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200 Chapter 4: Applications of Differentiation
4.1 Extrema on an Interval
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____ 1. Find the value of the derivative (if it exists) of the function at the
extremum point .
a.
b.
c.
d. 0
e. 1
____ 2. Find the value of the derivative (if it exists) of at the indicated
extremum.
4.1 Extrema on an Interval
201
a. is undefined. b.
c.
d.
e.
____ 3. Find the value of the derivative (if it exists) of the function at the
extremum point .
0
does not exist
____ 4. Find all critical numbers of the function .
critical numbers:
critical numbers:
critical numbers:
critical numbers:
no critical numbers
202
Chapter 4: Applications of Differentiation
____
5.
Find any critical numbers of the function
.
a. 0
b.
c.
d.
e.
____
6.
Find all critical numbers of the function
,
.
a.
b.
c.
d.
e.
____
7.
Find all critical numbers of of the function
,
.
a.
b.
c.
d.
e.
4.1 Extrema on an Interval
203
____ 8.
Locate the absolute extrema of the function
on the closed interval
.
absolute maximum:
absolute minimum:
absolute maximum:
absolute minimum:
absolute maximum:
absolute minimum:
absolute maximum:
absolute minimum:
absolute maximum:
absolute minimum:
____ 9. Locate the absolute extrema of the function on the closed
interval .
absolute max: ; no absolute min
no absolute max or min
no absolute max; absolute min:
d.
absolute max:
; absolute min:
e.
absolute max:
; absolute min:
____ 10. Locate the absolute extrema of the function on the closed interval
.
absolute max: ; absolute min:
no absolute max; absolute min:
c.
absolute max:
; absolute min:
d.
absolute max:
; no absolute min
e.
no absolute max or min
204 Chapter 4: Applications of Differentiation
____ 11. Locate the absolute extrema of the given function on the closed interval .
a.
absolute max: (0, 0); absolute min
b.
absolute max:
; absolute min (0, 0)
c.
absolute max:
; absolute min
d.
absolute max:
; no absolute min
e.
no absolute max; absolute min:
____ 12.
Locate the absolute extrema of the function
on the closed interval
.
a.
The absolute maximum is
, and it occurs at the critical number x = 0.
The absolute minimum is
, and it occurs at the left endpoint
.
b.
The absolute maximum is
, and it occurs at either endpoint
.
The absolute minimum is 0, and it occurs at the critical number x = 0.
The absolute maximum is , and it occurs only at the left endpoint .
The absolute minimum is 0 and it occurs at the critical number x = 0.
d.
The absolute maximum is
, and it occurs at the critical number x =
0.
The absolute minimum is
, and it occurs at the right endpoint
.
e.
The absolute maximum is
, and it occurs only at the right endpoint
The absolute minimum is 0 and it occurs at the critical number x = 0.
____ 13. Locate the absolute extrema of the function on the closed interval
.
a. absolute max: (0,1) ; absolute min:
4.1 Extrema on an Interval
205
b.
absolute max:
; absolute min: (0, 1)
c.
absolute max:
; no absolute min
no absolute max; absolute min: (0, 1)
absolute max: (0,1); no absolute min
____
14.
Locate the absolute extrema of the function
on the closed interval
. Round your answers to four decimal places.
a.
left endpoint:
right endpoint:
absolute maximum
critical points:
absolute minimum
b.
left endpoint:
absolute minimum
right endpoint:
absolute maximum
c.
critical points: none
left endpoint:
absolute minimum
right endpoint:
absolute maximum
critical points:
d.
left endpoint:
absolute minimum
right endpoint:
absolute maximum
e.
critical points: none
left endpoint:
right endpoint:
absolute maximum
critical points:
absolute minimum
____
15.
Locate the absolute extrema of the function
on the closed interval
.
a.
The absolute
is
, and it occurs at the left endpoint
.
The absolute
is
, and it occurs at the right endpoint
.
b.
The absolute
is
, and it occurs at the right endpoint
.
The absolute
is
, and it occurs at the left endpoint
.
c.
The absolute
is
, and it occurs at the left endpoint
.
206
Chapter 4: Applications of Differentiation
The absolute
is
, and it occurs at the right endpoint
.
d.
The absolute
is
, and it occurs at the right endpoint
.
The absolute
is
and it occurs at the left endpoint
.
e.
The absolute
is
, and it occurs at the left endpoint
.
The absolute
is
, and it occurs at the right endpoint
.
____ 16. Sketch the graph of the function
and locate the absolute extrema of the function on the interval .
a.
left endpoint:
absolute minimum
right endpoint:
absolute maximum
b.
left endpoint:
absolute minimum
right endpoint:
absolute maximum
c.
left endpoint:
absolute minimum
right endpoint:
absolute maximum
d.
left endpoint:
absolute minimum
right endpoint:
absolute maximum
e.
left endpoint:
absolute minimum
right endpoint:
absolute maximum
____ 17.
Use a graphing utility to graph the function
and locate the absolute
extrema of the function on the interval
.
absolute minimum:
no absolute maximum
absolute minimum:
no absolute maximum
absolute minimum:
no absolute maximum
absolute minimum:
absolute maximum:
absolute maximum:
absolute minimum:
4.1 Extrema on an Interval
207
____ 18.
Use a graphing utility to graph the function
and locate the
absolute extrema of the function on the interval .
a.
left endpoint:
absolute minimum
right endpoint:
absolute maximum
b.
critical points: none
left endpoint:
absolute minimum
right endpoint:
absolute maximum
c.
critical points: none
left endpoint:
absolute minimum
right endpoint:
critical points:
absolute maximum
d.
left endpoint:
absolute maximum
right endpoint:
absolute minimum
e.
critical points: none
left endpoint:
absolute minimum
right endpoint:
absolute maximum
critical points:
____ 19.
Use a computer algebra system to graph the function
and determine all
absolute extrema on the interval .
absolute maximum:
absolute minimum:
absolute maximum:
absolute minimum:
absolute maximum:
absolute minimum:
absolute maximum:
absolute minimum:
absolute maximum:
absolute minimum:
208 Chapter 4: Applications of Differentiation
4.1 Extrema on an Interval
Answer Section
4.2 Rolle’s Theorem and the Mean Value Theorem 209
4.2 Rolle's Theorem and the Mean Value Theorem
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____ 1. Determine whether Rolle's Theorem can be applied to on the closed interval
If Rolle's Theorem can be applied, find all values of c in the open interval such that
Rolle's Theorem applies; c = 6, c = 5
Rolle's Theorem applies; c = 4, c = 6
Rolle's Theorem applies; c = 5
Rolle's Theorem applies; c = 4
Rolle's Theorem does not apply
____ 2. Determine whether Rolle's Theorem can be applied to the function on the
closed interval [–1,3]. If Rolle's Theorem can be applied, find all values of c in the open interval (–
1,3) such that
Rolle's Theorem applies; c = 1
Rolle's Theorem applies; c = 2
Rolle's Theorem applies; c = 0
Rolle's Theorem applies; c = –1
Rolle's Theorem does not apply
____ 3. Determine whether Rolle's Theorem can be applied to the function
on the closed interval If Rolle's Theorem can be applied, find all
numbers c in the open interval such that
a.
Rolle's Theorem applies;
b.
Rolle's Theorem applies;
c.
Rolle's Theorem does apply;
d.
Rolle's Theorem applies;
e. Rolle's Theorem does not apply
210 Chapter 4: Applications of Differentiation
____ 4. Determine whether Rolle's Theorem can be applied to the function
on the closed interval . If Rolle's Theorem can be applied, find all
numbers c in the open interval
such that
.
a.
Rolle's Theorem applies;
b.
Rolle's Theorem applies;
c.
Rolle's Theorem applies;
d.
Rolle's Theorem applies;
e.
Rolle's Theorem does not apply
____ 5. Determine whether Rolle's Theorem can be applied to on the closed
interval If Rolle's Theorem can be applied, find all values of c in the open interval
such that
c = 8
c = 12, c = 11
c = 11, c = 8
c = 12
Rolle's Theorem does not apply
____ 6. Determine whether Rolle's Theorem can be applied to the function on the
closed interval . If Rolle's Theorem can be applied, find all numbers c in the open
interval such that
Rolle's Theorem applies; c = 0
Rolle's Theorem applies;
Rolle's Theorem applies;
Rolle's Theorem applies;
Rolle's Theorem does not apply.
4.2 Rolle’s Theorem and the Mean Value Theorem 211
____ 7. The ordering and transportation cost C for components used in a manufacturing
process is approximated by where C is measured in thousands of dollars and x
is the order size in hundreds. According to Rolle's Theorem, the rate of change of the cost must be 0 for
some order size in the interval Find this order size. Round your answer to three decimal
places.
9.657 components
6.828 components
8.000 components
6.000 components
4.828 components
____ 8. The height of an object t seconds after it is dropped from a height of 550 meters is Find
the average velocity of the object during the first 7 seconds.
34.30 m/sec
–34.30 m/sec
–49.00 m/sec
49 m/sec
–16.00 m/sec
____ 9. The height of an object t seconds after it is dropped from a height of 250 meters is
Find the time during the first 8 seconds of fall at which the instantaneous
velocity equals the average velocity.
32 seconds
19.6 seconds
6.38 seconds
4 seconds
2.45 seconds
____ 10. Determine whether the Mean Value Theorem can be applied to the function
on the closed interval [3,9]. If the Mean Value Theorem can be applied, find all
numbers c in the open interval (3,9) such that .
MVT applies; c = 6
MVT applies; c = 7
MVT applies; c = 4
MVT applies; c = 5
MVT applies; c = 8
212 Chapter 4: Applications of Differentiation
____ 11. Determine whether the Mean Value Theorem can be applied to the function on
the closed interval [0,16]. If the Mean Value Theorem can be applied, find all numbers c
in the open interval (0,16) such that .
a.
MVT applies;
b. MVT applies; 4
c.
MVT applies;
MVT applies; 8
MVT does not apply
____ 12. Determine whether the Mean Value Theorem can be applied to the function
on the closed interval If the Mean Value Theorem can be applied,
find all numbers c in the open interval such that .
MVT applies;
MVT applies;
MVT applies;
MVT applies;
____ 13.
A company introduces a new product for which the number of units sold S is
where t is the time in months since the product was introduced. Find the
average value of
during the first year.
a.
b.
c.
d.
e.
4.2 Rolle’s Theorem and the Mean Value Theorem 213
____ 14. A company introduces a new product for which the number of units sold S is
where t is the time in months since the product was introduced. During what
month does equal the average value of during the first year?
October
July
December
April
March
____ 15. A plane begins its takeoff at 2:00 P.M. on a 2200 -mile flight. After 12.5 hours, the plane
arrives at its destination. Explain why there are at least two times during the flight when the speed
of the plane is 100 miles per hour.
By the Mean Value Theorem, there is a time when the speed of the plane must equal the
average speed of 303 mi/hr. The speed was 100 mi/hr when the plane was accelerating
to 303 mi/hr and decelerating from 303 mi/hr.
By the Mean Value Theorem, there is a time when the speed of the plane must equal the
average speed of 152 mi/hr. The speed was 100 mi/hr when the plane was accelerating
to 152 mi/hr and decelerating from 152 mi/hr.
By the Mean Value Theorem, there is a time when the speed of the plane must equal the
average speed of 88 mi/hr. The speed was 100 mi/hr when the plane was accelerating to
88 mi/hr and decelerating from 88 mi/hr.
By the Mean Value Theorem, there is a time when the speed of the plane must equal the
average speed of 117 mi/hr. The speed was 100 mi/hr when the plane was accelerating
to 117 mi/hr and decelerating from 117 mi/hr.
By the Mean Value Theorem, there is a time when the speed of the plane must equal the
average speed of 176 mi/hr. The speed was 100 mi/hr when the plane was accelerating
to 176 mi/hr and decelerating from 176 mi/hr.
____ 16. Which of the following functions passes through the point and satisfies
?
a.
b.
c.
d.
e.
214 Chapter 4: Applications of Differentiation
____ 17. Find a function f that has derivative and with graph passing through
the point
(5,6). a.
b.
c.
d.
e.
4.2 Rolle’s Theorem and the Mean Value Theorem 215
4.2 Rolle's Theorem and the Mean Value Theorem
Answer Section
216 Chapter 4: Applications of Differentiation
4.3 Increasing and Decreasing Functions and the First Derivative Test
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____ 1. Use the graph of the function given below to estimate the open
intervals on which the function is increasing or decreasing.
a.
increasing on
; decreasing on
b.
increasing on
; decreasing on
c.
increasing on
; decreasing on
d.
increasing on
; decreasing on
e.
increasing on
; decreasing on
____ 2. Identify the open intervals where the function is increasing or
decreasing.
decreasing on
increasing on
d. decreasing on ; increasing on
e.
4.3 Increasing and Decreasing Functions and the First Derivative Test
217
____
3.
Identify the open intervals where the function
is increasing or
decreasing.
a.
decreasing:
; increasing:
b.
decreasing on
c.
increasing:
; decreasing:
d.
increasing:
; decreasing:
e.
increasing:
; decreasing:
____
4.
Identify the open intervals on which the function
is
increasing or decreasing.
a.
increasing on
decreasing on
b.
decreasing on
increasing on
c.
increasing on
decreasing on
d.
increasing on
decreasing on
e.
decreasing on
increasing on
____
5.
Find the critical number of the function
.
a.
b.
c.
d.
e.
____ 6.
Find the open interval(s) on which
is increasing or decreasing.
a.
increasing on
decreasing on
b.
increasing on
decreasing on
c.
increasing on
decreasing on
d.
increasing on
decreasing on
e.
increasing on
decreasing on
218
Chapter 4: Applications of Differentiation
____
7.
Find the relative extremum of
by applying the First
Derivative Test.
relative minimum:
relative maximum:
relative minimum:
relative minimum:
relative maximum:
____ 8. For the function :
Find the critical numbers of f (if any);
Find the open intervals where the function is increasing or decreasing; and
Apply the First Derivative Test to identify all relative extrema.
Then use a graphing utility to confirm your results.
a.
(a)
x = 0 ,
(b)
increasing:
; decreasing:
(c)
relative max:
; relative min:
b.
(a)
x = 0 ,
(b)
decreasing:
; increasing:
(c)
relative min:
; relative max:
c.
(a)
x = 0 ,
(b)
increasing:
; decreasing:
(c)
relative max:
; no relative min.
d.
(a)
x = 0 ,
(b)
increasing:
; decreasing:
(c)
relative max:
; relative min:
e.
(a)
x = 0 ,
(b)
decreasing:
; increasing:
(c)
relative min:
; relative max:
4.3 Increasing and Decreasing Functions and the First Derivative Test
219
____ 9. For the function :
Find the critical numbers of f (if any);
Find the open intervals where the function is increasing or decreasing; and
Apply the First Derivative Test to identify all relative extrema.
Use a graphing utility to confirm your results.
a.
(a)
(b)
increasing:
; decreasing:
(c)
relative max:
b.
(a)
(b)
increasing:
; decreasing:
(c)
relative max:
c.
(a)
(b)
decreasing:
; increasing:
(c)
relative min:
d.
(a)
(b)
decreasing:
; increasing:
(c)
relative min:
; relative max:
e.
(a)
(b)
decreasing:
; increasing:
(c)
relative min:
____
10.
Find the open interval(s) on which the function
is increasing in the
interval Round numerical values in your answer to three decimal places.
increasing on:
increasing on:
increasing on:
increasing on:
increasing on:
