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

322 Chapter 4 Applications of Differentiation
19. y − x3
x 3 1 1
20. y − x 3
x 2 2
21. y − sx 2 3dsx 22. y − sx 2 4ds 3 x
23. y − sx 2 1 x 2 2 24. y − sx 2 1 x 2 x
x
25. y −
sx 2 1 1
27. y − s1 2 x 2
x
26. y − xs2 2 x 2
x
28. y −
sx 2 2 1
29. y − x 2 3x 1y3 30. y − x 5y3 2 5x 2y3
31. y − s 3 x 2 2 1 32. y − s 3 x 3 1 1
33. y − sin 3 x 34. y − x 1 cos x
35. y − x tan x, 2y2 , x , y2
36. y − 2x 2 tan x, 2y2 , x , y2
37. y − sin x 1 s3 cos x, 22 < x < 2
38. y − csc x 2 2sin x, 0 , x ,
39. y − sin x
1 1 cos x
40. y − sin x
2 1 cos x
41. y − arctanse x d 42. y − s1 2 xde x
43. y − 1ys1 1 e 2x d
44. y − e 2x sin x, 0 < x < 2
45. y − 1 x 1 ln x 46. y − e2 x 2 e x
47. y − s1 1 e x d 22 48. y − e x yx 2
49. y − lnssin xd 50. y − lns1 1 x 3 d
51. y − xe 21yx 52. y − ln x
x 2
21S 53. y − e arctan x 54. y − tan x 2 1
x 1 1D
55. In the theory of relativity, the mass of a particle is
m 0
m −
s1 2 v 2 yc 2
where m 0 is the rest mass of the particle, m is the mass
when the particle moves with speed v relative to the
observer, and c is the speed of light. Sketch the graph of m
as a function of v.
56. In the theory of relativity, the energy of a particle is
length, and h is Planck’s constant. Sketch the graph of E as a
function of . What does the graph say about the energy?
57. A model for the spread of a rumor is given by the equation
pstd −
1
1 1 ae 2kt
where pstd is the proportion of the population that knows the
rumor at time t and a and k are positive constants.
(a) When will half the population have heard the rumor?
(b) When is the rate of spread of the rumor greatest?
(c) Sketch the graph of p.
58. A model for the concentration at time t of a drug injected
into the bloodstream is
Cstd − Kse 2at 2 e 2bt d
where a, b, and K are positive constants and b . a. Sketch
the graph of the concentration function. What does the
graph tell us about how the concentration varies as time
passes?
59. The figure shows a beam of length L embedded in concrete
walls. If a constant load W is distributed evenly along its
length, the beam takes the shape of the deflection curve
y − 2
W
24EI x 4 1 WL
12EI x 3 2 WL2
24EI x 2
where E and I are positive constants. (E is Young’s modulus
of elasticity and I is the moment of inertia of a crosssection
of the beam.) Sketch the graph of the deflection
curve.
y
0
W
L
60. Coulomb’s Law states that the force of attraction between
two charged particles is directly proportional to the product
of the charges and inversely proportional to the square of
the distance between them. The figure shows particles with
charge 1 located at positions 0 and 2 on a coordinate line and
a particle with charge 21 at a position x between them. It
follows from Coulomb’s Law that the net force acting on the
middle particle is
Fsxd − 2 k
x 2 1
k
sx 2 2d 2 0 , x , 2
where k is a positive constant. Sketch the graph of the net
force function. What does the graph say about the force?
E − sm 0 2 c 4 1 h 2 c 2 y 2
where m 0 is the rest mass of the particle, is its wave
+1
0
_1
x
+1
2
x
7et0405x60
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