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\hskip36pc \= Math.\ 270-05\\
\>Fall, 2005\\
\>R. B. Kearfott
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\begin{center}
\bf Second Examination \\
\it Monday, September 26, 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. Each entire problem is worth 25 points.
You may keep this exam sheet.}

\Large
\begin{enumerate}
   \item  \label{prob:sketch-derivative}
          Sketch a graph of the derivative of the
          function in Figure~\ref{fig:sketch-derivative}.
          \begin{figure}[h]
            \centerline{\includegraphics[scale=1.0]{tst2fig1.eps}}
            \caption{\label{fig:sketch-derivative} The function $f$
            for problem~\protect{\ref{prob:sketch-derivative}}}
          \end{figure}
   \item Use the definition of limit and algebraic manipulations to
         evaluate the following limits.
         $$\textrm{(a)}\quad
            \lim_{h\rightarrow0} \frac{\frac{1}{x+h} - \frac{1}{x}}{h}
            \qquad\textrm{(b)}\quad
            \lim_{h\rightarrow0} \frac{(x+h)^3 - x^3}{h}
         $$
   \item Draw the graph of a function $f(x)$ such that $f(0)=1$,
         $f'(0) = 1$, $f''(x)< 0$ for $x>0$,  $f'(1)=0$, and
         $f(2)=0$.
   \item Compute $df/dt$ algebraically, for the following functions
         $f$. (Assume that all letters occurring except for $f$ and
         $t$ represent constant numbers.)
         $$\textrm{(a)}\quad
              f(t) = -\frac{1}{2}g t^2 + v_0 t + s_0
           \qquad\textrm{(b)}\quad
              f(t) = t^3 + 3t^2 + 3t + 1
         $$
\end{enumerate}
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