

(b) No

(c) (−1, 0)

61. (a) See the Student Solutions Manual.

(b)

(c)

63. ν ≈ 3750 m/s

65. See Student Solutions Manual.

67. See Student Solutions Manual.

69. Answers will vary.

71. See Student Solutions Manual.

73.

(a) v(t) = 2t cos(ωt)i + 2t sin(ωt)j + t²v_d and a(t) = 2 cos(ωt)i + 2 sin(ωt)j + 4tv_d + t²a_d

(b) 2 cos(ωt)i + 2 sin(ωt)j + 4tv_d

75. See Student Solutions Manual.

77. r = e^2t, where θ = t; v(t) = 2ru_r + ru_θ; a(t) = 3ru_r + 4ru_θ

Section 11.5

1. True

2. False

11. r(t) = (e^t − e)i + (t − t ln t)j + t²k

13. r(t) = (3 − 2 cos t)i + (sin t − 1)j + tk

15. r(t) =

17. v(t) = −32tk, υ(t) = 32t, and r(t) = −16t²k

19. v(t) = (sin t + 1)i + (1 − cos t)j, v (t) = and r (t) = (t − cos t + 1)i + (t − sin t + 1)j

21. v(t) = i − 9.8tk, v (t) = and r(t) = ti + (5 − 4.9t²)k

23. v(t) = (2 − e^−t)i + (t + 1)j, v (t) = and r (t) =

25. The range ≈ 23,895 m, the time of flight ≈ 53 s, and the greatest height reached ≈3449 m.

27. (a) x = and y =

(b) Range ≈ 724.6 m

(c) Time of flight ≈ 7.8 s

(d)

29. (a) The projectile travels approximately 152.5 feet up the hill.

(b) The projectile is in the air for approximately 1.9 seconds.

31. (a) v(t) = 5ti + (−9.8t + 3)j and v(t) =

(b) r(t) =

(c) ≈0.612 s

(d)

33. v₀ ≈ 54.521 ft/s

35. The initial speed of the ball is approximately 114 ft/s. It took the ball approximately 5 s to reach the vines.

37. (a) Range ≈ 114.342 ft

(b) ≈93 ft

39. See Student Solutions Manual.

41. See Student Solutions Manual.

43. See Student Solutions Manual.

45. See Student Solutions Manual.

Section 11.6

1. ≈79.479 days

3. Answers will vary.

5. (a)See Student Solutions Manual.

(b) M ≈ 2.0 × 10³⁰kg

7. See Student Solutions Manual.

Review Exercises

1. (a) All real numbers.

(b)

(c) r′(2) = 4i + 3j

3. (a) All real numbers.

(b)

(c) r′(0) = i + 2k

7. Continuous

9. r′ (t) = −2 sin ti − 3 sin tj + k and r″ (t) = −2 cos ti − 3 cos tj

11. [f(t) · g(t)]′ = 4t − 2 sin(2t) cos t − cos(2t) sin t − 5 cos t and

[f(t) × g(t)]′ = (−2 sin(2t) sin t + cos(2t) cos t − 5 sin t)i + (−10 − sin t − t cos t)j + (cos t − t sin t + 4t sin(2t) − 2 cos(2t))k