Name: ___________________________________ Date: ______________
1. The average weight in pounds of American men in their sixties (in 1979) as a function of
their heights in inches is given in the following table. The formula that expresses the
weight w in terms of the height h is given by w = _____+_____h
height (h)
weight (w)
68
164
69
170
70
176
71
182
72
188
73
194
2. A furniture moving company charges a fixed amount plus a charge for each pound that
they move. A person who shipped 90 pounds of furniture was charged $420, while
someone else was charged $660 to ship 170 pounds.
A. Write a function that represents the moving cost, C, in terms of pounds, x, and fixed
cost.
B. Suppose the company changes their rates. They increase the per pound charge by $1
but cut the fixed amount they charge by half. What is the new function that represents
the new moving cost, D?
C. Will someone who ships 170 pounds pay more or less with the new rates than they
would have with the original rates?
3. Do you expect the average rate of change in the number of cases of polio in the U.S.
since 1950 to be positive or negative?
4. Values for g(x) are given in the following table. Does it appear that g(x) is concave up or
concave down?
x
g(x)
1
100
2
90
3
81
4
73
5
66
Page 1
6
60
5. Consider the following graph. Between point B and point C, the graph is: (mark all that
apply)
A)
B)
C)
D)
increasing
decreasing
concave up
concave down
6. Suppose that
is the price per unit (in dollars) of widgets which will induce
producers to supply q thousand widgets to the market, and suppose that
is the
price per unit at which consumers will buy q thousand units. Which is larger, S(150) or
S(100)?
A) S(100)
B) S(150)
7. Suppose that
is the price per unit (in dollars) of widgets which will induce
producers to supply q thousand widgets to the market, and suppose that
is the
price per unit at which consumers will buy q thousand units. If
and
, what do you predict about the future selling price of widgets (currently at
$10)?
A) It will fall.
B) It will rise.
8. A town has 2400 people initially. Find the formula for the population of the town, P, in
terms of the number of years, t, if the town grows by 65 people a year.
A)
B)
C)
D)
Page 2
9. A town has 800 people initially. Find the formula for the population of the town, P, in
terms of the number of years, t, if the town shrinks at an annual rate of 6% a year.
A)
B)
C)
D)
10. The demand and supply curves for a certain product are given in terms of price, p, by
and
.
What is the equilibrium price?
A) $3
B) $58
C) $28
D) $200
11. Solve
for t. Round to two decimal places.
12. Find a formula for the exponential function partially defined in the following table.
Round any constants to 3 decimal places.
x
0
10
5
20
10
?
15
?
20
?
13. The number of bacteria in milk grows at a rate of 10% per day once the milk has been
bottled. When milk is put in the bottles, it has an average bacteria count of 500 million
per bottle.
A. Write an equation for
B. Graph
, the number of bacteria t days after the milk was bottled.
. Label the axes and intercepts.
Page 3
14. Each of the curves in the following figure represents the balance in a bank account at time
t after a single deposit at time t = 0. Assuming continuously compounded interest,
which curve represents the smallest initial deposit?
Page 4
15. In nature, the populations of two animals, one of which preys upon the other (such as
foxes and rabbits) are observed to oscillate with time, and are found to be well
approximated by trigonometric functions. The population of foxes is shown in the
graph below. Find the amplitude.
16. If
defines the function graphed in the following figure, then A =
_____, B =_____, and C =_____.
17. At high tide, the water level is 8 feet below a certain pier. At low tide, the water level is
28 feet below the pier. Assuming sinusoidal behavior, let
= the water level relative
to the pier, at time t (in hours). At t = 0 the water is -18 feet and falling until it reaches
the first low tide at t = 3. Give a formula for
.
18. Given the equation
, find and simplify
Page 5
.
19. The graphs of
.
and
are given in the following figure. Estimate
A) –7
B) –15
C)
D) 15
20. The following figure is the graph of
, the cumulative number of customers
served in a certain store during business hours one day, as a function of the hour of the
day. About when was the store the busiest?
A)
B)
C)
D)
11am
1pm
3pm
5pm
21. Write the function
in the form
Page 6
. Round a to 3 decimal places.
22. The graph of
is shown below. 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.
B.
C.
D. slope AB
23. Given the following data about the function f, estimate
x
3.0
8.2
3.2
9.5
3.4
10.5
3.6
11.0
E. 1
F. 0
.
3.8
13.2
24. A certain function f is decreasing and concave down. In addition,
and
. Which of the following are possible values for
? Select all that apply.
A) 4
B) 5
C) 6
D) 7
Page 7
25. From the following graph, estimate
A)
B)
C)
D)
.
–4
–3
–2
–1
26. Consider the two functions shown below.
A.
B.
A) The function in graph A is the derivative of the function in graph B.
B) The function in graph B is the derivative of the function in graph A.
C) Neither function is the derivative of the other.
Page 8
27. Let
be the temperature in degrees Celsius at a height h (in meters) above the surface
of the earth. Which of the following gives the rate of change of temperature with respect
to a height at 40 meters above the surface of the earth, in degrees per meter?
A)
B)
C) h such that
D) h such that
28. To study traffic flow along a major road, the city installs a device at the edge of the road
at 4:00 am. The device counts the cars driving past, and records the total periodically.
The resulting data is plotted on a graph, with time (in hours) on the horizontal axis and
the number of cars on the vertical axis. The graph is shown below. It is a graph of the
function
= Total number of cars that have passed by after t hours. When is the
traffic flow the greatest?
A)
B)
C)
D)
At t=6 hours.
At t=3 hours.
At t=4 hours.
At t=5 hours.
Page 9
29. To study traffic flow along a major road, the city installs a device at the edge of the road
at 4:00 am. The device counts the cars driving past, and records the total periodically.
The resulting data is plotted on a graph, with time (in hours) on the horizontal axis and
the number of cars on the vertical axis. The graph is shown below. It is a graph of the
function
= Total number of cars that have passed by after t hours. Estimate
.
A)
B)
C)
D)
1000
1300
1600
1900
Page 10
30. Consider the function f sketched in the following figure. Do you expect
to
be positive, negative, or zero?
31. Consider the function f sketched in the following figure. Do you expect
positive, negative, or zero?
Page 11
to be
32. Consider the function f sketched in the following figure. Do you expect
positive, negative, or zero?
to be
33. A company graphs
, 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?
Page 12
34. Cost and revenue functions for a certain chemical manufacturer are given in the following
figure. How much is the revenue from the sale of 10 tons?
A)
B)
C)
D)
$3,200
$4,500
$3.20
$4.50
35. Cost and revenue functions for a certain chemical manufacturer are given in the following
figure. When does Revenue=Cost?
A)
B)
C)
D)
14 tons
7 tons
22 tons
27 tons
Page 13
36. 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)
B)
C)
D)
$100/ton
$320/ton
$5,200/ton
$4,500/ton
37. Cost and revenue functions for a certain chemical manufacturer are given in the following
figure. What is the current sale price?
A)
B)
C)
D)
$4,300/ton
$8,300/ton
$320/ton
$500/ton
Page 14
38. Cost and revenue functions for a certain chemical manufacturer are given in the following
figure. Should the company increase production beyond 20 tons?
A) yes
B) no
39. 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)
B)
C)
D)
7
14
25
22
Page 15
40. At a production level of 2000 for a product, marginal revenue is $3.75 per unit and
marginal cost is $4.00 per unit. Do you expect maximum profit to occur at a production
level above or below 2000?
A) below
B) above
Page 16
41. Sketch a graph with the following conditions:
and
.
42. 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)
Songs Purchased (in millions)
A) Estimate
B) Use
0
1
100
5
177
10
275
25
with the appropriate units.
to estimate
to 2 decimal places.
43. Find the first derivative of
.
A)
B)
C)
D)
Page 17
366
50
485
100
642
200
856
500
1077
1000
44. Find the first derivative of w = x2 + ax.
A) 2x + a
B) 2x
C) (2+ a)x
45. Find the first derivative of
.
A)
B)
C)
D)
46. The equation for the tangent line to the curve
_____.
47. The curve
points? Select all that apply.
A) 1
B) -1
when x = 7 is y = _____x +
has a horizontal tangent at which of the following
C)
D)
E) 0
48. A tomato is thrown from the top of a tomato cart its distance from the ground, in feet, is
modeled by the equation
where t is measured in seconds and the
initial height of the cart is
feet.
(A) At what time is the tomato at its maximum height?
(B) What is the maximum height?
(C) What is the initial velocity of the tomato (at t = 0)?
Page 18
49. Consider the function
_____ < x < _____.
. We know that
50. A power function of the form
a?
has
51. The value of a car is falling at 10% per year so that if
in dollars, its value after t years is given by
depreciating after 4 years?
A)
dollars per year
B)
is concave down when
-3/4 and
-3/16. What is
is the purchase price of the car
. How fast is the car
dollars per year
C)
dollars per year
D)
dollars per year
52. The population of Ghostport has been declining since the beginning of 1800. The
population, in thousands, is modeled by
, where t is measured in years.
At what rate was the population declining at the beginning of 2000?
53. Find the first derivative of
.
A)
B)
C)
D)
Page 19
54. Find the derivative of
.
A)
B)
C)
D)
55. Find the derivative of
.
A)
B)
C)
D)
56. Consider the function
, where a and b are constants. Find
.
57. The population of Mexico in millions is described by the formula
, where
t is the number of years after 1980. In the year 2015, the population will be increasing at
the rate of _____ million people per year. Round to 2 decimal places.
58. The population of Mexico in millions is described by the formula
, where
t is the number of years after 1980. How many years will it take for the population to
double? Round to 2 decimal places.
59. The population of Mexico in millions is described by the formula
, where
t is the number of years after 1980. How many years will it take before the population
is increasing at a rate of 8 million people per year? Round to 2 decimal places.
Page 20
60. What is the equation of the line tangent to the curve
at the point above x =
2? Leave your coefficients in fraction form, such as "a/b". They can be improper
fractions.
61. Find
for
.
for
.
A)
B)
C)
D)
62. Find
A)
B)
C)
D)
63. A linear approximation of
valid for x near 4 is given by
A) True
B) False
Page 21
64. The following table gives values for two functions f and g and their derivatives. What is
?
x
f
g
f'
g'
-1
3
1
-3
2
0
3
2
-2
3
1
1
2.5
-1.5
2
2
0
3
-1
2.5
3
1
4
1
3
65. The following table gives values for two functions f and g and their derivatives. What is
? Round to 2 decimal places.
x
f
g
f'
g'
-1
3
1
-3
2
0
3
2
-2
3
1
1
2.5
-1.5
2
2
0
3
-1
2.5
3
1
4
1
3
66. The following table gives values for two functions f and g and their derivatives. What is
?
x
f
g
f'
g'
-1
3
1
-3
2
0
3
2
-2
3
1
1
2.5
-1.5
2
2
0
3
-1
2.5
Page 22
3
1
4
1
3
67. A demand curve for a product has the equation
, where p is price and q is
quantity. What is the marginal revenue as a function of the quantity sold?
A)
B)
C)
D)
68. Find the derivative of
A)
B)
C)
D)
.
69. Find the derivative of
A)
B)
C)
D)
.
70. Compute
for
.
A)
B)
C)
D)
71. The size of an impala population is represented by the function
,
where t is time in months since the beginning of the year and
is measured in
thousands. After 4 months, the population is __________(increasing/decreasing) at a
rate of _____ thousand per month. Round to 2 decimal places.
Page 23
72. The number of hours, H, of daylight in Madrid as a function of the date is given by the
formula
, where t is the number of days since the
beginning of the year. What are the units of
?
73. The number of hours, H, of daylight in Madrid as a function of the date is given by the
formula
, where t is the number of days since the
beginning of the year. What is
? Round to 3 decimal places.
74. The first derivative of
is
.
A) True
B) False
75. The first derivative of
A) True
B) False
76. The first derivative of
A) True
B) False
77. The first derivative of
A) True
B) False
78. The first derivative of
A) True
B) False
is
is
.
.
is
.
is
Page 24
.
79. Differentiating
gives
.
A) True
B) False
80. Over which of the following intervals is the function
A)
B)
81. Given the following table, let
x
. Find
0
1
1
3
2
5
3
4
2
-1
0
1
3
4
1
-1
1
3
2
4
82. Given the following table, let
x
. Find
0
1
1
3
2
5
3
4
2
-1
0
1
3
4
1
-1
1
3
2
4
83. Find a value of a such that the function
Page 25
increasing?
.
.
has a critical point at
.
84. The following is a graph of
true?
A)
B)
C)
D)
. Which of the following statements about
changes sign at , , and
changes sign at
and
has a local maximum or minimum at
has a local maximum or minimum at
,
, and
and
85. The graph of the function
is:
A. increasing and concave up on what interval?
B. increasing and concave down on what interval?
C. decreasing and concave upon what interval?
D. decreasing and concave down on what interval?
Page 26
are
86. Estimate the inflection points of
A.
!B.
if the following graph is the graph of
!C.
List inflection points in increasing order of x-coordinates; separate each point with a
comma.
87. If
of
A) 0
for
, which of the following are local and/or global extrema
?
B)
C)
D)
E)
F)
G)
Page 27
88. If
A) 0
for
, which of the following are inflection points of
B)
C)
D)
E)
89. The quantity of a medication in the bloodstream t hours after it is ingested is given, in
mg, by
. What is the maximum quantity of the medication in the
bloodstream?
A) 110 mg
B) 300 mg
C) 815 mg
D) 150 mg
90. In the function y = 5sin (x) + 5, in the interval from 0
does the function contain a global maximum?
A)
, at which value(s) of x
and
B) 0 and
C)
only
D) 6
E) 3
91. With x people aboard, a South African airline makes a profit of (1000-4x) rands per
person for a specific flight. How many people would the airline prefer to have on
board?
Page 28
?
92. Given the following table of production quantities with their corresponding marginal
revenue and marginal cost, estimate the production level that maximizes profit.
q
MR
MC
0
100
40
10
100
75
20
100
100
30
100
120
40
100
150
50
100
190
93. If the total revenue and total cost (in dollars) are given by
?
What quantity of gadgets, to the nearest whole number should be produced to maximize
profit? What is the maximum profit?
94. A factory produces a product that sells for $11. They currently produce 2400 items per
month, at an average cost of $5 per item. The marginal cost at this level is $4. Assume
that the factory can sell all the items that it produces.
A. What is the profit at this production level?
B. Would increasing production increase or decrease average cost?
Page 29
95. The graph of a cost function is given in the following figure. Estimate the value of q at
which average cost is minimized.
Page 30
96. Which average cost function corresponds to the total cost function shown in the
following figure?
A)
B)
C)
Page 31
D)
97. The average cost per item to produce q items is given by
A. What is the total cost,
.
, of producing q items?
B. What is the marginal cost, MC, of producing q items?
C. At what production level does marginal cost equal average cost?
98. Given the cost function
and the demand function
, find the value of q (to the nearest whole number) for which revenue is
a maximum.
99. The elasticity for a good is E=1.6. What is the effect on demand of a 4% price increase?
A) 6.4% decrease
B) 6.4% increase
C) 2.5% decrease
D) 2.5% increase
100. The demand curve for a product is
.
A. Find the elasticity of demand (to three decimal places) at a price of p=14.
B. Is demand elastic or inelastic at this price?
101. An amusement park finds that when it charges $13 for an all-day pass, attendance is
about 3300 per day. When it charges $18 , attendance is about 2900 per day. Is daily
revenue higher at a price of $13 or a price of $18?
Page 32
102. The demand for doughnuts at a bakery is given by
number of doughnuts sold at a price of p dollars each.
, where q is the
A. Find the elasticity of demand to two decimal places if the price is $0.60.
B. Will revenue be increased by raising or lowering the price?
103. Raising the average price of an entree at a restaurant from $13 to $14 reduces the number
of customers per day from 400 to 375.
A. What is the elasticity of demand to two decimal places for entrees at a price of $13?
B. Would raising the price from $13 to $14 increase or decrease the profit?
104. A disease is released into a small town. The number of people infected is modeled by
the equation
. What is the population at t=0 of this disease?
105. A disease is released into a town. The number of people, in thousands infected is
modeled by the equation
. How many people are infected after
hours?
106. A disease is released into a town. The number of people infected each day is modeled
by the equation
. Estimate when
at this time.
Page 33
Estimate the value of n
107. A biologist found that the number of Drosophila fruit flies, N(t), assumes the following
growth pattern if the food source is limited:
.
A. How many fruit flies were there in the beginning (to the nearest fly)?
B. At what time was the population increasing most rapidly (to the nearest day)?
C. At what rate does the number of fruit flies increase after 5 days (to the nearest fly
per day)?
108. The following table gives the number of students who have joined a new school club t
days after it was formed.
A. Estimate the value of t where concavity changes in this function.
B. Use your answer from part (a) to estimate the maximum membership in the club.
t (days)
P (number of students)
1
4
2
9
3
18
4
36
5
70
6
135
7
183
8
218
9
243
10
254
109. The following table shows the total sales, in thousands, since a new DVD was released.
A. Estimate the point of diminishing returns.
B. Using your answer from part (A), predict the total possible sales for the DVD.
week
sales
0
0
1
9
2
24
3
60
4
141
Page 34
5
294
6
506
7
691
8
807
110. The dose response curve in the following figure is given by
, where R is percent
of maximum response and x is the dose of the drug in mg. The inflection point is at
(10,40) and
. Would
be greater or less than 9 for values of x less than
10?
111. The drug concentration curve for a drug after t hours is given by
ng/ml.
The minimum effective concentration is 10 ng/ml. Is the drug effective at t = 2 hours?
112. The peak concentration of 15 ng/ml for a drug occurs 2.5 hours after a 6 mg dose is
administered. Sketch a graph that represents the concentration, C, as a function of time,
t.
Page 35
113. The following three equations are graphed in the figure. Which graph corresponds to
equation C?
A.
!!B.
!!C.
114. Does the maximum value of the surge function
increased and b is held constant?
115. For
increase or decrease when a is
, determine a critical point.
116. The average cost per item to produce q items is given by
0. What is the marginal cost, MC, of producing q goods?
A)
for q>
B)
C)
D)
117. The average cost per item to produce q items is given by
for q>
0. At what point q (to two decimal places) does marginal cost equal average cost?
Page 36
118. Rank the following products from 1 to 4 according to their elasticity, with 1 being the
highest.
A. high performance automobile
B. cellular phone
C. laundry detergent
D. movie theater tickets
119. The number of students in a high school who have heard a rumor about new graduation
requirements (as a function of time) is most likely to be modeled by which of the
following types of functions?
A) logistic
B) periodic
C) surge
D) linear
E) exponential
Page 37
Answer Key - prac_final
1A. –244
1B. 6
2. A.
B.
C. more
negative
concave up
A, C
B
A
A
C
C
–2.01
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13A. A.
13B. B.
14.
15.
16A.
16B.
16C.
IV
300
4
1/5
2
17.
18.
19. A
20. B
21.
Page 38
22A.
22B.
22C.
22D.
22E.
22F.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
A. 6
B. 3
C. 2
D. 4
E. 5
F. 1
5
B
B
A
B
C
A
negative
positive
positive
10
A
A, D
B
C
A
D
A
Page 39
41.
42A. 3.1348
42B.
43.
44.
45.
46A.
46B.
47.
48A.
48B.
2639.50 million songs
D
A
B
–14
53
C, D, E
48C.
49A.
49B.
50.
51.
52.
53.
54.
55.
56.
57.
0
0.625
3
B
0.036788 thousand people per year or 36.8 people per year.
C
B
C
4.65
Page 40
58.
59.
60.
61.
62.
63.
64.
65.
66.
67.
68.
69.
70.
71A.
71B.
72.
73.
74.
75.
76.
77.
78.
79.
80.
81.
82.
83.
84.
85.
25.67
55.06
A
B
True
5
–0.33
5
B
B
C
A
decreasing
0.91
hours/day
0.039 hours/day
True
True
True
True
False
True
A
1
16
–1
A, D
A.
B.
C.
D.
86. A. (2.5,-1)
B. (1,1), (4,-3)
87.
88.
89.
90.
91.
C. (0,0), (2,0), (6,0)
A, C, E
D
A
C
125
Page 41
92.
93.
94A.
94B.
95.
96.
97.
20
1350, $1572.50
A. $14,400
B. decrease
20
B
A.
B.
C. 10
98. 1714
99. A
100. A. 1.289
B. elastic
101. 18
102. A. 1.11
B. lowering
103. A. 0.81
104.
105.
106.
107.
B. increase
1
2444 people
520 days, 100 people
A. 13
B. 9
C. 27
108. A. 6
B. 270
109. A. 506,000
B. 1,012,000
110. less
111. yes
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112.
113. II
114. increase
115.
116. D
117. 0.70
118. A. 1
B. 3
C. 4
D. 2
119. A
Page 43