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\def\header{{\hfill Exam 4 \hfill A. Miller \hfill Spring 2001
                                       \hfill Math 217\hfill}}
\markboth\header\header
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\probno=0
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       \string\qquad \string\\ \string\hline}}

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       \string\qquad \string\\ \string\hline}}

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\begin{document}

\bigskip

Show all work. {\bf Circle your answer.}

No books, no notes, no
electronic devices at all.

Solutions will be posted shortly after the exam:
www.math.wisc.edu/$\sim$miller/m217

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

\begin{center}
{\bf Hand in to your TA -- Circle your DIScussion section (column one):} \\
\end{center}
\begin{center}
\begin{tabular}{l|l|l|l}
DIS 321 & 7:45   TR  & 901  Van Vleck & Forray, Susan Jane     \\
DIS 322 & 8:50   TR  & 6224 Soc Sci   & Bisgard, James William \\
DIS 324 & 9:55   TR  & 123  Ingraham  & Bisgard, James William \\
DIS 326 & 11:00  TR  & B235 Van Vleck & Leete, Jennifer A.     \\
DIS 327 & 11:00  TR  & 113  Ingraham  & Lu, Tianshi            \\
DIS 328 & 12:05p TR  & B235 Van Vleck & Leete, Jennifer A.     \\
DIS 329 & 12:05p TR  & 6228 Soc Sci   & Lu, Tianshi            \\
DIS 330 & 1:20p  TR  & B321 Van Vleck & Forray, Susan Jane     \\
DIS 332 & 2:25p  TR  & B135 Van Vleck & Hur, Youngmi           \\
DIS 334 & 3:30p  TR  & B329 Van Vleck & Hur, Youngmi           \\
\end{tabular}
\end{center}





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\begin{center} \Large
 \begin{tabular}{||c|c|c||} \hline\hline
  Problem & Points & Score \\  \hline\hline
  \input{exam.tbl}
  \hline \hline
 \end{tabular}
\end{center}

\immediate\openout\examTBLFile=exam.tbl

\setcounter{page}{0}

\prob{10} % Exam 1 problem 4
Establish the formula
$$\sum_{k=1}^n 2k=2+4+\cdots+2n= n + n^2$$
for every positive integer $n$, by
showing
\bigskip
\par (a) the formula is correct for $n=1$, and
\bigskip
\bigskip
\par (b) if true for $n$, the formula is also true for $n+1$.

\answerx See Exam 1 problem 4.
\par\bigskip (a) $\sum_{k=1}^1 2k=2=1^2+1$
\par\bigskip (b) $\sum_{k=1}^{n+1}2k= \sum_{k=1}^{n}2k\;\;+2(n+1)=
n^2+n+2(n+1)=n^2+3n+2$
\par and $(n+1)^2 +(n+1)=n^2+2n+1+n+1=n^2+3n+2$  and so
$$\sum_{k=1}^{n+1}2k= (n+1)^2 +(n+1)$$

\prob{10} % Exam 3 problem 3
Find  all $\theta$ which solve the equation:
$$2\sin(\theta)\cos(\theta)\;-\;2\sin(\theta)\;-\;\cos(\theta)\;+\;1=0$$
\answer See Exam 3 problem 3.  $(2\sin\theta -1)(\cos\theta -1)=0$
\par $\theta=2\pi n$ or
\par $\theta={\pi\over 6}+2\pi n$ or
\par $\theta={5\pi\over 6}+2\pi n$
\par for $n=0,1,-1,2,-2,\ldots$

\prob{10} % P 4.3 27
Find $x$ if $\log_x(3x)={2\over 3}$
\answer $x={1\over 27}$

\prob{10} % P 4.5  42
Solve the logarithmic equation for $x$.
$$\ln(x)+\ln ({{x+2}})=\ln(3)$$
\answer  $x=1$


\prob{10} % C 6.9  14
Find $\int \; t \;2^{t^2}\;dt$.
\answer ${1\over{2\ln(2)}}2^{t^2}+C$

\prob{10} % C 6.10  9
Find the derivative of $y=10^{\log_2(x)}$
\answer ${\log_2(10)\over x}10^{\log_2(x)}$ or
${\ln 10\over \ln 2}{1\over x}e^{{\ln 10\over \ln 2}(\ln x) } $


\finishtbl
\printanswers

\end{document}