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

196 Chapter 3 Differentiation Rules
3.3 Exercises
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1–16 Differentiate.
1. f sxd − x 2 sin x 2. f sxd − x cos x 1 2 tan x
3. f sxd − e x cos x 4. y − 2 sec x 2 csc x
5. y − sec tan
6. tsd − e stan 2 d
7. y − c cos t 1 t 2 sin t 8. f std − cot t
e t
x
9. y −
10. y − sin cos
2 2 tan x
11. f sd − sin
1 1 cos
13. y − t sin t
1 1 t
12. y − cos x
1 2 sin x
14. y − sin t
1 1 tan t
15. f sd − cos sin 16. f std − te t cot t
17. Prove that d scsc xd − 2csc x cot x.
dx
18. Prove that d ssec xd − sec x tan x.
dx
19. Prove that d dx scot xd − 2csc2 x.
20. Prove, using the definition of derivative, that if
f sxd − cos x, then f 9sxd − 2sin x.
21–24 Find an equation of the tangent line to the curve at the
given point.
21. y − sin x 1 cos x, s0, 1d 22. y − e x cos x, s0, 1d
23. y − cos x 2 sin x, s, 21d 24. y − x 1 tan x, s, d
25. (a) Find an equation of the tangent line to the curve
y − 2x sin x at the point sy2, d.
(b) Illustrate part (a) by graphing the curve and the tangent
line on the same screen.
26. (a) Find an equation of the tangent line to the curve
y − 3x 1 6 cos x at the point sy3, 1 3d.
(b) Illustrate part (a) by graphing the curve and the tangent
line on the same screen.
27. (a) If f sxd − sec x 2 x, find f 9sxd.
(b) Check to see that your answer to part (a) is reasonable by
graphing both f and f 9 for | x | , y2.
28. (a) If f sxd − e x cos x, find f 9sxd and f 99sxd.
(b) Check to see that your answers to part (a) are reasonable
by graphing f , f 9, and f 99.
29. If Hsd − sin , find H9sd and H99sd.
30. If f std − sec t, find f 0sy4d.
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31. (a) Use the Quotient Rule to differentiate the function
f sxd − tan x 2 1
sec x
(b) Simplify the expression for f sxd by writing it in terms
of sin x and cos x, and then find f 9sxd.
(c) Show that your answers to parts (a) and (b) are
equivalent.
32. Suppose f sy3d − 4 and f 9sy3d − 22, and let
tsxd − f sxd sin x and hsxd − scos xdyf sxd. Find
(a) t9sy3d (b) h9sy3d
33–34 For what values of x does the graph of f have a horizontal
tangent?
33. f sxd − x 1 2 sin x 34. f sxd − e x cos x
35. A mass on a spring vibrates horizontally on a smooth
level surface (see the figure). Its equation of motion is
xstd − 8 sin t, where t is in seconds and x in centimeters.
(a) Find the velocity and acceleration at time t.
(b) Find the position, velocity, and acceleration of the mass
at time t − 2y3. In what direction is it moving at that
time?
equilibrium
position
0
36. An elastic band is hung on a hook and a mass is hung on the
lower end of the band. When the mass is pulled downward
and then released, it vibrates vertically. The equation of
motion is s − 2 cos t 1 3 sin t, t > 0, where s is measured
in centi meters and t in seconds. (Take the positive direction
to be downward.)
(a) Find the velocity and acceleration at time t.
(b) Graph the velocity and acceleration functions.
(c) When does the mass pass through the equilibrium
position for the first time?
(d) How far from its equilibrium position does the mass
travel?
(e) When is the speed the greatest?
37. A ladder 10 ft long rests against a vertical wall. Let be the
angle between the top of the ladder and the wall and let x be
the distance from the bottom of the ladder to the wall. If the
bottom of the ladder slides away from the wall, how fast
does x change with respect to when − y3?
38. An object with weight W is dragged along a horizontal
plane by a force acting along a rope attached to the object.
x
x
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