%% Jarre & Wong, PM
\newif\ifprelim\prelimfalse
\def\whichexam{Exam II}
\def\examdate{October 26, 1999}
\def\correctanswers{%
\ll 1 3
\ll 2 2
\ll 3 1
\ll 4 4
\ll 5 2
\ll 6 3
\ll 7 5
}
\magnification1200\footline{\ifnum\pageno=1 \hfil\else\hfil\number\pageno\hfil\fi}
\font\small=cmr7
\def\newpage{\vfill\eject}
\newdimen\pspacewidth
\newcount\credit\credit=6
\newcount\pspace\pspace=0
\newcount\pc\pc=1
\hfuzz=5pt
\let\xxsum=\sum
\edef\sum{\displaystyle\xxsum}

\newdimen\picH\newdimen\picV
\def\picture #1 by #2 (#3 scaled #4){
\picV=#2 \divide\picV by1000 \multiply\picV by#4 
\picH=#1 \divide\picH by1000 \multiply\picH by#4
\vtop to \picV{
      \hrule width \picH height 0pt depth 0pt
      \vfill
      \special{picture #3 scaled #4}}}

\newcount\mc
\def\newchoice#1{\ifnum\pc<3 \newpage\else\ifprelim\else\newpage\fi\fi\null%
\vskip10pt\centerline{{\bf#1}}\mc=0 %
\ifnum\pc<3\credit=5 \else\credit=11 \fi}
\newif\ifmultichoice
\def\problemspacing{\ifprelim\vskip10pt\headline{\hfill}\else%
\ifmultichoice%
\ifnum\mc>0 \ifodd\pc\relax\newpage\null\vskip10pt\else\vfill\fi\fi%
\else%
\ifnum\mc>0 \newpage\null\vskip10pt\fi\fi%
\mc=1 \fi}

\def\newp#1.{\problemspacing%
\noindent{\bf\number\pc.}(\number\credit\ pts.)
\global\advance\pc by1}

\newbox\Aa\newbox\Ab\newbox\Ac
\newbox\Ad\newbox\Ae
\newbox\aa\newbox\ab\newbox\ac
\newbox\ad\newbox\ae

\newdimen\Aspace
\newcount\Ax
\newcount\tmpPC
\def\rememberPC{\global\tmpPC=\pc\global\advance\tmpPC by-1 }

\hfuzz=20pt

\def\five{\global\Ax=5 \global\Aspace=.2\hsize\relax\global\advance\Aspace by-20pt}
\def\three{\global\Ax=3 \global\Aspace=.3\hsize\relax\global\advance\Aspace by-20pt}
\def\two{\global\Ax=2 \global\Aspace=.5\hsize\relax\global\advance\Aspace by-20pt}
\def\one{\global\Ax=1 \global\Aspace=\hsize\relax\global\advance\Aspace by-20pt}

\def\AN#1#2#3#4#5{%
\setbox\Aa=\hbox to\Aspace{#1\hfil}%
\setbox\Ab=\hbox to\Aspace{#2\hfil}\setbox\Ac=\hbox to\Aspace{#3\hfil}%
\setbox\Ad=\hbox to\Aspace{#4\hfil}\setbox\Ae=\hbox to\Aspace{#5\hfil}}

\def\ANp#1(a)#2(b)#3 (c)#4 (d)#5 (e)#6\Pra{%
\AN{#2}{#3}{#4}{#5}{#6}\Pra%
}

\newdimen\Ls\Ls=10pt
\def\pra{\ifcase\Ax \or\praone\or\pratwo\or\prathree\or \or\prafive\fi}

\def\prafive{\vskip\Ls\noindent(a)\quad\box\Aa(b)\quad\box\Ab%
(c)\quad\box\Ac(d)\quad\box\Ad%
(e)\quad\box\Ae}

\def\prathree{\vskip\Ls\noindent(a)\quad\box\Aa(b)\quad\box\Ab%
(c)\quad\box\Ac\vskip\Ls\noindent(d)\quad\box\Ad%
(e)\quad\box\Ae}

\def\pratwo{\vskip\Ls\noindent(a)\quad\box\Aa(b)\quad\box\Ab%
\vskip\Ls\noindent(c)\quad\box\Ac(d)\quad\box\Ad%
\vskip\Ls\noindent(e)\quad\box\Ae}

\def\praone{\vskip\Ls\noindent(a)\quad\box\Aa\vskip\Ls\noindent(b)\quad\box\Ab%
\vskip\Ls\noindent(c)\quad\box\Ac\vskip\Ls\noindent(d)\quad\quad\box\Ad%
\vskip\Ls\noindent(e)\quad\box\Ae}

\def\resetboxes{\setbox\Aa=\box\aa\setbox\Ab=\box\ab\setbox\Ac=\box\ac%
\setbox\Ad=\box\ad\setbox\Ae=\box\ae}
\def\switchbox#1#2{\ifcase#1 \relax\or\setbox\aa=\box#2\or\setbox\ab=\box#2\or%
\setbox\ac=\box#2\or\setbox\ad=\box#2\or\setbox\ae=\box#2 \fi}

\def\Pra#1#2#3#4#5{\switchbox{#1}{\Aa}\switchbox{#2}{\Ab}%
\switchbox{#3}{\Ac}\switchbox{#4}{\Ad}\switchbox{#5}{\Ae}%
\resetboxes%
\ifcase\Ax \or\praone\or\pratwo\or\prathree\or \or\prafive\fi%
\immediate\write16{\number\pc - \number#1}%
}


\def\dd#1#2{{{\textstyle d\> #1}\over{\textstyle d#2}}}
 \def\frac#1#2{{{\textstyle#1}\over{\textstyle#2}}}
\def\Lim{\displaystyle\lim}
\def\Int{\displaystyle\int}


\def\frac#1#2{{\displaystyle{{\textstyle#1}\over{\textstyle#2}}}}

\long\def\boxit#1#2#3{\vbox{\hrule\hbox{\vrule\kern#2
\vbox{\hsize=#3\kern#2#1\kern#2}\kern#2\vrule}\hrule}}
\newcount\pc
\def\pr{\vskip10pt\noindent\number\pc.\global\advance\pc by 1}
\def\dd#1#2{{{\textstyle d\> #1}\over{\textstyle d#2}}}

\def\rhs#1#2{\par\noindent\hbox to\hsize{\hfill\hbox to#1{#2\hfill}}}
\def\ul#1{\vrule width#1 height -3pt depth3.1pt}
\def\UL#1#2{\hbox to0pt{\qquad #2\hss}\ul{#1}}
\newif\ifnotmarkanswers\notmarkanswerstrue
\newcount\ansmark
\newdimen\AA\AA=20pt
\def\ll#1 #2{\tabalign {#1.}&
\global\ansmark=#2 \relax%
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\ifnum\ansmark=1 \relax($\bullet$)\else(a)\fi&
\ifnum\ansmark=2 \relax($\bullet$)\else(b)\fi&
\ifnum\ansmark=3 \relax($\bullet$)\else(c)\fi&
\ifnum\ansmark=4 \relax($\bullet$)\else(d)\fi&
\ifnum\ansmark=5 \relax($\bullet$)\else(e)\fi&
\hskip\AA\cr}

\def\marked{\pageno=1 %
\global\notmarkanswersfalse
\headline{\vbox{\rhs{3in}{Name:\ \ul{3in}\hss}\vskip10pt%
\rhs{3in}{Instructo:\ \UL{2.3in}{Bullwinkle}\hss}%
}}
\answersheet}



%%%%%%%
\def\+{\tabalign}
\long\def\exam#1{%
\pc=1 \pageno=1 \pspace=0 %
\headline{\vbox{\rhs{3in}{Name:\ \ul{3in}\hss}\vskip10pt%
\rhs{3in}{Instructor:\ \UL{2.3in}{#1}\hss}%
}}
\long\gdef\answersheet{
\null\vskip10pt
\centerline{\whichexam}
\centerline{\examdate}
\vskip10pt

\itemitem{$\bullet$}
The Honor Code is in effect for this examination.
All work is to be your own.
\itemitem{$\bullet$} No calculators.
\itemitem{$\bullet$} The exam lasts for one hour.
\itemitem{$\bullet$} Be sure that your name is on every page in case
pages become detached. 
\itemitem{$\bullet$} Be sure that you have all 11 pages of the test.
\vskip5pt
\centerline{Good Luck!}
\vskip4pt
\def\HH{\hskip.75in}
\settabs\+9.\qquad &\HH&\HH&\HH&\HH&\HH&\cr
\hskip.5 in\boxit{\hbox to 3.8in{\vbox{\obeylines\baselineskip20pt
\noindent
PLEASE MARK YOUR ANSWERS WITH AN X, not a circle!
\correctanswers
}\hss}}{10pt}{5in}
%%\vskip10pt

\settabs\+Total multiple choice: &\hskip 2in\cr
\hfill{{\vbox to0pt{\boxit{\vbox{\obeylines\hsize=3in\baselineskip26pt
DO NOT WRITE IN THIS BOX!
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\tabalign\hfill8.&\quad\ul{1in}\cr
\tabalign\hfill9.&\quad\ul{1in}\cr
\tabalign\hfill10.&\quad\ul{1in}\cr
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\tabalign\hfill12.&\quad\ul{1in}\cr
\tabalign\hfill13.&\quad\ul{1in}\cr

\tabalign\hfil {\bf Total:}&\quad\ul{1in}\cr
}}{10pt}{1in}\vss}}}}
\answersheet
\multichoicetrue
\newchoice{Multiple Choice}
\five

\newp 1.
If \ $x+\sin y = xy$ \ then at the point $x=0$ and $y=0$, the value of
$\displaystyle{dy\over dx}$ is

\ANp
(a) $0$
(b) $1$
(c) $-1$
(d) $2$
(e) $1\over2$ 
\Pra12345

\newp 2.
Suppose the derivative of a function $y=f(x)$ is
$y^\prime=x^2(x+2)(x-2)$. Then,

\two
\ANp
(a)$x=-2$ is a local minimum
(b) $x=2$ is a local minimum
(c) $x=2$ is a local maximum 
(d) $x=0$ is a local minimum
(e) $x=0$ is a local maximum 
\Pra12345

\newp 3.
A thin circular plate is heated in the oven.
Its radius is increasing at a rate of 1 inch per hour.
At what rate is the area of the plate increasing when 
the radius is 10 inches?
\five
\ANp
(a) $20\pi$
(b) $10\pi$
(c) $2\pi$
(d) $\pi$
(e) $100\pi$
\Pra12345

\newp 4.
Let $f(x)= x^2-9$ for $0\le x\le 5$. Then the absolute maximium
occurs at $x=$
\three
\ANp
(a) $-3$ 
(b) $0$ 
(c) $3$ 
(d) $5$ 
(e) \hbox to 0pt{there is no absolute maximum\hss}
\Pra12345

\two
\newp 5.
The asymptotes of the function 
$$f(x)={x^2-x-2\over x^2+3x+2}={(x+1)(x-2)\over (x+1)(x+2)}$$
are
\ANp
(a) $x=-1$ and $x=-2$ 
(b) $x=-2$ and $y=1$
(c) $x=2$ and $y=-1$
(d) $x=1$ and $x=2$
(e) $x=-1$, $x=-2$, and $y=1$ 
\Pra12345


\newp 6.
Find $dy$ for $y=\cos(4-x^2)$
\ANp
(a) $dy = y^\prime$
(b) $dy = -\sin(4-x^2)\ dx$
(c) $dy = 2x\sin(4-x^2)\ dx$
(d) $dy = \displaystyle{dx\over -\sin(4-x^2)}$
(e) $dy = 2 \cos(4-x^2)\sin(4-x^2)\ dx$
\Pra12345

\newp 7.
The first iteration of Newton's method for solving $x^3+x^2+1=0$ starting at $x_0=-2$ gives
\ANp
(a) $x_1={3\over 8}$
(b) $x_1=-{3\over 8}$
(c) $x_1=-{13\over 8}$
(d) $x_1=-{15\over 8}$
(e) $x_1=-{19\over 8}$
\Pra12543
\multichoicefalse
\newchoice{Partial Credit}

\newp 8.
Let $y=f(x)=5x^{2/5}-2x$ with
$y^\prime=2(x^{-3/5}-1)$ and $y^{\prime\prime}=-{6\over 5}x^{-8/5}$.
\item{}
Find all critical points
\item{}
List the intervals where $f$ is increasing / decreasing.
\item{}
List the intervals where $f$ is concave up / concave down
\item{}
List all local maxima and local minima, or say so if there are none
\item{}
List all inflection points, or say so if there are none



\newp 9.
Let $f(x)=\displaystyle {20x\over 4+x^2}$.
Then, all derivatives of $f$ are defined for all $x\in (-\infty,\infty)$.
The following information is assumed to be known - we do not want you
to verify it!
$$
f^\prime(x) < 0 \quad \hbox{for} \quad x\in(-\infty,-2) 
\quad \hbox{and} \quad x\in(2,\infty),
\hskip10pt f^\prime(x) > 0 \quad \hbox{for} \quad x\in(-2,2) ,
$$
$$
f^{\prime\prime}(x) < 0 \quad \hbox{for} \quad x\in(-\infty,- 6) 
\quad \hbox{and} \quad x\in( 0,6 ),
$$$$
f^{\prime\prime}(x) > 0 \quad \hbox{for} \quad x\in(- 6,0) 
\quad \hbox{and} \quad x\in( 6,\infty ).
$$
\centerline{The only asymptote of $f$ is $y=0$.}
\vskip8pt
\noindent
Evaluate $f$ at critical points and inflection points
and graph the function.

\vtop{\vskip1in\picture{6.81in} by {6.93in} ({blankgraph} scaled 600)}


\newp 10.
Car $A$ and car $B$ are approaching the intersection ``$C$''
of two streets intersecting at a right angle.
Car $A$ is going South at 45 mph, car $B$ is heading West at 30
mph. We denote the angle $\angle (C,B,A)$ by $\theta$, the distance 
from $C$ to $B$ by $x$, and the distance from $C$ to $A$ by $y$.
Then, $\tan \theta = {y\over x}$.
At what rate is the angle $\theta$ changing when car $A$ and
car $B$ are both 1 mile from the intersection?
\vskip-40pt\vtop{\picture{6.81in} by {6.93in} ({cars} scaled 300)}
\vskip-24pt\hskip50pt$\theta$


\newp 11.
Let $f(x)=12-x^2$ for $x\in[-\sqrt{12},\sqrt{12}]$.
The graph of $f$ and the $x$-axis bound a region.
Find the area of the rectangle $R$ with largest area 
which can be inscribed in this region.
You may use the fact that the rectangle is symmetric to the $y$-axis.

\vtop to 2in{
\vtop to 0pt{\hbox to 0pt{\picture{6.81in} by {6.93in} ({area} scaled 300)\hss}\vss}
\vtop to 0pt{\vskip90pt\hbox to 0pt{$-\sqrt{12}$\hskip 1.9in$\sqrt{12}$\hss}\vss}
\vfil}

\newp 12.
About how accurately should you measure a variable $t$ to insure that
your calculation of the area of the rectangle with sides $a=t$, and $b=2t$ 
is within $5\%$ of its true value?

\global\credit=10
\newp 13. 
Find the value or values of $c$ guaranteed by the Mean Value Theorem applied to
the function $f(x)=x^3$ on the interval $[1,2]$.

\vfill\eject}

\exam{Dwyer}
\exam{Cholak}
\exam{Taylor}
\exam{Wong}
\exam{Cao}
\exam{Jarre}
\exam{Nollet}
%\exam{Bullwinkle}
\marked{}
\end
