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\def\header{{\hfill Exam 1 \hfill A. Miller \hfill Fall 2013
                                  \hfill Math 221\hfill}}
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Circle your answer. \par\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               Start
Number                Time
 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{8} State and prove the product rule of differentiation.
Use complete sentences and correct grammar.
Assume that the reader is an intelligent
person who knows the rules of limits and
the definition of the derivative but nothing more.

\prob{8} % wed 9-25 4.5 : 2 and extra.tex
Give a direct proof from the definition of derivative that
$$\frac{d}{dx}\left(\frac1x\right)=-\frac1{x^2}$$
No points will be given for quoting the power rule.


\prob{8} % 1.7:4 
A function $y=f(x)$ which satisfies the property that for every $x$
that $y=f(x)$ is
the minimum value $y$ satisfying the equation
$$y^2-xy+(x-1)=0$$
Give an explicit definition of $f$, i.e., a formula for
$f$.

\answer  $y=\min(1,x-1)$

\prob{8} %   1.7:11

Find numbers $a<b$
such that the function $$f(x)=\sin(2x)$$
with domain restricted to $a\leq x\leq b$ has an inverse
and $b-a$ is as big as possible.
Give a formula for inverse function for $f$,
$f^{-1}(x)$, using the arcsine function.

\answer $a=-\frac\pi4$, $\;\;b=\frac\pi4$, 
$\;\;f^{-1}(x)=\frac12\arcsin(x)$

\def\pr{\prime}

\prob{8} % 3.4:9
Use the $\epsilon-\delta$ definition to prove the following limit:
$$\lim_{x\to 9}\;\; \sqrt{x}\;=\; 3$$

\answer $\delta=\min(\epsilon,1)$ works.  The one is to
insure we don't take the square root of a negative number.
$$|\sqrt{x}-3|=\left|\frac{{(\sqrt{x}-3)}{(\sqrt{x}+3})}{{\sqrt{x}+3}}\right|=
\frac{|x-9|}{{|\sqrt{x}+3}|}\leq \frac{|x-9|}{{3}}\leq\delta\leq\epsilon $$

\prob{8} % 3.16 : 4
Find
$$\lim_{t\to 0} \;\;\frac{3t}{\sin(2t)}$$

\answer $\frac32$

\prob{8} % moodle sep 18
Find the following:
$$\lim_{t\to\infty} \frac{(At^3+ 4)^2}{(At^2 -1)^3}$$
$A$ is a constant which may be zero.

\answer If $A=0$, then the limit is $-16$.  If $A\neq 0$ then
the limit is $\frac1A$. 



\prob{8} % 3.14:20
For what values of $a$ and $b$ is it true that $f$ is
a continuous function?

$$f(x)=\left\{
\begin{array}{ll}
x+a     & \mbox{ if $x\leq 1$ }\\
x^2+b  & \mbox{ if $1<x\leq 2$ }\\
x+3     & \mbox{ if $2<x$ }\\
\end{array}\right.$$

\answer $a=b=1$



\prob{8} %  Mon 9-30 4.12 : 15
Find the first and second derivative of
$$f(x)=\sin^2(x)$$

\answer second derivative is $2\cos^2(x)-2\sin^2(x)$.

\prob{8} % Wed 10-02 extra problem
For $p=\cos(\cos(\cos \theta))$ find  $\frac{dp}{d\theta}$.

\answer $-\sin(\cos(\cos(\theta))) \sin(\cos(\theta))\sin(\theta)$

\prob{8}  % Related rates  last problem
A crystal in the shape of a cube dissolves in acid so that the volume of
the cube decreases by 5 cubic centimeters per minute. 
How fast is the edge of the cube changing when the volume is 1000
cubic centimeters?

\answer $-\frac1{60}$ centimeters per minute or decreasing
by $\frac1{60}$ centimeters per minute.

\prob{6} %  Fri 9-27 4.8 :11
Find the derivative of
$$f(x)=\sqrt{\frac{1+x}{2-x}}$$

\answer $-\frac12\sqrt{\frac{2-x}3}$

\prob{2} % 2.6 :7
A bottle is being filled with soda.  The volume in ounces
of soda at time $t$ in seconds is $B(t)$.  What units does the
derivative $B^\pr(t)$ have?

\begin{enum}
\item ounces
\item ounces per second
\item seconds 
\item seconds per ounce
\item bottles of soda
\end{enum}

\xprob{2} % 3.14:5
Find 
$\lim_{x\to -3} \;\; (2x+5)^{2013}$

\begin{enum}
\item $2013$
\item $(-3)^{2013}$
\item $\infty$
\item $-1$
\item $-\infty$
\item The limit is undefined.
\end{enum}

\xprob{2} %  1.2  
Suppose  $x$ is the repeating decimal $$.47474747\cdots$$  Find
integers $p$ and $q$ such that $x=\frac{p}q$.

\begin{enum}
\item $\frac{47}{100}$
\item $\frac{99}{47}$
\item $\frac47$
\item $\frac{47}{99}$
\item This is impossible.
\end{enum}


 \printanswers

\finishtbl
\end{document}



