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\begin{tabbing}
\hskip36pc \= Math.\ 270-05\\
\>Fall, 2005\\
\>R. B. Kearfott
\end{tabbing}
\begin{center}
\bf First Examination \\
\it Monday, September 12, 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.}

\begin{enumerate}
   \item  In 1626, Peter Minuit purchased the island of Manhattan (on
          behalf of the Dutch West India Company) from
          the indigenous residents for trinkets worth approximately
          $\$24.00$. Suppose that, instead, Mr. Minuit had put his money
          in the stock market, with an average return of
          $10\%$  per year,
          \begin{enumerate}
             \item Derive a function $A(t)$ that expresses the
                  amount of money $A$ that Mr. Minuit (or his heirs)
                  would have in their account $t$ years after 1626.
             \item What is $A(1776-1626)$?
             \item What is $A(2005-1626)$?
             \item Do you think Peter Minuit would have been better
                   off purchasing Manhattan or putting his money in
                   the stock market?  For comparison, the present total
                   worth of goods and services produced in all of
                   the United States per year is approximately
                   $\$11,750,000,000,000=11.75\times10^{12}\textrm{
                   dollars}$.
             \item Do you think this reasoning is appropriate, when
                   applied over such a long period of time?  What
                   assumptions that we used when we wrote down
                   $A(t)$ might have been violated?
          \end{enumerate}
   \item Arrange the following in order of increasing rate of
         growth, as $x\rightarrow\infty$.
         $$\begin{array}{rlrl}
         \textrm{(a)} &f(x) = \sqrt{x} &
         \textrm{(b)} &f(x) = \log{x} \\
         \textrm{(c)} &f(x) = x^4 &
         \textrm{(d)} &f(x) = e^{5x} \\
         \textrm{(e)} &f(x) = x^2 &
         \textrm{(f)} &f(x) = x^3 + 1,000,000,000,000x + 1 \\
         \textrm{(g)} &f(x) = .0000000000001e^x &
         \textrm{(h)} &f(x) = 1,000,000,000e^{0.0000000001x}  \\
         \textrm{(i)} &f(x) = \displaystyle\frac{x^2+1}{x^2-2x+1}
         \end{array}$$
   \item Solve the following equation for $t$.
         $$2^{24} = 10^t.$$
   \item Provide numbers equal to the following limits, if they
         exist, but say so if the limit does not exist.
         $$\begin{array}{rlrlrl}
         \textrm{(a)} &\ds\lim_{x\rightarrow\infty} \frac{x^2-1}{x^2+1} &
         \textrm{(b)} &\ds\lim_{x\rightarrow1} \frac{x^2-1}{x^2+1} &
         \textrm{(c)} &\ds\lim_{x\rightarrow\infty} \sin(x)
         \\ \\
         \textrm{(d)} &\ds\lim_{x\rightarrow\pi/2} \tan(x) &
         \textrm{(e)} &\ds\lim_{x\rightarrow\infty} \frac{\ln(x)}{x} &
         \textrm{(f)} &\ds\lim_{x\rightarrow\infty} \frac{x}{\ln{x}
         }
         \end{array}$$

\end{enumerate}
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