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\begin{tabbing}
\hskip36pc \= Math.\ 455-01\\
\>Spring, 2005\\
\>R. B. Kearfott
\end{tabbing}
\begin{center}
\bf First Examination \\
\it Thursday, February 24, 2005
\end{center}

\noindent {\scriptsize {\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. This exam is closed book. You may leave after handing in
your exam paper, but be sure to check your answers carefully.}

\begin{enumerate}

\item (20 points) \label{prob:poly-approx}
 Find a polynomial that approximates
      $$f(x) = \left\{\begin{array}{cl}
                       \frac{\sin(x)}{x} &x\ne 0 \\
                        1                & \mbox{otherwise.}
               \end{array}\right\}$$
to within an absolute error of $10^{-5}$ on the interval
$x\in[-0.1,0.1]$.

\item (40 points) Consider:
      \begin{equation}\label{eq:sineprime-quotient}
          \frac{f(x+h)-f(x)}{h},
      \end{equation}
       where $f$ is as in problem~\ref{prob:poly-approx}.
      \begin{enumerate}
         \item Compute the quantity using
            (i) three significant digit rounded decimal arithmetic;
            (ii) all the digits in your calculator.
         In the first case, recall that the number of significant digits
         is the number of leading non-zero digits.  Also, round
         $f(x+h)$ and $f(x)$ properly before subtracting.
         \item Assuming the representation computed from your
               calculator is exact,
               compute the absolute and relative error in
               the above calculation.
         \item Using $f(0)=1$ and
               $f([0.1]\in[0.998,0.999]$, use interval
               arithmetic to obtain a lower bound and an upper
               bound (i.e.\ an enclosure) of the exact quotient.
      \end{enumerate}

\item (20 points)  Answer:
      $$|\cos(x) - 1| = \bigoh{?}\qquad\mbox{as $x\rightarrow0$}.$$
      Show your computations.

\item (20 points) Use Newton's method to compute an approximation to
      the smallest positive $x$ such that $x=\cos(x)$.  Iterate
      until the first four digits do not change.
      ({\em Hint: You may wish to graph the two functions to obtain
         an idea of what the initial guess should be.})

\end{enumerate}

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