Assignment
Math Practice Test DSM Part 1B Answer Key
-
University:
Cornell University -
Course:
MATH 1110 | Calculus I Academic year:
2023
-
Views:
103
Pages:
9
Author:
Nikolai Harris
Section I Part B
Directions: Solve each of the following problems, using available space for scratch work.After
examining the form of the choices, decide which is the best of the choices given. Do not spend
too much time on any one problem. Calculators may be used on this part of the exam.
In this exam:
(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/is assumed to be tµe set of all real
numbers x for which f(x) is a real number.
(3) The inverse1 of a trigonometric function f may be indicated using the inverse function
notation 1- or with the prefix "arc" (e.g., sin- 1x = arcsin x)
29. The volume of the solid formed by revolving the region bounded by the graphs of y = 9 and
y = (x-3) 2 about the line y = 9 is given by which of the following integrals?
r
)----r( (92 -(x-3) 4 ) dx
_0
@"J.'
(9-(x -3) 2 ) 2 dx
_59,v1rl6 ((x-3)
3
�
4
-
92 ) dx
(9-(x-3) 2) dx
4
2
�( ((x-3) -9) dx
D ; � k � wi � r ::: q - (-x -1 \
l
f�
(
1-(x -1/ )\,(�
Answer
[[] 30. Let f be the function given by f ( x) = tan x and let g be the function given by g ( x) = x2. At
what value of x in the interval O ::; x ::; 7r do the graphs off and g have parallel tangent lines?
(A) 0
N
'f}
(B) 0.660
X
@2.083
� 1'(x)
-= �
lws x)
·1
/),R/4
1
'12 -: 7..x
x -:;:;;
=- � '{x)
( E) 2.207
(D) 2.194
( � ::: J)p� ,J�)
i X :: J_ 'l<
�
��
-i.oi3
Answer
31. Let f(t) = J fort> 0. For what value oft is f'(t) equal to the average rate of change of/ on
the closed interval [a, b] ?
�
0
�f
(C)
MOU � ��
-I :. -:L
l?i tJ,.b
-t 7. -:: tvb
t�
a
1
- lab
II
::
(D)
1
lab
1(�)-1({}-)
b- A
(E)
Ii(¼-!)
( �+
Cl �e \n¾()
Answer 32. Let/be the function given by f(x) = 3 + fx cos (t2) dt. What is the least positive number a,
0
for which f'(a) = 0?
i
(A ) 1
81.253
I ( } -:
�
0+
(D) 1.772
(C) 1.571
COS X�
(E) 3.142
"2
)< := 1J:
2
x: =
vf �
L 2,3
Answer
33. Let R (t) represent the �at which water is leaking out of a tank, where t is measured in
hours. Which of the following expressions represents the total amount of water in gallons that
leaks out in the first three hours?
(A) R(3)-R(O)
(3)-R(O)
(B) R 30
�' R(t)dt
3
(D) L R'(t) dt
(E)
½L3 R(t)
dt
s �(t)
J;t
r
��1��1 °'�
Yu
(,., /);b
-� � ,{, �
� 1
tfu.�t<.it �W),¼�
� .,,w,. N
� ll-< {o-{a,(
)MA,�.
'1
Answer 34. Which of the following gives the total area of the region enclosed by the graph of the
polar curve r = 1 + cos 0 ? IJ •• - = ..!... ( 2 n-(
1 A.
1_
1_
�
'-').
Pun O fll,t.tQ\ ut V\MJ.
� Q\l'I
f'(x)changes sign atx = c. - N>, .f{>c) �
1
�
1f "'-6
•
��-
[a,IJ
□
Answer 1
36. Let R(x) be the radius of a round pipe that drains water from a dam, where x is measured in
30' 000
feet from the dam. Which choice best explains the meaning of 1r
( R ( x) )2 dx ?
0,000
(A) The amount of water in square feet that the pipe can hold in the section from 10,000 to
30,000 feet from the dam.
' Mhe amount of water in cubic feet that the pipe can hold in the section from 10,000 to
Vio,ooo feet from the dam.
( C) The amount of water in cubic feet flowing through the pipe from 10,000 to 30,000 feet.
(D) The amount of water in cubic feet in any 20,000-foot section of the pipe.
(E) The rate of which water flows through the pipe in the section that is 10,000 to 30,000 feet
from the darn.
-rr ( 1,6c))'" �
Juk
·rrJ {,i.,) lJ. • \fol
1ot'
'1> '1.t).)
'1 f'1"'
�
-0.3 -0.2 -0.1
0.1
0
0.2
0.3
2.002 2.008 2.018
2
f(x) 2.018 2.008 2.002
g(x)
1
1
2
2
2
1
2
h(x) 1.971 1.987 1.997 undefined 1.997 1.987 1.971
-.;t,,;. ,"( ,..;,.
□
Answer
X
37. The table above gives the values of three functions, f, g, andhnear x = 0. Based on the values
given, for which function does it appear that the limit as x approaches zero is 2 ?
(A) /only
(B) g only
(C) honly
@tandhonly
(E) f, g, andh
□
Answer z
dy
1t 38. If dx = � and x = 1 when y = 1, then y =
(A) �x 2 + 12
J � -"'d • J
½ � .. =
✓
(D) - ✓ 2x 3+ 1
✓
2x +1
@ 3
(C)
,l�A - (')(
(
,
-i-C
J.,1 ,. =-
x 3 j3
-
2,
l
3
&
J ...
l ,.. (. -, C
.L
½ ':1 1. J
'1 i. !=_J
3
'j
V
-=
k
tf! c.
i
)(
)c
=-t
3 +l
'
l
._
,L
�
□
Answer
i)(1 .. ,
a
y
__,. y = f'(x)
39. The figure above shows the graph of the derivative of a polynomial function f.
How many points of inflection does the graph of/have?
;.
...:
(A) None
(B) One
• (C) Two
(D) Three
□
Answer y
5
4-t------------:--3
2
1
-+--+---+--+--1-1--+--+---+---+--+---+-�x
1 2 3 4 5 6 7 8 9 10 11 12
-1
e
40. As shown in the figure above, the function f(x) consists of a line segment from (0, 4) to (8, 4)
and one-quarter of a circle with a radius of 4. What is the average (mean) value of this function
on the interval [0, 12] ?
(A) 2
""'
-'----[;()c\ h
3.714
v,-1� :.
(C) 6.855
12.-0
0
(D) 22.283
(E) 44.566
: Ti ( 32 + �)
� fi_ ( 3 2 _. Yrr)
� 3.; ·�
□
Answer
41. The graph of the third-degree Maclaurin po nomial that approximates sin (x) intersects the
graph of y = x2 at how many points?
(A) None
(B) One
(C) T
Answer
@] 42. The amount, A (t), of a certain item produced in a factory is given by
A (t) = 4000 + 4B(t- 3)- 4(t- 3)3
where t is the number of hours of production since the beginning of the workday at 8:00am.
At what time is the rate of productiQll,at its maximum?
(A) 9:00am
A '{t)
(B) 10:00am
@11:00am
(D) 12:00noon
(E) 1:00 pm
l So
A'' l �)
.
A.1n �
It ' 1 l +)
or R"'l-1\
�" r
43. The derivative of 4x 2 cos (x) is
-=-
tM-xi mi i.t
'i, - 1 2 l t -i t
' --
□
Answer
(A) -Bx sin (x)
eYBx cos (x)- 4x2 sin (x)
( C) Bx cos (x) + 4x2 sin (x)
(D) -8x cos (x) -_4x2 sin (x)
(E) -8x cos (x) + 4x2 sin (x)
Answer
[[] 44. A population increases according to the equation P(t) = 6000- 5500e- 0•159' for t 2: 0,
t measured m 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
l, M
�000 - S'S'OOt - 0 ·•s1 �:
t .at 0o
(B) Third
@Fourth
(D) Eighth
(E) Twenty-ninth
l,ooo
Y / -: �001), S'S"'OO -e •o.,�4'
Y 1 � 3 000
t?
f"i "'-"
,,+. of
ihWYJ�cA,' elk ,
{ 3, i, JoDO)
't(Ml.,.; "1 ,f.(,.
0
Answer
I/le, � I'.
y
35
30
13
25
�vt
11
boY<5 uf
-�&�
20
15
f'(x)
�
ll
10
5
0
1
2
4
3
5
6
7
-5
Note:This is the graph of f'(x), NOT the graph of f(x).
45. Let/be a differentiable function on the closed interval [0,7]. The graph of f'(x) is shown
above. If/(2) = 10, which of the following best approximates the maximum value of f(x)?
(A) 30
r
(B) 50
�)( fo�
��l-;...it,
(C) 70
f('i) wiU
f '{x)
@uo
(D) 90
OCM4,r'
HI. ..
A,+
)( :.�.
Answer
5
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