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

268 Chapter 3 Differentiation Rules
65. At what points on the curve y − sin x 1 cos x,
0 < x < 2, is the tangent line horizontal?
66. Find the points on the ellipse x 2 1 2y 2 − 1 where the
tangent line has slope 1.
67. If f sxd − sx 2 adsx 2 bdsx 2 cd, show that
f 9sxd
f sxd
− 1
x 2 a 1 1
x 2 b 1 1
x 2 c
68. (a) By differentiating the double-angle formula
cos 2x − cos 2 x 2 sin 2 x
obtain the double-angle formula for the sine function.
(b) By differentiating the addition formula
sinsx 1 ad − sin x cos a 1 cos x sin a
obtain the addition formula for the cosine function.
69. Suppose that
f s1d − 2 f 9s1d − 3 f s2d − 1 f 9s2d − 2
ts1d − 3 t9s1d − 1 ts2d − 1 t9s2d − 4
(a) If Ssxd − f sxd 1 tsxd, find S9s1d.
(b) If Psxd − f sxdtsxd, find P9s2d.
(c) If Qsxd − f sxdytsxd, find Q9s1d.
(d) If Csxd − f stsxdd, find C9s2d.
70. If f and t are the functions whose graphs are shown, let
Psxd − f sxdtsxd, Qsxd − f sxdytsxd, and Csxd − f stsxdd.
Find (a) P9s2d, (b) Q9s2d, and (c) C9s2d.
71–78 Find f 9 in terms of t9.
y
1
0
1
71. f sxd − x 2 tsxd 72. f sxd − tsx 2 d
73. f sxd − ftsxdg 2 74. f sxd − tstsxdd
75. f sxd − tse x d 76. f sxd − e tsxd
77. f sxd − ln | tsxd |
79–81 Find h9 in terms of f 9 and t9.
79. hsxd − f sxdtsxd
f sxd 1 tsxd
81. hsxd − f stssin 4xdd
g
f
x
78. f sxd − tsln xd
80. hsxd −Î f sxd
tsxd
;
;
82. (a) Graph the function f sxd − x 2 2 sin x in the viewing
rectangle f0, 8g by f22, 8g.
(b) On which interval is the average rate of change larger:
f1, 2g or f2, 3g?
(c) At which value of x is the instantaneous rate of change
larger: x − 2 or x − 5?
(d) Check your visual estimates in part (c) by computing
f 9sxd and comparing the numerical values of f 9s2d
and f 9s5d.
83. At what point on the curve y − flnsx 1 4dg 2 is the tangent
horizontal?
84. (a) Find an equation of the tangent to the curve y − e x that
is parallel to the line x 2 4y − 1.
(b) Find an equation of the tangent to the curve y − e x that
passes through the origin.
85. Find a parabola y − ax 2 1 bx 1 c that passes through the
point s1, 4d and whose tangent lines at x − 21 and x − 5
have slopes 6 and 22, respectively.
86. The function Cstd − Kse 2at 2 e 2bt d, where a, b, and K are
positive constants and b . a, is used to model the concentration
at time t of a drug injected into the bloodstream.
(a) Show that lim t l ` Cstd − 0.
(b) Find C9std, the rate of change of drug concentration in
the blood.
(c) When is this rate equal to 0?
87. An equation of motion of the form s − Ae 2ct cosst 1 d
represents damped oscillation of an object. Find the velocity
and acceleration of the object.
88. A particle moves along a horizontal line so that its coordinate
at time t is x − sb 2 1 c 2 t 2 , t > 0, where b and c
are positive constants.
(a) Find the velocity and acceleration functions.
(b) Show that the particle always moves in the positive
direction.
89. A particle moves on a vertical line so that its coordinate at
time t is y − t 3 2 12t 1 3, t > 0.
(a) Find the velocity and acceleration functions.
(b) When is the particle moving upward and when is it
moving downward?
(c) Find the distance that the particle travels in the time
interval 0 < t < 3.
(d) Graph the position, velocity, and acceleration functions
for 0 < t < 3.
(e) When is the particle speeding up? When is it slowing
down?
90. The volume of a right circular cone is V − 1 3 r 2 h, where
r is the radius of the base and h is the height.
(a) Find the rate of change of the volume with respect to
the height if the radius is constant.
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