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\pagestyle{myheadings}
\def\la{\langle}  \def\ra{\rangle}
\def\header{{\hfill Exam 2 \hfill A. Miller \hfill Fall 99
                                       \hfill Math 211\hfill}}
\markboth\header\header
\newcount\probno

\probno=0
\def\prob{\advance\probno by 1  \vfill\newpage  % \par
    \noindent\the\probno . }

\newwrite\ans
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  \immediate\write\ans{}
  \immediate\write\ans{\string\bigskip \the\probno . }
  \copytoblankline}
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\def\printanswers{\newpage\begin{center} Answers \end{center}
    \immediate\closeout\ans\input answers}

\begin{document}

\bigskip

\begin{center}
{%\Large
No calculators.  No notes.  No books. \\
Show all your work.  Write clearly and carefully.  \\
Little partial credit will
be given for the correct answer with incorrect or no work.  \\
Cross out any
work or answers which you do not want graded.  \\
Circle your answer.\\
Hand in your exam to your TA.
}
\end{center}

\par\vskip .25in
\begin{center}
Name\makebox[3in]{\hrulefill} \\
\end{center}

\begin{center}
Circle your TA section: \\
\end{center}

\begin{verbatim}
211 002 321     Lang, Mike           7:45  TR  B333 Van Vleck
211 002 322     Lang, Mike           8:50  TR   202 Bascom
211 002 324     Rhee, Kyoungwon      9:55  TR   301 Educ Sci
211 002 326     Rhee, Kyoungwon      11:00 TR  3331 Sterling
211 002 327     Hartwig, Brian       12:05 TR   346 Birge
211 002 328     Lee, Byungyoon       12:05 TR   341 Van Vleck
211 002 329     Lee, Byungyoon       1:20  TR   341 Van Vleck
211 002 331     Hartwig, Brian       2:25  TR   114 Ingraham
211 002 332     Kethireddypally      2:25  TR   223 Ingraham
211 002 334     Kethireddypally      3:30  TR  B325 Van Vleck
\end{verbatim}

\bigskip

\begin{center} \Large
 \begin{tabular}{||c|c|c||} \hline\hline
  Problem & Points & Score \\  \hline\hline
%  \input{exam.tbl}
%  \hline \hline
1& 4 & \qquad\\ \hline
2& 4 & \qquad\\ \hline
3& 4 & \qquad\\ \hline
4& 4 & \qquad\\ \hline
5& 4 & \qquad\\ \hline
6& 4 & \qquad\\ \hline
7& 4 & \qquad\\ \hline
8& 4 & \qquad\\ \hline
9& 4 & \qquad\\ \hline
10& 4 & \qquad\\ \hline
Total & 40& \hskip 2em\relax \\ \hline
 \end{tabular}
\end{center}

\setcounter{page}{0}


% 4.3-31
\prob Find $dy\over dx$ by logarithmic differentiation:
$$y={{ x(x-2)(x+3)}\over {(x-4)(x+5)}}$$
\answer ${{ x(x-2)(x+3)}\over {(x-4)(x+5)}}
\left({1\over{x}}+{1\over{x-2}}+{1\over{x+3}}-{1\over{x-4}}-
{1\over{x+5}}\right)$

% 4.4-35
\prob A manufacturer experiences weekly costs of
$$C(x)= 4000+500x+e^{5x}$$
in producing $x$ items.  If the items can be sold for \$1000 each, find
the weekly production level $x$ that maximizes profits.
\answer $x={1\over 5}\ln(100)$

% 4.5-29
\prob When a foreign substance is introduced into the body, the body's
defense mechanisms move to break down the substance and excrete it.  The
rate of excretion is usually proportional to the concentration in the body,
and the half-life of the resulting exponential decay is referred to as the
{\it biological half-life} of the substance.
\par If, after 8 hours, 10\% of a
massive dosage of a substance has been excreted by the body, what is the
biological half-life of the substance?
\answer ${\ln({1\over 2})}\over {{1\over 8}\ln({9\over 10})}$

%5.6-23
\prob Find the area bounded by the curves $y=x^3$ and $y=x^2$.
\answer $1\over 12$

%5.7-5
\prob Find the consumer's surplus for the demand function,
$$p=D(x)={8\over {x+1}}$$
when the {\bf price} is \$2.
\answer $8\ln(4)-6$

%5.7-31
\prob An investor wishes to find an annuity capable of producing
a continuous revenue stream of \$50000 per year for 10 years,
beginning 15 years from now.  What is the present value of this revenue
stream, assuming an interest rate of 6\%?
\answer ${50000\over (-.06)}\left(e^{-(.06)25}-e^{-(.06)15}\right)$

% 5.8-25
\prob The population of the earth is presently 6 billion people.
This number is expected to grow exponentially to 12 billion in the
next 30 years.  What will be the average population of the earth
during this 30 year period?
\answer $6\over\ln(2)$ billion

% 6.3-35
\prob Find the maximum and minimum values of
$$f(x)=x+cos(x)$$
in the interval $[0,\pi]$.
\answer minimum $1$ at $x=0$, maximum $\pi-1$ at $x=\pi$.

% 6.4-21
\prob Find
$$\int^{\sqrt{\pi/2}}_0 \;\; x\;\sin^4(x^2)\;\cos(x^2)\;dx$$
\answer ${1\over 10}$

\prob  Prove the Fundmental Theorem of Calculus:
\par Let $f$ be continuous
on $[a,b]$ and let $F$ be any antiderivative for $f$,
i.e., $F^\prime(x)=f(x)$ for all $x$.
Then
$$\int^b_a f(x)dx = F(b)-F(a)$$
\answer Textbook page 408

\printanswers

\end{document}