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University Of Connecticut *
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Course
1132Q
Subject
Mathematics
Date
Apr 3, 2024
Type
Pages
16
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University of Connecticut
Department of Mathematics
Math 1132
Final Exam Practice
Spring 2022
Name:
Signature:
TA Name:
Class Time:
Read This First!
•
Read each question carefully.
•
The
short answer
questions (starting from problem 11) will
only
receive credit by
providing clear work using techniques learned in this class.
•
No books, formula sheets, calculators, smart watches, or other references are permitted.
•
No personal scrap paper is allowed.
If you want some additional paper to do some
work, it will be provided by your instructor. But scrap paper will
not
be graded.
MATH 1132Q Formula Sheet
Trigonometric identities
.
sin
2
θ
+ cos
2
θ
= 1
sin 2
θ
= 2 sin
θ
cos
θ
cos
2
θ
=
1 + cos 2
θ
2
1 + tan
2
θ
= sec
2
θ
cos 2
θ
= cos
2
θ
-
sin
2
θ
sin
2
θ
=
1
-
cos 2
θ
2
Approximate integration formulas
.
Midpoint Rule
.
M
n
= Δ
x
(
f
(
x
1
) +
f
(
x
2
) +
· · ·
+
f
(
x
n
))
, where
x
i
=
1
2
(
x
i
-
1
+
x
i
)
The error bound in the Midpoint Rule is
|
E
M
| ≤
K
(
b
-
a
)
3
24
n
2
, where
K
is chosen so that
|
f
00
(
x
)
| ≤
K
for
a
≤
x
≤
b
.
Trapezoidal Rule
.
T
n
=
Δ
x
2
(
f
(
x
0
) + 2
f
(
x
1
) + 2
f
(
x
2
) +
· · ·
+ 2
f
(
x
n
-
1
) +
f
(
x
n
))
The error bound in the Trapezoidal Rule is
|
E
T
| ≤
K
(
b
-
a
)
3
12
n
2
, where
K
is chosen so that
|
f
00
(
x
)
| ≤
K
for
a
≤
x
≤
b
.
Simpson’s Rule
(For
even
n
).
S
n
=
Δ
x
3
(
f
(
x
0
) + 4
f
(
x
1
) + 2
f
(
x
2
) + 4
f
(
x
3
) +
· · ·
+ 2
f
(
x
n
-
2
) + 4
f
(
x
n
-
1
) +
f
(
x
n
))
The error bound in Simpson’s Rule is
|
E
S
| ≤
K
(
b
-
a
)
5
180
n
4
, where
K
is chosen so that
|
f
(4)
(
x
)
| ≤
K
for
a
≤
x
≤
b
.
Two Maclaurin series
For
-
1
< x <
1,
ln(1 +
x
) =
∞
X
n
=1
(
-
1)
n
-
1
n
x
n
=
x
-
x
2
2
+
x
3
3
-
x
4
4
+
· · ·
and
arctan(
x
) =
∞
X
n
=0
(
-
1)
n
2
n
+ 1
x
2
n
+1
=
x
-
x
3
3
+
x
5
5
-
x
7
7
+
· · ·
Taylor’s Inequality
.
If
|
f
(
n
+1)
(
x
)
| ≤
M
for
|
x
-
a
| ≤
d
then
|
f
(
x
)
-
T
n
(
x
)
| ≤
M
|
x
-
a
|
n
+1
(
n
+ 1)!
for
|
x
-
a
| ≤
d
.
Math 1132
Final Exam Practice
Problem 1:
Indicate which of the following statements are true or false by circling T (true)
or F (false).
(i)
The trapezoidal rule for
Z
5
2
√
x dx
with
n
= 5 will be an overestimate.
T
F
(ii)
For the error bound in the Midpoint Rule approximation to
Z
3
1
dx
x
, the
T
F
smallest value of
K
that we can use is 2
/
27.
(iii)
The improper integral
Z
∞
1
dx
√
x
converges.
T
F
(iv)
The sequence
(
-
1)
n
n
2
n
2
+ 1
for
n
≥
1 converges.
T
F
(v)
The series 4
-
8
3
+
16
9
-
32
27
+
· · ·
+ (
-
1)
n
-
1
2
n
+1
3
n
-
1
+
· · ·
converges to 12.
T
F
(vi)
If lim
n
→∞
a
n
= 0 then
∞
X
n
=1
a
n
converges.
T
F
(vii)
ln(2
.
5) =
∞
X
n
=1
(
-
1)
n
-
1
(1
.
5)
n
n
.
T
F
(viii)
For all
x
, sin(
x
2
) =
∞
X
n
=0
(
-
1)
n
x
2
n
+2
(2
n
+ 1)!
.
T
F
(ix)
The orthogonal trajectories of the curves
y
=
kx
4
satisfy
dy
dx
=
x
4
y
.
T
F
(x)
The polar coordinates (
-
1
,
0) and (1
, π
) describe the same point.
T
F
Copyright c 2022 UConn Department of Mathematics.
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Math 1132
Final Exam Practice
Problem 2:
Which trig substitution should we use to integrate
Z
√
x
2
-
4
x
dx
?
(A)
x
= 2 sec
θ, dx
= 2 sec
2
θ dθ
(B)
x
= tan
θ, dx
= sec
2
θ dθ
(C)
x
=
1
2
sec
θ, dx
=
1
2
sec
θ
tan
θ dθ
(D)
x
=
1
2
tan
θ, dx
=
1
2
sec
2
θ dθ
(E)
x
= 2 sec
θ, dx
= 2 sec
θ
tan
θ dθ
Problem 3:
To evaluate
Z
x
+ 1
x
3
+ 9
x
dx
using the method of partial fractions, what is the
right expression for the integrand (the function being integrated)?
(A)
A
x
3
+
B
9
x
(B)
A
x
+
B
(
x
+ 3)
2
(C)
A
x
+
Bx
+
C
(
x
+ 3)
2
(D)
A
x
+
B
x
2
+ 9
(E)
A
x
+
Bx
+
C
x
2
+ 9
Copyright c 2022 UConn Department of Mathematics.
Page 2 of 14
Math 1132
Final Exam Practice
Problem 4:
For which of the following series can we apply the Integral Test directly?
I.
∞
X
n
=1
1
(
n
+ 2)
2
II.
∞
X
n
=1
(
-
1)
n
n
1
.
1
III.
∞
X
n
=3
ln(
n
)
n
(A) I only
(B) II only
(C) III only
(D) I and II only
(E) I and III only
Problem 5:
What is the smallest
N
so that the Alternating Series Remainder Theorem
guarantees the remainder
R
N
of the
N
-th partial sum of
∞
X
n
=1
(
-
1)
n
-
1
2
n
satisfies
|
R
N
| ≤
1
10000
?
(A) 4999
(B) 5000
(C) 5001
(D) 10000
(E) 10001
Copyright c 2022 UConn Department of Mathematics.
Page 3 of 14
Math 1132
Final Exam Practice
Problem 6:
If
f
(
x
) =
∞
X
n
=0
(
-
1)
n
x
6
n
+4
(2
n
+ 1)!
for all
x
then find the Maclaurin series for
f
0
(
x
).
(A)
∞
X
n
=0
(
-
1)
n
(6
n
+ 4)!
(2
n
+ 1)!
x
6
n
+3
(B)
∞
X
n
=0
(
-
1)
n
(6
n
+ 4)
(2
n
+ 1)!
x
6
n
+3
(C)
∞
X
n
=0
(
-
1)
n
x
6
n
+3
(6
n
+ 4)(2
n
+ 1)!
(D)
∞
X
n
=0
(
-
1)
n
x
6
n
+3
(6
n
+ 4)!
(E)
∞
X
n
=0
(
-
1)
n
x
6
n
+5
(6
n
+ 5)(2
n
+ 1)!
Problem 7:
Which is the general solution to the differential equation
dy
dx
=
-
y
+ 1
x
2
?
(A)
y
=
e
1
/x
-
1 +
C
(B)
y
=
Ae
1
/x
-
1
(C)
y
=
1
x
-
1 +
C
(D)
y
=
-
1
3
x
3
+
x
+
C
(E)
y
=
Ae
1
/x
Copyright c 2022 UConn Department of Mathematics.
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Math 1132
Final Exam Practice
Problem 8:
Select the integral that is the arc length of
y
=
e
3
x
for 0
≤
x
≤
1.
(A)
Z
1
0
(1 +
e
3
x
)
dx
(B)
Z
1
0
√
1 +
e
6
x
dx
(C)
Z
1
0
p
1 +
e
9
x
2
dx
(D)
Z
1
0
√
1 + 9
e
6
x
dx
(E)
Z
1
0
p
1 + 9
e
9
x
2
dx
Problem 9:
Choose the polar coordinates for the indicated intersection point
P
of the
curves
r
=
√
2
-
cos
θ
and
r
= cos
θ
shown below.
y
x
-
1
1
-
1
1
-
2
-
3
P
(A)
√
2
,
π
4
(B)
√
2
,
-
π
4
(C)
1
√
2
,
π
4
(D)
1
√
2
,
-
π
4
(E)
√
2
,
7
π
4
Copyright c 2022 UConn Department of Mathematics.
Page 5 of 14
Math 1132
Final Exam Practice
Problem 10:
The curve
r
= cos(2
θ
) is graphed below. Select the definite integral equal to
the area inside the loop crossing the positive
x
-axis.
y
x
(A)
Z
π/
4
-
π/
4
cos
2
(2
θ
)
dθ
(B)
Z
π/
4
0
cos
2
(2
θ
)
dθ
(C)
1
2
Z
π/
4
0
cos
2
(2
θ
)
dθ
(D)
1
2
Z
π/
3
0
cos
2
(2
θ
)
dθ
(E)
Z
π/
2
0
cos
2
(2
θ
)
dθ
Copyright c 2022 UConn Department of Mathematics.
Page 6 of 14
Math 1132
Final Exam Practice
Problem 11:
(a) Compute
Z
(
x
+ 3)
e
-
2
x
dx
using integration by parts.
Show all of your work to receive
credit
.
Final Answer:
(b) Find the value of
Z
∞
0
(
x
+ 3)
e
-
2
x
dx
if it converges, or say it diverges.
Show all of your
work to receive credit
.
Final Answer:
Copyright c 2022 UConn Department of Mathematics.
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Math 1132
Final Exam Practice
Problem 12:
Compute the following using partial fractions.
Show all of your work to
receive credit
.
(a)
Z
2
x
-
5
x
2
-
x
-
6
dx
Final Answer:
(b)
Z
x
+ 1
x
(
x
-
4)
dx
Final Answer:
Copyright c 2022 UConn Department of Mathematics.
Page 8 of 14
Math 1132
Final Exam Practice
Problem 13:
Compute
Z
cos
2
x dx
.
Show all of your work to receive credit
.
Final Answer:
Problem 14:
For the alternating series
∞
X
n
=1
(
-
1)
n
-
1
n
2
determine the smallest
N
so that the
Alternating Series Remainder Theorem guarantees the remainder
R
N
for the
N
-th partial
sum satisfies
|
R
N
| ≤
1
10
4
.
Show all of your work to receive credit
.
Final Answer:
Copyright c 2022 UConn Department of Mathematics.
Page 9 of 14
Math 1132
Final Exam Practice
Problem 15:
Determine whether the following series are absolutely convergent, condition-
ally convergent, or divergent.
Show all of your work to receive credit
.
a)
∞
X
n
=2
1
n
(ln
n
)
3
Final Answer:
b)
∞
X
n
=3
n
4
+ 2
n
n
2
+ 5
n
Final Answer:
c)
∞
X
n
=1
(
-
1)
n
-
1
√
n
2
+ 3
Final Answer:
Copyright c 2022 UConn Department of Mathematics.
Page 10 of 14
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Math 1132
Final Exam Practice
Problem 16:
Find the radius of convergence
and
interval of convergence for the series
∞
X
n
=1
(
-
1)
n
-
1
n
3
n
(
x
-
2)
n
.
Show all of your work to receive credit
.
Final Answer:
Copyright c 2022 UConn Department of Mathematics.
Page 11 of 14
Math 1132
Final Exam Practice
Problem 17:
a) Find the 2nd-degree Taylor polynomial
T
2
(
x
) centered at 4 for
√
x
.
Show all of your work
to receive credit
.
Final Answer:
b) Use Taylor’s inequality to give an upper bound on
|
√
x
-
T
2
(
x
)
|
in terms of
x
when
|
x
-
4
| ≤
1. You do
not
have to simplify your final expression.
Show all of your work to
receive credit
.
Final Answer:
Copyright c 2022 UConn Department of Mathematics.
Page 12 of 14
Math 1132
Final Exam Practice
Problem 18:
Solve the differential equation
dy
dx
=
x
2
y
2
with
y
(0) = 1.
Show all of your
work to receive credit
.
Final Answer:
Problem 19:
Find the equation for the tangent line to the parametric curve
x
=
t
2
-
3
t
,
y
=
√
t
when
t
= 9.
Show all of your work to receive credit
.
Final Answer:
Copyright c 2022 UConn Department of Mathematics.
Page 13 of 14
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Math 1132
Final Exam Practice
Problem 20:
Below are graphs of
r
= 3 sin
θ
and
r
= 1 + sin
θ
.
x
y
r
= 3 sin
θ
r
= 1 + sin
θ
a) Determine polar coordinates for the points where the curves cross besides the origin.
Final Answer:
b) Set up, but do
not
evaluate, an integral for the area of the region
inside
r
= 3 sin
θ
and
outside
r
= 1 + sin
θ
.
Final Answer:
Copyright c 2022 UConn Department of Mathematics.
Page 14 of 14