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\noindent
\vfil \noindent
\large
\hfil Math 126 D - Spring 2011 \hfil \pp
\hfil Mid-Term Exam Number One\hfil  \pp
\hfil April 26, 2011 \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 
 

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{\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 Find the equation of the plane which contains the point $(3,2,1)$ and the line of intersection of the planes
\[
x+2y-z=5
\]
and
\[
2x-y+4z=1.
\]

\newp

\item Consider the curve $C$ defined by the vector function $\vec{r}(t)=\langle t^3,t^2,\frac{1}{t} \rangle$.

Find the curvature of $C$ at the point $(8,4,\frac{1}{2})$.

\newp

\item A particle moves in three-dimensional space. Its position at time $t>0$ is given by
the vector function
\[
\vec{r}(t) = \langle 3t^2,\frac{2}{t},5t+2\rangle .
\]

\begin{enumerate}

\item At what value(s) of $t$ is the speed of the particle a minimum?

\vfil

\item Find the tangential component of this particle's acceleration vector at time $t=1$.

\end{enumerate}

\newp

\item Find all points on the curve 
\[
x = 3t^2+2t, \,\,\, y=5t^2-t
\]
at which the tangent line to the curve passes through the point $(0,-4)$.

You only need to specify the $t$ values of these points, not their coordinates.

\newp

\item Find the area of the triangle with vertices $(0,1,2)$, $(-1,2,-1)$ and $(3,2,1)$.

\end{enumerate}


\end{document}


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