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\begin{document}

\noindent
\vfil \noindent
\large
\hfil Math 126 C - Spring 2010\hfil \pp
\hfil Mid-Term Exam Number Two\hfil  \pp
\hfil May 13, 2010 \hfil \pp
\normalsize
\vfil

\medskip
\hfil Name: \hrulefill \hrulefill \hspace{0.5in} Student ID no. : \hrulefill

\vfil

\hfil Signature: \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 & 20 &      \\ \hline
4 &  10 &    \\ \hline
Total & 50 & \\ \hline 
\end{tabular}
}
\end{center}
\vfil
\begin{itemize}
\item Complete all questions. 
\item You may use a scientific calculator during this
examination; graphing calculators and other electronic devices
are not allowed and should be
turned off for the duration of the exam.

\item If you use trial-and-error, a guess-and-check method, or numerical
approximation
when an exact method is available, you will not receive full credit.

\item You may use one double-sided, 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 A particle moves along a curve in the $xy$-plane so that its position vector is
\[
\vec{r}(t) = \langle t + \cos t, t- \sin t \rangle
\]
for $t\ge 0$.  Assume $t$ is in seconds, and coordinates are in centimeters.

\begin{enumerate}
\item Find the speed of the particle at time $t=\pi$.
\vfil
\item There are infinitely many times $t$ when the velocity vector and the acceleration vector for this particle
are orthogonal.  Give one of these times.
\vfil
\end{enumerate}

\newp

\item Find the curvature of the curve 
\[
x= t^2, y= 1-t, z=1-t^2
\]
at the point $t=3$.

\newp

\item Let 
\[
f(x,y)=   \frac{1}{x}+\frac{1}{y} + x + y.
\]
\begin{enumerate}
\item Find a point on the surface $z=f(x,y)$ where the tangent plane is parallel to the plane
$48x+6y+2z=7$.
\vfil
\item Find and classify all critical points of the surface $z = f(x,y)$.
\vfil

\end{enumerate}


\newp

\item Let $R$ be the region in the first quadrant of the $xy$-plane bounded by 
$y=6-x$, $y=6-2x$, and the $x$-axis.

Express the volume of three-dimensional space lying above $R$ and below the
surface
\[
z=xy
\]
as \textbf{one} iterated double integral.



\end{enumerate}

\end{document}