Math 1220-002
Midterm #2
Spring 2012
November 5, 2012
Midterm #2: Improper Forms & Infinite Series
Instructions: You may use a calculator to assist you with any arithmetic, a writing
utensil, and your 3x5 reference card that you have prepared. No other materials
are permitted. Please show all of your work to the level of detail that is specified
in the instructions for that particular problem. You MUST write your final answer
in the space provided to receive full credit. The point value of each problem is
specified in the problem statement. Most of the credit will be awarded for doing
the calculus correctly and showing the appropriate steps. A correct final answer
with no work supporting it will receive very few points!! You may turn in your
exam and leave when you have finished.
Recommendation: You do not need to work on these problems in order. I would
suggest first answering the ones you find the easiest. Then go back and work on
the harder ones at the end. Also, if you finish early I would suggest double
checking all of your work, just in case you made any very simple mistakes.
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Spring 2012
November 5, 2012
Math 1220-002
Midterm #2
Use L’Hopital’s Rule or a related method in order to evaluate the following limits. Write
your final answers in the boxes below but show your work in the space below each
problem. (15 points)
2.
(a)
sin(x) tan (x)
x-40
xsin(x)
—
urn
=
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:
()
=
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tvr
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cs6)
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(b)
7x
urn
xi
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7x
i 7 1( i4)
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7
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(c)
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urn [ln(5x
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=
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Spring 2012
November 5, 2012
Math 1220-002
Midterm #2
Evaluate the following improper integrals. Write your final answers in the boxes below
3.
but show your work in the space below each problem. If the integral goes to infinity or
negative infinity write “DNE” or “WTF” in the answer box. (10 points)
(a)
r
5
571
dx=
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2
J_4+x
2:
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(b)
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x
J
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5L/rrj
dx=
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a
J_- t—c
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j
L-
r
—
u2cJu
I-f
(JZ
+c
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ma1()/
Spring 2012
November 5, 2012
Math 1220-002
Midterm 442
Find the exact values of the following infinite sums and partial sums. Write your final
4.
answer for each problem in the boxes but show all of your work in the space below. (20
points)
HINT: All of the following sums are either geometric or collapsing. For collapsing sums, try to
identify the pattern.
(a)
V5k
2k+1
3
L
k=1
± f5k
7— 32H
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(b)
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4/
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s
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(c)
10
1 k+1
+3(i)
((3kI
5
=
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(±)
‘
)/3
.
(d)
+
100
k)
)
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fl)
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Spring 2012
November 5, 2012
Math 1220-002
Midterm #2
Use either the integral test, the ratio test, or the comparison test determine whether
5.
the following series converge or diverge. If necessary you may also use a combination of these
tests. Write your final answer for each problem in the boxes but show all of your work in the
space below. (15
points)
(a)
V5k
Lk!
k= 1
I
I
Cn::,
5—
ç4:
I
lVl
c
(b)
Z
k=1
2—,-
k+1
2 + 2k
k
k?
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—
LJ
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(c)
1n(k)
k
k=1
I
(lV-c
k(
r
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Spring 2012
November 5, 2012
Math 1220-002
Midterm #2
Use any method we have learned so far in class to determine whether the following
6.
absolutely convergent, conditionally convergent, or divergent. Write your final
are
series
answer for each problem in the boxes but show all of your work in the space below. (20 points)
(a)
f_fl k
k+1
fk
k=1
;
H
I
‘‘Jr-rJfrI
f
(k-. 2 k”
_k1i:
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k
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f rV’ 5
0
fr- + I
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(b)
:
k=1
k sin (k)
3 + 1
k
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(v
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ln(1)
k
14
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