3.1 The Derivative and the Tangent Line Problem
109
3.1 The Derivative and the Tangent Line Problem
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____ 1. Find the slope of the line tangent to the graph of the function at the point
.
a.
b.
c.
d.
e.
____ 2. Find the slope of the line tangent to the graph of the function at the point
.
a.
b.
c.
d.
e.
____ 3. Find the derivative of the function by the limit process.
a.
b.
c.
d.
e.
____ 4. Find the derivative of the function by the limit process.
a.
b.
c.
d.
e.
110
Chapter 3: Differentiation
____
5.
Find the derivative of the following function
using the limiting
process.
a.
b.
c.
d.
e.
____
6.
Find the derivative of the following function using the limiting process.
–4
____ 7. Find the derivative of the following function using the limiting process.
a.
b.
c.
d.
e.
____ 8. Find the derivative of the following function using the limiting process.
a.
b.
c.
d.
e.
3.1 The Derivative and the Tangent Line Problem
111
____ 9.
Find the derivative of the following function using the limiting process.
a.
b.
c.
d.
e.
____
10.
Find the derivative of the function
using the limiting process.
a.
b.
c.
d.
e.
____
11.
Find the derivative of the function
by the limit process.
a.
b.
c.
d.
e.
112
Chapter 3: Differentiation
____
12.
Find an equation of the tangent line to the graph of the function
at
the point
.
a.
b.
c.
d.
e.
____ 13. Find an equation of the tangent line to the graph of the function at
the point
.
a.
b.
c.
d.
e.
____
14.
Find an equation of the line that is tangent to the graph of the function
and parallel to the line
.
a.
b.
c.
d.
e.
____
15.
Find an equation of the line that is tangent to the graph of
and parallel to the given
line.
a.
b.
c. and d.
and
3.1 The Derivative and the Tangent Line Problem
113
____ 16.
Find an equation of the line that is tangent to the graph of the function
and parallel to the line .
a.
b.
c.
d.
e.
____ 17. The graph of the function f is given below. Select the graph of .
a. d.
114 Chapter 3: Differentiation
b. e.
c.
____ 18. Identify the graph which has the following characteristics.
Graph 1 Graph 2
3.1 The Derivative and the Tangent Line Problem
115
Graph 3 Graph 4
Graph 2
Graph 3
Graph 1
Graph 4
none of the above
____ 19. Use the alternative form of the derivative to find the derivative of the function at
.
a.
b.
c.
d.
e.
____ 20. Use the alternative form of the derivative to find the derivative of the function at
.
a.
b.
c.
d.
e.
116
Chapter 3: Differentiation
____
21.
Describe the x–values at which the graph of the function
given below
is differentiable.
is differentiable at .
is differentiable everywhere except at .
is differentiable everywhere except at .
d.
is differentiable on the interval
.
e.
is differentiable on the interval
.
3.1 The Derivative and the Tangent Line Problem
117
3.1 The Derivative and the Tangent Line Problem
Answer Section
3.2 Basic Differentiation Rules and Rates of Change
119
3.2 Basic Differentiation Rules and Rates of Change
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____ 1. Find the derivative of the function.
a.
b.
c.
d.
e.
____ 2. Find the derivative of the function.
a.
b.
c.
d.
e.
____ 3. Use the rules of differentiation to find the derivative of the function .
a.
b.
c.
d.
e.
120 Chapter 3: Differentiation
____ 4. Find the derivative of the function .
a.
b.
c.
d.
e.
____ 5. Use the rules of differentiation to find the derivative of the function .
a.
b.
c.
d.
e.
____ 6. Find the derivative of the function .
a.
b.
c.
d.
e.
____ 7. Find the derivative of the function .
a.
b.
c.
d.
e.
3.2 Basic Differentiation Rules and Rates of Change
121
____ 8.
Find the slope of the graph of the function at the given value.
when
a.
b.
c.
d.
e.
____ 9. Find the slope of the graph of the function at the given value.
when
a.
b.
c.
d.
e.
____ 10. Find the slope of the graph of the function at the given value.
when
a.
b.
c.
d.
e.
122
Chapter 3: Differentiation
____
11.
Find the slope of the graph of the function at the given value.
at
a.
b.
c.
d.
e.
____
12.
Find the derivative of the function
.
a.
b.
c.
d.
e.
____
13.
Find the derivative of the function
.
a.
b.
c.
d.
e.
3.2 Basic Differentiation Rules and Rates of Change
123
____ 14.
Find the slope-intercept equation of the line tangent to the graph of
when
a.
b.
c.
d.
e.
____ 15. Determine all values of , (if any), at which the graph of the function has a
horizontal tangent.
The graph has no horizontal tangents.
____ 16. Determine all values of , (if any), at which the graph of the function has a
horizontal tangent.
The graph has no horizontal tangents.
____ 17. Determine all values of , (if any), at which the graph of the function has a
horizontal tangent.
The graph has no horizontal tangents.
124 Chapter 3: Differentiation
____ 18. Suppose the position function for a free-falling object on a certain planet is given by
. A silver coin is dropped from the top of a building that is 1370 feet
tall. Determine the velocity function for the coin.
a.
b.
c.
d.
e.
____ 19. Suppose the position function for a free-falling object on a certain planet is given by
. A silver coin is dropped from the top of a building that is 1372 feet
tall. Determine the average velocity of the coin over the time interval .
–113 ft/sec
80 ft/sec
112 ft/sec
–112 ft/sec
–80 ft/sec
____ 20. Suppose the position function for a free-falling object on a certain planet is given by
. A silver coin is dropped from the top of a building that is 1372 feet tall.
Find the instantaneous velocity of the coin when .
–96 ft/sec
–32 ft/sec
–20 ft/sec
–144 ft/sec
–48 ft/sec
____ 21. Suppose the position function for a free-falling object on a certain planet is given by
. A silver coin is dropped from the top of a building that is 1372 feet tall. Find
the time required for the coin to reach ground level. Round your answer to the three decimal places.
2.849 sec
10.273 sec
10.693 sec
3.087 sec
10.091 sec
3.2 Basic Differentiation Rules and Rates of Change
125
____ 22.
Suppose the position function for a free-falling object on a certain planet is given by
. A silver coin is dropped from the top of a building that is 1370 feet tall. Find
velocity of the coin at impact. Round your answer to the three decimal places.
–286.705 ft/sec
–138.492 ft/sec
–111.041 ft/sec
–276.984 ft/sec
–261.984 ft/sec
____ 23.
A ball is thrown straight down from the top of a 300-ft building with an initial
velocity of
–12 ft per second. The position function is
. What is the velocity
of the ball after 4 seconds?
The velocity after 4 seconds is –76 ft per second.
The velocity after 4 seconds is –116 ft per second.
The velocity after 4 seconds is –140 ft per second.
The velocity after 4 seconds is –52 ft per second.
The velocity after 4 seconds is –280 ft per second.
____ 24. A projectile is shot upwards from the surface of the earth with an initial velocity of 108
meters per second. The position function is .
What is its velocity after 7 seconds?
The velocity after 7 seconds is 181.7 meters per second.
The velocity after 7 seconds is –142.3 meters per second.
The velocity after 7 seconds is 39.4 meters per second.
The velocity after 7 seconds is 73.7 meters per second.
The velocity after 7 seconds is –176.6 meters per second.
____ 25.
The volume of a cube with sides of length s is given by
. Find the rate of
change of volume with respect to when
centimeters.
648 cm2
216 cm2
36 cm2
108 cm2
72 cm2
126 Chapter 3: Differentiation
3.2 Basic Differentiation Rules and Rates of Change
Answer Section
3.2 Basic Differentiation Rules and Rates of Change
127
128 Chapter 3: Differentiation
3.3 Product and Quotient Rules and Higher-Order Derivatives
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____
1.
Find the derivative of the algebraic function
.
a.
b.
c.
d.
e.
____
2.
Use the Product Rule to differentiate
.
a.
b.
c.
d.
e.
____
3.
Find the derivative of the function
.
a.
b.
c.
d.
e.