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\def\header{{\hfill Final \hfill A. Miller \hfill Spring 2004
                                       \hfill Math 222\hfill}}
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    \string& \qquad \string\\ \string\hline}
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%%% \outer\def\answer{ {\par\bigskip Circle your answer.} \par\medbreak

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\begin{document}

\bigskip

Show all work.  
Simplify your answers.  
Circle your answer.

You may use a cheat sheet: one $8 {1\over 2} \times 11$ sheet of paper with
anything written on both sides.

No books, no calculator, no cell phones, no pagers, no electronic devices. 


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

\bigskip
Hand in to your TA.  Circle your TA's name:

{\large
\begin{tabbing}
xxxxxxxxx \=Ben Ellison xxxxxxxxxxxxxxx \=          Derek Moffitt \kill
\> Adam Berliner \>        Dan McGinn \\
\> Ben Ellison \>          Derek Moffitt \\
\> Jon Godshall \>         Richard Oberlin \\
\> E. Alec Johnson \> \\
\end{tabbing}
}

\bigskip

\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

\bigskip
 
Solutions will be posted shortly after the exam:
www.math.wisc.edu/$\sim$miller/m222

\setcounter{page}{0}

\newpage 

\def\i{{\bf i}}
\def\j{{\bf j}}
\def\k{{\bf k}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\prob{8} % 18.2 - 
Find the general solution of the differential equation:
$$y^{\prime\prime}-4y=e^x$$

\answer $y=-{1\over 3}e^x+C_1e^{2x}+C_2e^{-2x}$.

\prob{8} % 12.6 - 9
Find the polar coordinates of the point with Cartesian coordinates:
$$(-{\sqrt{3}\over 2},{\sqrt{3}\over 2})$$

\answer $r=\sqrt{3}$, $\theta={3\over 4}\pi$

\prob{8} % 13.3 - 13
Find a vector that has the same direction as $3\i - 4\j$ but twice
its length. 

\answer $6\i - 8\j$.

\prob{8} % 14.2 - 7
Find the $\cos(\theta)$ where
$\theta$ is the angle between the vectors $\i-\j+\k$ and
$2\i+\j+\k$.

\answer $\cos(\theta)={2\over\sqrt{18}}={\sqrt{2}\over 3}$

\prob{8} % 8.4 - 45  - exam 1
Find $$\int e^{x}\cos(x)\;dx$$

\answer ${1\over 2}(e^x\cos(x)+ e^x\sin(x))+C$

\prob{8}  % 9.4 -  29 - exam 1
Evaluate this improper integral or show that it diverges:
$$\int_1^e {dx\over x\ln(x)}$$

\answer It diverges $\ln(\ln(x))]^e_1=\infty$.

\prob{8}  % 10.4 - 25 - exam 1
Determine whether the following series converges or diverges and indicate
which test you use.
$${1\over 2\sqrt{3}}+{1\over 3\sqrt{4}}+{1\over 4\sqrt{5}}+\ldots$$

\answer It converges by comparison to the convergent series
$\sum {1\over n^{3/2}}$.

\prob{8}  % 10.6
Find the power series $\sum_{n=0}^\infty a_nx^n$ for the function
$$f(x)={1\over {1-4x^2}}$$
and determine the interval where it converges.

\answer $\sum 4^nx^{2n}$.  Converges for $-{1\over 2}<x< {1\over 2}$.
 

\prob{8} % 13.4 - 33
In the plane a curve ${\bf r}(t)$ satisfies 
$${\bf r^{\prime\prime}}(t)= \i\;+\;e^t\;\j$$
$${\bf r}^{\prime}(0)=2\i\;+\j$$
$${\bf r}(0)=\i\;+\j$$ 
Find the curve ${\bf r}$.

\answer ${\bf r}(t)=({t^2\over 2}+2t+1)\i\;\;+\;\; e^t\j$

\prob{8} % 14. 4 - 17
Find the equation, $ax+by+cz=d$, 
of the plane that contains the intersecting lines:
$${\bf L}_1(t)=(1,1,1)\; +\; t(1,-2,1)$$
$${\bf L}_2(t)=(2,-1,2)\;+\; t(-1,1,1)$$ 

\answer $ (-1,1,1)\times (1,-2,1)= (3,2,1)$ so equation is
\par $3x+2y+z=d$.  Plug in $(1,1,1)$ to find out $d=6$. 
Hence equation is $3x+2y+z=6$.

\prob{8} % 14.5 - 29 
Find the curvature $\kappa$ for the curve
$${\bf r}(t)= \cos(t)\;\i\;\;+\;\; \sin(t)\;\j\;\;+\;\; t\;\k$$
as a function of $t$.

\answer  $\kappa={1\over 2}$

\prob{8} % 12.3 - 13
A tangent line to the ellipse $x^2+2y^2=1$ intersects the
$y$-axis at $(0,2)$.  Find the $y$-coordinate of the point of tangency.

\answer Let $(x,y)$ be the point of tangency.  Then the tangent
line has slope
$${y-2}\over x$$
By implicit differentiation of $x^2+2y^2=1$ we get that
$$x+2y{dy\over dx}=0 \mbox{ hence } {dy\over dx}={x\over{-2y}}$$
Equating gives
$${{y-2}\over x}= {x\over{-2y}} 
\;\;\;\;\;\;\;\;\;\;\;(-2y)(y-2)=x^2=1-2y^2$$
$$4y=1\;\;\;\;\;\;\;\;\;\;\;\;
y={1\over 4}$$


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\prob{4}  
Use this page for scratch paper or for any comments you would like to
make about the course.


(You will get four points whether or not you write anything -
I just wanted the exam to add up to 100.)



\finishtbl

\printanswers

\end{document}


