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{\Large {Mathematics 2D, Fall 1997}}\\
\large Review for Final Exam
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\begin{enumerate}

\item Limits and Continuity:
Any exam problem will be similar like the one on the Midterm, see also
the homework problems from Section~14.3.

\item  Maxima and Minima of Functions:
Find Extrema values of a function $f$ by finding points at which all
partial derivatives are zero.  (See homework problems from
Section~14.5.)  Careful:
\begin{itemize}
\item An extremum may occur on the boundary of the domain of $f$.  You
may have to investigate the edges \emph{and} corners separately.
(See, for example, p.~782 no.~41.)
\item Not every critical point is an extremum, there can be inflection
points and saddles.  Use ``circumstantial evidence'' to classify the
critical points.
\end{itemize}

\item Linear Approximations:
Be able to find the equation for a plane tangent to a surface at a
given point.  Note that the gradient vector is a convenient way for
doing so, see p.~804 Example~5.  

\item Chain Rule:
Review the questions from the Midterm exam, and related problems. 

\item Directional Derivatives and the Gradient Vector:
Definition and geometrical interpretation.  In particular, see p.~807
no.~42.

\item Lagrange Multipliers and Constrained Optimization
Problems:
See homework problems from Section~14.9, and also some of the
Chapter~14 review questions.

\item Change of Variables in double and triple integrals:
Know the transformation formula (topic of this week), and be able to
apply it to simple examples (e.g.\ change from Cartesian to polar
coordinates).  The transformation formula is very important, therefore
I expect you to memorize it.  

Know how to change variables into polar, cylinder, and spherical
coordinates.

\item Change of the order of integration:
Here you should always sketch the region of integration, even if you
are not explicitly asked to do so in the exam (it is very hard to do
the problems otherwise).

Practice also some examples that involve triple integrals.

\item Evaluation of multiple integrals:
Most likely you will not be told which technique to use.  If direct
evaluation is difficult, you should consider
\begin{itemize}
\item changing the order of integration,
\item changing to polar/cylinder/spherical coordinates.
\end{itemize}
Practice to ``see'' which method works best with examples from the
Chapter~15 review (no. 1--6, 15).  Unfortunately the book always gives
hints as to which methods to use.  You should try to shuffle these
with other integrals, so that you learn to spot the ``right'' method.

\item Know how to compute area, volume, total mass, center of mass,
moments of inertia, and surface area by multiple integration.

There are plenty of problems in the homework, as well as in the
Chapter~15 review.  

\end{enumerate}

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