Chapter 2
44.
Every day the Undergraduate Office of Admissions receives inquiries from eager high
school students (e.g. “Please send me an application”,etc.) They keep a running count
of the number of inquiries received each day, along with the total number received until
that point. Below is a table of weekly figures from about the end of August to about the
end of October of a recent year. One of these columns can be interpreted as a rate of
change. Which one is it?
A)
“Week of”
B)
“Inquiries That Week”
C)
“Total for Year”
difficulty: medium section: 2.3
45.
The cost of extracting T tons of ore from a copper mine is
()fT
dollars. What are the
units for
‘( )fT
?
A)
dollars/ton
B)
tons/dollar
difficulty: easy section: 2.3
46.
The cost of extracting T tons of ore from a copper mine is
()fT
dollars. Would you
expect
‘( )fT
to be positive or negative?
A)
positive
B)
negative
difficulty: easy section: 2.3
Chapter 2
47.
The following table shows the number of oranges sold in one month,
()fp
, against the
price per bag, p (in cents). Find an approximation for
‘(750)f
. Use the nearest right–
hand value to make your estimate.
750
800
850
900
950
()fp
50,000
48,000
44,000
37,000
29,000
48.
The following table gives the wind chill factor (
F
) as a function of the wind speed
(miles/hour) when the air temperature is 20
F
. What is the derivative of wind chill
with respect to wind speed when the air temperature is 20
F
and the wind speed is 20
miles per hour? Use the nearest right-hand value to make your estimate.
Wind speed (mph)
5
10
15
20
25
Wind chill factor (
F
)
16
3
-5
-10
-15
49.
Let
()th
be the temperature in degrees Celsius at a height h (in meters) above the
surface of the earth. Which of the following gives the temperature in degrees Celsius at
a height of 1000 meters?
A)
(1000)t
B)
‘(1000)t
C)
h such that
( ) 1000th =
D)
h such that
‘( ) 1000th=
Chapter 2
50.
Let
()th
be the temperature in degrees Celsius at a height h (in meters) above the
surface of the earth. Which of the following gives the height, in meters, at which the
rate of change of temperature with respect to height is 25 degrees per meter?
A)
(25)t
B)
‘(25)t
C)
h such that
( ) 25th =
D)
h such that
‘( ) 25th=
51.
Let
()Lr
be the amount of lumber, in board-feet, produced from a tree of radius r
(measured in inches). Which of the following gives the number of board-feet obtained
from a tree of radius 8 inches?
A)
(8)L
B)
‘(8)L
C)
r such that
( ) 8Lr =
D)
r such that
‘( ) 8Lr=
52.
Let
()Lr
be the amount of lumber, in board-feet, produced from a tree of radius r
(measured in inches). Which of the following gives the radius (in inches) of a tree that
produces 150 board-feet of lumber ?
A)
(150)L
B)
‘(150)L
C)
r such that
( ) 150Lr=
D)
r such that
‘( ) 150Lr =
53.
Let
()Ct
represent the dollar amount charged per hour by a computer consultant to a
client when they sign a contract t hours of work. The consultant gives a discount to the
client if the contract is increased by 10 hours. Estimate the amount charged per hour
when the client orders 130 hours of work.
Chapter 2
54.
Let
()Ct
represent the dollar amount charged per hour by a computer consultant to a
client when they sign a contract t hours of work. The consultant gives a discount to the
client if the contract is increased by 10 hours. Interpret the following statements.
A)
(10) 100C=
.
B)
5
(10) 10
C−
=
.
Part A:
The consultant charges $100 for 10 hours of work.
Part B:
The cost per hour will go down by $5 for the next 10 hours added to the
contract.
55.
The noise level, N, in decibels, of a rock concert is given by
()N f d=
, where d is the
distance in meters from the concert speakers. Which of the following gives the rate of
change, in decibels per meter, of noise 600 meters away from the speakers?
A)
(600)f
B)
‘(600)f
C)
d such that
( ) 600fd=
D)
d such that
‘( ) 600fd=
56.
The noise level, N, in decibels, of a rock concert is given by
()N f d=
, where d is the
distance in meters from the concert speakers. Which of the following gives the
distance, in meters, away from the speakers at which the noise is 80 decibels?
A)
(80)f
B)
‘(80)f
C)
d such that
( ) 80fd=
D)
d such that
‘( ) 80fd=
Chapter 2
57.
The population of a certain town is given by the function
()Pt
where t is the number of
years since the town was incorporated. If
‘( )Pt
is constant for t >185, what will
(200)P
be if
(185) 46,000P=
and
‘(185) 225P=
?
Ans:
49,375
difficulty: easy section: 2.3
58.
The following table gives the wind chill factor (
F
) as a function of the wind speed
(miles/hour) when the air temperature is 20
F
. What are the units for the derivative
of wind chill with respect to wind speed when the air temperature is 20
F
?
Wind speed (mph)
5
10
15
20
25
Wind chill factor (
F
)
16
3
-5
-10
-15
A)
mph/
F
B)
F
/mph
difficulty: easy section: 2.3
59.
The graph of
()fx
is shown in the following figure. Give an estimate for
‘(4)f
A)
–10
B)
10
C)
–20
D)
20
graphically. difficulty: medium section: 2.2
Chapter 2
Page 26
60.
The graph of
()fx
is shown in the following figure. Is
‘‘(4.5)f
positive, negative, or
zero?
A)
zero
B)
positive
C)
negative
61.
Suppose the graph of f is in the figure below. Is f (B) positive, negative, or zero?
Ans:
positive
difficulty: easy section: 1.1
graphically. difficulty: easy section: 2.4
Page 27
62.
Suppose the graph of f is in the figure below. Is
(A)f
positive, negative, or zero?
63.
Suppose the graph of f is in the figure below. Is
(A)f
positive, negative, or zero?
Chapter 2
64.
A function f satisfies the following conditions:
(2) 10f=
,
(8) 7f=
,
‘(2) 0f
,
‘(8) 0f
, and
‘‘( ) 0fx
for
28x
. Which of the following are possible values
for
(5)f
? Select all that apply.
A)
4
B)
8
C)
14
graphically. difficulty: medium section: 2.4
65.
Suppose a function is given by the following table of values. Estimate the
instantaneous rate of change of f at x=1.7, and use this estimate to find the equation for
the tangent line to f at x=1.7. The line is y = _____x+_____. Use the nearest right-
hand value to make your estimate.
x
1.1
1.3
1.5
1.7
1.9
2.1
()fx
12
15
21
23
24
25
Part A:
5
Part B:
14.5
difficulty: medium section: 2.2
66.
Suppose a function is given by the following table of values. Is
”f
most likely
positive or negative at x=1.9?
x
1.1
1.3
1.5
1.7
1.9
2.1
()fx
12
15
21
23
24
25
A)
positive
B)
negative
graphically. difficulty: easy section: 2.4
Chapter 2
67.
Could the function on the right be the second derivative of the function on the left?
Ans:
no
Learning Objectives: Understand what the second derivative conveys graphically.
difficulty: easy section: 2.4
68.
The cost of mining a ton of coal is rising faster every year. Suppose
()Ct
is the cost of
mining a ton of coal at time t. Which of the following must be positive? Select all that
apply.
A)
()Ct
B)
‘( )Ct
C)
‘‘( )Ct
69.
The cost of mining a ton of coal is rising faster every year. Suppose
()Ct
is the cost of
mining a ton of coal at time t. Which of the following must be increasing? Select all
that apply.
A)
()Ct
B)
‘( )Ct
C)
‘‘( )Ct
Chapter 2
70.
The cost of mining a ton of coal is rising faster every year. Suppose
()Ct
is the cost of
mining a ton of coal at time t. Which of the following must be concave up? Select all
that apply.
A)
()Ct
B)
‘( )Ct
C)
‘‘( )Ct
71.
Consider the following graph. In region III,
‘( )fx
is ________ (positive/negative) and
‘‘( )fx
is ________ (positive/negative).
Chapter 2
72.
Consider the function f sketched in the following figure. Do you expect
(4) (2)
2
ff−
to be positive, negative, or zero?
73.
Consider the function f sketched in the following figure. Do you expect
‘(2)f
to be
positive, negative, or zero?
Chapter 2
74.
Consider the function f sketched in the following figure. Do you expect
‘‘(4)f
to be
positive, negative, or zero?
Chapter 2
Page 33
75.
Using the graph of
()fx
, at x=D is
2
2
dy
dx
positive, negative or zero?
76.
Write the Leibniz notation for the first and second derivatives of the given function and
include units.
“The amount of paint, G, in gallons, is a function of time m, in minutes”
Learning Objectives: Understand what the second derivative conveys graphically.
difficulty: medium section: 2.4
77.
Let
()St
represent the number of students enrolled in school in the year t. If
enrollment is decreasing steadily, then
/ds dt
_____0 and
22
/d s dt
_____0. (Enter
“<“,”>”, or “=”)
Part A:
<
Part B:
=
B C E
A
D
-5
-4
-3
-2
-1
1
2
3
4
5
-5 -4 -3 -2 -1 1 2 3 4 5
x
y
Ans:
negative
Learning Objectives: Understand what the second derivative conveys graphically.
difficulty: easy section: 2.4
Chapter 2
78.
A driver obeys the speed limit as she travels past different towns in the order A, B, C.
In town A, the speed limit is 35 mph. In town B, the speed limit is 60 mph, and in town
C the speed limit is 55 mph. It always takes her two minutes to reach the new speed
limit when she passes by a new town. If
()St
represents the driver’s position at time t,
then is
‘‘( )St
for the first two minutes she is passing town B positive or negative?
79.
A company graphs
‘( )Ct
, the derivative of the number of pints of ice cream sold over
the past ten years. Out of t=1,2,4,8, and 10, in what year was C ”(t) greatest?
Chapter 2
80.
Cost and revenue functions for a certain chemical manufacturer are given in the
following figure. How much does it cost to produce 10 tons?
A)
$4,500
B)
$3,200
C)
$4.50
D)
$3.20
marginal revenue. difficulty: easy section: 2.5
Chapter 2
81.
Cost and revenue functions for a certain chemical manufacturer are given in the
following figure. When does Revenue=Cost?
A)
14 tons
B)
7 tons
C)
22 tons
D)
27 tons
marginal revenue. difficulty: easy section: 2.5
Chapter 2
82.
Cost and revenue functions for a certain chemical manufacturer are given in the
following figure. Marginal revenue at 20 tons is about how much?
A)
$100/ton
B)
$320/ton
C)
$5,200/ton
D)
$4,500/ton
Chapter 2
83.
Cost and revenue functions for a certain chemical manufacturer are given in the
following figure. What is the current sale price?
A)
$4,300/ton
B)
$8,300/ton
C)
$320/ton
D)
$500/ton
marginal revenue. difficulty: easy section: 2.5
84.
Cost and revenue functions for a certain chemical manufacturer are given in the
following figure. Should the company increase production beyond 25 tons?
A)
no
B)
yes
marginal revenue. difficulty: medium section: 2.5