MAC 2312
Exam 3A
Spring 2016
A. Sign your bubble sheet on the back at the bottom in ink.
B. In pencil, write and encode in the spaces indicated:
1) Name (last name, first initial, middle initial)
2) UF ID number
3) Section number
C. Under “special codes” code in the test ID 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 answer sheet, for “Test Form Code”, encode A.
• B C D E
E. 1) This test consists of 13 multiple choice questions, ranging from two points to four
points in value, plus two sheets (three pages) of free response questions worth 27
points. The test is counted out of 70 points, and there are seven bonus points
available.
2) The time allowed is 90 minutes.
3) You may write on the test.
4) Raise your hand if you need more scratch paper or if you have a problem with your
test. DO NOT LEAVE YOUR SEAT UNLESS YOU ARE FINISHED WITH
THE TEST.
F. KEEP YOUR BUBBLE SHEET COVERED AT ALL TIMES.
G. When you are finished:
1) Before turning in your test check carefully for transcribing errors. Any mistakes you leave in are there to stay.
2) You must turn in your scantron and tearoff sheets to your discussion leader or exam
proctor. Be prepared to show your picture I.D. with a legible signature.
3) The answers will be posted in Canvas within one day after the exam. Your discussion leader will return your tearoff sheet with your exam score in discussion.
Your score will also be posted in Canvas within one week of the exam.
NOTE: Be sure to bubble the answers to questions 1−13 on your scantron.
Questions 1 − 12 are worth 4 points each.
1. Find the Maclaurin series of f (x) =
x
and determine its radius of
(3 − x)2
convergence R.
∞
X
nxn
1
a. f (x) =
and
R
=
3n+1
3
n=1
b. f (x) =
∞
X
nxn
and R = 3
c. f (x) =
n+1
3
n=1
d. f (x) =
e. f (x) =
∞
X
n=0
n=1
∞
X
n=1
n−1
−nx
3n+1
∞
X
−nxn
and R =
3n+1
and R =
1
3
−nxn
and R = 3
3n+1
1
3
X
2. Suppose that
cn (x − 2)n converges for x = 4 and diverges for x = −2.
Which of following must be correct?
a.
X
cn (−2)n converges
b.
X
cn 4n diverges
c.
X
cn 3n converges
d.
X
cn (−1)n converges
e.
X
n
5
cn
diverges
2
3. Find the sum of
a. 2π
π3 π5 π7 π9
−
+
−
+ ···.
2!
4!
6!
8!
b. π
d. −π
c. 0
2A
e. −2π
4. Find the interval of convergence of
∞
X
(−1)n
n=2
1 1
a. − ,
2 2
1 1
b. − ,
2 2
d. [−3, −2]
x
5. Express the integral
0
d.
∞
X
(−1)n e2 xn
n=0
∞
X
n=1
c. [−3, −2)
e. (−3, −2]
Z
a.
n ln n
(2x + 5)n .
e2+t − e2
dt as a power series.
t
∞
X
e2 xn
b.
n · n!
n=0
n · n!
∞
X
e2 xn+1
c.
(n + 1)n!
n=0
∞
X
e2 xn
e.
n · n!
n=1
(−1)n e2 xn
n · n!
6. If f (x) = sin(x2 ), find f (102) (0).
a. −
102!
205!
b.
102!
205!
c. −
102!
51!
102!
51!
d.
7. Consider the parametric curve
C : x(t) = 2t + 3, y(t) = et + e−t , 0 ≤ t ≤ 2.
Find the length of the curve.
a. e2 −
d. e −
1
e2
1
e
b. e2 +
e. e +
1
e2
c. e2
1
e
3A
e.
205!
102!
8. Let C be the curve x(t) = t3 −3t2 , y(t) = 2t3 +6t2 . Which of the following
is/are true about the curve C?
P. The curve C has a horizontal tangent at t = 0.
Q. The curve C has a vertical tangent at t = 2.
R. A point on C is moving in the direction
to 2.
a. P and Q
b. Q only
d. Q and R
e. P, Q and R
as t increases from 0
c. P and R
9. Let R be the region enclosed by the given curves:
y = cos x, y = sin 2x, x = 0, x =
π
2
Set up an integral for the area of the region R.
Hint: Sketch the region R.
π/2
Z
(cos x − sin 2x) dx
a. Area =
0
π/2
Z
(sin 2x − cos x) dx
b. Area =
0
π/6
Z
π/2
Z
(cos x − sin 2x) dx +
c. Area =
0
(sin 2x − cos x) dx
π/6
π/6
Z
Z
π/2
(sin 2x − cos x) dx +
d. Area =
(cos x − sin 2x) dx
0
Z
π/6
π/3
Z
π/2
(cos x − sin 2x) dx +
e. Area =
0
(sin 2x − cos x) dx
π/3
4A
10. Determine the graph of the polar equation
r = 2 cos θ + 2 sin θ
by converting it to a Cartesian equation first.
a. A circle with center (1,1) and radius 2
√
b. A circle with center (1,1) and radius 2
c. A circle with center (−1, −1) and radius 2
√
d. A circle with center (−1, −1) and radius 2
e. A circle with center (0,0) and radius 2
11. Set up an integral for the area of the region that lies inside r =
and outside r = cos θ.
Z π
1 √
1
2
Area =
( 3 sin θ) dθ −
(cos θ)2 dθ
π/3 2
π/3 2
Z 2π √
Z π/2
1
1
Area =
( 3 sin θ)2 dθ −
(cos θ)2 dθ
π/6 2
π/6 2
Z π
Z 2π √
1
1
2
Area =
( 3 sin θ) dθ −
(cos θ)2 dθ
π/6 2
π/6 2
Z π/2
Z π
1
1 √
Area =
( 3 sin θ)2 dθ −
(cos θ)2 dθ
π/6 2
π/6 2
Z π
Z π/2
1 √
1
2
Area =
( 3 sin θ) dθ −
(cos θ)2 dθ
π/3 2
π/3 2
Z
a.
b.
c.
d.
e.
2π
5A
√
3 sin θ
b
Z
12. If
a
1
(−1 + 2 sin θ)2 dθ presents the area of the shaded region R
2
below, find a and b.
r = −1 + 2 sin θ
R
a. a =
11π
and b = 2π
6
b. a =
2π
and b = π
3
c. a =
5π
and b = 2π
3
d. a =
3π
and b = 2π
2
e. a =
5π
and b = π
6
13. (2 points) Write the Cartesian point (1, −1) in polar coordinates.
√
π
a. − 2, −
4
√ π d.
2,
4
√ 7π
b. − 2,
4
√ 5π
e.
2,
4
6A
c.
√ 3π
− 2,
4
MAC 2312 Exam 3A, Part II Free Response
Name:
Section #:
SHOW ALL WORK TO RECEIVE FULL CREDIT
1
.
x2
(a) Find the Taylor series for f (x) centered at a = 3.
1. (7 points) Let f (x) =
f (x) =
(Express your answer using summation notation.)
(b) Determine the radius of convergence of the series in part (a).
radius of convergence -
7A
1
1
2. (10 points) Let the parametric curve C be x(t) = t − , y(t) = t +
(t 6= 0).
t
t
(a) When is the curve C decreasing?
C is decreasing on
.
(b) Sketch the curve C and label the direction of the curve.
5
(c) Set up an integral for the area of the region bounded by the curve C and the line y = .
2
Z
A=
dt
8A
3. (10 points) Let r = 1 − 2 cos(2θ).
(a) Solve for r = 0, 0 ≤ θ ≤ 2π.
θ=
(b) Sketch the polar curve.
π
2
r
2π
3
π
3
π
6
5π
6
π
π
2π
−4
θ
−2
2
0
4
7π
6
11π
6
4π
3
3π
2
5π
3
(c) Set up an integral for the area of the region enclosed by the loop that sits on the
positive y-axis.
Z
A=
dθ
9A
Name:
UF ID #:
University of Florida Honor Pledge:
On my honor, I have neither given nor received unauthorized aid doing this
exam.
Signature:
FR Scores
1
2
3
FR Total
/7
/10
/10
/25