3.7 Related Rates
185
3.7 Related Rates
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____
1.
Assume that x and y are both differentiable functions of t . Find
for the equation
.
a.
b.
c.
d.
e.
____
2.
Assume that x and y are both differentiable functions of t. Find
for the equation
.
a.
b.
c.
d.
e.
____
3.
A point is moving along the graph of the function
such that
centimeters per second. Find
when x = .
a.
b.
186 Chapter 3: Differentiation
c.
d.
e.
____ 4. A point is moving along the graph of the function such that =
centimeters per second. Find when .
a.
b.
c.
d.
e.
____ 5.
Find the rate of change of the distance
between the origin and a moving point on
the graph of
if
centimeters per second.
a.
b.
c.
d.
e.
3.7 Related Rates
187
____
6.
The radius, r, of a circle is decreasing at a rate of
centimeters per minute.
Find the rate of change of area, A, when the radius is .
a.
sq cm/min
b.
sq cm/min
c.
sq cm/min
d.
sq cm/min
e.
sq cm/min
____
7.
The radius r of a sphere is increasing at a rate of
inches per minute. Find the rate of
change of the volume when r =
inches.
a.
b.
c.
d.
e.
____
8.
A spherical balloon is inflated with gas at the rate of
cubic centimeters per
minute. How fast is the radius of the balloon increasing at the instant the radius is
centimeters?
a.
b.
c.
d.
e.
188
Chapter 3: Differentiation
____
9.
All edges of a cube are expanding at a rate of centimeters per second. How fast is
the volume changing when each edge is centimeters?
a.
b.
c.
d.
e.
____ 10.
A conical tank (with vertex down) is
feet across the top and
feet deep. If
water is flowing into the tank at a rate of
cubic feet per minute, find the rate of change of the
depth of the water when the water is
feet deep.
a.
b.
c.
d.
e.
____ 11. A ladder feet long is leaning against the wall of a house (see figure). The base of the
ladder is pulled away from the wall at a rate of feet per second. How fast is the top of the ladder
moving down the wall when its base is feet from the wall? Round your answer to two decimal
places.
3.7 Related Rates
189
ft/sec
ft/sec
ft/sec
ft/sec
ft/sec
____ 12. A ladder feet long is leaning against the wall of a house (see figure). The base of
the ladder is pulled away from the wall at a rate of feet per second. Consider the triangle formed by
the side of the house, the ladder, and the ground. Find the rate at which the area of the triangle is
changed when the base of the ladder is feet from the wall. Round your answer to two decimal
places.
a.
b.
c.
d.
e.
190
Chapter 3: Differentiation
____
13.
A ladder
feet long is leaning against the wall of a house (see figure). The base of
the ladder is pulled away from the wall at a rate of feet per second. Find the rate at which the angle
between the ladder and the wall of the house is changing when the base of the ladder is feet from
the wall. Round your answer to three decimal places.
rad/sec
rad/sec
rad/sec
rad/sec
rad/sec
____ 14. A man 6 feet tall walks at a rate of feet per second away from a light that is 15
feet above the ground (see figure). When he is feet from the base of the light, at what rate is the
tip of his shadow moving?
a.
3.7 Related Rates
191
ft/sec
ft/sec
c.
ft/sec
d.
ft/sec
e.
ft/sec
____ 15.
A man 6 feet tall walks at a rate of
feet per second away from a light that is 15
feet above the ground (see figure). When he is feet from the base of the light, at what rate is the
length of his shadow changing?
a.
ft/sec
b.
ft/sec
c.
ft/sec
d.
ft/sec
e.
ft/sec
192
Chapter 3: Differentiation
____
16.
An airplane is flying in still air with an airspeed of
miles per hour. If it is
climbing at an angle of
, find the rate at which it is gaining altitude. Round your answer to four
decimal places.
a.
b.
c.
d.
e.
3.7 Related Rates
193
3.7 Related Rates
Answer Section
194 Chapter 3: Differentiation
3.8 Newton’s Method
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____
1.
Complete two iterations of Newton‘s Method for the function
using the
initial guess
. Round your answers to four decimal places.
a.
b.
c.
d.
e.
____
2.
Complete two iterations of Newton’s Method for the function
using the
initial guess
. Round all numerical values in your answer to four decimal places.
a.
b.
c.
d.
3.8 Newton’s Method 195
e.
____ 3. Use Newton´s Method to approximate the zero(s) of the function
accurate to three decimal places.
a.
b.
c.
d.
e.
____ 4. Use Newton’s Method to approximate the zero(s) of the function
accurate to three decimal places.
a.
b.
c.
d.
e.
____ 5. Use Newton’s Method to approximate the zero(s) of the function
accurate to three decimal places.
a.
b.
c.
d.
e.
____ 6. Approximate the positive zero(s) of the function to three decimal
places. Use Newton’s Method and continue the process until two successive approximations differ by
less than 0.001.
a.
b.
c.
d.
e.
196
Chapter 3: Differentiation
____
7.
Use Newton´s Method to approximate the x-value of the indicated point of
intersection of the two graphs accurate to three decimal places.Continue the process until
two successive approximations differ by less than 0.001. [Hint: Let .]
a.
b.
c.
d.
e.
____ 8. Use Newton’s Method to approximate the x-value of the indicated point of
intersection of the two graphs accurate to three decimal places.Continue the process until
two successive approximations differ by less than 0.001. [Hint: Let .]
3.8 Newton’s Method 197
a.
b.
c.
d.
e.
____ 9. Apply Newton’s Method to approximate the x-value of the indicated point of
intersection of Continue the process until two successive approximations
differ by less than 0.001. [Hint: Let .] Round your answer to three decimal places.
a.
b.
c.
d.
e.
____ 10.
Approximate the fixed point of the function
between
to two
decimal places. [A
of a function f is a value of x such that
.]
a.
b.
c.
d.
e.
198 Chapter 3: Differentiation
____ 11. Suppose that the total number of arrests T (in thousands) for all males ages 14 to 27
in 2006 is approximated by the model where x is
the age in years (see figure). Approximate the two ages to one decimal place that had total arrests of
thousand.
a.
b.
c.
d.
e.
____ 12. A manufacturer of digital audio players estimates that the profit for selling a
particular model is where P is the profit in dollars and
x is the advertising expense in 10,000’s of dollars (see figure). Find the smaller of two advertising
amounts that yield a profit P of $ . Round your answer to the nearest dollar.
a.
b.
c.
d.
e.
3.8 Newton’s Method 199
3.8 Newton’s Method
Answer Section