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\hskip36pc \= Math.\ 301-02\\
\>Summer, 2005\\
\>R. B. Kearfott
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\begin{center}
\bf Fourth Examination \\
\it Monday, July 25, 2005
\end{center}

\noindent {\small {\bf Instructions:} This exam should be done on
your own paper. Your name should be on each sheet and on the back of
the last sheet; the answers should appear written carefully and in
order. If in doubt, show intermediate steps: Full credit may not be
given, even for correct answers, unless work is arranged clearly and
explained. This exam is closed book. You may leave after handing in
your exam paper, but be sure to check your answers carefully. You
may keep this exam sheet. Each entire problem is worth 25 points.}

\Large
\begin{enumerate}
   \item Compute the Taylor polynomial of degree 5 for
         $f(x) = \cos(x)$, expanded about $x=\pi/2$.
   \item  \label{prob:compute-taylor-coeffs}
          \begin{enumerate}
             \item Write down the degree 4 Taylor polynomial for $f(x)$,
                   centered at $x=0$, where
                   $$f(x) = \left\{\begin{array}{rl}
                    \displaystyle
                    \frac{\sin{x}}{x} & \mbox{if $x\ne0$}, \\
                    1 & \mbox{if $x=0$.}\end{array}\right.
                   $$
             \item Is this Taylor polynomial an overestimate or
                   underestimate for $f$, for $-\pi/2\le x \le \pi/2$?
          \end{enumerate}
   \item What degree Taylor polynomial should be used to ensure that
         $f(x)$ is approximated with an error of at most $10^{-10}$
         for $|x|\le 1$, for $f$ as in
         Problem~\ref{prob:compute-taylor-coeffs}?
   \item Compute the radius of convergence of the Taylor series for
         $$f(x) = \ln(1+3x),$$ expanded about $x=0$.
\end{enumerate}
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