% LaTex2e
\documentclass[12pt]{article}
\usepackage{amssymb,latexsym}

%%%%%%% \usepackage{anysize}
%\marginsize{left}{right}{top}{bottom}
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%%%%%% \marginsize{2cm}{2cm}{1cm}{3cm}

\pagestyle{myheadings}
\def\header{{\hfill Final \hfill A. Miller \hfill Fall 2010
                                       \hfill Math 222\hfill}}
\markboth\header\header

\newcount\probno
\newcount\total
\newwrite\examTBLFile

\probno=0

\def\message{$\;\;\;$ Show all of your work. 
Circle your answer. \par\bigskip}
\def\message{}

\def\nextpage{\vfill \newpage}
%\def\nextpage{}

\def\prob#1{\advance\probno by 1 \par\bigskip
    \nextpage
    \noindent\the\probno . (#1 pts)
    \message
    \advance\total by #1
    \immediate\write\examTBLFile{\the\probno
       \string& #1 \string&
       \string\qquad \string\\ \string\hline}}

%  \def\prob#1{\advance\probno by 1 \par  \bigskip   % \vfill \newpage
%      \noindent\the\probno . % (#1 pts)
%      \advance\total by #1
%      \immediate\write\examTBLFile{\the\probno
%         \string& #1 \string&
%         \string\qquad \string\\ \string\hline}}

\def\finishtbl{
    \immediate\write\examTBLFile{Total \string& \the\total
    \string& \qquad \string\\ \string\hline}
     }

\newwrite\ans
\immediate\openout\ans=answers
\outer\def\answer{\par\medbreak
  \immediate\write\ans{}
  \immediate\write\ans{\string\bigskip \the\probno . }
  \copytoblankline}
\outer\def\answerx{\par\medbreak
  \immediate\write\ans{}
  \immediate\write\ans{\string\bigskip \the\probno . }
  \copytoblankline}
\outer\def\putanswer{\par\medbreak
  \immediate\write\ans{}
  \copytoblankline}
\def\copytoblankline{\begingroup\setupcopy\copyans}
\def\setupcopy{\def\do##1{\catcode`##1=12}\dospecials
     \catcode`\|=12\obeylines}
{\obeylines \gdef\copyans#1
  {\def\next{#1}%
  \ifx\next\empty\let\next=\endgroup%
  \else\immediate\write\ans{\next}\let\next=\copyans\fi\next}}
\def\printanswers{\newpage\begin{center} Answers \end{center}
    \immediate\closeout\ans\input answers}

% \def\answer{\par\bigskip\bigskip\bigskip\noindent Answer: \par}
% \outer\def\answer{ {\par\bigskip Circle your answer.} \par\medbreak

\newenvironment{enum}
{\begin{list}{(\alph{enumi})}
{\usecounter{enumi}\setlength{\rightmargin}{\leftmargin}}}
{\end{list}}

\newenvironment{Enum}
{\begin{list}{(\Alph{enumi})}
{\usecounter{enumi}\setlength{\rightmargin}{\leftmargin}}}
{\end{list}}

\def\pr{\prime}
\def\vs{\vspace{1in}}

\def\conj(#1){\overline{#1}}

\def\vtor(#1,#2,#3){
\left(\begin{array}{r}
#1 \\
#2 \\
#3 \\
\end{array}\right)}



\begin{document}

\bigskip

\begin{center}
Instructions
\end{center}

No notes, no books, no calculators, no cell phones, no pagers,
no electronic devices of any kind.

Show all of your work. Circle your answer.

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

\bigskip 


\bigskip

\begin{minipage}[top]{2.5in}
Circle your section number below. 

Hand in to your TA.

\begin{verbatim}
Section
Number
 321  7:45  Zhao, Jie
 332  9:55  McMahon, Kayla
 322  8:50  Zhao, Jie
 328  1:20  Emrah, Elnur
 331  9:55  Kim, Yoosik
 324 11:00  McMahon, Kayla
 325 12:05  Wang, Kejia
 326 12:05  Emrah, Elnur
 327  1:20  Wang, Kejia
 330  2:25  Kim, Yoosik
\end{verbatim}
\end{minipage}
\begin{minipage}{2in}
\begin{center} \Large
 \begin{tabular}{||c|c|c||} \hline\hline
  Problem & Points & Score \\  \hline\hline
   \input{exam.tbl}
  \hline \hline
 \end{tabular}
\end{center}
\end{minipage}

\immediate\openout\examTBLFile=exam.tbl

\bigskip

Solutions will be posted shortly after the exam:

www.math.wisc.edu/$\sim$miller

\def\numx(#1){#1\par}

\setcounter{page}{0}
\newpage

%%%%%%%%% exam 1 %%%%%%%%%%%%%%%%%%%%%

\prob{12} 
Find
$$\int x\sqrt{1+x^2}\;dx$$

\answer $\frac12(1+x^2)^{\frac32}+C$

\prob{12}
Find
$$\int x^2 e^x \;dx$$

\answer $x^2e^x-2xe^x+2e^x+C$

\prob{12} %  124
Find
$$\int {1+x^3\over x(1+x)}\;\;dx$$

\answer $\ln|x|+\frac{\;\;x^2}2-x+C$

\prob{12}
Use the degree 1 Taylor polynomial for the appropriately chosen
function to estimate the value of $\sqrt{9.1}$.  Find
an upper bound for the absolute value of the error
$|\epsilon|$.

\answer $\sqrt{9.1}\approx\frac{181}{60}$ 
$\;\;\;|\epsilon|\leq \frac{1}{21600}$

%%%%%%%%%%% exam 2 %%%%%%%%%%%%%%%%%%%%

\prob{12} % 298
Find the general solution of
$$\frac{dy}{dx} = 1+y^2$$

\answer $y=\tan(x+C)$


\prob{12} 
Find the general solution of 
$$\frac{dy^2}{dx^2}+2\frac{dy}{dx}+y=x+1$$

\answer $y=(x-1)+(C_1+C_2\,x)\,e^{-x}$

%%%%%%%%%% post exam 2 %%%%%%%%%%%%%%%

\prob{12} % 378
\begin{itemize}
\item $||\vec{a}||=5$
\item $||\vec{b}||=6$, $\;\;$ and 
\item $||\vec{a}-\vec{b}||=7$.
\end{itemize}

What is the dot product: $\vec{a}\cdot \vec{b}$?

\answer $6$

\prob{12} % 392
Find the area of the triangle $ABC$ with corner points:
\begin{itemize}
\item $A=(1,2,0)$ 
\item $B=(2,3,1)$
\item $C=(1,5,4)$
\end{itemize}

\answer $\frac12 \sqrt{26}$

\prob{12} 
Find the angle $\theta$ at the corner $B$ in the triangle $ABC$
with corner points:
\begin{itemize}
\item $A=(1,2,0)$
\item $B=(2,3,1)$
\item $C=(1,5,4)$
\end{itemize}

\answer $\arccos(\frac{-4}{\sqrt{42}})$


\prob{12} % 404-5
Sketch the curve
$$\vec{x}(t)=(2+\cos t)\;\vec{i}\;\;+\;\;(3+\sin t)\;\vec{j}$$

\answer This a circle of radius one centered at $(2,3)$.


\prob{12}
Find the parametric equation $\vec{L}(t)=\vec{a}+t\vec{b}$
for the tangent line to the curve:
$$\vec{\rho}(t)=( t+1)\;\vec{i}\;\;+\;\;(t^{3})\;\vec{j}
\;\;+\;\;(t^2)\;\vec{k}$$
at $t=1$.

\answer $\vec{L}(t)=\vtor(2,1,1)+t\vtor(1,3,2)$

\prob{12}  % 418
Find the length of the curve
$$\;\vec{\gamma}(t)=(\cos t)\;\vec{i}\;\;+\;\;(t^{\frac32})\;\vec{j}
\;\;+\;\;(\sin t)\;\vec{k}$$
for $0\leq t\leq 1$.

\answer $\frac8{27}((\frac{13}{4})^{\frac32}-1)$

\def\tf{\par\bigskip\noindent
 {  {T $\;$ F $\;\;\;\;\;\;\;$}}}
\def\message{}

\prob{6} Circle T or F for True or False. No penalty for guessing.

True means that the equation holds for all three dimensional vectors.

\tf $\vec{b}\times \vec{a}=-(\vec{a}\times \vec{b})$
\tf $\vec{b}\cdot \vec{a}=\vec{a}\cdot \vec{b}$
\tf $\vec{a}\cdot (\vec{b}+\vec{c})=
(\vec{a}\cdot \vec{b})+(\vec{a}\cdot \vec{c})$
\tf $\vec{a}\times (\vec{b}+\vec{c})=
(\vec{a}\times \vec{b})+(\vec{a}\times \vec{c})$
\tf $(\vec{a}+\vec{b})+\vec{c}=\vec{a}+(\vec{b}+\vec{c})$
\tf $(\vec{a}\times\vec{b})\times\vec{c}=\vec{a}\times(\vec{b}\times\vec{c})$

\answer The last one is False.

\printanswers

\finishtbl
\end{document}



