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\newcommand{\dsps}{\displaystyle}
\newcommand{\pp}{\par \noindent}
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\begin{document}

\noindent
\vfil \noindent
\large
\hfil Math 126 C - Spring 2009 \hfil \pp
\hfil Mid-Term Exam Number Two\hfil  \pp
\hfil May 14, 2009 \hfil \pp
\normalsize
\vfil
\medskip
\hfil Name: \hrulefill \hrulefill \hrulefill
%\hspace{0.5in}
% Section: \hrulefill
\vfil
\begin{center}
{\large
\begin{tabular}{||c|c|r||} \hline 
 1 & 10 &\hspace{10mm} \hfil\\ \hline 
 2 & 10 &       \\ \hline
 3 & 10 &      \\ \hline
4 &  10 &    \\ \hline
5 & 10 & \\ \hline
Total & 50 & \\ \hline 
\end{tabular}
}
\end{center}
\vfil
\begin{itemize}
\item Complete all questions. 
\item You may use a scientific, non-graphing calculator during this
examination.  
Other electronic devices are not allowed, and should be
turned off for the duration of the exam.
\item If you use a trial-and-error or guess-and-check method
when an algebraic method is available, you will not receive full credit.
\item You may use one hand-written 8.5 by 11 inch page of notes.
\item Show all work for full credit.  
\item You have 50 minutes to complete the exam. 
\end{itemize}
\vfil
.

\newp

\begin{enumerate}

\item The curve $y=x^x$ has one local extremum.  Find the curvature at that point.

\newp

\item An object moves so that its position at time $t$ is given by
\[
\vec{r}(t) = \langle t^2, t^3-4t, 0 \rangle.
\]

A portion of its path is shown below.

\epsfig{file=fig1.ps, 
 width=8cm,
 angle=270 }


Find all times when the object's velocity vector is orthogonal to its acceleration vector. 

\newp

\item Let 
\[
z = xe^y + y \sin x + \frac{x}{y}.
\]

\begin{enumerate}

\item Find $\dsps \frac{\partial z}{\partial x}$.

\vfil

\item Find $\dsps \frac{\partial z}{\partial y}$.

\end{enumerate}

\newp

\item You wish to build a four-sided box like the one shown in the figure,
with three rectangular sides perpendicular to a rectangular base.

You want the box to have a volume of 100 cubic centimeters.  

If all sides are to be made from the same thin material, what dimensions
will minimize the amount of material used?

Be sure to justify your answer using the Hessian (i.e., the second derivatives test).

\epsfig{file=box01.eps, 
 width=6cm,
 angle=0 }

\newp

\item Find the volume of the solid under the surface $z=xy^2$ and above
the triangle with vertices $(0,0)$, $(0,5)$ and $(2,3)$.

\end{enumerate}

\end{document}


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