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\begin{tabbing}
\hskip36pc \= Math.\ 270-05\\
\>Fall, 2005\\
\>R. B. Kearfott
\end{tabbing}
\begin{center}
\bf Final Examination \\
\it Wednesday, December 7, 2005 10:15AM to 12:45 PM
\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 16 points, and 4 points are free. You may
keep this exam sheet.}

\begin{enumerate}
   \item Radioactive carbon 14, produced by interaction of
         carbon dioxide in the air with cosmic rays, is absorbed by
         all living things, and the proportion of carbon 14 to
         regular carbon 12 in plants and animals is almost constant
         while the plant or animal is alive.  However, after the
         plant or animal dies, the carbon 14 in its tissue decays
         exponentially, with a half-life of 5700 years.
         \begin{enumerate}
            \item Write down an equation for the percentage of
                  carbon 14 left in the tissue after $t$ years.
            \item At an archaeological site in New Mexico, charcoal
                  from an ancient fire is determined to have $29\%$
                  of the original carbon 14 remaining.
                  Approximately how long ago did the fire burn?
         \end{enumerate}
   \item Find a value of the constant $k$ such that
         $$\lim_{x\rightarrow1}\frac{x^2-kx+4}{x-1}$$
         exists.
   \item An object is oscillating at the end of a spring.  Its
         position, in centimeters, relative to a fixed point, is
         given as a function of time $t$ in seconds by
         $$y = y_0 \cos(2\pi\omega t), \quad\textrm{with $\omega$
           a constant.}
         $$
         \begin{enumerate}
            \item Find an expression for the velocity and
                  acceleration of the object.
            \item Show that the function $y$ satisfies the
                  differential equation
                  $$\frac{d^2 y}{dt^2} + 4\pi^2 \omega^2 y = 0.$$
         \end{enumerate}
   \item A spherical cell is growing at a constant rate of $400$
         cubic micrometers per day.  At what rate is its radius
         increasing when the radius is 10 micrometers?
   \item Find parametric equations for a circle of radius 3 centered
         at the point $(1,2)$ and traced counterclockwise.
   \item \begin{enumerate}
         \item Graph $f(x) = x(x+1)(x-2)$.
         \item Find the total area between the graph and the
               $x$-axis between $x=-1$ and $x=2$.
         \item Find $\ds\int_{-2}^1 f(x)\/dx$ and interpret it in
               terms of areas.
         \end{enumerate}
\end{enumerate}
\end{document}
