MAC 2312
April 24, 2003
Final Exam
Prof. S. Hudson
Name
Show all your work and explain your reasoning. Don’t use your own paper, a calculator
or book. You may ask for paper or for clariﬁcations.
1) [5pts] Find the average value of
f
(
x
) =
x
4
on the interval [0
,
2].
2) [10pts] Approximate ln 2 two ways (you may leave your answers as fractions):
A) Give a Taylor series for ln(1 +
x
) and set
x
= 1 (this should produce a familiar
series). Then ﬁnd the 3rd partial sum. Give an error estimate for this answer.
B) Give the integral deﬁnition of ln(
x
) and set
x
= 2. Estimate the integral using
Simpson’s rule with
n
= 2. Don’t do an error estimate.
3) [5 pts each] Compute (or explain why no answer exists):
R
tan
2
x
sec
2
x dx
R
x
2
e
x
dx
R
4
0

x

2


1
/
2
dx
=
R
+
∞
e
dx
x
ln
x
4) [5pts] Use a famous power series to ﬁnd the McLaurin series for
f
(
x
) =
x
sin(
x
2
).
5) [10pts] Find the interval of convergence of the series
∞
X
n
=1
3
n
x
n
n
3
6) [10pts] Find the area of the the region inside
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 Summer '08
 Storfer
 Calculus, Taylor Series, dx, error estimate, Prof. S. Hudson

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