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\def\header{{\hfill Exam 2 \hfill A. Miller \hfill Fall 2001
                                       \hfill Math 213\hfill}}
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\begin{document}

\bigskip

Show all work.  Circle your answer.

No books, no notes, no calculator, no cell phones, no pagers, no
electronic devices at all.

Solutions will be posted shortly after the exam:
www.math.wisc.edu/$\sim$miller/m213

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

\begin{center}
{\bf Circle your DIScussion section (column one):} \\
\end{center}

TA: Youngsuk Lee

\begin{center}
\begin{tabular}{l|l|l}
 DIS 301 & 8:50  T   & 6322 SOC SCI   \\
 DIS 302 & 8:50  R   & 215 INGRAHAM   \\
 DIS 303 & 9:55  T   & 225 INGRAHAM   \\
 DIS 304 & 9:55  R   & 495 VAN HISE   \\
\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}

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\prob{5} % 9.2
Find all first and second order partial derivatives of the function:
$$f(x,y)=e^{x+y^2}$$
\answer $f_x=e^{x+y^2}=f_{xx}$, $f_y=2ye^{x+y^2}=f_{xy}=f_{yx}$,
$f_{yy}=2e^{x+y^2}+4y^2e^{x+y^2}$.

\prob{5} % 8.3
Find the average of the function $$f(x)={1\over{x\ln(x)}}$$
on the interval $[2,5]$.
\answer  Substitue $u=\ln(x)$ and $du={1\over x}dx$.
$${1\over {5-2}}\int^5_2 {1\over{x\ln(x)}}dx=
{1\over {3}}(\ln(\ln(5)) -\ln(\ln(2)))$$

\prob{6} % 9.1
Graph the first octant of the plane determined by the equation:
$$x+3y+2z = 6$$
\answer The plane intersects the three axis at
$(6,0,0)$ and $(0,2,0)$ and $(0,0,3)$


\prob{8} % 8.4
Determine whether the integral below converges or diverges and
find its value if it converges.
$$\int_1^\infty {2x\over{(x^2+1)^3}}\;dx$$
\answer Converges to $1\over 8$.
Substitute $u=x^2+1$ and $du=2x dx$.


\prob{8} % 8.3
The function $$f(x)=2000e^{-.01x}$$ represents a flow of money in
dollars per year over 3 years.  Assume 5\% per year compounded continuously. 
Find
\par (a) the present value
\par (b) the accumulated amount after 3 years.
\answer \par (a)
$\int_0^3 e^{-.05x}2000e^{-.01x}dx=2000\int_0^3e^{-.06x}dx=
{2000\over .06}(1-e^{-.18})$
\par (b) $(e^{.15})$ times part (a)
% 5490.993
% 6379.6236

\prob{10} % 9.3
Find the critical points of the function and classify each as
either saddle points or relative (or local) maximums or minimums.
$$f(x,y)=2x^2-4xy+y^4+2$$
\answer Saddle at $(0,0)$, loc mins at $(1,1)$ and $(-1,-1)$.

\prob{8} % 8.2
The graph of the function $y=\sqrt{x}$ for $x$ such that $0\leq x\leq k$
is rotated around the $x$-axis, i.e. $y=0$.  The volume of
the solid of rotation is $8\pi$.  What is $k$?
\answer $$\int ^k_0\pi y^2\;dx=
\int ^k_0\pi x\;dx=\pi {k^2\over 2}=8\pi$$ so $k=4$.


\finishtbl
\printanswers

\end{document}




