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\begin{tabbing}
\hskip36pc \= Math.\ 301-02\\
\>Summer, 2005\\
\>R. B. Kearfott
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\begin{center}
\bf Third Examination \\
\it Wednesday, July 13, 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 16 points, and 4 points are free.}

\begin{enumerate}
   \item  A cylindrical tank with height 30 feet and radius 10 ft
          sits on a tower so that the bottom of the tank is 75 feet
          above the ground.  Suppose that water is pumped from the
          ground to fill the tank.  How much work, in ft.-lb.,
          does it take to fill the tank?  (Water weighs about 62.4
          pounds per cubic foot.)
   \item  Suppose price at which $q$ widgets can be sold is $c(q) =
          1500-0.1(\sqrt{q})$, whereas the price at which producers are
          willing to sell $q$ widgets is $p(q) = 220 + 0.1\sqrt{q}$.
          \begin{enumerate}
             \item Find the equilibrium quantity and equilibrium
                   price.
             \item Compute the consumer surplus and the producer
                   surplus.
          \end{enumerate}
   \item Suppose a probability density of a non-negative quantity
         $x$ is given by $p(x) = \frac{1}{4}e^{-.25x}$ for $x\ge 0$ and
         $p(x)=0$ for $x < 0$.
         \begin{enumerate}
            \item Write down an expression for the cumulative
                  probability $P(x)$.
            \item What is the probability that the quantity is
                  between $1$ and $2$?
            \item What is the mean?
            \item What is the median?
         \end{enumerate}
   \item Suppose you put $100$ at the beginning of each month into
         a savings account that earns $0.25\%$ interest per month,
         paid at the end of the month.
         \begin{enumerate}
            \item Using the ``$\sum$" notation, write down the
                  amount you will have in the account after the
                  beginning of month $n+1$ (starting with $n=0$ for
                  the initial month).
            \item Write down this amount in closed form.
            \item How much will you have after 5 years?
         \end{enumerate}
   \item Does
         $$\sum_{i=0}^\infty \frac{(-1)^i}{i!}$$
         converge?  State why.  \newline
         \textit{Note: $i!$ is the product
         of the first $i$ integers, where $0!$ is defined to be
         $1$.}
   \item Compute the radius of convergence of each of the
         following.
         $$\mbox{(a)}\quad\sum_{i=0}^\infty \frac{x^i}{i!}
           \qquad\mbox{(b)}\quad\sum_{i=1}^\infty
           \frac{(-1)^{i-1}}{i} x^i$$
\end{enumerate}
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