Topics for the 1st Midterm
Math 222, “Spring” 2009
Topics covered on this exam
The first midterm will be on the material covered in sections 1–16 of the lecture notes.
•
Integration.
You should be able to use the following methods to compute integrals.
–
Substitution
–
Integration by parts
–
Reduction formulas: know how to use them if you are given one, and know how to derive one.
–
Partial fractions.
•
Taylor polynomials.
–
You should know how to compute
T
a
∞
f
(
x
) (i.e. give a formula for the coefficient of
x
n
) by
computing
f
(
n
)
(
a
).
–
Use partial fractions to compute Taylor series of simple rational functions.
–
Use Lagrange’s formula for the remainder to estimate the error term
R
n
f
(
x
) if
x
and
n
are
explicitly given, or “for

x
 ≤
0
.
2”.
–
Understand the “littleoh” notation.
–
Use a combination of “littleoh”, “standard Taylor series”, algebra of Taylor polynomials (in
cluding “
T
n
[
f
0
(
x
)] = derivative of
T
n
+1
[
f
(
x
)]”) to efficiently compute Taylor polynomials.
Theory Questions
The following questions may not necessarily appear on the midterm, but answering them will help you find
gaps in your understanding of the material.
(1) State the Fundamental Theorem of Calculus.
(2) The integral
R
1
0
4(
x
+
a
)
3
d
x
contains two variables,
x
and
a
. Which of these is a dummy variable
and why?
(3) State and derive the formula for integration by parts.
(4) Explain to a friend what the partial fraction expansion of a rational function looks like and how you
find it.
(5) What is the Taylor polynomial of degree
n
of a function
f
(
x
) at the point
a
?
(6) State the Geometric Sum Formula.
(7) What is the difference between a Taylor and a Maclaurin polynomial?
(8) What is the remainder term in a Taylor polynomial?
(9) State Lagrange’s formula for the remainder term in a Taylor polynomial.
(10) What does it mean for a function
f
(
x
) to be
o
(
x
n
) as
x
→
0?
Sample Problems for the 1st Midterm
These are problems from 222 midterms that were given in the past, and they are relevant to the coming first
midterm. Other old exams are available on the web, but be warned that the syllabus changes every year,
so that problems which were on the 1st midterm 2 years ago may show up on this year’s 2nd midterm, and
vice versa
. Don’t prepare for last year’s midterm!
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