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Sullivan 04 apcalc4e 45342 ch02 166 233 5pp August 7, 2023 12:54

Section 2.5 • The Derivative of the Trigonometric Functions 223

2.5 Assess Your Understanding

Concepts and Vocabulary
d
′′
1. True or False cos x = sin x In Problems 39–50, find y .
dx
39. y = sin x 40. y = cos x
d
2. True or False tan x = cot x 41. y = tan θ 42. y = cot θ
dx
43. y = t sin t 44. y = t cos t
d 2
3. True or False sin x = −sin x
PAGE
x
x
dx 2 221 45. y = e sin x 46. y = e cos x
d π π 47. y = 5 sin u − 4 cos u 48. y = 6 sin u + 5 cos u
4. True or False sin = cos
dx 3 3
49. y = a sin x + b cos x 50. y = a sec θ + b tan θ
Skill Building
In Problems 51–56:
In Problems 5–38, find y . ′ (a) Find an equation of the tangent line to the graph of f at the
indicated point.
PAGE 2
219 5. y = x − sin x 6. y = cos x − x (b) Graph the function and the tangent line.

π 1
7. y = tan x + sin x 8. y = cos x − tan x 51. f (x) = sin x at (0, 0) 52. f (x) = cos x at ,
3 2
9. y = 3 sin θ − 2 cos θ 10. y = 4 tan θ + sin θ π
53. f (x) = tan x at (0, 0) 54. f (x) = tan x at , 1
4
11. y = sin x cos x 12. y = cot x tan x π √
55. f (x) = sin x + cos x at , 2
4
π
PAGE 2
220 13. y = t cos t 14. y = t tan t 56. f (x) = sin x − cos x at , 0
4
In Problems 57–60:
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x
x
221 15. y = e tan x 16. y = e sec x
(a) Find all points on the graph of f where the tangent line is
horizontal.
17. y = π sec u tan u 18. y = πu tan u
(b) Graph the function and the horizontal tangent lines
on the interval [−2π, 2π].
cot x csc x
19. y = 20. y =
x x PAGE
220 57. f (x) = 2 sin x + cos x 58. f (x) = cos x − sin x
2
2
21. y = x sin x 22. y = t tan t 59. f (x) = sec x 60. f (x) = csc x
√ √
23. y = t tan t − 3 sec t 24. y = x sec x + 2 cot x Applications and Extensions
In Problems 61 and 62, find the nth derivative of each function.
sin θ x
25. y = 26. y =
1 − cos θ cos x 61. f (x) = sin x 62. f (θ) = cos θ
π π
sin t tan u cos + h − cos
27. y = 28. y = 2 2
1 + t 1 + u 63. What is lim ?
h→0 h
PAGE sin x cos x sin(π + h) − sin π
219 29. y = 30. y = 64. What is lim ?
e x e x
h→0 h
PAGE
sin θ + cos θ sin θ − cos θ 222 65. Simple Harmonic Motion The signed distance s (in meters) of
31. y = 32. y =
sin θ − cos θ sin θ + cos θ an object from the origin at time t (in seconds) is modeled by the
1
sec t csc t position function s(t) = cos t.
33. y = 34. y = 8
1 + t sin t 1 + t cos t
(a) Find the velocity v = v(t) of the object.
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221 35. y = csc θ cot θ 36. y = tan θ cos θ (b) When is the speed of the object a maximum?
(c) Find the acceleration a = a(t) of the object.
1 + tan x csc x − cot x (d) When is the acceleration equal to 0?
37. y = 38. y =
1 − tan x csc x + cot x (e) Graph s, v, and a on the same screen.
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