Chapter 2
85.
Cost and revenue functions for a certain chemical manufacturer are given in the
following figure. To maximize profit, how many tons should the company produce?
A)
7
B)
14
C)
25
D)
22
Ans: D Learning Objectives: Understand cost, marginal cost, revenue and
86.
To produce 250 items the total cost is $4600 and the marginal cost is $11. Estimate the
cost of producing 500 items.
Ans:
$7350
Learning Objectives: Understand cost, marginal cost, revenue and marginal revenue.
difficulty: medium section: 2.5
87.
To produce 250 items the total cost is $4700 and the marginal cost is $15. Which
estimate is more likely to be accurate, one for producing 251 items, or one for
producing 500 items?
Ans:
251
Learning Objectives: Understand cost, marginal cost, revenue and marginal revenue.
difficulty: easy section: 2.5
Chapter 2
88.
The world’s only manufacturer of left-handed widgets has determined that if q left-
handed widgets are manufactured and sold per year at price p, then the cost function is
8000 50Cq=+
, and the manufacturer’s revenue function is
R pq=
. The
manufacturer also knows that the demand function for left-handed widgets is
2000 25qp=−
.
A. Write the profit function
π
in terms of price p.
B. Sketch the profit function to determine what price yields the largest profit. What is
that price?
89.
The graph of a cost function is given in the following figure. Which item costs the
most to produce?
A)
The 300th item
B)
The 100th item
C)
The 200th item
Chapter 2
90.
Cost and revenue functions are graphed in the first figure. What does the second figure
show?
A)
Total profit
B)
Marginal cost
C)
Marginal revenue
marginal revenue. difficulty: medium section: 2.5
Chapter 2
91.
Given the following table, find
π(3)
.
q
0
1
2
3
4
5
6
7
()Rq
0
3
6
9
12
15
18
21
()Cq
3
5
7
8
9
11
14
18
92.
Given the following table, find MC(2).
q
0
1
2
3
4
5
6
7
()Rq
0
3
6
9
12
15
18
21
()Cq
3
5
7
8
9
11
14
18
93.
The following table gives the cost and revenue, in dollars, for different production
levels, q. What are the fixed costs?
q (units)
0
1000
2000
3000
4000
5000
()Rq
(dollars)
0
4000
8000
12,000
16,000
20,000
()Cq
(dollars)
1700
5000
8000
10,000
15,000
24,000
94.
The following table gives the cost and revenue, in dollars, for different production
levels, q. For what value of q is profit maximized?
q (units)
0
1000
2000
3000
4000
5000
()Rq
(dollars)
0
3000
6000
9000
12,000
15,000
()Cq
(dollars)
1200
4000
6000
7000
11,000
19,000
Chapter 2
95.
A newspaper headline recently read , ” Taxes are increasing at an decreasing rate”.
This says that the second derivative is negative.
A)
True
B)
False
section: 2.5
96.
Your friend Herman operates a neighborhood lemonade stand. He asks you to be his
financial advisor and wants to know how much lemonade he can make with the $3.47 he
happens to have on hand. The only information he can give you is that once last month
he spent $2 and made 19 glasses of lemonade, and another time he spent $5 and made
83 glasses of lemonade. Create a linear cost function,
()Cq
, giving the cost in dollars
of making q glasses of lemonade. How many full cups of lemonade can Herman make
with this model?
Ans:
47
difficulty: hard section: 2.5
97.
Your friend Herman operates a neighborhood lemonade stand. . Last month he spent
$2 and made 19 glasses of lemonade, and another time he spent $5 and made 83 glasses
of lemonade. You decide to use this data to create a linear cost function,
()Cq
, giving
the cost in dollars of making q glasses of lemonade. If lemonade sells for $0.15 per
glass, how many glasses must he sell to break even?
Ans:
12
section: 2.5
98.
Your friend Herman operates a neighborhood lemonade stand. He asks you to be his
financial advisor and wants to know how much lemonade he can make with the $3.12 he
happens to have on hand. The only information he can give you is that once last month
he spent $2 and made 19 glasses of lemonade, and another time he spent $5 and made
83 glasses of lemonade. You decide to use this data to create an exponential cost
function,
()Cq
, giving the cost in dollars of making q glasses of lemonade. How many
full cups of lemonade can Herman make with this model?
Ans:
32
section: 2.5
Chapter 2
Page 44
99.
At a production level of 2000 for a product, marginal revenue is $3.50 per unit and
marginal cost is $3.00 per unit. Do you expect maximum profit to occur at a
production level above or below 2000?
A)
above
B)
below
100.
The graph of
()fx
is shown in the following figure. Arrange the following values in
order from smallest to largest by placing a “1” by the smallest, a “2” by the next
smallest, and so forth.
A.
‘( )fA
B.
‘( )fB
C.
‘( )fC
D.
‘( )fD
E.
‘( )fE
1
3
4
2
5
difficulty: medium section: 2 review
difficulty: medium section: 2.5
Chapter 2
Page 45
101.
A table of values is given for
()fx
.
A. Is
‘( )fx
positive or negative?
B. Is
‘‘( )fx
positive or negative?
C. Approximate
‘(4)f
by averaging the approximations from either side.
x
3
3.5
4
4.5
5
5.5
6
()fx
17
27
34
38
41
43
44
102.
The function
()y f x=
is graphed below.
A. Is
‘(3)f
positive, negative, or zero?
B. Is
‘‘(3)f
positive, negative, or zero?
Chapter 2
Page 46
103.
The function
()y f x=
is graphed below. Which is larger,
‘(5)f
or
‘(4)f
?
104.
The function
()y f x=
is graphed below. Which is larger,
‘‘(3)f
or
‘‘(6)f
?
Chapter 2
105.
The following table gives the number of passenger cars, in millions, in the United
States, C, as a function of years, t. We have
()C f t=
. Is f ”(t) positive or negative?
t (year)
1940
1950
1960
1970
1980
C (# of cars, in millions)
27.5
40.3
61.7
89.3
121.6
Ans:
positive
difficulty: medium section: 2 review
106.
The following table gives the number of passenger cars, in millions, in the United
States, C, as a function of years, t. We have
()C f t=
. Estimate
‘(1950)f
. Use the
nearest right-hand value to make your estimate.
t (year)
1940
1950
1960
1970
1980
C (# of cars, in millions)
27.5
40.3
61.7
89.3
121.6
A)
2.14 million cars/year
B)
40.3 million cars/year
C)
44.6 million cars/year
D)
3.71 million cars/year
graphically. difficulty: medium section: 2 review
107.
Given the following data about the function, f, use estimates of
‘(3.75)f
and
‘(4.25)f
to estimate
‘‘(4)f
. Use the nearest right-hand value to make your estimate.
x
3
3.5
4
4.5
5
5.5
6
()fx
10
8
7
4
2
0
-1
Ans:
difficulty: medium section: 2 review
108.
Given the following data about the function, f, use an approximation of the tangent line
at x=4.5 to estimate
(4.75)f
.
x
3
3.5
4
4.5
5
5.5
6
()fx
10
8
7
4
2
0
-1
Ans:
3
difficulty: medium section: 2 review
Chapter 2
109.
There is a population of
()Pt
thousand bacteria in a culture at time t hours after the
beginning of an experiment. You know that
(10) 15P=
,
‘(10) 0.4P=
, and
‘‘(10) 0.008P=
. Using these values, make a prediction for
(10.5)P
.
A)
15.2
B)
15.4
C)
15.6
D)
15.8
difficulty: hard section: 2 review
110.
There is a population of
()Pt
thousand bacteria in a culture at time t hours after the
beginning of an experiment. You know that
(10) 25P=
,
‘(10) 0.3P=
, and
‘‘(10) 0.008P=
. Using these values, make a prediction for
‘(10.5)P
A)
0.302
B)
0.304
C)
0.306
D)
0.308
difficulty: hard section: 2 review
Chapter 2
111.
The first figure shows the graph of the derivative of a function. Could the second
figure be the original function?
Chapter 2
112.
Sketch a graph with the following conditions:
( ) > 0fx
and
( ) > 0fx

.
-5
-4
-3
-2
-1
1
2
3
4
5
-5 -4 -3 -2 -1 1 2 3 4 5
x
y
Chapter 2
Page 51
113.
Which point has a slope of –0.5 ?
114.
In 2007, Apple’s iTunes music store sold 2 billion songs. The number of iTunes songs
purchased (in millions) is shown on the following chart, S(t), where time is measured in
days since Apple iTunes sold 1 million songs (March 15, 2003).
Time (in days)
0
100
177
275
366
485
642
856
1077
1396
Songs Purchased
(in millions)
1
5
10
25
50
100
200
500
1000
2000
B
D F
E
AC
-2
-1
1
2
3
4
1 2 3 4 5 6 7
x
y
Chapter 2
115.
There is a function used by statisticians, called the error function, which is written
y=erf(x). Suppose you have a statistical calculator, which has a button for this function.
Playing with your calculator, you discover the following:
x
erf(x)
1
0.29793972
0.1
0.03976165
0.01
0.00398929
0
0
Using this information alone, give an estimate for erf ‘(0), accurate to 2 decimal places.
Ans:
0.40
116.
Estimate the value of
‘( )fx
for the function
( ) 10x
fx=
.
A)
1
2.303 (10)x
x−
B)
1
(10)x
x−
C)
10(10)x
D)
2.303(10)x
Ans: D Learning Objectives: Estimate the derivative of a function given
numerically. difficulty: medium section: 2 review
117.
Assume that f and g are differentiable functions defined on all of the real line. It is
possible that
0f
everywhere,
‘0f
everywhere, and
‘‘ 0f
everywhere.
A)
True
B)
False
Ans: B Learning Objectives: Understand what the derivative conveys graphically.
difficulty: medium section: 2 review
118.
Assume that f and g are differentiable functions defined on all of the real line. f can
satisfy:
‘‘ 0f
everywhere,
‘0f
everywhere, and
0f
everywhere.
A)
True
B)
False
Ans: A Learning Objectives: Understand what the second derivative conveys
graphically. difficulty: medium section: 2 review
Chapter 2
119.
Assume that f and g are differentiable functions defined on all of the real line. f and g
can satisfy:
‘( ) ‘( )f x g x
for all x and
( ) ( )f x g x
for all x.
A)
True
B)
False
difficulty: easy section: 2 review
120.
Assume that f and g are differentiable functions defined on all of the real line. If
‘( ) ‘( )f x g x=
for all x and if
00
( ) ( )f x g x=
for some
0
x
, then
( ) ( )f x g x=
for all x.
A)
True
B)
False
graphically. difficulty: easy section: 2 review
121.
Assume that f and g are differentiable functions defined on all of the real line. If
‘‘ 0f
everywhere and
‘0f
everywhere then
lim ( )
xfx
→+
= −
.
A)
True
B)
False
graphically. difficulty: medium section: 2 review
122.
Assume that f and g are differentiable functions defined on all of the real line. If
‘0f
everywhere and
0f
everywhere then
lim ( )
xfx
→+
=
.
A)
True
B)
False
graphically. difficulty: medium section: 2 review
123.
Let
2
( ) 4f x x=+
. Derive an exact formula for the derivative function
‘( )fx
by
computing algebraically the limit of a difference quotient.
difficulty: easy section: 2 review
Chapter 2
124.
Let
2
( ) 2f x x=+
. Write an equation for the line tangent to the graph of
2
( ) 2f x x=+
at the point where x = 4.
difficulty: hard section: 2 review
125.
Approximate to 3 decimal places (with a difference quotient and a calculator) the
derivative of
21x+
at x = 1.
Ans:
0.577
126.
Find
‘( )fx
algebraically by using the limit definition if
1
() 2
fx x
=
+
.
A)
2
1
( 2)x+
B)
2
1
( 2)x
−
+
C)
1
2x+
D)
1
difficulty: hard section: 2 review
127.
Using a calculator, estimate the derivative of
( ) cos( )f x x=
at x = 0. Make sure your
calculator is set to radians.
Ans:
0
section: 2 review
128.
Using a calculator, estimate the derivative of
( ) sin( )f x x=
at x =
. Make sure your
calculator is set to radians.
Ans:
section: 2 review
Chapter 2
129.
Give the difference quotient approximation to 2 decimal places of
‘(4)f
where
3
( ) 5f x x=+
.
130.
A. Give a difference quotient approximation (to one decimal place) of
‘(3)f
where
2
( ) 16f x x=+
.
B. Find the equation of the line tangent to the graph of
()fx
at the point where x = 3.
B.
131.
Find the derivative of
2
( ) 3 2 4g x x x= + −
at x = 3 algebraically.
132.
Find the derivative of
3
( ) 2m x x=
at x = 2 algebraically.