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\noindent
\vfil \noindent
\large
\hfil Math 126 C - Autumn 2010 \hfil \pp
\hfil Mid-Term Exam Number Two\hfil  \pp
\hfil November 23, 2010 \hfil \pp
\hfil Answers \hfil \pp
\normalsize

There were two versions of the exam.

\medskip

Version A - In problem 1, $f(x,y)=\frac{9}{4}xy^2+y^3-x$.

\begin{enumerate}

\item There are two critical points: $(-4/9,2/3)$ and $(4/9,-2/3)$ and
they are both saddle points.

\item (a) $\dsps \frac{1}{2}e^4-\frac{3}{2}e^3-\frac{1}{2}e+\ln 2 -\frac{9}{8}$
(b) $\dsps \frac{1}{4} \sin 64$

\item $4 \pi$

\item $t = \frac{1}{2} \sin^{-1} \frac{2}{3.3} \approx 0.32554929$

\item (a) $z=5x-4y+8$ (b) There are infinitely many such pairs.  One pair is
$(1,1,0)$ and $(5,0,5)$.

\end{enumerate}

\bigskip

Version B - In problem 1, $f(x,y)=\frac{1}{4}xy^2+y^3-x$.

\begin{enumerate}

\item There are two critical points: $(-12,2)$ and $(12,-2)$ 
and they are both saddle points.

\item (a) $\dsps \frac{1}{2}e^4-\frac{3}{2}e^3-\frac{1}{2}e+\ln 2 -\frac{9}{8}$      
(b) $\frac{1}{6} \sin 144$

\item $\pi$

\item $t = \frac{1}{2} \sin^{-1} \frac{2}{4.05} \approx 0.25824277$

\item (a) $z=7x-3y+8$ (b) There are infinitely many such pairs.  One pair is
$(0,0,0)$ and $(1,\frac{-10}{3},3)$.

\end{enumerate}

\end{document}


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