%% Denzler
\magnification1200\footline{\ifnum\pageno=1 \hfil\else\hfil\number\pageno\hfil\fi}
\def\whichexam{Exam I}
\def\examdate{February 15, 2001}
\def\numberpages{10}
\def\numberofproblems{14}
\def\multiplechoicemax{5}
\def\partialcreditmax{12}
%%
\newif\ifprelim
\prelimfalse
\prelimtrue
%%
\gdef\f(#1){\smfrac{1}{\cos(#1)}}
\gdef\f(#1){\sec(#1)}
\gdef\smmfrac#1#2{{{#1}\over{#2}}}
%%
%%%%%  The macro \correctanswers
%%%%%  sets the number of partial credit problems automatically.
%%%%%  We assume that the first answer listed after the problem is
%%%%%  the correct one.
%%%%%  The permutation indicates in which slots to put the answers
%%%%%  (so the first number in the permutation locates the correct answer!) 
\def\correctanswers{%
\ll 1 42315
\ll 2 41532
\ll 3 53214
\ll 4 31245
\ll 5 14532
\ll 6 24315
\ll 7 31245
\ll 8 52431
\ll 9 23451
\ll 10 12453
}

\def\rhb#1{\noindent\hbox to\hsize{\hfil#1}}

\def\longans#1{\vtop{\noindent\hsize=6.25 true in #1}}
\def\arctan{{\rm arctan\>}}

\long\def\Ans#1#2€#3€#4€#5€#6€#7\endp{
\ifcase#1\relax\or\one\or\two\or\three\or\four\or\five\fi
\AN{#2}{#3}{#4}{#5}{#6}\Pra}

\def\dxd{\displaystyle}

\input epsf.def
\magnification1200\footline{\ifnum\pageno=1 \hfil\else\hfil\number\pageno\hfil\fi}
\newdimen\picV
\newdimen\picH
\def\px#1{\setbox0=\hbox{\noindent\epsfxsize=\picH \epsfbox{:problem09:#1}}%
\picV=\ht0 %
\vbox to 0pt{\vskip-20pt\box0 \vss}\vrule width 0pt height 0pt depth\picV%
}

\newcount\cnumberofproblems
\cnumberofproblems=\numberofproblems
\global\advance\cnumberofproblems by 1 

\font\ld=cmbx12
\def\endp{\relax}
\newcount\totalproblems\totalproblems=\numberofproblems
\newtoks\ta\newtoks\tb
\long\def\rightappenditem#1\to#2{%
\ta={\\{#1}}\tb=\expandafter{#2}%
\xdef#2{\the\tb\the\ta}}
\def\leftremove#1\to#2{\expandafter\lopoff#1\lopoff#1#2}
\long\gdef\lopoff\\#1#2\lopoff#3#4{\gdef#4{#1}\gdef#3{#2}}

\magnification1200\footline{\ifnum\pageno=1 \hfil\else\hfil\number\pageno\hfil\fi}
\def\loop#1\repeat{\def\body{#1}\iterate}
\def\iterate{\body\let\next=\iterate\else\let\next=\relax\fi\next}
\let\Correctanswers=\correctanswers
\def\correctanswers{\global\numbermultiplechoice=0 \gdef\list{}\Correctanswers}
\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=\multiplechoicemax \else\credit=\partialcreditmax \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}

\long\def\newp#1.#2\endp{\problemspacing%
\noindent\vbox{\noindent{\bf\number\pc.}(\number\credit\ pts.)
\global\advance\pc by1#2\endp}\vfil}


\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\four{\global\Ax=4 \global\Aspace=.25\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}}

\newdimen\Ls\Ls=10pt

\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\prafour{\vskip\Ls\noindent(a)\quad\box\Aa(b)\quad\box\Ab%
(c)\quad\box\Ac(d)\quad\box\Ad\vskip\Ls\noindent%
(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\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}

\newcount\pcc
\def\mooseAntlers#1#2#3#4#5@{\gdef\pA{#1}\gdef\pB{#2}\gdef\pC{#3}\gdef\pD{#4}\gdef\pE{#5}}
\def\Pra{\leftremove\list\to\moose%
\expandafter\mooseAntlers\moose@%
\switchbox{\pA}{\Aa}\switchbox{\pB}{\Ab}%
\switchbox{\pC}{\Ac}\switchbox{\pD}{\Ad}\switchbox{\pE}{\Ae}%
\resetboxes%
\ifcase\Ax \or\praone\or\pratwo\or\prathree\or\prafour\or\prafive\fi%
\pcc=\pc\advance\pcc by -1 %
\immediate\write16{Problem \number\pcc - Correct Answer \number\pA}%
}


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

\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\newcount\numbermultiplechoice\newcount\numberpartialcredit
\newdimen\AA\AA=20pt
\def\ll#1 #2#3#4#5#6{\tabalign {#1.}&
\global\ansmark=#2 \relax%
\ifnotmarkanswers \global\ansmark=0 \fi%
\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\global\advance\numbermultiplechoice by1 
\rightappenditem{#2#3#4#5#6}\to\list%
}

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



%%%%%%%
\def\eat#1{}
\def\exam#1{\ifprelim\Exam{Bullwinkle}\else\expandafter\EExam#1,@\fi}
\def\EExam#1,#2@{\Exam{#1}\def\xx{#2}%
\ifx\empty\xx\let\next=\eat\else\let\next=\EExam\fi\next#2@}

\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 \numberpages\ 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

\global\numberpartialcredit=\numbermultiplechoice
\global\advance\numberpartialcredit by 1
\settabs\+Total multiple choice: &\hskip 2in\cr
\hfill{{\vbox to0pt{\boxit{\vbox{\obeylines\hsize=3in\baselineskip26pt
DO NOT WRITE IN THIS BOX!
\tabalign Total multiple choice: &\quad\ul{1in}\cr
\loop\ifnum\numberpartialcredit< \cnumberofproblems
\tabalign\hfill\number\numberpartialcredit.&\quad\ul{1in}\cr\global\advance\numberpartialcredit by 1
\repeat

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

\def\kg{{\rm kg}}
\def\log{{\rm log}}
\newp 1.
The function $x^x$ has a global minimum on the interval $x>0$.
(You need not check this.) This minimum value occurs at
\Ans5
$x=\frac1e$€
$x=\frac12$€
$x=\ln2$€
$x=\ln e-1$€
$x=e$€
\endp

\newp 2.
$\displaystyle\frac{d}{dx} \ln(\ln x) =\  ?$
\Ans5
$\frac{1}{x\ln x}$€
$\frac1{\ln x}$€
$\frac x{\ln x}$€
$\frac{\ln x}{\ln(\ln x)}$€
$\frac{\ln x}x$€
\endp

\newp 3.
$\Int_0^{\pi/3}\ \tan x\ dx\ =$
\Ans5
$\ln2$€
$\ln\frac12$€
$3$€
$4$€
$\sqrt3$€
\endp

\newp 4.
On the teeter-totter with weights as indicated you need to place one further mass of
$2\kg$ in such a way as to balance the beam. 
At what distance from the {\sl left}{--}end is the correct location?

\epsfbox{teeter.eps}%

\Ans4
$1.5$€
$0.5$€
$1.0$€
$2.0$€
There is no such distance.\hss€
\endp

\newp 5.
Suppose you know the following results about a differentiable function:
\itemitem{$\bullet$} its inverse function $g$ exists
\itemitem{$\bullet$} $f(2)=4$ and $f^\prime(2)=3$.
Which of the following is certain to be true?
\Ans5
$g^\prime(4)=\frac13$€
$g^\prime(2)=\frac34$€
$g^\prime(2)=\frac13$€
$g^\prime(4)=\frac12$€
$g^\prime(3)=\frac14$€
\endp

\newp 6.
If $y=\frac12\bigl(e^x+e^{-x}\bigr)$, then
$\Int^2_{-2}\sqrt{1+\biggl(\frac{dy}{dx}\biggr)^2\ }\ dx\ =$
\Ans5
$e^2-e^{-2}$€
$e^2+e^{-2}$€
$e^2-1$€
$e^2$€
$2e^2$€
\endp

\newp 7.
$\frac{d}{dx} 4\>\arctan e^x \big\vert_{x=0}\ =$
\Ans5
$2$€
$0$€
$1$€
$3$€
$4$€
\endp

\newp 8.
Which of the following is an integrating factor for the expression
$$y^\prime +(\cot x) y$$
\noindent
i.e. which function $v$ below satisfies
$(vy)^\prime=vy^\prime+v(\cot x)y$.
\Ans5
$\sin x$€
$\sin y$€
$\ln\sin x$€
$e^{\sin x}$€
$1$€
\endp

\newp 9.
Which of the following statements is true for general $x>0$
\Ans3
$10^{\ln x}=x^{\ln 10}$€
$10^{\ln x}=\ln 10^x$€
$10^{\ln x}=e^{\log_{10}x}$€
$10^{\ln x}=(\ln x)^{10}$€
$10^{\ln x}=x^{\log_{10}e}$€
\endp

\newp 10.
$$\frac d{dx} (\arccos x)^2\ =$$
\Ans4
$\frac{-2\arccos x}{\sqrt{1-x^2\ }}$€
$\frac{-2x}{\sqrt{1-x^2\ }}$€
$\frac{\sin x}{\cos^2x}$€
$\frac{-2x}{\sqrt{1-x^4\ }}$€
$\arccos \frac{x}{\sqrt{1-x^2\ }}$€
\endp

\multichoicefalse
\newchoice{Partial Credit}
\def\credit{13}
\newp 11.

%\epsfscale=1600 
\hskip80pt\epsfbox{graph.eps}%

\item{(a)} Find the area of the boomerang shaped (shaded) region.
\item{(b)} Find the moment about the axis of your choice.
\item{(c)} Use symmetry to find the coordinates of the center of mass.

\noindent{\bf Hint for (a) and (b):}
The integral for each answer will be in two pieces.
\endp

\newp 12.
Evaluate the integral
$\Int_{-1}^{0} \frac{1}{x^2+2x+2}\ dx $
\endp

\def\credit{12}
\newp 13.
Find $y(1)$ if $y$ is the unique function which satisfies 
$$y^\prime=\sqrt{1-y^2\ }\>\cdot\> x\hskip10pt, \hskip10pt y(0)=0$$
\endp

\newp 14.
Find the general solution to the differential equation
$$y^\prime\ =\ \frac{2y}{x}+x\hskip10pt (x>0)$$
\endp

\vfill\eject}


\exam{}\marked{}
\end
