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\begin{document}

\noindent
\vfil \noindent
\large
\hfil Math 126 C - Spring 2008 \hfil \pp
\hfil Mid-Term Exam Number One\hfil  \pp
\hfil April 24, 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 & 20 &\hspace{10mm} \hfil\\ \hline 
 2 & 20 &       \\ \hline
 3 & 10 &      \\ \hline
4 &  20 &    \\ \hline
5 & 10 & \\ \hline
Total & 80 & \\ \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 Let $\dsps f(x)=\frac{1}{3-\frac{1}{4}x}$.

\begin{enumerate}
\item Find the second-order Taylor polynomial $T_2(x)$ for $f(x)$
based at $b=0$.

\vfil

\item Give a bound on the error
\[
|f(x)-T_2(x)|
\]
for $-0.5 \le x \le 0.5$.
\end{enumerate}


\newp

\item Let 
\[
h(x) = \int_0^x t \cos t^3 \, dt
\]

\begin{enumerate}
\item Find the first four non-zero terms of the Taylor series based at $b=0$ for $h(x)$.

\vfil

\item Use your answer to part (a) to compute $h^{(20)}(0)$.

\end{enumerate}

\newp

\item Find the first three non-zero terms of the Taylor series for 
\[
g(x) = \frac{x}{2+x^2}
\]

\newp

\item Consider the polar curve defined by the equation
\[
r = \theta (12- \theta)
\]
for $0 \le \theta \le 12$.  The curve is shown in the figure below.

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

\begin{enumerate}
\item Find the slope of the tangent line to this curve at $\theta=\frac{\pi}{2}$.

\vfil \vfil

\item Find the value of $\theta$ corresponding to the self-intersection 
point indicated by the arrow.

\end{enumerate}


\newp

\item There are infinitely many unit vectors which are orthogonal to the vector
$\langle 3, -2, 4 \rangle$.  Give an example of one of them.





\end{enumerate}

\end{document}


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