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Math 2065
Review Exercises for Exam II
The syllabus for Exam II is Sections 2.2 { 2.6 of Chapter 2 and Sections 3.1 { 3.4 of Chapter 3.
You should review all of the assigned exercises in these sections. In addition, Section 2.1 contains
the main formulas for computation of Laplace transforms, and hence, while not explicitly part
of the exam syllabus, it will be necessary to be completely conversant with the computational
techniques of Section 2.1. For this reason, some of the exercises related to this section from the
last review sheet are repeated here. A Laplace transform table will be provided with the test.
Following is a brief list of terms, skills, and formulas with which you should be familiar.
²
Know how to use all of the Laplace transform formulas developed in Section 2.1 to be able
to compute the Laplace transform of elementary functions.
²
Know how to use partial fraction decompositions to be able to compute the inverse Laplace
transform of any proper rational function. The key recursion algorithms for computing
partial fraction decompositions are Theorem 1 (Page 108) for the case of a real root in the
denominator, and Theorem 1 (Page 117) for a complex root in the denominator. Here are
the two results:
Theorem 1
(
Linear Partial Fraction Recursion).
Let
P
0
(
s
)
and
Q
(
s
)
be polynomials.
Assume that
a
is a number such that
Q
(
a
)
6
= 0
and
n
is a positive integer. Then there is
a unique number
A
1
and polynomial
P
1
(
s
)
such that
P
0
(
s
)
(
s
¡
a
)
n
Q
(
s
)
=
A
1
(
s
¡
a
)
n
+
P
1
(
s
)
(
s
¡
a
)
n
¡
1
Q
(
s
)
:
The number
A
1
and polynomial
P
1
(
s
)
are computed as follows:
A
1
=
P
0
(
s
)
Q
(
s
)
ﬂ
ﬂ
ﬂ
ﬂ
s
=
a
and
P
1
(
s
) =
P
0
(
s
)
¡
A
1
Q
(
s
)
s
¡
a
:
Theorem 2
(
Quadratic Partial Fraction Recursion).
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 Winter '09

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