Chapter 5
49.
A shop is open from 9am-7pm. The function
()rt
Part A:
1
Part B:
20
difficulty: medium section: 5.4
graphed below gives the rate at
which customers arrive (in people/hour) at time t. Suppose that the salespeople can
serve customers at a rate of 80 people per hour.
A. People have to start waiting in line before being served at about _____ o’clock.
B. The number of people in line when the line is the longest is about _____.
50.
The Ethnic food line at the Cougar Eat can serve customers at the rate of about 30 per
hour. From 10 am until 4 pm one day, the rate R at which customers entered then line
was about
2
( ) 45 5( 3)R t t= − −
customers per hour at t hours past 10 am. About what
time did a waiting line form?
A)
11:15 am
B)
11:00 am
C)
11:30 am
D)
11:45 am
total change. difficulty: medium section: 5.4
Chapter 5
51.
The Ethnic food line at the Cougar Eat can serve customers at the rate of about 30 per
hour. From 10 am until 4 pm one day, the rate R at which customers entered then line
was about
2
( ) 45 5( 3)R t t= − −
customers per hour at t hours past 10 am. About when
was the waiting line the longest?
A)
3:00 pm
B)
2:45 pm
C)
2:30 pm
D)
2:15 pm
52.
The Ethnic food line at the Cougar Eat can serve customers at the rate of about 30 per
hour. From 10 am until 4 pm one day, the rate R at which customers entered then line
was about
2
( ) 45 5( 3)R t t= − −
customers per hour at t hours past 10 am. About how
many customers were served between 10 am and 4 pm that day? Round to the nearest
whole number.
Ans:
163
Learning Objectives: Interpret the definite integral of rate of change as total change.
difficulty: medium section: 5.4
53.
After a foreign substance is introduced into the blood, the rate at which antibodies are
made is given by
2
() 1
t
rt t
=+
thousands of antibodies per minute, where time t is
measured in minutes and
05t
. If there are no antibodies in the blood at t = 0, how
many antibodies are there after 5 minutes? Round to the nearest whole number.
Ans:
1629
Learning Objectives: Interpret the definite integral of rate of change as total change.
difficulty: medium section: 5.4
54.
The rate of growth of the net worth of a company is given by
2
2000 12t−
dollars per
year t years after its formation in 1990. How much did it increase in value between
1990 and 2005?
Ans:
$16,500
Learning Objectives: Interpret the definite integral of rate of change as total change.
difficulty: medium section: 5.4
Chapter 5
55.
A reagent is cooling in a laboratory instrument. Explain in words what
60
0( ) 25F t dt
means if
()Ft
is the temperature of the reagent in degrees Fahrenheit and t is time in
minutes.
The amount by which the reagent cooled in the first 60 minutes is 25
difficulty: medium section: 5.4
56.
The table below gives the average rate of monthly U.S. field production of crude oil for
each decade from the 1920s through the 2000s (based on estimates calculated from the
US Energy Information Administration). Use this data to estimate the total US field
production from the 1920s to the 2000s. Note that there are 120 months in a decade.
Production is given in millions of barrels per month.
Decade
1920s
1930s
1940s
1950s
1960s
1970s
1980s
1990s
2000s
Production
57
97
136
216
260
280
248
215
174
A)
188,160,000,000 barrels
B)
94,050,000,000 barrels
C)
221,352,000,000 barrels
D)
72,350,000,000 barrels
section: 5.4
57.
A large scale commercial bakery makes cream filling for snack cakes. The bakery puts
a new machine into production. The machine ramps up gradually, increasing cream
filling production at a rate of
0.21
250 x
e
pounds per day over the first week. How many
pounds of cream filling does the new machine make during the first week of
production?
A)
3987
B)
176
C)
1469
D)
4386
Chapter 5
58.
On a recently discovered planetoid, acceleration due to gravity is 13 feet/ sec2. While
building a research habitat, a hammer is dropped from the top of a tower and hits the
ground in 15 seconds. Because the hammer is dropped, its initial velocity is 0. Use a
graph of the velocity function to determine the height of the tower.
A)
1463 feet
B)
1125 feet
C)
1170 feet
D)
1755 feet
total change. difficulty: medium section: 5.4
59.
According to the European Journal of Clinical Pharmacology, the half-life of the ACE
inhibitor lisinopril increases from 12 hours in people with normal kidney function to 24
hours in people with mild kidney failure. A group of patients takes a 5-mg dose of
lisinopril at 6 am. The rate that lisinopril decreases in the bloodstream is given by
.058
0.29 t
Ae
−
=−
for normal kidney function and
.028
0.14 t
Be
−
=−
for impaired kidney
function.
After 16 hours, what is the difference of the amount of lisinopril in the bloodstream
between a person with normal kidney function and a person with impaired kidney
function?
Ans:
1.22 mg
initial value and a rate of change. difficulty: medium section: 5.4
60.
A water line made of PVC decays and eventually breaks. The rate that water flows into
the street from the break is given by the function
9
11 , 0 11
() 9, 11
tt
rt t
=
, in gallons
per hour. Use a calculator or graph to determine how many gallons of water have been
lost from the water line break after 13 hours.
A)
67.5 gallons
B)
166.5 gallons
C)
93.5 gallons
D)
112.5 gallons
Chapter 5
Page 24
61.
If the velocity function
()vt
is measured in feet per second and t gives time in seconds,
what are the units of measurement for
12
2()v t dt
?
A)
feet
B)
feet/second
C)
feet/second2
D)
seconds/foot
62.
The marginal cost function for a manufacturing company is given by
2
‘( ) 10 30C q q q= − +
dollars per box, where q is the number of boxes manufactured.
If
(0) 15C=
, find the total cost of manufacturing 10 boxes. Round to the nearest
dollar.
Ans:
$148
section: 5.5
63.
The marginal cost in dollars per unit of producing q units is given in the following table.
Estimate the total variable cost to produce 25 units.
q
0
5
10
15
20
25
()Cq
9
7
6
8
10
13
Ans:
$210.00
section: 5.5
64.
Suppose
(0) 5F=
and
‘( ) 2 14F x x=−
. Then
()Fx
has a local _______
(maximum/minimum) on
09x
at x = _____.
Part A:
minimum
Part B:
7
section: 5.5
difficulty: easy section: 5.5
Chapter 5
65.
Suppose
(0) 5F=
and
‘( ) 2 2F x x=−
. Use a calculator to calculate
(1)F
.
66.
The graph of
‘( )fx
is shown in the following figure. Given that
(0) 10f=
, find
(50)f
.
67.
The graph of f is shown in the following figure. Find
(3)F
if
‘Ff=
and
(0) 2F=
.
Chapter 5
68.
If
()rt
represents the rate at which a country’s debt is growing, then the increase in its
debt between 1980 and 1985 is given by
A)
(1985) (1980)
1985 1980
rr−
−
B)
(1985) (1980)rr−
C)
1985
1980
1()
5r t dt
D)
1985
1980 ()r t dt
E)
1985
1980
1‘( )
5r t dt
69.
The graph of
”f
is shown below. If f is increasing at x = -1, which of the following
must be true? Choose all that apply.
A)
‘(2) ‘(4)ff=
B)
‘(4) ‘( 1)ff−
C)
‘(4) 0f
D)
(5) (6)ff=
Chapter 5
70.
32
3
9π
92
x dx
−−=
.
A)
True
B)
False
analyze features of a function given information about the function’s derivative.
difficulty: medium section: 5.5
71.
If a function is concave up, then the left-hand Reimann sums are always less than the
right-hand Reimann sums with the same subdivisions, over the same intervals.
A)
True
B)
False
analyze features of a function given information about the function’s derivative.
difficulty: easy section: 5.5
72.
If
( ) 0
b
af x dx =
and f is continuous, then f must have at least one zero between a and b
(assume
ab
).
A)
True
B)
False
73.
If
‘( ) 2f t t=
is a production rate, measured in items per hour, then how many items
were produced from hour 2 to hour 6?
Ans:
32
a function given information about the function’s derivative. difficulty: easy
section: 5.5
Chapter 5
74.
A local business produces souvenirs for the tourist trade. The business has fixed costs
of $6 thousand, and it costs an additional $9.93 thousand in variable costs to produce 10
thousand souvenirs. A consultant told the business that their marginal cost function is
2
( ) 1.2 0.08C q q
=−
dollars per thousand souvenirs. What will it cost to increase their
production to 19 thousand souvenirs?
A)
$ 2,735
B)
$ 2,582
C)
$ 1,397
D)
$ 2,324
difficulty: medium section: 5.5
75.
The graph below shows a marginal cost function,
()Cq
$ per item. If the fixed cost is
$900, estimate the total cost of producing 250 items.
A)
$ 2850
B)
$ 1950
C)
$1530
D)
$2950
2
4
6
8
10
12
14
50 100 150 200
x
y
Chapter 5
76.
Water is flowing into a container at an increasing rate, as shown in the following table.
Give a lower estimate for the total number of gallons of water in the container after 30
minutes.
time (minutes)
0
5
10
15
20
25
30
rate (gal/min)
5
7
9
12
15
19
24
Ans:
335
Learning Objectives: Estimate the total change given information about a rate of
change. difficulty: easy section: 5 review
77.
The following table gives the rate, in cubic centimeters, that air is leaking from a
balloon t seconds after it is inflated. Let
()rt
be that rate. What is the meaning of
15
0()r t dt
?
t
0
5
10
15
20
()rt
14
11
9
8
7
A)
The rate in cubic centimeters per second that air is leaking out of the balloon after
15 seconds.
B)
The total number of cubic centimeters of air that have leaked out of the balloon
after 15 seconds.
C)
The number of seconds it takes for 15 cubic centimeters of air to leak out of the
balloon.
D)
The number of seconds it takes for the rate air is leaking out of the balloon to be
15 cubic centimeters per minute.
Ans: B Learning Objectives: Interpret the definite integral of rate of change as
total change. difficulty: medium section: 5 review
78.
The following table gives the rate
()rt
, in cubic centimeters, that air is leaking from a
balloon t seconds after it is inflated. Estimate
20
0()r t dt
.
t
0
5
10
15
20
()rt
14
11
9
8
7
A)
192.5 cubic centimeters
B)
49 cubic centimeters
C)
245 cubic centimeters
D)
75.5 cubic centimeters
Ans: A Learning Objectives: Estimate a definite integral from a table, graph, or
formula. difficulty: easy section: 5 review
Chapter 5
79.
Use the following figure to estimate
10
0()f x dx
(average left- and right-hand sums).
80.
Use the following figure to find the value of
c()
af x dx
Chapter 5
81.
Use the following figure to find the value of
c()
af x dx
Ans:
13
axis minus area below the x-axis. difficulty: medium section: 5 review
82.
The marginal cost function of producing a particular product is given by
‘( ) 1000 20C q q=−
, where q is quantity. If the fixed costs are $3000, what is the total
cost to produce 10 items?
Ans:
$12,000
section: 5 review
83.
The marginal cost function of producing a particular product is given by
‘( ) 1000 20C q q=−
, where q is the number of items produced. If the fixed costs are
$4000 and the items are sold for $700 each, what is the break even point?
Ans:
40
section: 5 review
Chapter 5
84.
Data for a function G is given in the following table. Estimate
‘(1)G
.
x
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
()Gx
0.00
0.01
0.03
0.04
0.06
0.10
0.15
0.30
0.50
0.72
1.00
85.
Data for a function G is given in the following table. Estimate
‘‘(0.5)G
.
x
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
()Gx
0.00
0.01
0.03
0.04
0.06
0.10
0.15
0.30
0.50
0.72
1.00
86.
Data for a function G is given in the following table. Estimate
0.9
0()G x dx
(to 3
decimal places). Average upper and lower sums.
x
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
()Gx
0.00
0.01
0.03
0.04
0.06
0.10
0.15
0.30
0.50
0.72
1.00
Chapter 5
Page 33
87.
The following figure shows the rate of growth of two cities, with
()
A
ft
being the
growth of City A after t years and
()
B
ft
being the growth of City B after t years. If
the two cities have the same population at t = 0, 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.
4
0()
A
f t dt
B.
4
0()
B
f t dt
C.
6
0()
A
f t dt
D.
6
0()
B
f t dt
Chapter 5
88.
Below is a graph of the rate r in arrivals per minute at which students line up for
breakfast. The first people arrive at 6:50 am and the line opens at 7:00 am. The line
serves students at a constant rate of 20 students per minute. Estimate the length of the
line at 7:10.
A)
240
B)
340
C)
440
D)
540
section: 5 review
Chapter 5
89.
Below is a graph of the rate r in arrivals per minute at which students line up for
breakfast. The first people arrive at 6:50 am and the line opens at 7:00 am. The line
serves students at a constant rate of 20 students per minute. Estimate the rate at which
the line is growing in length at 7:14
A)
3 people per minute
B)
4 people per minute
C)
8 people per minute
D)
24 people per minute
section: 5 review
Chapter 5
90.
Below is a graph of the rate r in arrivals per minute at which students line up for
breakfast. The first people arrive at 6:50 am and the line opens at 7:00 am. The line
serves students at a constant rate of 20 students per minute. Estimate the length of time
a person who arrives at 7:00 has to stand in line.
A)
1.5 minutes
B)
3.5 minutes
C)
7.5 minutes
D)
11.5 minutes
section: 5 review
Chapter 5
91.
Below is a graph of the rate r in arrivals per minute at which students line up for
breakfast. The first people arrive at 6:50 am and the line opens at 7:00 am. The line
serves students at a constant rate of 20 students per minute. Estimate the time at which
the line disappears.
A)
7:10 am or earlier
B)
7:20 am
C)
7:30 am
D)
7:40 am or later
above the x-axis minus area below the x-axis. difficulty: medium
section: 5 review
92.
An air conditioning unit is switched on in an
80
F room. The room is cooling off at a
rate of
( ) 2(0.8)t
rt =
degrees F per minute, with t in minutes after the unit was turned
on. Set up an appropriate integral and evaluate it with a calculator to find the
temperature of the room after 10 minutes. Round to the nearest degree.
Learning Objectives: Evaluate a definite integral using technology.
difficulty: medium section: 5 review
Chapter 5
93.
Suppose
()ft
is given by the following graph. If
0
( ) ( )
x
F x f t dt=
, what is
(1)F
?
94.
If
3
0( ) 7f x dx =
,
3
0( ) 3g x dx =
, and
5
3( ) 12f x dx =
, find the value of
3
0( ( ) ( ))f x g x dx+
.
95.
If
3
0( ) 9f x dx =
,
3
0( ) 4g x dx =
, and
5
3( ) 13f x dx =
, find the value of
3
0(2 ( ) 5 ( ))f x g x dx−
.
96.
If
3
0( ) 10f x dx =
,
3
0( ) 2g x dx =
, and
5
3( ) 15f x dx =
, find the value of
5
0()f x dx
.