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{\Large {Mathematics 2C, Spring 1997}}\\
\large Review for Final Exam
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\begin{enumerate}

\item Limits of Sequences:  There will be a strong emphasis on
computing limits by looking at the power series of the functions (as
on the second midterm exam).  See homework from Section~11.9.

\item Taylor's formula and the expression for the remainder in
Taylor's formula.  You should be able to compute the Taylor series of
a given function $f$, and---in a simple case---be able to prove that
the series converges to $f$.  What you have to prove is that the
remainder converges to zero.  Try this for a simple function, for
example $f(x)=\sin x$.

\item Recursively defined Sequences: See Midterm I, or p.~583 no.~41.

\item Find limit of a given series (if it converges).  Look at
Midterm~I, Question 4 as well as p.~585 Example~3.

\item Test a given series for convergence.  Possible tests are:
\begin{description}
\item[Integral Test:] The most sensitive, but sometimes difficult to
apply.

\item[Comparison Test:] Use to compare the given series with a known
case, or to obtain a simpler series before using the integral test.

\item[Ratio Test:] Often easy to apply, but works only if the terms in
the series decay at least as fast as the terms in a geometric
series. Thus it cannot be used to test convergence of the $p$-series,
for example.  Use the ratio test to find the interval of convergence
of a power series (it will typically fail on the endpoints of the
interval and you will have to use one of the other tests).

\item[Root Test:] A close relative of the ratio test.

\item[Alternating Series Test:] The name says when it applies.  It
will never give you a statement about absolute convergence (as, for
example, the ratio test does)!

\item[$n$th Term Test for Divergence:] If the sequence of terms in the
series does not converge to zero, the series cannot converge.

\end{description}
Test your knowledge with the Chapter~11 review problems.  Be aware of
the difference between absolute and conditional convergence.

\item Power Series Computations: Products of Power Series (see
Question~1 on Midterm~II), tricks like on p.~632 under the red box up
to the heading ``Binomial Series'', and p.~635 Examples~7 and~8
(termwise differentiation and integration).

\item Parametric curves: What are they? How can you eliminate the
parameter?  How can you get some idea about how the graph looks like?
The example is likely to be in three dimensions, but the computations
are essentially as in two dimensions.

\item Integral computations: Remember your Math 2B (area under a
curve, arc length, volume of a solid of revolution by method of cross
section, volume of a solid of revolution by method of cylindrical
shells, surface area of a solid of revolution) and know how to use
substitution to use these integrals for parametric curves.  See
homework from Section~12.2.

\item Vector algebra in two and three dimensions.  Important concepts:
length of a vector, dot product, cross product.  Rules for the cross
product (red box on p.~702).  Likely exam question:  Find an equation
for a plane in some specific characterization (see your lecture
notes), find the angle between two planes, or the angle between a
plane and a line.

\item Cylinders and Quadratic Surfaces: Maybe.

\item Cylindrical and Spherical Coordinates: See homework problems (in
particular those among questions 11--30) from Section~13.7.

\end{enumerate}

General notes:

\begin{itemize}

\item No calculators.  A list of infrequently used formulae will be
supplied, if necessary (integrals, binomial series etc.).

\item If you want to leave from the exam early, you must do so at least 15
minutes before the end of the exam.

\item Remember the \emph{Give Up} option.

\end{itemize}

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