Sample Examination I
Section I Part A
Direction" Solve each of the following problems ,I using available space for scratchwork. After
examining the form of the choices, decide which is the best of the choices given. Do not spend
too much time on anyone problem. Calculators may NOT be used on this part of the exam.
In this test:
Unless otherwise specified, the domain of a function f is assumed to be the set of all real
numbers x for which f(x) is a real number.
1.
=
If f(x)
(A)
x2 - 9 is continuous at x
x+3
= -3,
then f( -3)
=
3
(B) -3
(C)
0
(D)
6
(E) -6
Answer
D
2.
The graph of y = 3x2
(A)
(0,0) only
(B)
(1,2) only
-
x3 has a relative maximum at
(C) (2,4) only
(D)
(4, -16) only
(E) (0,0) and (2,4)
Answer
D
1
2
3.
Sample Examination I
A particle moves in the xy-plane so that its velocity vector at time t is v(t) = (t2,sin71"t)
and the particle's position vector at time t = 0 is (1,0). What is the position vector of the
particle when t = 3 ?
(A) (9,~)
(B) (10,~)
(C) (6, -271")
(D) (10,271")
(E) (10,2)
Answer
D
(A) Undefined
(B) 2ln2 - 2
(C) 2ln2 - 1
(D) 2ln2
(E)
-00
Answer
D
5.
(A) 0
(B) 1
(C) -1
(D)
l...
10
(E) --
1
10
Answer
D
Sample Examination I
6.
lim
e: -dt
cos t
11
1
x .....•
o
3
x
t
(A) -cos 1
is
(B) cas 1
(C) - sin 1
(D) sin 1
(E) nonexistent
Answer
D
7.
If f(x)
=
J4 sin x
+ 2,
then f'(O) =
(A) -2
(B)
0
(C)
J2
(D)
J2
2
(E)
1
Answer
D
8.
. t h e sum 0 f t he senes
. -3 - -3
Wh at IS
2 8
(A) ~
7
(B) ~
8
+ -3 - - 3 + ....?
32
128
(C)~
5
(D) ~
8
(E) 2
Answer
D
4
9.
Sample Examination I
The equation of the tangent line to the curve x2
+ y2
= 169 at the point (5, -12)
is
(A) 5y - 12x = -120
(B) 5x - 12y
=
119
(C) 5x - 12y
=
169
(D) 12x
+ 5y = 0
(E) 12x
+ 5y
= 169
Answer
D
10.
E
The figure above shows the graph of the velocity of a moving object as a function of time.
At which of the marked points is the speed the greatest?
(A)
A
(B)
B
(C)
C
(D)
D
(E)
E
l
1
Answer
D
Sample Examination I
00
11.
What are all values of x for which the series
(_l)n
L -l--x
n=2
n
5
converges?
nn
(A) -e < x ~ e
(B) -1 ::;x
<
(C) -e ::;x
<e
(D) -1
1
< x ::; 1
(E) -1 ::;x ::; 1
Answer
D
12.
J
-==l=
J4_X2
dx=
(A) Arcsin ~
(B)
2J 4 -
+C
X2 + C
(C) Arcsin x + C
(D) J4 - X2
+C
(E) ~ Arcsin ~
+C
Answer
D
6
13.
Sample Examination I
If the graph of f(x) = 2x2
(A)
+ l5.x
has a point of inflection at x = -1, then the value of k is
1
(B) -1
(C)
2
(D) -2
(E)
0
Answer
D
14.
If
J
xsec2
(A) tanx
X
dx = I(x)
+ In I cos z] + C,
then I(x) =
(C) xtanx
(D) x2 tanx
(E) tan2 x
Answer
D
l
1
j
Sample Examination I
15.
7
-
Which of the following is an equation of the line tangent to the curve with parametric
equations x = 3e-t, y = 6et at the point where t = 0 ?
2x
+y
- 12 = 0
(B) - 2x
+y
- 12 = 0
(A)
(C)
2x
(D) -2x
(E)
+y -
6 =0
+y -
6=0
+y
2x
=0
Answer
D
16.
J
dx
2x2
+ 3x + 1 =
(A) 2 In
2X
1
+ 1I +C
x+1
I
;>' I + C
I
I+ C
(B) In (2: :
(C) In x + 1
2x + 1
(D) In / ~:
(E) Inl(x
11/ + C
+ 1)(2x + 1)1 + C
Answer
D
8
17.
Sample Examination I
If x = sin
t and y
= cos2
t, then
d2~ at
dx
(A) 0
t = ~ is
2
(C) -~
(D) -2
(E) 2
Answer
D
18.
.
If y = x(ln X)2,
then
dy
dx =
(A) 3(ln X)2
(B) (lnx)(2x
+ In x)
(C) (Inx)(2
+ lnx)
(D) (lnx)(2
+ xlnx)
(E) (lnx)(l
+ lnx)
Answer
D
6
19.
If
10
(x2
-
2x
+ 2)
dx
is approximated
z -axis, then the approximation
by three inscribed
rectangles
of equal width on the
is
(A) 24
(B) 26
(C) 28
(D) 48
(E) 76
Answer
D
Sample Examination
20.
I
9
A particle moves on the z -axls so that at any time t its velocity v( t) = sin 2t subject to
the condition x(O) = 0 where x(t) is the position function. Which of the following is an
expression for x(t) ?
(A)
(B)
(C)
(D)
cos2t
+~
-! sin 2t + ~
-!cos 2t
-!cos 2t + !
(E) -!cos2t
- ~
Answer
D
21.
Which of the following is the equation for ~~ whose slope field is shown below for -3
and -10
< y < 10?
(A)
dy
-=y
(B)
dy
-=e
x
(C)
dy
.
dx = smx
(D)
(E)
<x <3
dx
dx
dy = x2 _ 9
dx
dy
3
= 3x -9x
dx
-
Answer
D
10
Sample Examination I
22.
.
sin(x2)
The Taylor series for
2
x
(A)
(B)
f
(_1)kx2k+1
k=O (2k
f
+ I)!
(_1)kx2k
k=O (2k
+ I)!
(_1)kx2k+l
(C)
E
(D)
1
00
(_1)kx2k-l
-+L
x k=1 (2k - 1)!
00
(E)
f
centered at x = 0 is
(2k)!
(_1)kx4k
k=O (2k
+ 1)!
Answer
D
l
b
23.
If the length of a curve y
= f(x)
from x
=a
to x
= b is given
by L
=
Je2X
+ 2e + 2 dx,
X
then f(x) may be
(A)
2e2x
+ 2ex
(B)
2x
le
2
+ 2ex + 2x
(C)
eX-x+3
(D)
e"
(E)
eX+x-2
+1
Answer
D
Sample Examination I
24.
The maximum value of f(x) = 2x3
-
9x2
+ 12x
11
- 1 on [-1,2J is
(A) 0
(B) 1
(C) 2
(D) 3
(E) 4
Answer
D
25.
Let R(t) represent the rate at which water is leaking out of a tank, where t is measured in
hours. Which of the followingexpressions represents the total amount of water that leaks out
in the first three hours?
(A)
(B)
R(3) - R(O)
R(3) - R(O)
3-0
3
(C)
10
(D)
10
(E)
!10
R(t) dt
3
R' (t) dt
3
R(t) dt
Answer
D
i·
12
Sample Examination
I
1
26.
[4
t- 2
il
(t
(A)
1~1 (t + 1) (t _ 4)
+ l)(t
_ 4) dt is found by using which of the limits below?
t- 2
4
x
[4
(B) li
I,
x-+rr+
14
(C) lim
x-+4-
x
1
i
t- 2
(t
+ l)(t
(t
+ l)(t
x-+l
(E)
lim
x
x
x-+4 -
1
dt
- 4)
t- 2
dt
- 4)
t- 2
4
(D) lim
(t
dt
+ l)(t
- 4)
dt
t-2
dt
(t + 1) (t - 4)
Answer
D
27.
The average value of the function f(x)
(A)
-1. sin(2)
(B)
-i
= cos(~x)
on the closed interval [-4,0]
is
2
sin(2)
(C)
~ cos(2)
(D)
i
(E)
~ sin(2)
sin(2)
Answer
D
Sample Examination
28.
If n is a positive integer, then
.
1[
1
(1/ )
n 1+
n
lim -
n-+oo
1
+ 1 + (12/ n) + ... + 1 + (/)
n ti
I
13
] can be
expressed as
2
1
(B) --1 dx
/ 1
(C)
12
(D)
r
I,
(E)
2
X
+
X
dx
_2_ dx
x
+1
21
1
- dx
1 X
Answer
D
14
Sample Examination I
Section I Part B
Directions: Solve each of the following problems, using available space for scratchwork.
After
examining the form of the choices, decide which is the best of the choices given. Do not spend
too much time on anyone problem. Calculators may be used on this part of the exam.
In this test:
(1) The exact numerical value of the correct answer does not always appear among the choices
given. When this happens, select from among the choices, the number that best approximates
the exact numerical value.
(2) Unless otherwise specified, the domain of a function
numbers x for which f(x) is a real number.
29.
f
is assumed to be the set of all real
The volume of the solid formed by revolving the region bounded by the graph of y = (x - 3)2
and the coordinate axes about the x-axis is given by which of the following integrals?
3
(A)
7r
10
(B)
7r
103 (x -
(x - 3)2 dx
3)4 dx
3
(C) 27r
10 (x -
(D) 27r
1
(E) 27r
10
3)2 dx
3
x(x - 3)2 dx
3
x(x - 3)4 dx
Answer
D
30.
.
1im
x-+-3
+ 3x
+ 6x + 9
x2
VX2
(A) -3
.
IS
(B) -1
(C) 1
(D) 3
(E) nonexistent
Answer
D
Sample Examination I
31.
15
The cost C of producing x items is given by C(x) = 20,000 + 5(x - 60)2. The revenue R
obtained by selling x items is given by R(x) = 15,000 + 130x. The revenue will exceed the
cost for all x such that
< 46
(A)
0< x
(B)
a > 46
(C)
x
(D)
46 < x < 100
(E)
z
< 100
> 100
Answer
D
32.
x
0
1
2
3
4
f(x)
20
19.5
18
15.5
12
5
6
7.5 2
7
8
9
10
-4.5
-12
-20.5
-30
Some values of a continuous function are given in the table above. The Trapezoidal Rule
approximation for Ido f(x) dx is
(A)
30.825
(B)
32.500
(C)
33.325
(D)
33.333
(E)
35.825
Answer
D
16
Sample Examination I
33.
For which pair of functions f(x)
f(x)
g(x)
(A)
eX
x2
(B)
eX
ln z
(C)
ln z
eX
(D)
x
ln z
(E)
3x
2x
and g(x)
below, will the lim f((X))
x-+oo 9 x
=
O?
Answer
D
34.
Which of the following gives the area of the region enclosed by the graph of the polar curve
r = 1 + cos O?
(A)
7r
!a (1
(B)
fa7r(1 + coSB)2 dO
(C)
27r
lao (1 +cosO)
(D)
27r
lao (1
(E)
21 la0 (1 + cos2 0) dO
+ cos" 0)
+ cos 0)2
dO
dO
dO
27r
Ii
J.••
Answer
D
Sample Examination
I
17
35.
A
1000
500
1500
2000
The figure above shows a road running in the shape of a parabola from the bottom of a hill
at A to point B. At B it changes to a line and continues on to C. The equation of the road is
ax2
R(x) =
{
from A to B
'
bx +c,
from B to C
B is 1000 feet horizontally from A and 100 feet higher. Since the road is smooth, R'(x) is
continuous. What is the value of b?
(A)
0.2
(B)
0.02
(C)
0.002
(D)
0.0002
(E)
0.00002
Answer
D
(A) 1
(B) 2
1
(C) 1 _ yj
(E) Does not converge
2
Answer
D
18
Sample
Examination
I
37.
x
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
j(x)
2.018
2.008
2.002
2
2.002
2.008
2.018
g(x)
1
1
1
2
2
2
2
h(x)
1.971
1.987
1.997
undefined
1.997
1.987
1.971
The table above gives the values of three functions, j, g, and h near x = O. Based on
the values given, for which of the functions does it appear that the limit as x approaches
zero is 2?
(A)
j
(B)
9 only
(C)
h only
(D)
j
(E)
t, g, and h
only
and
h only
Answer
D
38.
If ~~ = xy2 and x
(A)
(B)
=
1 when y = 1, then y =
x2
-2
-3
x2
(C)
(D)
(E)
Answer
D
Sample Examination
39.
The area of the region enclosed by the graphs of y
=
2
e<x
)
-
2 and y
=
vi4 -
X2
I
19
is
(A) 2.525
(B) 4.049
(C) 4.328
(D) 5.050
(E) 6.289
Answer
D
40.
If I(x) = l(x2 - 12)(x2 + 4)1, how many numbers
conclusion of the Mean Value Theorem?
in the interval
-2
~ x ~ 3 satisfy the
(A) None
(B) One
(C) Two
(D) Three
(E) Four
Answer
D
20
41.
Sample Examination I
On which of the following intervals does the polynomial
approximate the function y = cos(x)?
(A)
[0,3]
(B)
[0,5]
(C)
[2,4]
(D)
[3,6]
(E)
[6,7]
y =
_3;
+X
-r-
(2x~7r)3
best
Answer
D
42.
The amount A(t) of a certain item produced in a factory is given by
A(t) = 4000
+ 48(t
- 3) - 4(t - 3)3
where t is the number of hours of production since the beginning of the workday at 8:00 am.
At what time is the rate of production increasing most rapidly?
(A)
8:00 am
(B)
10:00 am
(C)
11:00 am
(D)
12:00 noon
(E)
1:00 pm
~
\
t
Answer
D
Sample Examination
43.
At how many points on the curve y = 4x5
pass through the origin?
(A)
One
(B)
Two
(C)
Three
(D)
Four
(E)
Five
-
3x4
+ 15x2 + 6 will
the line tangent
I
21
to the curve
Answer
D
44.
A population
grows according to the equation pet) = 6000 - 5500e-O.159t for t 2: 0,
t measured in years. This population will approach a limiting value as time goes on. During
which year will the population reach half of this limiting value?
(A) Second
(B) Third
(C) Fourth
(D) Eighth
(E) Twenty-ninth
Answer
D
22
Sample Examination I
45.
Y
-,
35
/
30
j
25
\
V
15
10
o
\
/
20
5
V
/
/
1
\
\
\
/
L.-J
.:
V
f' (x)
\
2
3
4
5
6
7
x
\
-5
Note: This is the graph of f'(x),
graph of f(x).
NOT the
Let f be a differentiable function for all x. The graph of f'(x) is shown above. If f(2)
which of the following best approximates the maximum value of f(x)?
(A)
30
(B)
50
(C)
70
(D)
90
(E)
llO
= 10,
I
i
1
Answer
I
D I
\
t
~
Sample
SECTION
II - FREE-RESPONSE
GENERAL
Examination
I
23
QUESTIONS
INSTRUCTIONS
You may wish to look over the problems before starting work on them, since it is not expected
that everyone will be able to complete all parts of all problems. All problems are given equal
weight, but the parts of a particular problem are not necessarily given equal weight. The
problems are printed in the booklet and in the green insert. When you are told to begin,
open your booklet, carefully tear out the green insert, and start work.
•
You should write all work for each part of the problem in the space provided for
that part in the booklet. Be sure to write clearly and legibly. If you make an
error, you may save time by crossing it out rather than trying to erase it. Erased
or crossed-out work will not be graded.
•
Show all your work. Indicate clearly the methods you use, because you will
be graded on the correctness of your methods as well as the accuracy of your
final answers. Correct answers without supporting work may not receive credit.
Justifications require mathematical
(noncalculator) reasons.
•
Your work must be expressed in standard mathematical
notation rather than
calculator syntax. For example, J15 X2 dx may not be written as fnInt(X2, X, 1,5).
•
Unless otherwise specified, answers (numeric or algebraic) need not be simplified.
If your answer is given as a decimal approximation, it should be correct to three
places after the decimal point.
•
Unless otherwise specified, the domain of a function
all real numbers x for which f(x) is a real number.
SECTION
II PART A: QUESTIONS
f
is assumed to be the set of
1,2,3
A GRAPHICS CALCULATOR IS REQUIRED FOR SOME PROBLEMS
PROBLEMS IN TI-HS SECTION OF THE EXAM.
OR PARTS OF
You are permitted to use your calc' ilator to solve an equation, find the derivative
of a function at a point, or calculate the value of a definite integral. However, you
must clearly indicate the setup of your problem, namely the equation, function,
or integral you are using. If you use other built-in features or programs, you must
show the mathematical steps necessary to produce your results.
YOU HAVE 45 MINUTES
FOR PART A.
DO NOT GO ON TO PART B UNTIL YOU ARE TOLD TO DO SO.
SECTION
II PART B: QUESTIONS
YOU MAY NOT
USE A GRAPHICS
YOU HAVE 45 MINUTES
4,5,6
CALCULATOR
IN TI-IIS PART OF THE EXAM.
FOR PART B.
YOU MAY USE SOME OF THIS TIME TO WORK ON PART A. IF YOU WORK O~
PART A, YOU MAY NOT USE A GRAPHICS CALCULATOR.
24
Sample Examination
I
Section II Part A: Graphing Calculator MAY BE USED.
1.
Let f be the function given by f(x)
I
by f (x) =
(a)
x2
+ 6x + 8
+ 3)2
(x
for all x
of
=
2
8
x -3 for all x
x+
of
-3 and whose derivative is given
-3.
In the viewing window provided below, sketch the graph of f.
10
-20
Viewing Window
[-10,10] by [-20,10]
(b)
Find the range of f. Use f'ex)
to justify your answer.
(c)
Find
(d)
Explain what your answer to (c) tells about the graph of f(x).
(a)
In the viewing window provided below, sketch the graph of f.
lim f'(x).
x--±oo
10
10
-20
Viewing Window
[-10,10] by [-20,10]
Sample Examination
(b)
Find the range of f. Use f'(x)
to justify your answer.
(c)
Find
(d)
Explain what your answer to (c) tells about the graph of f(x).
lirn f'(x).
x-+±oo
I
25
I
Sample Examination I
30
Section II Part B: Graphing
4.
Calculator
MAY NOT BE USED.
I'(x)
4
Note: This is the graph of the derivative of
not the graph of I.
I,
The figure above shows the graph of I', the derivative of I. The domain of I is -4 S; x S; 4.
The derivative of I is an even function and /'( -3) = /'(3) = O.
1
(a)
For what value of x in the interval
Justify your answer.
i
(b)
For what value of x in the interval -4 S; x S; 4 does / have a relative minimum?
your answer.
(c)
For what values of x is the graph of
(d)
If 1(0) = 0, find the value of [:
(a)
For what value of x in the interval
Justify your answer.
I
-4
S; x S; 4 does / have a relative
concave downward?
Use
I'
f
maximum?
t
Justify
I
to justify your answer.
I(x) dx. Justify your answer.
-4
::; x ::; 4 does
I
have a relative maximum?
I
l
I
t
Sample Examination I
31
(b)
For what value of x in the interval -4 ~ x ~ 4 does I have a relative minimum? Justify
your answer.
(c)
For what values of x is the graph of I concave downward? Use I' to justify your answer.
(d)
If 1(0) = 0, find the value of
[aa I(x)
dx. Justify your answer.
SAMPLE EXAMINATION
1. E
2. C
3. B
4. B
5. A
16. D
17. 0
19. B
20. D
6. B
21. E
7. C
22. E
8. C
23. E
31. D
32. B
18. C
33. C
34. B
35. A
36. B
37. D
38. B
Ia.
b. y :.::::-8 or 11 2 -4
-3:
c. 1
9. C
24. E
39. D
I
10. D
25. C
40. D
11. D
26. E
41. D
12. A
27. E
42. C
13. C
28. E
43. A
14. C
29. B
44. C
15. A
30. E
45. E
d. The graph of f(:r) looks like a line with a 810pe of one (as :r -+ too)
~·,,··0
-I
f~ t
8
b. A( ,;" t, 1) B (_t, 1)
e. A d. 90th""kJ ••••" ow"ng
u"....d•• "he
"""e <ate
-·1 .
3'a. 17211d d&y
6. 731
.!la. x::. 3
=
b. x
·-3
267,546
c
c. ~-2
<
b. pet)
~./
x
=
minutes
<0
and 2
<x S4
6eO.9t~
~.l_+{J:2eO.9t
C.
~.~
d. 0
30 000
,
.>:
•.•....
»->:
00.. when
;t;_C::--:::i. and x
;.•...~
=
Ij
c. 6
d. hex)
=-
rr
I
;I
+ 3x +
6
.:1
l