BARUCH COLLEGE
MATH 2003
SPRING 2006
MANUAL FOR THE UNIFORM FINAL EXAMINATION
The final examination for Math 2003 will consist of two parts.
Part I:
This part will consist of 25 questions similar to the
questions that appear in Problem Section A and Section M (Matrices).
No calculator will be allowed on this part.
Part II:
This part will consist of 10 questions.
The graphing calculator is allowed on this part.
At least 6 of the questions will be similar to questions
that appear in Problem Sections C and CM (Calculator allowed).
The remaining 4 questions will be similar to questions
that appear in all Sections: A, M, C and CM.
(Note that all 10 questions might come from Section C.)
There may be a few new problem types on the exam that are not similar
to the problems in this review. If such problems appear, they will
be similar to problems that you have seen during the semester.
GRADING: Each question will be worth 3 points.
Anyone who gets 34 or 35 questions correct will be assigned a grade of 100.
No points are subtracted for wrong answers.
CONTENTS OF THIS MANUAL:
Page 2 shows the sample questions that correspond to each section of the current text.
(When a section has been covered in class, the list indicates the problems that can
be used in studying for the exam that includes that section during the semester.)
Problem Section A. (Problems for which a calculator may not be used.)
Problem Section M. (Problems on matrices for which a calculator may not be used.)
Problem Section C. (Problems for which a calculator should be used.)
Problem Section CM. (Problems on matrices for which a calculator can be used.)
Answers to the problems.
1
Math 2003
Textbook Sections Corresponding to Sample Uniform Final Exam Questions
Textbook: Precalculus and Elements of Calculus, Warren B. Gordon, Walter O. Wang, Baruch College,
CUNY
Section
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
2.1
2.2
2.3
2.4
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.10
3.11
Problems
9, 26, 35, 76, 112, 113, 119, 153
16, 22, 31, 33, 61 (also from 0.5), 64, 69, 73, 86, 139, 152
71, 142, 150, 164, C8
20, 27, 28, 101, 102, 103, 143, 146, C5, C15
29, 67, 144, 145, 154, 155, C9
90, 104, 105, 108, 147, 165, C12, C16, C17, C23, C25
30, 32, 34, 59, 63, 70, 74, 75, 100, 106, 107, 109, 110, 111, 130, 132, 133, 136, 149, 168, C2,
C10, C28
62, 156, 157, C26, C27, C32
M12, M14, M18
M3, M4, M5, M6, M9, M10, M15, M16, M21, M25, CM2, CM4
M7, M8, M13, M17, M20, M22, CM3, CM5
M1, M2, M11, M19, M23, M24, CM1, CM6
8, 36, 37, 60, 139, 158, C6, C18, C24
10, 12, 46, 47, 88, 89, 93, 95, 120, 121, 151, C11
1, 2, 3, 4, 17, 38, 39, 40, 42, 45, 51, 72, 78, 84, 85, 114, 115, 116, 117, 124, 125, 137, 138,
141, 159, 160, 166, C13, C20, C21, C35
5, 21, 23, 41, 43, 44, 68, 79, 80, 81, 82, 83, 99, 118, 134, 135, 140, 141, 163, C1, C7
6, 7, 13, 48, 50, 52, 58, 87, 92, 128, 129, 131, 162, C34
14, 53, 54, 94, 122, 161, C18, C19
11, 25, 49, 65, 66, 91, 123, 127, 148, C3, C14, C29, C30, C33
15, 18, 55, 57, 96, 126
19, 24, 56, 77, 86, 97, 98
C31
2
SECTION A
1. Find
a) 3
b) 4
c) 0
d) 1
2. Given
a) 0
find
b) 1
c) -1
if it exists.
d) it does not exist
3. At x = 1 the function
e) 2x
is not continuous because
a) f(1) does not exist
b)
d) f(1) = 1
e)
does not exist
4. In order to make the function
a) f(1) = 0
e) ! (does not exist)
b) f(1) = -1
c)
and f(1) exist but
continuous at x = 1 we must define
c) f(1) = 2
d) f(1) = 1
e) f(1) = x
5. Find the horizontal asymptote for the graph of
a) y = 0
b) y = 1
c) y = 3
d) y = 2
e) y = -1
6. Find the derivative of
a)
b)
c)
d)
e)
7. Find the derivative of
a) 3x2 - 6x - 9
b) x2 + 4x - 11
c) x2 - 6x - 9
d) 2x - 5
3
e) 3x2 - 14x + 9
8. If f(x) = -x2 + x, which of the following will calculate the derivative of f(x)?
a)
b)
c)
d)
e)
9. Which of the following is the equation of the line that has x-intercept (3,0) and y-intercept (0,-4)?
a) 4x + 3y = 12
b) 3x - 4y = 0
c) 3x - y = 13
d) 3x - 4y = 25
e) 4x - 3y = 12
10. Find the equation of the line tangent to the graph of y = 2x2 - 2x +3 at the point (1, 3).
a) y = 2x - 2 b) y = 4x2 - 6x + 5
c) y = 2x + 1 d) 2x + y = 5 e) y = 2x + 2
11. The position equation for a moving body is
for t " 0.
Find the time when the velocity is 0.
a) 1
b) 3/2
c) 0
d) 5/6
e) 2/3
12. Find all of the points at which the graph of f (x) = x4 - 4x + 5 has horizontal tangent lines.
a) (2,0) only
b) (1,0) only c) (1,2) and (-1,10)
13. The derivative of
a)
a)
e) (1,2) only
is
b)
14. The derivative of
d) (1,0) and (2,13)
c)
d)
e) 2(2x - 6)(x2 + x)
is
b) -9x2 (1 - 3x3)
c)
d)
4
e) -9x2
15. Find dy/dx by using implicit differentiation if xy + y = 5x2.
a)
b)
c)
d) 10x - 1
e) 10x - y
16. Given f(x) = 3x - 2 and g(x) = 5x2 + 1, find the composite function f(g(x))
a) -10x2 + 3x - 2
b) 15x3 - 10x2 + 3x - 2
17.
c) 15x2 + 1
d) 45x2 - 60x + 21
e) 15x3 - 2
has a hole, that is a missing point where the limit exists. What are the
coordinates of the hole?
a) (5, -4/3)
b) (5, -19)
c) (9/2, 5)
d) (-5, -11/19) e) (-4/3, 9/2)
18. What is the slope of the line tangent to xy3 = 8 at the point (1, 2)?
a) -2/3
b) 1
c) 12
d) 8
e) -8
19. At what rate is the area of a circle increasing if the radius is increasing at the rate of 10 feet per minute
when the radius is 6 feet long? (A = ! r2)
a) 2! ft2/min
b) 60 ft2/min
c) 120! ft2/min
d) 12! ft2/min
20. A factory has a profit function
what production level will the factory break even?
a) 10
b) 20
c) 30
21. The graph of
22. If
a) -12
for a production level of x units. For
d) 6000
e) 0
has the horizontal asymptote:
a) y = 1/2
b) y = 1/4
c) y = 0
e) It does not have a horizontal asymptote
d) y = 4
and
b) -9
then
=
c) -1/2
d) 2/3
e) 3
c) – 4
d)
e)
23. Evaluate the limit:
a) 0
b) 4
e) 120 ft2/min
5
24. The area of a square is increasing at the rate of 7 square inches per minute. At what rate is the length of
one of the sides increasing at the moment when the side has length 4?
a) 7/4
b) 8
c) ½
d) 7/16
e) 7/8
25. Let the profit function for a particular item be P(x) = -10x2 + 160x - 100. Use the marginal profit
function to approximate the increase in profit when production is increased from 5 to 6.
a) 30
b) 40
c) 55
d) 60
e) 70
26. The equation of the straight line through the point (-6, 4) with x-intercept 3 is:
a)
b)
c)
d)
e)
27. Describe the solution(s) to the equation
a) One rational solution
d) Two complex solutions
b) Two rational solutions
e) Cannot be determined
28. The axis of symmetry of
a) -4
b) y = 5
:
c) Two irrational solutions
is:
c) y = 2
d) x = -4
e) x = 2
29. Find the center and radius of the circle given by (x + 3)2 + (y – 2)2 = 6
a) Center (3, -2); radius 36
d) Center (-3, 2); radius 36
b) Center (-3, 2); radius
e) Center (9, 4); radius 6
c) Center (3, -2); radius
30. Find a possible formula for the rational function with the following properties:
Zeros at x = -3 and x = 2
Vertical asymptotes at x = -5 and x = -7
Horizontal asymptote at y = -2
a)
b)
d)
e)
c)
6
31. For the function
A)
and
B)
and
C)
and
D)
, which of the following are possibilities for p(t) and q(t)?
and
a) A and B only
b) A and C only
32. The zeros of
c) B, C and D only
d) C only
e) A, B, C, D
are
a) x = 5/3 only
b) x = 5/3 and x = 2
e) x = -5/3 and x = 2
c) x = 2 only
d) x = -5/3 only
33. Find f(2, -3) for f (x, y) = 5x2 - 4xy + y3
a) 17
b) (-3, 2)
c) -31
d) (3, -2)
e) 23
34. The graph to the right could be the graph of the function:
a)
b)
c)
d)
e)
35. The distance from the origin to the midpoint of the line segment joining P(5, 5) and Q(1, 3) is
a) 25
b)
c) 20
36. Given f (x) = 7x2, find
a) 14x + 7"x
b) 14x + ("x)2
d) 5
e)
.
c) 7"x
7
d) 1
e) 14x + 7("x)2
37. Find
9a) 0
.
b) 5x + 2
c) 5x
38. What is
a) !
e) undefined
c) 2
d) 11
e) 3
c) !
d) 5/4 e) -5/4
?
b) 0
39. What is
a) 5
d) !
?
b) 0
40. The intervals on which
a) (-!, !)
d) (-2, 2)
(all real x)
(-2 < x < 2)
41. Find
a) 1
is continuous are
b) (-4, 4)
(-4 < x < 4)
e) (-!, -2) # (-2, 2) # (2, !)
c) (-!, -4) # (4, 8)
(all real x $ -2, 2)
(x < -4 or 4 < x < 8)
.
b) 1/4
c) -3/8
d) !
e) -!
42. Find the intervals on which f (x) = x2 - 2x + 1 is continuous.
a) (0, 2)
d) (0, !)
(0 < x < 2)
(x > 0)
b) (-!, !)
(all real x)
c) (0, 1)
(0 < x < 1)
e) (-4, 4) # (4, !) (-4 < x < 4 or x > 4)
43.
has as vertical asymptotes
a) x = -1 and x = 2
b) x = 1 only
44.
has as a horizontal asymptote
a) y = 1/3
b) y = ½
c) y = 3
c) x = 0 only
d) y = 0
d) x = -1 only
e) y = 3/2
8
e) x = 1 and x = -2
45. Find
a) 1
.
b) !
c) -!
d) 2/3
e) 5/6
46. Find the derivative of
a)
b)
c)
47. The slope of the line tangent to
a) 16
b) 3
d)
e)
at (4, 1) is
c) 1/4
d) -1/4
e) 1
c) 3
d) -3
e) -1
48. Find f %(-1) for
a) 5
b) -5
49. What is the average rate of change of f (x) = x2 - 4x + 1 on the interval [-1, 1]?
a) -4
b) 4
c) -6
d) -2
e) 2
50. At which of the following points is the slope of the tangent to
a) (-1, -2)
b) (1, ½)
51. The function
a) x = 0
c) (1, 0)
d) (-3, ½)
equal to ½?
e) (3, ½)
is differentiable for all values of x except
b) 0 < x < 1
c) -1 < x < 0
d) x < 0
e) x < -1
52. If g(x) = (x + 1)(x2 - 3x + 1), then the slope of the line tangent to the graph of g(x) at the point where
x = -2 is
a) 0
b) 2
c) -3
d) -11
e) 18
53. The derivative of f (x) = (x - 5x2)7 is
a) 7(x - 5x2)6
b) 7(1 - 10x)6
c) 7(1 - 10x)(x - 5x2)6
9
d) 70x(x - 5x2)6
e) (1 - 10x)6
54. The derivative of
a)
is
b)
c)
d)
e)
55. Use implicit differentiation to find dy/dx when x = 1 and y = -2 for x3 + 2xy + y2 = 1.
a) -1/2
b) -1
c) 0
d) 1/6
e) undefined
56. The radius r of a circle is increasing at the rate of 3 inches per minute. What is the rate of change in the
area when r = 20 inches?
a) 6! inch/min
b) 400! in2/min
c) 40! in2/min
d) 60! in2/min
e) 120! in2/min
57. If xy = 3, find dy/dt when x = 6 and dx/dt = 4.
a) 1/3
*b) -1/3
c) 0
d) ½
e) undefined
58. The derivative of f (x) = x(4x - 7)4 is
a) (4x - 7)4
b) 4x(4x - 7)3
c) 16x(4x - 7)3
d) (20x - 7)(4x - 7)3
e) 17x(4x - 7)3
59. Suppose u(x) is a polynomial of degree 6. What is the minimum number of zeros u(x) could have?
a) 0
b) 1
c) 2
d) 4
60. Which is the best choice for describing
a)
e) 6
?
,
b)
is approximately equal to
c)
is the best approximation of the tangent line to f(x)
d)
gives the slope of the secant line connecting a fixed point and a variable point
e) At the fixed point
,
61. The domain of
a)
as x gets very small
measures the rate of change of
with respect to x.
is
b)
c) (-7, 7)
10
d) [-7, 7]
e)
62. The following tables give values of functions f, g, u and v. Determine which of these functions could be
linear.
x
f(x)
x
g(x)
x
u(x)
x
v(x)
50
0.10
1
5
0
50
-3
5
55
0.11
2
4
100
100
-1
1
60
0.12
3
5
200
150
0
-1
65
0.13
4
4
300
200
3
-7
70
0.14
5
5
a) u and g only
b) f and v only
63. Given
c) g only
d) f only
e) f, u, and v only
the zeros of f are:
a) x = -5/2, 3
b) x = -3, 5/2
64. Let
c) x = -5/2, 0, 3
d) x = -3, 0, 5/2
c)
d)
e) x = -3, -5/2, 0
. Then
a)
b)
e)
65 and 66. The height in feet above the ground of a ball thrown upwards from the top of a building is given
by
s = -16t2 + 160t + 200, where t is the time in seconds
65. What is the average velocity of the ball between 2 and 4 seconds?
a) 96
b) 64
c) 32
d) 128
e) 515
66. What is the instantaneous velocity of the ball at t = 2 seconds?
a) 96
b) 64
c) 32
d) 128
e) 515
67. The y-axis is tangent to a circle centered at (5, 2). Find the equation of the circle
a) x2 - 10x + y2 - 4y + 29 = 5
d) (x + 5)2 + (y + 2)2 = 5
b) x2 - 10x + y2 - 4y + 29 = 25
e) (x + 5)2 + (y + 2)2 = 25
11
c) x2 - 4x + y2 - 10y + 29 = 4
68. Evaluate
a) 0
`
b)
c)
d) 20
e) DNE
69. Suppose that f (x, y) = 30 - (y2 + 4x). Find f (3, -2).
a) 29
b) 38
c) 16
d) 13
e) 14
70. Give a possible formula for the polynomial graphed on the right:
a)
b)
c)
d)
e)
71 and 72. For problems 71 and 72, use the function
71. The coordinates of the y-intercept of the graph of f(x) are:
a) (0 2)
b) (-2, 0)
c) (1/3, 0)
d) (0, -1)
e) the graph has no y-intercept
72. In order to make f(x) continuous on its entire domain, one must:
a) redefine f(-2) = 0 b) redefine f(0) = -1 c) redefine f(0) = 2
e) f(x) cannot be made continuous by redefining one point
73. Find two functions f and g such that
a)
d)
and
b)
and
d) redefine f(1/3) = 0
where
and
e)
c)
and
12
.
and
74. Find a possible formula for the rational function with the following properties:
Zeros at x = -4 and x = 3
Vertical asymptote at x = -2
Horizontal asymptote at y = 4
“Hole” (removable discontinuity) at x = -1
a)
b)
c)
d)
e)
75. Suppose m(x) is a polynomial of degree 5. What is the maximum number of zeros m(x) could have?
a) 0
b) 1
c) 4
d) 5
e) 6
76. If the point (-4, 7) lies on the graph of the equation
a) 1/2
b) -3
c) 5
d) 3/2
77. For x3 + y2 = 9, find
at the point (2, -1) if
a) -3
d) 8
b) 1/6
c) 13/4
b) x > 0
= -2
is discontinuous when
c) x = 2
79. The function
a) t = 1 only
e) 9
e) -12
78. The function
a) x = 0
, then the value of b must be
d) x > 2
e) never
has vertical asymptotes
b) t = -1 only c) t = 3 only
d) t = 1 and t = 3 only
e) t = -1, t = 1 and t = 3
80. Find ALL of the vertical asymptotes for
a) x = ± 4
b) x = ± 3
c) y = 1
d) y = 9/16
13
e) x = ±3, ±4
81. Find the horizontal asymptote of
a) y = 5
b) y = 7
c) y = 0
d) y = 3/8
e) y = 5/7
82. Find the horizontal asymptote of
a) y = 0
b) y = 7/5
c) y = -7/3
d) y = 7
e) there is no horizontal asymptote
b) 3/2
c) !
d) -7/2
e) 0
b) 1
c) -1
d) 0
e) !
83. Find
a) 1/8
84. Find
a) 2
85. Find
a) 1
b) -6
c) !
86. Given
d) x
e) -3
Determine
a)
b)
c)
d)
e)
87. Find the derivative of
a)
b)
.
c)
d)
e)
88. Find the slope of the line tangent to the graph of f (x) = 2x2/3 - 2x + 9 at the point (8, 1).
a) 11
b) -11/6
c) 1
d) -4/3
e) -8
89. Determine the x coordinate of the point(s), if any, at which the function
a horizontal tangent.
a) x = 0
b) x = -3, 5
c) x = 3, -5
d) No such points
e) x = -15, 1
14
has
90. Determine the average cost of producing 50 items if the total cost is given by the function
where x is the number of units produced.
a) 95,000
b) 9,500
c) 1,900
d) 190
e) Not enough information is given to determine the average cost.
91. Find the average rate of change of the function f (x) = -x3 + 7x + 1 on the closed interval
[1, 4].
a) -14
b) 7
c) -6
d) -42
e) 4
92. Find the slope of the line tangent to the graph of the function f (x) = x2(2x + 5) at x = 1.
a) 4
b) 16
c) 14
d) 10
e) 0
93. What is the EQUATION of the line tangent to y = 4x3 at (2, 32)?
a) y = 12x2 - 16
b) y = 12x2
c) y = 48
d) y = 12x + 8
94. Find the derivative of
a)
95. Given
a) (0,5)
.
b)
b) (-7,2)
e) y = 48x - 64
c)
d)
e)
. At what point will the curve have zero slope?
c) (2, 0)
d) (2, -7)
e) (0,2)
96. If 5x2 - 3xy + y = 2, find dy/dx by using implicit differentiation and evaluate dy/dx at the point
(0, 2).
a) -2
b) -5/4
c) 0
d) 6
e) Does not exist.
97. The side of a square is increasing at the rate of 2 feet per minute. Find the rate at which the area is
increasing when the side is 7 feet long.
a) 28 ft2/min b) 14 ft2/min c) 49 ft2/min d) 28! ft2/min e) 49! ft2/min
98. If y = 3x3 - 6x + 4, find dx/dt when x = 2 and dy/dt = 60.
a) ½
b) 2
99. The graph of the function
c) 4
d) 30
e) 1800
has as its vertical asymptote(s)
a) x = -1 only b) x = 1 only c) x = 1 and x = -1
d) No vertical asymptotes
15
e) y = 1 only
100. Which of the following represents the graph of an odd function?
a)
b)
d)
e)
c)
101. A cosmetics company has analyzed that their profit depends on the percent of their sales that they spend
on advertising, according to the formula Profit
where x is the percent of their
sales spent on advertising. What percent of sales spent on advertising will produce the largest Profit?
a) 45%
b) 25%
c) 15%
d) 20%
e) 33 1/3%
102. The graph of f (x) = -x2 - 2x + 3 resembles
a)
b)
c)
d)
103. The function whose equation is
a) (2, 5)
b) (-2, 5)
c) (2, 9)
e)
has a graph which is a parabola whose vertex is:
d) (-2, -9)
e) (2, 1))
104. For a production level of x units of a commodity, the cost function in dollars is C = 200x + 4100. The
demand equation is p = 300 - 0.05x. What price p will maximize the profit?
a) $100
b) $250
c) $900
d) $1500
e) $6000
105. For a production level of x units of a commodity, the demand equation is p = 300 - 0.05x. What quantity
x will maximize revenue?
a) 3000
b) 6000
c) 0
d) 4000
e) 2500
106. Which of the following define functions which are even: A:
B:
a) A only
,
C:
b) B only
c) A and B
d) A and B and C
16
e) A and C only
107. Which of the following define functions which are odd: A:
B:
a) A only
C:
b) B only
c) A and B
d) A and B and C
108. For the demand function define by
for
a) 76
for
, the market equilibrium price is:
b) 70
c) 72
d) 28
109. The zeros of
a) 1 and -1
e)
e) A and C only
and the supply function
24
are:
b) -1 only
c) 1 only
d) 5 and 10
110. The horizontal asymptote of
a) x = 2
,
b) y = 2
c) x = -1
e) there are none
is:
d) y = -1
e) there is none
111. For the composite function defined by
, we could decompose the function into
. A possible decomposition for h(x) would be:
a)
and
c)
and
e)
and
b)
and
d)
and
112. The equation of the line parallel to
a)
b)
d)
e)
passing through the point (-5, 4) is:
c)
113. The equation of the line perpendicular to
a)
b)
d)
e)
passing through the point (-5, 4) is:
c)
114. Find
a) 0
b) 1
c) 3
d) 2
e) 3/2
17
115. Find
a) 4
.
b) 3.5
c) 2
d) 1
e) 0
116. Describe the intervals on which
a) All real x
b) All real x $ 3
is continuous.
c) All real x $ 2
d) All real x $ 1
117. Determine the values of x for which
a) All real x
b) All real x $ 4
e) All x in the interval (0,4)
is continuous.
c) All real x $ 1
d) All real x $ 1, 4
e) All x in the interval (0,5)
118. Find
a) 0
b) Does not exist (undefined)
c) 4
d) -10
e) -2
119. When the price is $40.00, a company can sell 300 chairs but for every 1.00 increase in price, the
demand drops by 3 chairs. Find the equation for the demand, x as a function of price, p.
a)
b)
d)
e)
c)
120. The equation of the line tangent to the curve y = 3x2 - 2x + 7 at the point (1, 8) is
a) y = 4x2
b) y = 6x
c) y = 4x + 4
d) y = 0
e) y = -4x + 8
121. If
then g%(x) is
a)
b)
c)
d)
122. Find the slope of the line tangent to
a) 3
b) 1/8
c) 6
e) 2x + x2
at (2, 4).
d) 2
e) 3/8
123. Find the values of x for which the rate of change with respect to x of y = 10x3 - 120x2 + 450x -130 is 0.
a) x = ± 300
b) x = 3, 5
c) x = 100
18
d) x = ± 3
e) x = 100, 500
124. Find
a) 8
b) -8
c) 0
d) Does not exist (undefined)
e) !
125. The function defined by
a) is continuous for real x
c) is discontinuous at x = 5 only
e) is discontinuous for 4 < x < 5
b) is discontinuous at x = 4 only
d) is discontinuous at x = 4 and x = 5
126. The slope of the line tangent to the curve x2 + y2 = 24 at the point (-4, 3) is
a) 4/3
b) -4/3
c) 3/4
127. If the demand equation is given by
a) x = 9
b) x = 2
c) x = 6
d) -3/4
e) 12
, then the marginal revenue is 0 when
d) x = 3
e) x = 5
128. Find the derivative of
a)
b)
c)
d)
e)
129. For what value(s) of x is the derivative of
a) x = 0 only
b) x = 2 only
130. For
a) x = 0
c) x = -2 only
and
b) x = 1
c) x = -1
equal to zero or undefined?
d) x = -2, 2 only
e) x = -2, 0, 2
Find all x such that
d)
x = 0 or x = 1
e) x = 0 or x = -1 or x = 1
131. Find the value of the derivative of y = 5x(2x - 4)4 at x = 3.
a) 1040
b) 560
c) 320
132. For the rational function
a) x = 2 and x = -2
b) y = 2 and y = -2
d) 160
e) 960
the horizontal asymptote is:
c) y = 4
19
d) x = 4
e) There is none
133. For the rational function defined by
a) x = 2 and x = -2
the vertical asymptotes are:
b) y = 2 and y = -2
c) y = 4
d) x = 4
e) There are none
134. Find
a) 0
b) -1/3
c) 2
d)
e)
135. Find
a) 0
b) -1/3
c) 2
d)
136. If
a) 4
e)
, find f(1, -6, 5).
b)
c) 9
d) 25
e) 7
b) 0
c) 5/8
d) !
e) Does not exist.
137. Find
a) 7/16
138. For what values of x is the function defined by
a) 1
b) -2 and 1
not continuous?
c) -5, -2 and 1
139. Evaluate
d) -1 and 2
for f (x) = x2 + 5x.
a) 7x + "x
b) 3x + "x
c) 5 + "x
d) 2x + 5 + "x
140. Find
a) 1/4
b) 4
c) 0
d) 1
e) !
b) !
c) 7
d) 3
e) Does not exist
141. Find
a) 0
e) -1, 1 and 2
20
e) 10x + "x
142. ABC Inc. buys a new truck for $32,000 The truck will have a scrap value of $8,000 after 8 years. If
they use the straight line method for depreciating the truck, find the value of the truck after 2 years.
a) 20,000
b) 22,000
c)
24,000
d) 26,000
e) 28,000
143. Find the maximum profit for the profit function defined by P(x) = -2x2 + 10x - 3.
a) 10
b) 19/2
c) (5 + !19)/2
d) 7/4
e) 67/8
144. Find the center of a circle given by the equation
a) (0, 0)
b) (-1, 1)
c) (1, -1)
d) (2, -2)
e) (-2, 2)
145. Find the radius of a circle given by the equation
a) 2
b) 4
c)
d)
0
e) 3
146. The vertex of the parabola
a) (4, 1)
b) (6, 37)
is:
c) (-6, 181)
147. For the functions defined by
which could be the supply function?
d) (2, -11)
for
a) f(x)
b) g(x)
c) either f(x) or g(x)
e) insufficient information to answer the question
e) (-2, 37)
and g(x) =
for
,
d) neither f(x) nor g(x)
148. The demand function and cost function for x units of a product are defined by
and
C = 0.65x + 400. Find the marginal profit when x = 100.
a) $2.35 per unit
b) $4.58 per unit
c) $193.50 per unit
149. The function defined by
d) $187.35 per unit
e) $3.65 per unit
moves (translates) the graph of
a) 3 units up and 8 units to the right
c) 3 units to the right and 8 units down
e) none of the other answers
b) 3 units down and 8 units to the left
d) 3 units to the left and 8 units up
150. If the Revenue function is defined by R(x) = 25x and the Cost function is defined by C(x) = 20x + 500,
the break even quantity will be:
a) 100
b) 40
c) 50
d) 1000
21
e) 200
151. Determine the x coordinate of the point(s), if any, at which the graph of
has a horizontal tangent.
a) x = -1, 9
b) None
152. Given
a) 84
c) x = 0, -1
and
b) 121
d) x = 0, 1, 9
e) x = 11
, find g(f(2))
c) 124
d) 127
e) 137
153. The population, P, of a town of 30,000 people grows at a constant rate of 2,000 people every year. Find
a formula for P as a function of time t (the number of years from the year the population was 30,000).
a) P = 30,000 + 2,000t b) P = 2,000 + 30,000t c) P = 30,000 - 2,000t
e) Not enough information is given to determine the function.
d) P = 2,000t - 30,000
154. The x coordinate of the center of the circle given by the equation
a) 5
b) -5
c) 3
d) -3
e) 4
155. The radius of the circle given by the equation
a) -5
b) 3
c) 8
is
is
d) 16
e) 4
156. For a linear regression between the variables, x = number of practice problems done the day before the
final exam and y = score on the final exam, you would expect the value of the correlation coefficient, r, to be
close to:
a) -2
b) -1
c) 0
d)
1
e) 2
157. For a linear regression between the variables, x = number of hours watching television the night before
the final exam and y = score on the final exam, you would expect the value of the correlation coefficient, r , to
be close to:
a) -2
b) -1
c) 0
d)
158. For the function defined by
a) 0
b) 1
c) 1/2
d)
1
e) 2
, find
e) 4
22
159. The function defined by
a) is continuous at x = 2
b) has a removable discontinuity at x = 2
c) has a non-removable discontinuity at x = 2 d) is differentiable at x = 2
e) has an asymptote at x = 2
160. The function defined by
a) is continuous at x = 2
b) has a removable discontinuity at x = 2
c) has a non-removable discontinuity at x = 2 d) is differentiable at x = 2
e) has an asymptote at x = 2
161. Find the slope of the line tangent to
a)
b)
c) 45
at x = 3.
d)
e)
162. Find the value of the derivative of
a) 120
b) 24
c) 16
at x = 5.
d) 40
e) 136
163. x = 2 is a vertical asymptote of y = f(x),
whose graph appears on the right.
Find
a) 0.4
d) !
b) 2
d) - !
c) 0
164. The EZ car rental company charges a fixed amount per day plus an amount per mile for renting a car.
For two different one-day trips Ed rented a car from EZ. He paid $70.00 for a 100 mile trip and $120.00 for a
350 mile trip. Determine the function for a one-day trip that EZ car rental uses if x represents the number of
miles driven.
a) f(x) = 0.2x + 50
d) f(x) = 0.02x + 70
b) f(x) = 0.02x + 50
e) f(x) = 0.2x + 190
c) f(x) = 0.02x + 70
23
165. Select the correct mathematical formulation of the following problem.
A rectangular area must be enclosed as shown. The sides labeled x
cost $20 per foot. The sides labeled y cost $10 per foot. If at most
$300 can be spent, what should x and y be to produce the largest area?
a) Maximize
b) Maximize
c) Maximize
d) Maximize
e) Maximize
xy if 40x + 20y = 300
xy if 20x + 10y = 300
xy if 2x + 2y = 300
2x + 2y if xy = 300
2x + 2y if 200xy = 300
x
y
y
x
Choice
166. Given
find the values of the limits shown.
(a)
(b)
(c)
(d)
(e)
-1
2
3
7
7 and 4
4
2
0
4
7 and 4
undefined
2
undefined
undefined
7 and 4
167. A woman 5 feet tall is walking away from a 12 ft. tall flagpole at a rate of 2 ft/sec. (See diagram) How
quickly is the length of her shadow changing?
a) decreasing at a rate of 6/5 ft/sec
b) increasing at a rate of 6/5 ft/sec
c) increasing at a rate of 2 ft/sec
d) decreasing at a rate of 10/7 ft/sec
e) increasing at a rate of 10/7 ft/sec
168. Select the correct mathematical formulation of the following problem.
A farmer wishes to construct 4 adjacent fields alongside a river as shown. Each field is x feet wide and y feet
long. No fence is required along the river, so each field is fenced along 3 sides. The total area enclosed by all
4 fields combined is to be 800 square feet. What is the least amount of fence required?
(a) Minimize xy if 4x + 5y = 800
(b) Minimize xy if 4x + 5y = 200
(c) Minimize 4x + 5y if xy = 800
(d) Minimize 4x + 5y if xy = 200
(e) Minimize 4(x + y) if xy = 800
24
SECTION M (MATRICES-NO CALCULATOR ALLOWED)
M1. If a linear system of equations does not have a single unique solution, then we may conclude that
a) it must have no solution.
b) it has exactly two solutions.
c) it must have an infinite number of solutions.
d) it either has no solutions or it has an infinite number of solutions.
e) it either has no solutions or it has exactly two solutions.
M2. The augmented matrix shown represents the solution of a system of equations in
the variables (x, y, z, w). What is the solution?
a)
b)
c)
d)
e) No solution
M3. When writing the system of equations shown in the form AX = B,
what should A be?
a)
b)
M4. Given A =
a)
b)
c)
and B =
c)
d)
3x - y
=8
-5x + 7y + 2z = 4
32y + z = 7
e)
, find A2B.
d)
e)
M5. If the dimensions of matrix A are 4 x 3, and the dimensions of matrix B are 3 x 1, what must the
dimensions of AB be?
a) 3 x 3
b) 4 x 1
c) 4 x 3
d) 3 x 1
e) 1 x 1
25
M6. If
a)
is represented by AX = B and A-1 =
b)
c)
d)
e)
M7. For the system of equations shown, what is the reduced row
echelon form that results from Gauss-Jordan row reduction?
a)
b)
d)
e)
M8. If A =
, find A-1
a)
b)
M9. If A =
, find A2.
a)
b)
, find the solution X
x - 2y + 2z + 7w = 3
x - 2y + 3z + 9w = 3
3x - 6y + 7z + 23w =8
c)
c)
d)
c)
e) Does not exist
d)
e)
M10. If A and B are matrices with inverses A-1 and B-1, respectively, then (AB)-1 equals
a) A-1B-1
b) (BA)-1
c) B-1A-1
d) BA
e) AB
26
M11. The system of linear equations on the right
has as its solution
a) only (1, 1, 1)
b) only (-1, 0, 0)
M12. If A =
, B=
a)
x + y - 3z = -1
y- z=0
-x + 2y
=1
c) only (3, 2, 2)
and C =
d) no solution
e) infinitely many solutions
, find AB - 5C.
b)
c) does not exist because AB does not exist.
d) does not exist because AB and 5C have different dimensions.
e) does not exist because all three matrices must have the same dimensions for AB - 5C to exist.
M13. If A =
, find the inverse of A, A-1.
a)
b)
c)
d)
e) Does not exist.
M14. Perform the following matrix operation:
a)
b)
.
c)
d)
27
e)
M15. Perform the following matrix operation:
a)
b)
M16. If A =
, find A2.
a)
b)
M17. If A =
, find the inverse, A-1.
a)
c)
d)
c)
e) Undefined
d)
b)
c)
e) Undefined
d)
e) Inverse does not exist
M18. Perform the matrix operations: 3
a)
b)
c)
M19. Solve the following system of equations:
d)
e)
x- y=8
2x + y = 7
x - 3y = 15
a) (10, 2)
b) (8, -3, -3) c) There are infinitely many solutions
d) No solution exists e) (5, -3)
M20. Find the reduced row echelon form that results
from using Gauss-Jordan row reduction.
a)
b)
4x + 2y = 6
6x + 3y = 9
c)
d)
28
e)
M21. Find the following product:
a)
b)
c)
d)
M22. The reduced row echelon form that results from
Gauss-Jordan row reduction of the system of
linear equations shown appears below.
e)
2x - 3y + z = -3
x - 2y - z = -9
3x - 4y + 3z = 3
Find all solutions to the system of equations.
a) (21, 15, 0)
b) (5z + 21, 3z + 15, z)
c) (1, 3, 4)
d) (21 - 5z, 15 - 3z, z) e) (-4, 0, 5)
M23. Solve the system of equations. x
+ z=3
x + y - 2z = 10
y - 3z = 8
a) (3, 7, 1)
b) (3 - z, 7 + 3z, 0)
c) (3 - z, 7 + 3z, z)
M24. Find all solutions to the system of equations.
a) (5 - w, -2 + w, -1 + w, w)
b) (5, -2, -1, 0)
d) (3 - z, 7 + 3z, 1)
x
+w=5
y-z
= -1
2x + y + w = 8
c) (1, -1, -1, 0)
M25. What is the (2, 3)-entry in the product
a) 0
b) -2
c) 17
e) No solution exists
d) (4, -1, 0, 1)
?
d) -5
e)-3
29
e) No solution exists
SECTION C
C1. Find
a) !
b) -!
c) -1
d) 2
e) 0
C2. Given
, write f(x) as a product of linear factors
a)
b)
d)
e)
c)
C3. The profit from manufacturing x items is given by P(x) = -0.5x2 + 46x - 10. Find the actual change in
profit (not an estimate) if the production is increased from 20 to 21 items.
a) $25.00
b) $25.50
c) $26.00
d) $710.00
e) $735.50
C4. Find the domain of the function
a)
b)
d)
e)
C5. Let
c)
. To the nearest hundredth the x-intercepts of the graph are
a) (0.63, -.02) only
d) (0.63, 0) only
b) (0.60, 0) and (0.67, 1)
e) (0.60, 0) and (0.67, 0)
c) (0.60, 0) only
C6. Find
a) 0
b) Does not exist (undefined)
c) x + 3
d)
b) Does not exist (undefined)
c) !
d) -!
e)
C7. Find
a) 0
30
e) 1
C8. A business purchases a piece of equipment for $15,250. The company can depreciate the equipment’s
value over its useful life of 10 years, at which time it will be sold as salvage, with an estimated value of $350.
Find the formula for a linear function that gives the value of equipment during its 10 years.
a) V = 15,250 - 850t
d) V = -1490t + 15,250
b) V = 17,750 - 11,050t
e) V = 1490t - 15,250
c) V = -15,250t + 1490
C9. Which of the following tables represent both the y-values on the circle
on EITHER SIDE of the point (-13, 34) and the corresponding y-values on the line tangent to the circle at
(-13, 34), using
?
a)
b)
d)
e)
c)
C10. The function
a) nowhere
b) x = 1 only
has vertical asymptotes at:
c) x = -3 only
d) x = 1 and x = 2
e) x = 2 and x = -3
C11. For what values of x is the slope of the line tangent to y = x3 + 11x2 - x + 17 equal to 0?
a) None
b)
C12. Let
c) 11/17
d) 0
e)
be a company’s total cost, in millions of dollars, for producing x thousand
cheeseburgers. Let
be the revenue, in millions of dollars the company receives for selling x
thousand cheeseburgers (x > 0). What is the lowest level of production, x, that allows the company to break
even?
a) 6000 cheeseburgers
d) 1000 cheeseburgers
b) 3000 cheeseburgers
e) 2000 cheeseburgers
c) 4000 cheeseburgers
31
C13. Find
a) 0
b) -12
c) -6
d) -!
e) undefined (does not exist)
C14. The cost in dollars of producing x units of a product is given by
for x " 0.
Determine the value of x when the marginal cost is 0.
a) 0.8
b) 1.4
c) 3.5
d) 4.2
e) 5.3
C15. Let
. There exists a value c such that f(c) = 10.
The digit in the tenths place of c is:
a) 1
b) 2
c) 4
d) 7
e) 9
C16 and C17. The following are a pair of supply and demand equations:
C16. Which of the following graphs BEST represents supply and demand as a function of the number
of units produced?
a)
d)
b)
c)
e)
C17. Find the point of market equilibrium for the demand and supply equations above.
a) (227.881, 12.565)
d) (1.697, 72.881)
b) (-16.315, 336.182)
e) (12.565, 227.881)
32
c) (-1.964, 73.857)
C18. Using the graphing calculator, graph
and the tangent line at (0, 0) in the window
xmin = -10 xmax = 10 xscl = 1 ymin = -2 ymax = 2 yscl = 1
a)
b)
d)
e)
c)
C19. Determine the intervals on which the function
a)
d)
is differentiable.
b)
e)
c)
C20. Find
a) Does not exist (undefined)
b) 0
c) 0.06
d) 1/72
e) 1/18
C21. Find
a) 5
b) -21/13
c) 4/3
d) 0
e) !
C22. The demand function for a product is
where x is the number of units of the
product sold and p is the price in dollars. Find the value of x for which the marginal revenue is one dollar per
unit.
a) 0
b) 53.93
c) 130.25
d) 131.3
33
e) No solution
C23. You want to choose one long-distance telephone company from the following options:
Company A charges $0.37 per minute
Company B charges $13.95 per month plus $0.22 per minute
Company C charges a fixed rate of $50 per month
Find the x-values (number of minutes you talk in one month to the nearest minute) for which Company B is
the cheapest
a)
b)
c)
d)
e)
C24. Evaluate
a)
b)
c)
d)
e) does not exist (undefined)
C25. An experimental power plant has a net power output
where t is the
number of days after start-up. For what approximate values of t will the net power output be negative?
a) t < -108.8
C26 and C27
b) t < 20.9
c) t < 108.8
d) 20.9 < t < 87.9
Use the data given:
x
-2
0
3
10
y
-5
-1
6
15
C26. Find the linear regression equation (round coefficients to 3 decimals):
a)
b)
c)
d)
e)
C27. Find the quadratic regression equation (round coefficients to 3 decimals):
a)
b)
c)
d)
e)
34
e) t > 400
C28. Find the zero of
a) -1.4198
that is between -2 and -1 (round to 4 decimals).
b) -1.4200
c) -1.4202
d) -1.4204
e) -1.4206
C29 and C30. The weekly profit that results from selling x units of a commodity weekly is given by
P = 50x - 0.003x2 - 5000 dollars.
C29.
Find the actual weekly change in profit that results from increasing production from 5000 units
weekly to 5015 units weekly.
a) $20.00
C30.
b) $300.00
c) $299.33
d) $298.00
e) $19.96
Use the marginal profit function to estimate the change in profit that results from increasing
production from 5000 units weekly to 5015 units weekly.
a) $20.00
b) $300.00
c) $299.33
C31. For the equation
decimals.
a) 1.18
d) $298.00
, use
b) 1.19
c) 1.20
d) 1.21
e) $19.96
and Newton’s Method to find
to 2
e) 1.22
C32. Find the linear regression equation for the data given (round coefficients to 3 decimals):
x
-2
1
3
7
y
8
5
1
-4
a)
b)
c)
d)
C33.
e)
The total profit that a company has obtained since it started business is given by
where t is the number of months since the company started
business. Find the average rate of change in profit with respect to time between 3 months after the
company started and 7 months after the company started.
a) 5997.26
C34.
b) -42,506.6
c) 2998.57
d) 1499.28
Find the equation of the straight line tangent to
a) None exists (false) b)
d) y = -11x + 24
e) 1499.31
at the point (3, -9).
c) y = -22x + 57
e) y = -11
35
C35.
Evaluate
a) 2.081
to three decimals
b) 2.091
c) 2.101
d) 2.111
e) 2.121
Matrix Exercises begin on the next page
36
SECTION CM (MATRICES-CALCULATOR ALLOWED)
CM1. Solve the following system of equations:
a) (0, 0, 1)
b) (x + 7z, y + 5z, z)
CM2. Given
c) no solution d) (x + 7z, y + 5z, 1) e) (x - 7z, y - 5z, 1)
,
and
a)
b)
d)
e)
, find
c)
CM3. Solve the following system of equations:
a) (-19/7, -9/7, 18/7) b) (1, 1, 1)
c) (-19k + 7, -9k + 7, 18k + 7) d) (-19, -9, 18)
37
e) no solution
CM4. and CM5. The following represents the system of equations AX = B
CM4. In order to solve the system, it is necessary to evaluate:
a)
b)
c)
d)
e)
CM5. The solution to the system above is:
a) (1, 1, 1)
b) (3, 9, -15)
c) (6, 5, -3)
d) (18, 15, -9)
e) no solution
CM6. Solve the following system of equations:
a) (2 + 45/28, k, 4/7, -1/28)
d) (2k + 45/28, k, 4/7, -1/28)
b) (2k + 45/28, 4/7, k, -1/28)
e) no solution
38
c) (45/28, -2, 4/7, -1/28)
ANSWERS
1. a
2. d
3. c
4. c
5. e
6. c
7. a
8. b
9. e
10. c
11. e
12. e
13. b
14. c
15. c
16. c
17. b
18. a
19. c
20. b
21. c
22. d
23. e
24. e
25. d
26. e
27. c
28. e
29. b
30. e
31. b
32. a
33. a
34. d
35. d
36. a
37. c
38. d
39. d
40. e
41. e
42. b
43. d
44. c
45. e
46. a
47. c
48. d
49. a
50. c
51. a
52. e
53. c
54. b
55. a
56. e
57. b
58.d
59. a
60. e
61. d
62. e
63. c
64. b
65. b
66. a
67. b
68. b
69. e
70. a
71. d
72. e
73. e
74. b
75. d
76. e
77. e
78. c
79. d
80. a
81. e
82. b
83. e
84. a
85. b
86. d
87. b
88. d
89. b
90. c
91. a
92. b
93. e
94. d
95. d
96. d
97. a
98. b
99. d
100. b
101. c
102. e
103. e
104. b
105. a
106. d
107. b
108. a
109. e
110. b
111. c
112. a
113. d
114. e
115. c
116. d
117. d
118. e
119. b
120. c
121. b
122. e
123. b
124. c
125. a
126. a
127. c
128. b
129. e
130. d
131. a
132. c
133. a
134. d
135. a
136. a
137. c
138. d
139. d
140. a
141. d
142. d
143. b
144. b
145. c
146. d
147. a
148. a
149. c
150. a
151. d
152. d
153. a
154. b
155. e
156. d
157. b
158. d
159. c
160. b
161. c
162. e
163. d
164. a
165. a
166. d
167. e
168. d
39
M1. d
M2. d
M3. a
M4. d
M5. b
M6. d
M7. e
M8. d
M9. b
M10. c
M11. e
M12. a
M13. d
M14. a
M15. a
M16. b
M17. c
M18. d
M19. d
M20. c
M21. c
M22. d
M23. e
M24. a
M25. d
C1. b
C2. c
C3. b
C4. e
C5. e
C6. e
C7. c
C8. d
C9. e
C10. d
C11. b
C12. b
C13. b
C14. b
C15. c
C16. b
C17. e
C18. b
C19. d
C20. e
C21. b
C22. c
C23. c
C24. b
C25. d
C26. c
C27. b
C28. e
C29. c
C30. b
C31. c
C32. a
C33. e
C34. d
C35. a
CM1. c
CM2. b
CM3. a
CM4. d
CM5. c
CM6. d