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\begin{document}

\noindent
\vfil \noindent
\large
\hfil Math 126 D - Spring 2011 \hfil \pp
\hfil Mid-Term Exam Number Two\hfil  \pp
\hfil May 17, 2011 \hfil \pp
\hfil Answers \hfil \pp

\normalsize

\medskip

\begin{enumerate}

\item There are three critical points: (0,0), which is a local minimum,
and $(\pm\sqrt{2},-1)$ which are both saddle points.

\item The tangent plane is $z=(e+1)x-1$.

\item The optimum box should have a square base with sides of length
$\dsps \frac{2}{\sqrt[3]{5}}$ meters, and height $\frac{5^{2/3}}{2}$ meters.

\item (a) $e^2-e$ (b) $-\frac{1}{6} \cos 64 + \frac{1}{6}$

\item $$\iint\limits_D xy \, dA = \int_{\pi/4}^{\pi/2} \int_1^2 r^3 \cos \theta \sin \theta
\, dr \, d\theta = \frac{15}{16}$$


\end{enumerate}


\end{document}


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