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

\item Limits and Continuity:

Compute the limit of a function $f(\boldsymbol{x})$ when
$\boldsymbol{x}$ is a vector.  (See homework problems from
Section~14.3.)  When can such limits fail to exist?  (See, for
example, p.~764 no.~27.)

\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.  Estimate errors of quantities derived
from measurements (p.~789 no.~31 and similar problems).

\item \textbf{Chain Rule:}

The chain rule needs a lot of practice, in abstract settings (e.g.,
derive the formula for implicit partial differentiation p.~795
eq.~(9), show that $\boldsymbol\nabla F$ is normal to the surface
$F(\boldsymbol{x})=0$, p.~797 no.~29) as well as in concrete cases
(some of the first problems in Section~14.7).

\item Directional Derivatives and the Gradient Vector:

Definition and geometrical interpretation.  In particular, see p.~807
no.~42.

\item \textbf{Lagrange Multipliers and Constrained Optimization
Problems:} 

Try to redo the homework problems from Section~14.5 which have
constraints by using the method of Lagrange Multipliers.  See homework
problems from Section~14.9, and maybe also some of the Chapter~14
review questions.

\item The Second Derivative Test in Two Variables:

Details TBA.

\end{enumerate}

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