5128_Ch03_pp098-184

156 Chapter 3 Derivatives
Explorations
76. The Derivative of sin 2x Graph the function y 2 cos 2x
for 2 x 3.5. Then, on the same screen, graph
sin 2x h sin 2x
y
h
for h 1.0, 0.5, and 0.2. Experiment with other values of
h, including negative values. What do you see happening as
h→0? Explain this behavior.
77. The Derivative of cos (x 2 ) Graph y 2x sin x 2 for
2 x 3. Then, on screen, graph
cos x h 2 cos x 2
y
h
for h 1.0, 0.7, and 0.3. Experiment with other values
of h. What do you see happening as h →0? Explain this
behavior.
Extending the Ideas
78. Absolute Value Functions Let u be a differentiable function
of x.
d u
(a) Show that u u.
d x u
(b) Use part (a) to find the derivatives of f x x 2 9 and
gx x sin x.
79. Geometric and Arithmetic Mean The geometric mean
of u and v is G uv and the arithmetic mean is
A u v2. Show that if u x, v x c, c a real
number, then
d G A
.
dx
G
Quick Quiz for AP* Preparation: Sections 3.4 –3.6
You should solve the following problems without using a
graphing calculator.
1. Multiple Choice Which of the following gives
dy/dx for y sin 4 (3x)? B
3. Multiple Choice Which of the following gives dy/dx for the
parametric curve x 3 sin t, y 2 cos t? C
(A) 3 2 cot t (B) 3 2 cot t (C) 2 3 tan t (D) 2 tan t (E) tan t
3
(A) 4 sin 3 (3x) cos (3x)
(B) 12 sin 3 (3x) cos (3x) 4. Free Response A particle moves along a line so that its
(C) 12 sin (3x) cos (3x)
(D) 12 sin 3 (3x)
position at any time t 0 is given by s(t) t 2 t 2, where
(E) 12 sin 3 s is measured in meters and t is measured in seconds.
(3x) cos (3x)
(a) What is the initial position of the particle? s(0) 2m
2. Multiple Choice Which of the following gives y for
y cos x tan x? A
(b) Find the velocity of the particle at any time t. v(t) s(t)
2t 1 m/s
(A) cos x 2 sec 2 x tan x (B) cos x 2 sec 2 x tan x (c) When is the particle moving to the right?
k
k k2
2 s 3
/2 s 2 s2
63. Velocity 2 5 m/sec
66. Acceleration d v
df ( x)
4
acceleration dt
dt
1 25 m/sec2 f(
d d
x) x
x t
(C) sin x sec 2 x
(D) cos x sec 2 x tan x (d) Find the acceleration of the particle at any time t.
(E) cos x sec 2 x tan x
(e) Find the speed of the particle at the moment when s(t) 0.
(c) The particle moves to the right when v(t) 0; that is, when 0 t 1/2.
(d) a(t) v(t) 2 m/s 2
(e) s(t) (t 1)(t 2), so s(t) 0 when t 2. The speed at that moment is
⏐v(2)⏐⏐3⏐ 3 m/s.
Answers to Section 3.6 Exercises
64. Acceleration d v
d v
d s
d v
v
dt
ds
dt
ds
k
(ks) k 2
50. Since the radius goes through (0, 0) and (2 cos t, 2 sin t), it has slope
2 s 2
given by tan t. But d y
d y/
dt
c os
t
cot t, which is the
k
65. Given: v
dx
dx/
dt
sin
t
s
negative reciprocal of tan t. This means that the radius and the tangent
are perpendicular.
dt
ds
dt
d s
acceleration: d v
d v
d s dv
v
f(x)f(x)
67. d T
du T
dL
du
T
gL kL k L g k 2