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\begin{document}

\noindent
\vfil \noindent
\large
\hfil Math 126 C - Spring 2008 \hfil \pp
\hfil Mid-Term Exam Number Two\hfil  \pp
\hfil May 22, 2008 \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
6 & 10 & \\ \hline
Total & 60 & \\ \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,
or read a numerical solution from a graph on your calculator,
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 Find the equation of the plane through the points
$(0,1,2)$, $(1,3,4)$, and $(5,5,5)$.

\newp

\item Let $L$ be the line
\[
x=1+2t, y=2+t, z=3+3t.
\]

\begin{enumerate}

\item Find the point of intersection of $L$ with the line
\[
x=19+t, y=13-\frac{1}{2}t, z=43-5t.
\]
\vfil

\item For what value of $a$ will the plane 
\[
ax+6y-7z=2
\]
\textit{not} intersect the line $L$?

\end{enumerate}

\newp

\item Consider the 3D curve defined by the position function
\[
\vec r(t) = \langle t, 4-t^2, \ln t \rangle.
\]

\begin{enumerate}

\item  Find the point on the curve 
at which the tangent vector is parallel to the vector $\langle 2, -1, 8 \rangle$.

\vfil

\item Find the parametric equations of the tangent line at the point you found in
part~(a).
\end{enumerate}

\newp

\item A particle is moving so that its position function is
\[
\vec{r}(t) = \langle t, t^2, \frac{1}{t^3} \rangle
\]

Find all times $t$ at which the tangential component of the particle's acceleration vector is
equal to zero.

\newp

\item Find and classify all critical points of the surface
\[
z=f(x,y) = xy^2+y-x.
\]

\newp

\item Find the points of maximum curvature on the curve $y=x^3$.



\end{enumerate}

\end{document}


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