James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

chapter 3 Review 267
11. The derivative of a polynomial is a polynomial.
12. If f sxd − sx 6 2 x 4 d 5 , then f s31d sxd − 0.
13. The derivative of a rational function is a rational function.
14. An equation of the tangent line to the parabola y − x 2
at s22, 4d is y 2 4 − 2xsx 1 2d.
tsxd 2 ts2d
15. If tsxd − x 5 , then lim
− 80
x l 2 x 2 2
EXERCISES
1–50 Calculate y9.
1. y − sx 2 1 x 3 d 4 2. y − 1
sx
2 1
s 5 x 3
3. y − x 2 2 x 1 2
sx
4. y − tan x
1 1 cos x
5. y − x 2 sin x 6. y − x cos 21 x
7. y − t 4 2 1
t 4 1 1
8. xe y − y sin x
9. y − lnsx ln xd 10. y − e mx cos nx
11. y − sx cos sx 12. y − sarcsin 2xd 2
13. y − e1yx
x 2
14. y − ln sec x
15. y 1 x cos y − x 2 y 16. y −S u 2 1
17. y − sarctan x
4
u 2 1 u 1 1D
18. y − cotscsc xd
19. y − tanS t
1 1 t 2D 20. y − e x sec x
21. y − 3 x ln x
23. y − s1 2 x 21 d 21
25. sinsxyd − x 2 2 y
27. y − log 5s1 1 2xd
29. y − ln sin x 2 1 2 sin2 x
31. y − x tan 21 s4xd
33. y − ln | sec 5x 1 tan 5x |
35. y − cots3x 2 1 5d
37. y − sinstan s1 1 x 3 d
22. y − secs1 1 x 2 d
24. y − 1ys 3 x 1 sx
26. y − ssin s x
28. y − scos xd x
30. y −
sx 2 1 1d 4
s2x 1 1d 3 s3x 2 1d 5
32. y − e cos x 1 cosse x d
34. y − 10tan
36. y − st lnst 4 d
38. y − arctansarcsin sx d
39. y − tan 2 ssin d 40. xe y − y 2 1
41. y −
sx 1 1 s2 2 xd5
sx 1 d4
42. y −
sx 1 3d 7 x 4 1 4
43. y − x sinhsx 2 d
44. y −
sin mx
x
;
;
;
;
45. y − lnscosh 3xd 46. y − ln Z
x 2 2 4
2x 1 5
Z
47. y − cosh 21 ssinh xd
49. y − cosse stan 3x d
51. If f std − s4t 1 1, find f 99s2d.
52. If tsd − sin , find t99sy6d.
53. Find y99 if x 6 1 y 6 − 1.
54. Find f snd sxd if f sxd − 1ys2 2 xd.
48. y − x tanh 21 sx
50. y − sin 2 scosssin x d
55. Use mathematical induction (page 72) to show that if
f sxd − xe x , then f snd sxd − sx 1 nde x .
t 3
56. Evaluate lim
tl 0 tan 3 s2td .
57–59 Find an equation of the tangent to the curve at the given
point.
57. y − 4 sin 2 x, sy6, 1d 58. y − x2 2 1
, s0, 21d
x 2 1 1
59. y − s1 1 4 sin x , s0, 1d
60–61 Find equations of the tangent line and normal line to the
curve at the given point.
60. x 2 1 4xy 1 y 2 − 13, s2, 1d
61. y − s2 1 xde 2x , s0, 2d
62. If f sxd − xe sin x , find f 9sxd. Graph f and f 9 on the same
screen and comment.
63. (a) If f sxd − xs5 2 x , find f 9sxd.
(b) Find equations of the tangent lines to the curve
y − xs5 2 x at the points s1, 2d and s4, 4d.
(c) Illustrate part (b) by graphing the curve and tangent lines
on the same screen.
(d) Check to see that your answer to part (a) is reasonable by
comparing the graphs of f and f 9.
64. (a) If f sxd − 4x 2 tan x, 2y2 , x , y2, find f 9 and f99.
(b) Check to see that your answers to part (a) are reasonable
by comparing the graphs of f , f 9, and f99.
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