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Sullivan 04 apcalc4e 45342 ch02 166 233 5pp August 7, 2023 12:54

224 Chapter 2 • The Derivative and Its Properties

66. Simple Harmonic Motion 70. Swinging Pendulum A simple
An object attached to a coiled pendulum is a small-sized ball swinging θ
spring is pulled down a from a light string. As it swings, the T, tension
distance d = 5 cm from its supporting string makes an angle θ with
equilibrium position and then 5 the vertical. See the figure. At an
released as shown in the angle θ, the tension in the string
figure. The motion of the is T = W , where W is the weight of
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Equ
object at time t seconds is 0 Equilibrium cos θ W, weight
simple harmonic and is d  5 the swinging ball.
modeled by d(t) = −5 cos t.
5 t t  0 (a) Find the rate of change of the tension T with respect to θ
when the pendulum is at its highest point (θ = θ max ).
(b) Find the rate of change of the tension T with respect to θ
when the pendulum is at its lowest point.
(a) As t varies from 0 to 2π, how does the length
of the spring vary? (c) What is the tension at the lowest point?
(b) Find the velocity v = v(t) of the object. 71. Restaurant Sales A restaurant in Naples, Florida, is very busy
(c) When is the speed of the object a maximum? during the winter months and extremely slow over the summer.
(d) Find the acceleration a = a(t) of the object. But every year the restaurant grows its sales. Suppose over the
next two years, the revenue R, in units of $10,000, is projected to
(e) When is the acceleration equal to 0?
follow the model
(f) Graph d, v, and a on the same set of axes.
R = R(t) = sin t + 0.3t + 1 0 ≤ t ≤ 12
67. Rate of Change A large, 8-ft-high decorative mirror is where t = 0 corresponds to November 1, 2024; t = 1 corresponds
placed on a wood floor and leaned against a wall. The weight to January 1, 2025; t = 2 corresponds to March 1, 2025; and so on.
of the mirror and the slickness of the floor cause the mirror
to slip. (a) What is the projected revenue for November 1, 2024;
March 1, 2025; September 1, 2025; and January 1, 2026?
(a) If θ is the angle between the top of the mirror and (b) What is the rate of change of revenue with respect to time?
the wall, and y is the distance from the floor to the top of
(c) What is the rate of change of revenue with respect to time for
the mirror, what is the rate of change of y with
January 1, 2026?
respect to θ?
(d) Graph the revenue function and the derivative
(b) In feet/radian, how fast is the top of the mirror slipping down function R = R (t).
′
′
π (e) Does the graph of R support the facts that every year the
the wall when θ = ?
4 restaurant grows its sales and that sales are higher during the
winter and lower during the summer? Explain.
68. Rate of Change The sides of 72. Polarizing Filters Polarizing filters transmit only light for
an isosceles triangle are sliding θ θ
1 cm 1 cm which the electric field oscillations are in a specific direction.
outward. See the figure. Light is polarized naturally by scattering off the molecules in the
(a) Find the rate of change atmosphere and by reflecting off many (but not all) types of
of the area of the triangle surfaces. If light of intensity I 0 is already polarized in a certain
with respect to θ. direction, and the transmission direction of the polarizing filter
π makes an angle with that direction, then the intensity I of the light
(b) How fast is the area changing when θ = ?
6 after passing through the filter is given by
2
Malus’s Law, I (θ) = I 0 cos θ.
69. Sea Waves Waves in deep water tend to have the symmetric
form of the function f (x) = sin x. As they approach shore,
however, the sea floor creates drag, which changes the shape of
the wave. The trough of the wave widens and the height of the
wave increases, so the top of the wave is no longer symmetric
with the trough. This type of wave can be represented by a
function such as
4
w(x) =
2 + cos x
/Getty Images
(a) Graph w = w(x) for 0 ≤ x ≤ 4π.
(b) What is the maximum and the minimum value of w?
(c) Find the values of x, 0 < x < 4π, at which w (x) = 0. nicolas
′
(d) Evaluate w near the peak at π, using x = π − 0.1, and near (a) As you rotate a polarizing filter, θ changes. Find the rate of
′
the trough at 2π, using x = 2π − 0.1.
change of the light intensity I with respect to θ.
(e) Explain how these values confirm a nonsymmetric wave (b) Find both the intensity I (θ) and the rate of change of the
shape. intensity with respect to θ, for the angles θ = 0 , 45 ,
◦
◦
◦
and 90 . (Remember to use radians for θ.)
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