MAC2311+Fall+2022+Exam+3+A-1
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Subject
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Date
Nov 24, 2024
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Uploaded by juliasshen
Calculus I: MAC2311
Name:
Fall 2022
Exam 3 A
Section:
11/16/2022
Time Limit: 90 Minutes
UF-ID:
Scantron Instruction
: This exam uses a scantron. Follow the instructions listed on this page to fill
out the scantron.
A. Sign your scantron
on the back
at the bottom in the white area.
B. Write
and code
in the spaces indicated:
1) Name (last name, first initial, middle initial)
2) UFID Number
3) 4-digit Section Number
C. Under
special codes
, code in the test numbers 3, 1:
1
2
•
4
5
6
7
8
9
0
•
2
3
4
5
6
7
8
9
0
D. At the top right of your scantron, fill in the
Test Form Code
as A.
•
B
C
D
E
E. This exam consists of 14 multiple choice questions and 5 free response questions. Make sure you
check for errors in the number of questions your exam contains.
F. The time allowed is 90 minutes.
G. WHEN YOU ARE FINISHED:
1) Before turning in your test check for
transcribing errors
. Any mistakes you leave in
are there to stay!
2) You must turn in your scantron to your proctor.
Be prepared to show your GatorID
with a legible signature.
Calculus I: MAC2311
Exam 3 A - Page 2 of 14
11/16/2022
It is your responsibility to ensure that your test has
19 questions
. If it does not, show it to your
proctor immediately. You will not be permitted to make up any problems omitted from your test after
the testing period ends. There are a total of 105 points available on this exam.
Part I Instructions
: 14 multiple choice questions.
Complete the scantron sheet provided with your
information and fill in the appropriate spaces to answer your questions. Only the answer on the scant-
ron sheet will be graded. Each problem is worth five (5) points for a total of 70 points on Part I.
1. Evaluate
lim
x
→
0
+
x
√
x
.
(
A
) 0
(
B
) 1
(
C
)
e
(
D
)
-∞
(
E
)
∞
2.
The volume of a cylinder is given by
V
=
πr
2
h
, where
V
,
r
, and
h
are functions of time,
t
.
If
dV
dt
(5) = 6
π
,
dh
dt
(5) = 1,
dr
dt
(5) =
-
1
4
, and
h
(5) = 2, what is the value of
r
(5)?
(
A
) 1
(
B
)
-
2
(
C
) 4
(
D
) 2
(
E
) 3
Calculus I: MAC2311
Exam 3 A - Page 3 of 14
11/16/2022
3. Which of the following statements must be true for
f
(
x
) =
1
2
x
2
-
x
-
2 ln(
x
)?
P.
f
(
x
) has exactly one critical number.
Q.
f
(
x
) is decreasing on (0
,
2).
R.
f
(
x
) is increasing on (2
,
∞
).
(
A
) None of P, Q, or R
(
B
) P and Q only
(
C
) Q and R only
(
D
) P and R only
(
E
) P, Q, and R
4. Suppose that a continuous function
f
(
x
) has horizontal tangent lines at
x
=
-
2,
x
= 0, and
x
= 1.
If
f
00
(
x
) = 3
x
2
+ 2
x
-
2, then according to the Second Derivative Test
f
(
x
) which of the following is
true for
f
(
x
)?
(
A
)
f
(
x
) has a local maximum at
x
= 1 and local minima at
x
=
-
2 and
x
= 0
(
B
)
f
(
x
) has local maxima at
x
=
-
2 and
x
= 0 and a local minimum at
x
= 1
(
C
)
f
(
x
) has a local maximum at
x
= 0 and a local minimum at
x
=
-
2 only
(
D
)
f
(
x
) has local maxima at
x
=
-
2 and
x
= 1 and a local minimum at
x
= 0
(
E
)
f
(
x
) has a local maximum at
x
= 0 and local minima at
x
=
-
2 and
x
= 1
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Calculus I: MAC2311
Exam 3 A - Page 4 of 14
11/16/2022
5. Consider the function
f
(
x
) where
f
(
x
) =
x
x
2
-
1
,
f
0
(
x
) =
-
(
x
2
+ 1)
(
x
2
-
1)
2
,
f
00
(
x
) =
2
x
(
x
2
+ 3)
(
x
2
-
1)
3
.
On what interval(s) is
f
(
x
) concave upward?
(
A
) (
-∞
,
-
1)
∪
(0
,
1)
(
B
) (
-
1
,
0)
∪
(0
,
∞
)
(
C
) (
-
1
,
0)
∪
(1
,
∞
)
(
D
) (
-∞
,
-
1)
(
E
) (
-∞
,
0)
6. Suppose that
f
(
x
) is continuous on (
-∞
,
∞
) and that the graph of its
derivative
y
=
f
0
(
x
) is
given below.
x
y
y
=
f
0
(
x
)
Where is
f
(
x
) decreasing?
(
A
) (
-∞
,
0)
(
B
) (
-
3
,
-
1)
(
C
) (
-∞
,
-
2)
(
D
) (0
,
∞
)
(
E
)
f
(
x
) is never decreasing
Calculus I: MAC2311
Exam 3 A - Page 5 of 14
11/16/2022
7. Let
y
= 3
x
2
+
x
+ 1. Suppose
x
increases from 1 to 1
.
1. Which of the following is Δ
y
-
dy
?
(
A
) 0
(
B
) 0
.
01
(
C
) 0
.
02
(
D
) 0
.
03
(
E
) 0
.
04
8. At which of the labeled points on the graph below is
f
00
(
x
)
> f
0
(
x
)
> f
(
x
)?
x
y
y
=
f
(
x
)
A
C
B
D
(
A
)
A
(
B
)
B
(
C
)
C
(
D
)
D
Calculus I: MAC2311
Exam 3 A - Page 6 of 14
11/16/2022
9. The base damage,
B
, dealt by a character in a certain MMORPG is given by
B
=
1
2
D
2
+ 3
SD
+
S
2
where
D
is the character’s Dexterity score and
S
is the character’s Strength score. If a character has
6 total points to assign between Strength and Dexterity, what combination of Strength and Dexterity
will maximize
B
?
(
A
)
S
= 2,
D
= 4
(
B
)
S
= 4,
D
= 2
(
C
)
S
= 1,
D
= 5
(
D
)
S
= 3,
D
= 3
(
E
)
S
= 5,
D
= 1
10.
Which of the following value(s) of
x
satisfy the conclusion of the Mean Value Theorem for
f
(
x
) =
x
3
-
2
x
2
-
4
x
+ 1 on the interval [0
,
2]?
(
A
)
x
= 0,
4
3
(
B
)
x
=
4
3
only
(
C
)
x
= 2
(
D
)
x
= 0 only
(
E
)
x
=
4
3
,
-
4
3
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Calculus I: MAC2311
Exam 3 A - Page 7 of 14
11/16/2022
11. Suppose
f
0
(
x
) =
x
3
-
2
x
2
-
3
x
. Which of the following must be
False
?
(
A
)
f
(
x
) has a local minimum at
x
=
-
1
(
B
)
f
(
x
) has a local minimum at
x
= 3
(
C
)
f
(
x
) has a local minimum at
x
= 0
(
D
)
f
(
x
) is increasing on (3
,
∞
)
12. What is the absolute maximum of
f
(
x
) =
e
sin(
x
)
on
π
4
,
3
π
4
?
(
A
) 1
(
B
)
e
(
C
)
e
√
2
/
2
(
D
)
e
√
3
/
2
(
E
)
e
1
/
2
Calculus I: MAC2311
Exam 3 A - Page 8 of 14
11/16/2022
13. Suppose
f
(
x
) is a differentiable function such that lim
x
→
0
f
(
x
) = 0 and lim
x
→
0
f
0
(
x
) = 1. Evaluate
lim
x
→
0
f
(
x
)
x
.
(
A
)
-
1
(
B
) 0
(
C
) 1
(
D
)
∞
(
E
)
-∞
14. Use the linear approximation of
f
(
x
) =
3
√
x
at
a
= 8 to approximate
3
√
7
.
76.
(
A
) 1
.
975
(
B
) 1
.
98
(
C
) 1
.
985
(
D
) 1
.
99
(
E
) 1
.
995
Calculus I: MAC2311
Exam 3 A - Page 9 of 14
11/16/2022
Calculus I: MAC2311
Name:
Fall 2022
Exam 3 A
Section:
11/16/2022
Time Limit: 90 Minutes
UF-ID:
Part II Instructions
: 5 free response questions. Neatly give a complete solution to each problem and
show all work and intermediate steps. We are grading the work and notation as well as the answer.
Each problem is worth seven (7) points. A total of 35 points is possible on Part II.
No credit will
given without proper work.
If we cannot read it and follow it, you will receive no credit for the
problem.
For Instructor Use Only:
FR 1
FR 2
FR 3
FR 4
FR 5
Total Points
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Calculus I: MAC2311
Exam 3 A - Page 10 of 14
11/16/2022
1. Use the linear approximation of the function
f
(
x
) =
p
4 + ln(
x
) at
x
= 1 to approximate the value
of
f
(1
.
4).
Calculus I: MAC2311
Exam 3 A - Page 11 of 14
11/16/2022
2. Sand pouring from a chute forms a conical pile whose radius is always equal to twice its height. If
the volume increases at a constant rate of 12
π
cubic feet per minute, at what rate is the height of the
pile changing when the radius is 2 feet? (The volume of a cone is given by
V
=
1
3
πr
2
h
).
Calculus I: MAC2311
Exam 3 A - Page 12 of 14
11/16/2022
3.
Find all value(s) of
x
which satisfy the conclusion of the Mean Value Theorem for the function
g
(
x
) =
ax
2
+
x
+
k
on the interval [0
,
1]. Assume that
a
and
k
are nonzero real numbers.
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Calculus I: MAC2311
Exam 3 A - Page 13 of 14
11/16/2022
4. Find the maximum area of an isosceles triangle inscribed in a semicircle of radius 6, if one vertex
of the triangle is on the midpoint of the diameter of the semicircle (A diagram is provided below). Be
sure to provide justification that your answer is a maximum value.
x
y
-
6
6
6
h
x
6
Calculus I: MAC2311
Exam 3 A - Page 14 of 14
11/16/2022
5. Sketch a graph of the function
f
(
x
) that has the following properties:
f
(
x
) is continuous on (
-∞
,
-
1)
∪
(
-
1
,
∞
)
f
(
x
) has a vertical asymptote
x
=
-
1
lim
x
→∞
f
(
x
) = 1 and
lim
x
→-∞
f
(
x
) = 2
f
(
x
) is increasing on (
-∞
,
-
1)
∪
(
-
1
,
2)
f
(
x
) is decreasing on (2
,
∞
)
f
(
x
) has a local maximum at (2
,
3) and no local minimums
f
(
x
) is concave upward on (
-∞
,
-
1)
∪
(4
,
∞
)
f
(
x
) is concave downward on (
-
1
,
4)
f
(
x
) has an inflection point (4
,
2)
x
y