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\def\header{{\hfill Final Exam \hfill A. Miller \hfill Fall 2013
                                  \hfill Math 221\hfill}}
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\newcount\probno
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\probno=0

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

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

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

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\def\printanswers{\newpage\begin{center} Answers \end{center}
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%  Answer: \par\bigskip\bigskip}
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\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.

Hand in to your TA.

\begin{verbatim}
Section
Number
 421 Cheng, Jingrui   07:45
 422 Cheng, Jingrui   08:50
 424 Lynch, John      09:55
 429 Lynch, John      01:20
 426 Powers, Michael  11:00
 427 Powers, Michael  12:05
 430 Soule, Michelle  01:20
 431 Soule, Michelle  02:25
 432 Charles, Zachary 07:45
 433 Charles, Zachary 02:25
\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

\setcounter{page}{0}
\newpage


\prob{12}
Find the area of the bounded region between
$y=1-x^2$ and $y=x+1$.

\answer $\frac16$

\prob{12}
Find the length of this curve:
$$x=2t^3+1 \;\;\; \;\;\; y=3t^2\;\;\;\;\;\;
   \;\;\; 0\leq t\leq 1$$

\answer $2(2^{\frac32}-1)$ or $4\sqrt{2}-2$.


\prob{12} % exam 1
$$\lim_{x\to 2} \;3x-1\;\; = \; 5$$
Show that this is true using an $\epsilon-\delta$ argument.

\answer $|(3x-1)-5|=3|x-2|\leq 3\delta=\epsilon$ if
$|x-2|\leq\delta$ and $\delta=\frac{\epsilon}3$.

\prob{12} % exam 1
$$\frac{d}{dx}(2x^2+1)=4x$$
Show that this is true directly from the definition of
derivative.  No points will be given for quoting rules of
differentiation.

\answer $\frac{f(x)-f(a)}{x-a}\to f'(a)$ as $x\to a$.
$$\frac{(2x^2+1)-(2a^2+1)}{x-a}=\frac{2(x^2-a^2)}{x-a}=
\frac{2(x-a)(x+a)}{x-a}=2(x+a)\to 4a \mbox{ as } x\to a$$

\prob{12} % exam 1
Find the derivatives of

\bigskip
$$(a)\;\;\;\;\;f(x)=\tan(1+x^2)$$

\vfill
$$(b)\;\;\;\;\;g(x)=2x^3+e^{\sin(x)}$$

\vfill

\answer (a) $(\sec(1+x^2))^2\, (2x)\;\;\;\;$ (b) $6x^2+e^{\sin(x)}\cos(x)$

\prob{12}  % exam 1 and 2
For what values of $a$ and $b$ is it true that $f$ is
a differentiable function?

$$f(x)=\left\{
\begin{array}{ll}
2x+1     & \mbox{ if $x\leq 1$ }\\
ax^2+b  & \mbox{ if $x>1$ }\\
\end{array}\right.$$

\answer $a=1$ and $b=2$


\prob{12}
A particle is constrained to move along the
graph of the equation $$y=x^3+x^2+x+1$$  For what values of $x$
are the $x$ and $y$ coordinates necessarily changing at the same rate?

\answer $x=0$ and $x=-\frac23$.  

\prob{14} %  5.12 
 $$f(x)=\frac{\ln(x)}{x}$$
Find intervals where it is increasing or decreasing.  Find
local maxima and minima. Find inflection points and intervals
where it is concave up or down. Find limits at $\infty$ and
$0^+$.  Sketch the graph.

\answer Critical point at $x=e$, inflection point at $x=e^{\frac32}$.

\prob{14} % exam 2 angenent exam2
You are designing a mural and you would like to have a margin
of 1 feet on the left, right, and top of the artwork, but none
on the bottom. If you allow 32 ft$^2$ for the area containing
the artwork itself, what dimensions should the artwork have if you
want to minimize the total area of the wall (i.e., of the artwork
and the margins.)

\answer Artwork 8 feet wide and 4 feet high. 



\prob{12} % 
Find the limit or show that it is not defined:

\bigskip
$$(a) \;\;\;\;\;\lim_{x\to 1} \;\frac{1-x}{\sqrt{x}-x}$$

\vfill
$$(b) \;\;\;\;\;\lim_{x\to \infty} \;\;(\sqrt{x^2+x}\;-\;x)$$

\vfill

\answer (a) $2\;\;\;$ (b) $\frac12$

\prob{13}
$R$ is the region below the curve $y=\frac1{\sqrt{1+x^2}}$
between $x=0$, $x=1$, and above the $x$-axis.
Equivalently 
$$R=\{(x,y)\;: 0\leq x \leq 1 \mbox{ and } 0\leq y\leq \frac1{\sqrt{1+x^2}}\}$$
Find the volume of the solid obtained by rotating $R$ around
the $x$-axis.

\answer $\frac{\pi^2}4$

\prob{13}
For the same $R$ region as in the previous problem:
$$R=\{(x,y)\;: 0\leq x \leq 1 \mbox{ and } 0\leq y\leq \frac1{\sqrt{1+x^2}}\}$$
Find the volume of the solid obtained by rotating $R$ around
the $y$-axis.

\answer $2\pi(\sqrt{2}-1)$


 \printanswers

\finishtbl
\end{document}



