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Exam III and Key
Aug 5, 2004
MAC 2312
S Hudson
1) (5pts) Find the Taylor polynomial of order n=2 centered at
a
= 1 for
f
(
x
) =
√
x
.
2) (5pts) Find the McLaurin Series (a=0) for
f
(
x
) = sin(
x
2
) by substituting into a known
series.
3) (10pts) Find the sum of the series (or write ”Diverges”):
a)
∑
∞
k
=0
π
+1
π
k
b)
∑
∞
k
=1
1
k
+2

1
k
+3
4) (40pts) Compute the integrals (or write ”diverges”). Show all your work.
a)
R
∞
1
ln(
x
)
x
2
dx
b)
R
+1

1
dx
x
2
c)
R
x
2
(
x
+2)
3
dx
d)
R
dx
√
9

4
x
2
e)
R
sec
4
x
tan
2
x dx
f)
R
1
x
3
+4
x
dx
g)
R
2
x
+5
x
2
+4
x
+5
dx
h)
R
ln(1 +
x
)
dx
1
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View Full Document5) (10pts) For each series, answer either Converges (C) or Diverges (D) and state which
“test” you are using.
a)
∑
∞
k
=1
2
k
k
2
+1
b)
∑
∞
k
=1
(

1)
k
k
2
k
+3
6) (10pts) Choose ONE proof, explain thoroughly:
a) Calculate
R
sec(
x
)
dx
, explaining each step.
b) State and prove the Divergence Test.
7) (20pts) Answer True or False:
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 Summer '08
 Storfer
 Calculus

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