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\large
\begin{tabbing}
\hskip23pc \= Math.\ 270-03\\
\>Spring, 1997\\
\>R. B. Kearfott
\end{tabbing}
\begin{center}
\bf First Examination \\
\it Thursday, Friday, February 14, 1997
\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. 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 20 points.}

\begin{enumerate}

\item \label{prob:sales-tax}
      The amount of sales tax collected in a certain city
      rose at a constant rate from 1960 until 1973.  Between
      1973 and 1982, the amount of sales tax collected increased
      exponentially.  Between 1982 and 1986, the amount decreased
      gradually.  From 1986 until 1990, the amount decreased
      exponentially.  From 1990 through 1996, the amount increased
      exponentially, and in 1996, the amount exceeded the amount
      collected in 1982.
      \begin{enumerate}
      \item State which of the graphs in
            Figure~\ref{fig:city_increase} best approximates the graph
            of the sales tax collected versus time in years.
      \item (important): Describe ,
            individually, why each of the other graphs is not a
            good representation.
      \end{enumerate}

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\caption{\label{fig:city_increase}The graphs for
problem~\protect{\ref{prob:sales-tax}}}
\end{figure}

\item A body of mass $m$ is falling downward with velocity $v$.
      Newton's Second Law of Motion, $F=ma$, says that the net
      downward force $F$ on the body is proportional to its
      downward acceleration $a$.  The net force $F$ consists of the
      force $F_g$ due to gravity, which acts downward, minus
      the air resistance $F_r$, which acts upward.  The force
      due to gravity is $mg$, where $g$ is a constant.  Assume that
      the air resistance is proportional to the velocity of the
      body.
      \begin{enumerate}
      \item Write an expression for the net force $F$ as a function
            of the velocity $v$.
      \item Write a formula giving $a$ as a function of $v$.
      \item Sketch $a$ against $v$.
      \end{enumerate}

\item In the middle of the nineteenth century, a prominent London
citizen remarked that ``If the number of horses in this city
continues to increase at the present rate, by 1950 all London streets
will be 6 feet deep in horse manure."  Assume that the statement
was made in 1850, that there were 50,000 horses in London then,
and that each horse is responsible for $1/500,000$ of a foot of horse
manure in the street.
   \begin{enumerate}
   \item Write down an exponential model for the amount of horse
         manure in the street as a function of time.  Call the
         rate constant $\alpha$, and let $t=0$ correspond to
         1850.
   \item If there were indeed 6 feet of horse manure in London
         in 1950, then compute the rate constant $\alpha$.
   \item With that $\alpha$, compute the percentage increase
         per year in the number of horses in London.
   \item What is wrong with applying such an exponential model
         over the period 1850--1950?
   \end{enumerate}

\item Each of the following functions tends to $\infty$ as
      $x\rightarrow\infty$, but they increase at different
      rates.  Rank them from slowest to fastest;  that is,
      list the one whose graph is on the bottom for large $x$
      first, then the one whose graph is next from the bottom,
      etc.
$$\begin{array}{lrcllrcl}
\qquad\mbox{(a)} & f(x) &=& e^{0.0001 x} &
\qquad\mbox{(b)} & f(x) &=& \sqrt{x} \\
\qquad\mbox{(c)} & f(x) &=& x^2 &
\qquad\mbox{(d)} & f(x) &=& \log(x) \\
\qquad\mbox{(e)} & f(x) &=& x^{1/3} &
\qquad\mbox{(f)} & f(x) &=& x^3 \\
\qquad\mbox{(g)} & f(x) &=& x^2 + e^{x} &
\qquad\mbox{(h)} & f(x) &=& e^{x^2}
\end{array}$$

\item The earth's orbital axis changes its orientation with
      respect to fixed stars, or {\em precesses} with a period of
      approximately 22,000 years.  A very rough model for the earth's
      climate assumes that the primary cause of ice ages is this
      precession.  The minimum temperature during the last ice
      age was roughly 11,000 years ago, so that, according to
      the rough model the earth is near the maximum global
      temperature today.  Assume that the temperature today
      is roughly $10^\circ$ centigrade warmer than it was 11,000
      years ago.
      \begin{enumerate}
      \item Write down a trigonometric model for the deviation
            from average temperature of the world, as a function
            of time $t$.  Take the present time to be $t=0$.
      \item According to this model, how many degrees cooler
            will the earth be 1000 years from now?  How many
            degrees cooler was the earth in the year 997 AD?
      \end{enumerate}
      (Note: A more realistic trigonometric model is based on
             the Milankovich theory, in which three different
             periodic processes affect climate.  Modern theories
             also include alterations to the atmosphere due to
             human activities.)

\end{enumerate}

\end{document}
