\def\whichexam{Exam II - 126 2000S}
\def\examdate{March 21, 2000}
\def\numberpages{11}
\def\numberofproblems{15}
%%
\newif\ifprelim
\prelimfalse
%\prelimtrue
%%
%%
%%%%%  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 51324
\ll  2 42351
\ll  3 14532
\ll  4 35214
\ll  5 32415
\ll  6 25413
\ll  7 15342
\ll  8 32154
\ll  9 21534
\ll 10 41352}

\def\arctan{{\rm arctan\>}}

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

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

\font\ld=cmbx12
\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\def\lopoff\\#1#2\lopoff#3#4{\def#4{#1}\def#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=5 \else\credit=10 \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\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 \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}\hss}\vskip10pt%
\rhs{3in}{Instructo:\ \UL{2.3in}{Bullwinkle}\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}


\two
\newp 1.
Solve the initial value problem $\displaystyle 
{dy\over dx} = {3x^2-1\over e^y}$, $y(1)=1$.
\LANp
\item
$y=\ln(x^3-x+e)$
\item
$y=\ln(x^3+x)+\ln(e/2)$
\item
$y=e^{x^3-2x+1}$
\item
$y=e^{x^3-x}$
\item
$y=\arctan({1\over \sqrt 3})+{\pi\over 4}$
\end{enumerate}


\three
\newp 2.
$$\int {x^2\over 1+x^2}dx =$$
\LANp
\item
$x-\arctan x + c$
\item
$\arctan{1\over x} + c$
\item
${x\over 2}\ln|1+x^2|+c$
\item
${1\over 2}\ln|1+x^2|+c$
\item
$\sec x\tan x + c$
\end{enumerate}

\five
\newp 3.
$$\int_4^5 {1\over x^2-5x+6}dx =$$
\LANp
\item
$\ln {4\over 3}$
\item
$\ln 20 - \ln 3$
\item
$\ln 5 - \ln 3$
\item
$\ln{5\over 4}$
\item
divergent
\end{enumerate}

\three
\newp 4.
$$\int(2x+1)e^x dx = $$
\LANp
\item
$2xe^x-e^x+c$
\item
${1\over 2}xe^{x^2+1}+c$
\item
$(x^2+x)e^x+c$
\item
$(x^2+x+c)e^x$
\item
divergent
\end{enumerate}

\two
\newp 5.
$$\int \ln(1-x) dx =$$
\LANp
\item
$(x-1)\ln(1-x)-x+c$
\item
$x\ln(1-x)-x+c$
\item
$(1-x)\ln(1-x)-(1-x)+c$
\item
${-2x\over 1-x^2}+c$
\item
$\ln(\arctan x^2)+c$
\end{enumerate}

\five
\newp 6.
$$\int_0^1 {2x dx\over \sqrt{1-x^2}}=$$
\LANp
\item
$2$
\item
$-4$
\item
$1$
\item
$1/2$
\item
divergent
\end{enumerate}

\newp 7.
If $a_1=-1$, $a_2=2$, and $a_{n+2}=a_{n+1}+a_n$ for $n\ge 1$
what is $a_5$?
\LANp
\item
$4$
\item
$-1$
\item
$2$
\item
$5$
\item
$-5$
\end{enumerate}


\newp 8.
Determine $\displaystyle\lim_{k\to \infty} 1+{2^k\over k!}$
\LANp
\item
1
\item
0
\item
$\infty$
\item
3/2
\item
$1/k!$
\end{enumerate}

\newp 9.
List all values of $x$ for which
$\sum_{n=1}^\infty \left({1\over x^2+3/4}\right)^n$
converges.
\LANp
\item
$|x|>1/2$
\item
$x>\sqrt{3/4}$
\item
$x \not = 0$
\item
$|x| \ge 1$
\item
$|x|\ge \sqrt3/2$
\end{enumerate}

\newp 10.
Determine $\displaystyle \sum_{n=1}^\infty\> \bigl((1/2)^n +
(1/4)^n\bigr)$
\LANp
\item
$4/3$
\item
$3/2$
\item
$3/4$
\item
$3$
\item
$9/4$
\end{enumerate}


\multichoicefalse
\newchoice{Partial Credit}

\newp 11.
Solve the equation 
$$t{dy\over dt} +2y=5t^3$$
with initial condition $y(1)=0$.

\newp 12.
Find just the partial fraction decomposition (not the integral) of

\noindent\hbox to\hsize{%
\vbox{\hsize=80pt$${2x^2-4x+1\over (x-1)^2(x-2)}=$$}\hfil}

\newp 13.
Using substitution, and then a trig substitution, find
%$\displaystyle \int_1^e {dy\over y\sqrt{1+(\ln y)^2}}$

\noindent\hbox to\hsize{%
\vbox{\hsize=80pt%
$$\int_1^e {dy\over y\sqrt{1+(\ln y)^2}}=$$}\hfil}


\newp 14.
Does $\displaystyle \int_1^\infty {(1+\sin x)\over x^{4/3}} dx$ converge
or diverge? Why?

\newp 15.
Determine the convergence of the sequence $\{a_n\}$ with
$\displaystyle a_n={1\over n^2}\int_1^n x\ dx$.

\vfill\eject}

\exam{}
\marked{}
\end