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\def\header{{\hfill Exam 2 \hfill A. Miller \hfill Fall 2006
                     \hfill Math 211 \hfill}}
\markboth\header\header
\newcount\probno
\newcount\total
\newwrite\examTBLFile

\probno=0
\def\prob#1{\advance\probno by 1
%      \ifodd\probno
%        \par\vfill\newpage
%      \else \vskip 3.8in \fi
    \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}
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  \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: }

% \outer\def\answer{ {\par\bigskip Circle your answer.} \par\medbreak
%  \def\myskip{\vskip 2in}
%  \def\np{\vfill\newpage}
%  \def\vf{\vfill}
%  \def\vs#1{\par\vskip #1in}

\def\blurb{
Detach pages 10-15 from your exam and take them with you.
Hand in only pages 0-9.  Put your answers to problems 9-15 on
page 9.  There is no partial credit for these problems and no work
need be shown. Do not hand in your scratch paper.}

\begin{document}



\bigskip

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

For problems 1-8: Put your answer in the box provided if there is a box. Show
every step of your solution.  Erase or mark out anything false in your work.
Simplify your answers as much as you can.  It may not be sufficient to have a
right answer to get full credit.  Some graders may be stricter than others.

\blurb

%  Use the penultimate page of the exam for scratch paper or ask a TA for
%  extra scratch paper.
% However you can bring to the exam
%    one  $8.5$ by $11$
% cheat sheet with anything you want written or printed on both sides.

Exam and answers will be posted shortly after the exam:
 www.math.wisc.edu/$\sim$miller

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

\bigskip
Circle the number (321-333) of your discussion section and hand in
to your TA:

\begin{verbatim}
 321    7:45    TR    B329 VAN VLECK  Kumar, Rohini
 322    8:50    TR    6322 SOC SCI    Kumar, Rohini
 323    9:55    TR    348 BIRGE       Mantilla Soler, Guillermo Arturo
 324   11:00    TR    1333 Sterling   Wang, Bing
 325   12:05p   TR    348 BIRGE       Wang, Bing
 326   12:05p   TR    B337 VAN VLECK  Potluri, Vijaya Kranthi
 327    1:20p   TR    B135 VAN VLECK  Mantilla Soler, Guillermo Arturo
 328    1:20p   TR    B123 VAN VLECK  Umarji, Pallavi Anand
 329    2:25p   TR    B211 VAN VLECK  Yin, Weidong
 330    2:25p   TR    B115 VAN VLECK  Umarji, Pallavi Anand
 331    3:30p   TR    B135 VAN VLECK  Yin, Weidong
 332    2:25p   TR    223 INGRAHAM    Christodoulopoulou, Konstantina
 333    3:30p   TR    B115 VAN VLECK  Christodoulopoulou, Konstantina
\end{verbatim}

\bigskip

\begin{center} \Large
 \begin{tabular}{||c|c|c||} \hline\hline
  Page & Points & Score \\  \hline\hline
%  \input{exam.tbl}
1& 8 & \qquad\\ \hline
2& 8 & \qquad\\ \hline
3& 8 & \qquad\\ \hline
4& 8 & \qquad\\ \hline
5& 8 & \qquad\\ \hline
6& 8 & \qquad\\ \hline
7& 8 & \qquad\\ \hline
8& 8 & \qquad\\ \hline
9-15& 36 & \qquad\\ \hline
Total & 100& \hskip 2em\relax \\ \hline
  \hline \hline
 \end{tabular}
\end{center}

\immediate\openout\examTBLFile=exam.tbl

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%   begin problems  %%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\setcounter{page}{0}

\def\text#1{\makebox(0,0)[cc]{#1}}
\def\answerbox{
\unitlength=1.36mm
\begin{picture}(60,30)
\put(30, 10){\framebox(80,18){\phantom{framebox}}}
\put(10,25){\text{Put your answer}}
\put(10,20){\text{ in the box.}}
\put(10,15){\text{Show work below.}}
\end{picture}
}

\prob{8} % 3.8-27

A fisherman is reeling in a fish he has caught at that rate of 3 feet  of
fishing line per second.  Assume that the fish is being pulled along the
surface of the water and the tip of the fishing pole is exactly 5 feet above
the surface of the water.  How fast is the fish moving along the surface when
the distance from the tip of the pole and the fish is 13 feet?    (You should
assume that the fish has enough drag to keep the line straight.)

\answerbox

\answer $13/4$

\prob{8} % 4.3-9

Sketch the graph of $$f(x)=x^3-3x^2+2$$ over the real line.  Label local
extreme points and inflection points.

\answer local max at $(0,2)$, local min at $(2,-2)$, and  inflection 
point at $(1,0)$.

\prob{8} % 4.5-15

A kennel owner has 80 feet of fencing and wants to fence off a rectangular
area which is divided into 3 adjacent identical rectangular subsections, as
shown in Figure \ref{kennel}.   What dimensions will maximize the area?

\def\text#1{\makebox(0,0)[cc]{#1}}

\unitlength=2.20mm
\begin{figure}
\begin{picture}(36,20)(-10,10)
\put(10,10){\line(0,1){20}}
\put(22,10){\line(0,1){20}}
\put(34,10){\line(0,1){20}}
\put(46,10){\line(0,1){20}}
\put(10,30){\line(1,0){36}}
\put(10,10){\line(1,0){36}}
\end{picture}
\caption{Kennel\label{kennel}}
\end{figure}

\answerbox

\answer $10$ by $20$

\prob{8} % 5.2-5
Find
$$\int \sqrt{2x-4}\;\;dx$$

\answerbox

\answer ${1\over 3}(2x-4)^{3\over 2}+C$

\prob{8} % 5.3-23
Find
$$\int {{x-5}\over{(x-2) (x+1)}}\;\; dx$$

\answerbox

\answer $2\ln|x+1|-\ln|x-2|+C$

\prob{8} % 6.1-13

The demand curve is $D(q)= {36\over{q+1}}$ and the supply curve is $S(q)=q+1$
for a certain commodity.  For each the price $p$ is written as a function of 
the quantity $q$.  Find the consumer surplus.

\answerbox

\answer $36\ln 6 - 30$

%producer surplus ${25\over 2}$.


\prob{8} % 6.2-7

Find the nominal annual interest rate, assuming interest is compounded
continuously, that must be paid by a bank so that an initial deposit
doubles in 9 years.


\answerbox

\answer ${1\over 9}\ln(2)$



\prob{8} % 6.3-21

Find the unique solution to the initial value
problem:
$$ {dy\over dx}= {xe^x\over 2y}, \;\;\;\;\; y(0)=1 $$

\answerbox

\answer $y=\sqrt{xe^x-e^x+2}$

%%%%%% multiple choice

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

\newpage

\blurb

\bigskip\bigskip\bigskip

\def\multi{\hfill a \hfill b \hfill c \hfill d \hfill e \hfill}

\begin{center} \Large
 \begin{tabular}{|c|c||c|} \hline\hline
  Problem & Circle the best answer \\  \hline\hline
9       & (a) 1 2 3 4 5      \\
\qquad  & (b) 1 2 3 4 5      \\
\qquad  & (c) 1 2 3 4 5      \\
\qquad  & (d) 1 2 3 4 5      \\
\qquad  & (e) 1 2 3 4 5      \\ \hline
10      & (a) True False       \\
\qquad  & (b) True False       \\
\qquad  & (c) True False       \\
\qquad  & (d) True False       \\
\qquad  & (e) True False       \\ \hline
11      & (m) \multi           \\
\qquad  & (M) \multi           \\ \hline
12      & (a) 1 2 3 4 5      \\
\qquad  & (b) 1 2 3 4 5       \\
\qquad  & (c) 1 2 3 4 5       \\
\qquad  & (d) 1 2 3 4 5       \\
\qquad  & (e) 1 2 3 4 5       \\ \hline
13      &    \multi            \\ \hline
14      &    \multi            \\ \hline
15      &    \multi            \\ \hline
\hline 
 \end{tabular}
\end{center}


Problems 9-15 are 5 points each except 11 which is 6 points.


\newpage

\blurb

\prob{5} % 4.1-29

Each of the graphs below is a differentiable function.
For each graph choose one of the conditions:

(1) $f^\prime(x)>0$ for all $x$ in the interval $(-\infty,0)$ and
    $f^\prime(x)>0$ for all $x$ in the interval $(0,\infty)$.

(2) $f^\prime(x)>0$ for all $x$ in the interval $(-\infty,0)$ and
    $f^\prime(x)<0$ for all $x$ in the interval $(0,\infty)$.

(3) $f^\prime(x)<0$ for all $x$ in the interval $(-\infty,0)$ and
    $f^\prime(x)>0$ for all $x$ in the interval $(0,\infty)$.

(4) $f^\prime(x)<0$ for all $x$ in the interval $(-\infty,0)$ and
    $f^\prime(x)<0$ for all $x$ in the interval $(0,\infty)$.

(5) None of above.



\bigskip
\bigskip

\begin{minipage}{2in}
\begin{center}
\includegraphics[scale=.2]{graph1} \\
(a) 1 2 3 4 5 
\end{center}
\end{minipage}
\begin{minipage}{2in}
\begin{center}
\includegraphics[scale=.2]{graph3} \\
(b) 1 2 3 4 5
\end{center}
\end{minipage}
\begin{minipage}{2in}
\begin{center}
\reflectbox{\includegraphics[scale=.2]{graph2}} \\
(c) 1 2 3 4 5
\end{center}
\end{minipage}

\bigskip
\bigskip
\bigskip
\bigskip

\begin{minipage}{3in}
\begin{center}
\includegraphics[scale=.2]{graph2} \\
(d) 1 2 3 4 5
\end{center}
\end{minipage}
\begin{minipage}{2in}
\begin{center}
\rotatebox{180}{\includegraphics[scale=.2]{graph1}} \\
(e) 1 2 3 4 5
\end{center}
\end{minipage}

\bigskip

\answer a3,b5,c4,d1,e2



\prob{5} % 4.2-33

Suppose $f(x)$ is a differentiable function on an open interval $(a,b)$.
Suppose that for some point $c$ with $a<c<b$ we have that $f^\prime(c)=0$.
Answer the following questions True or False:

\medskip
(a) If $f^{\prime\prime}(c)>0$, then $f(c)$ is a local minimum of $f$.

\medskip
(b) If $f^{\prime\prime}(c)<0$, then $f(c)$ is a local maximum of $f$.

\medskip
(c) If $f^{\prime\prime}(c)=0$, then $f(c)$ can be neither a
local maximum nor a local minimum of $f$.

\medskip
(d) If $f^{\prime\prime}(c)=0$, then $f(c)$ can be a local maximum,
a local minimum, or neither.

\medskip
(e) If $c$ is the only critical point of $f$ in the interval $(a,b)$ and
$f(c)$ is a local minimum of $f$, then $f(c)$ must be the global minimum
of $f$ in $(a,b)$.

\answer TTFTT


\newpage
\blurb

\prob{6} % 4.4-3
Find the global minimum $\;m\;$ and global maximum $\;M\;$ of the variable y,
where 
$$y=x+{1\over x}$$ 
and $x$ is in the interval $[2,4]$.

\medskip
(a) $m=1$

\medskip
(b) $m=2$

\medskip
(c) $m=2 {1\over 2}$

\medskip
(d) $m$ is none of above.

\medskip
(e) There is no global minimum.


\bigskip

\medskip
(a) $M=2$

\medskip
(b) $M=4$

\medskip
(c) $M=4 {1\over 4}$

\medskip
(d) $M$ is none of above

\medskip
(e) There is no global maximum.


\answer cc

\prob{5} % 5.1-27
Match the function $f(x)$ with one of its antiderivatives
$F(x)$

\bigskip

\begin{minipage}{2in}

(a) $f(x)=-{1\over x^2}$

\medskip

(b) $f(x)={1\over x}$

\medskip

(c) $f(x)= x$

\medskip

(d) $f(x)=e^x$

\medskip

(e) $f(x)= e^{x^2}$

\end{minipage}
\begin{minipage}{2in}
(1)  $F(x)= (.5)x^2 - 1$

\medskip

(2) $F(x)={1\over x}$

\medskip

(3)  $F(x)=e^x + 2$

\medskip

(4) $F(x)=\ln(x)$

\medskip

(5) None of above.

\end{minipage}

\answer a2, b4, c1, d3, e5

\newpage\blurb

\prob{5} % 5.4-23
The continuous function $f(x)$ has the properties that
$$\int_1^5 f(x)\;dx=10$$ 
and
$$\int_3^5 f(x)\;dx = 6.$$
What is $\int_1^3 f(x)\;dx$?

\medskip
(a)  2

\medskip
(b)  4

\medskip
(c) -4

\medskip
(d) None of above.

\medskip
(e) The value of the integral cannot be determined from the given information.

\answer b

\prob{5} % 5.5-37

Find the area $\;A\;$ between the graph of
$$f(x)={x\over{x+2}}$$ for $-1\leq x\leq 1$ and the $x-axis$.

\medskip
(a) $A=\int_{-1}^{1} {x\over{x+2}} \;dx $

\medskip
(b) $A=\int_{-1}^{1} -{x\over{x+2}} \;dx $

\medskip
(c) $A=\int_{-1}^{0} -{x\over{x+2}} \;dx + \int_{0}^{1} {x\over{x+2}} \;dx$

\medskip
(d) $A=\int_{-1}^{0} {x\over{x+2}} \;dx + \int_{0}^{1} -{x\over{x+2}} \;dx$

\medskip
(e) $A$ is none of above.

\answer c

\prob{5} % 5.6-37

Find the average value $A$ of $f(t)= t^2$ over the interval $0\leq t\leq 2$.

\medskip
(a) $A<0$

\medskip
(b) $0\leq A<1$

\medskip
(c) $1\leq A<2$

\medskip
(d) $2\leq A<3$

\medskip
(e) $3\leq A$

\answer c


\newpage
\phantom{Scratch}
\def\header{{Scratch}}
\markboth\header\header

\newpage
\phantom{Scratch}

\newpage
\phantom{Scratch}


\printanswers

\finishtbl

\end{document}



