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                                       \hfill Math 213\hfill}}
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\begin{document}

\bigskip

Show all work.  Circle your answer.

You may use your cheat sheet, one $8.5 \times 11$ inch paper with
anything you want written on either side.

Otherwise, no books, 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:} \\
\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}
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\prob{10} % exam 3  combines two problems from exam 1
Solve the differential equation:
$${dy\over dx}={\ln(x+1)\sqrt{y}\over{e^{\sqrt{y}}}}$$
Find an equation relating $x$ and $y$, you need not explicitely solve for
$y$.
\answer $$2e^{\sqrt{y}}=(x+1)\ln(x+1) -x+C$$

\prob{10} % exam 1
Find $$\int {\ln(x)\over x}dx$$
\answer $${(\ln(x))^2\over 2}+C$$

\prob{10} % exam 2
Find the critical points of the function and classify each as
either saddle points or relative (or local) maximums or minimums.
$$f(x,y)=x^2+4xy+y^2+6x+2$$
\answer Saddle at $(1,-2)$.

\prob{10} % exam 2
The graph of the function $y={x}^3$ for $x$ such that $0\leq x\leq 1$
is rotated around the $x$-axis, i.e. $y=0$.  Find the volume of
the solid of rotation.
\answer ${\pi\over 7}$

\prob{10} % 9.4 Lagrange - exam 3
A rectangular box with a square bottom and no top is to be built
so as to have volume 48 cubic inches.  The cost of the material for making
the square bottom is 3 cents per square inch.  The cost of the material
for making the four sides is 2 cents per square inch.  Find the dimensions
of the box, ie. bottom $x\times x$ and height $y$ which minimize the
cost.
\answer $4\times 4 \times 3$

\prob{10} % 10.3 Euler #6   exam 3
Use Euler's method to find an approximation solution to
 $${dy\over dx}=1+y \mbox { and } y=0 \mbox{ when } x=1$$
Use a step size of $h=\triangle x={1\over 10}=.1$ to find the approximate
value of $y$ when $x=1.2$.
\answer $.21$

\prob{10} % 9.6  Double integration #50 similar to exam 3.
Evaluate the double integral below.  The function below is impossible
to integrate symbolically as it is.

$$\int_0^1 \int_{y}^1\;e^{x^2}\;dx\; dy$$

\par (a) Draw the region $A$ over which the integration takes place. Shade
the region $A$.

\par (b) Describe $A$ in two different ways, i.e. $A=\{(x,y):\;???\;\}$

\par (c) Interchange the limits of integration.

\par (d) Integrate.

\answer (d) ${1\over 2}(e-1)$

\prob{10} % 12.2 Annuity a geometric series. #55
Tamara wants to buy a boat that she estimates will cost \$2,000 when she buys
it. How much money must she deposit at the beginning of each month in
order to have enough after one year to pay for her boat?  Assume
${1\over 2}$\%
interest per month compounded monthly
(6\% annual).  The first payment is made on Jan
1, 2002, the second on Feb 1, 2002, and so forth until the last payment is
made on Jan 1 2003, and the boat is bought on that day for \$2,000.
What is her monthly payment $p$?
\answer There are 13 payments.  For $i=.005$
$$p+p(1+i)+p(1+i)^2+\cdots+p(1+i)^{12}=2000$$
so $p({{1-x^{13}}\over{1-x}})=2000$ for $x=(1+i)$ and
hence $p=2000 ({{1-x}\over{1-x^{13}}})$

\prob{10} % 12.5
\par (a) Find the Taylor series for the function:
$$f(x)={4\over{4-x^2}}$$
\par (b) Determine the radius of convergence $r$,
converges for all $x$ with $-r<x<r$.

\answer $${4\over{4-x^2}}=1+{1\over 2}x^2+{1\over 4}x^4+\cdots =
\sum {1\over 2^n} x^{2n}$$
It converges for all $x$ such that $-2<x<2$, radius is $r=2$.



\prob{10} % new
\par (a) Find the Taylor polynomial of degree two near zero for the function:
$$f(x) =\sqrt{16+x}$$
\par (b) Use this polynomial to approximate
$$\sqrt{17}$$

\answer \par (a) $p(x)=4+{1\over 8}x-{1\over 512}x^2$
\par (b) $4\;{63\over 512}$


\finishtbl
\printanswers

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