\def\whichexam{Exam I - 126/S2000}
\def\examdate{February 15, 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 54321
\ll  2 52413
\ll  3 45213
\ll  4 32154
\ll  5 32451
\ll  6 12345
\ll  7 12354
\ll  8 21534
\ll  9 25413
\ll 10 51324
}

\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}

\five
\newp 1.
Let $f(x)=e^{(x^3)}$.
Find $\frac{df^{-1}}{dx}(e)$.
\ANp
(a) $\frac{1}{3e}$
(b) $\frac{1}{2e}$
(c) $\frac{3}{e}$
(d) $3e$
(e) $e^3$
\Pra

\three
\newp 2.
Compute
$\qquad\displaystyle \int^{\textstyle e^\pi}_{\textstyle 1}\frac{\ln x}{x}\>dx$
\ANp
(a)$\frac{\pi^2}{2}$
(b)$\frac\pi{e^\pi}$
(c)$e\pi-e$
(d)$\frac1e-\pi$
(e)$3\pi^2-2$
\Pra

\one
\newp 3.
Find the critical point(s) of $f(x)=x \ln x$ and determine whether they
are local minima, local maxima, absolute minima, absolute maxima or none of these.
If a point is an absolute max/min, the answer local max/min will receive no credit.
\ANp
(a)$x=e^{-1}$, a absolute minimum
(b)$x=e^{-1}$, a local maximum
(c)There are no local maxima or local minima
(d)$x=e^{-1}$, a absolute maximum
(e)$x=e^{-1}$, a local minimum
\Pra

\three
\newp4.
Find all solutions of the equation
$3^{(x^2)}=2$
\ANp
(a)$x=\pm\sqrt{{\rm log}_32}$
(b)$x=\pm\sqrt{{\rm log}_23}$
(c)$x=\smfrac12\sqrt{{\rm log}_32}$
(d)$x=\smfrac12\sqrt{{\rm log}_23}$
(e)$x=0$
\Pra

\two
\newp 5.
Let $f(x)=\frac{x^3 e^x}{(x^2+1)^5}$.
Find $\frac{df}{dx}$.
\ANp
(a)$\frac{x^3 e^x}{(x^2+1)^5}\biggl(\frac3x+1-10x(x^2+1)^{-1}\biggr)$
(b)$\frac{3x^2e^x+x^3e^x}{10x(x^2+1)^{4}}$
(c)$\frac{x^3e^x(x^2+1)^5-10x^4e^x(x^2+1)^4}{(x^2+1)^{10}}$
(d)$\frac{x^3e^x}{(x^2+1)^5}\biggl(\frac{(x^2+1)^5-5x^3e^x(x^2+1)^4}{(x^2+1)^{5}}\biggr)$
(e)$\frac{-3x^2e^x(x^2+1)^5(5x^3e^x(x^2+1)^4)}{(x^2+1)^{10}}$
\Pra

\five
\newp 6.
Calculate\quad $\Lim_{x\to 2^-}\>\frac{x^2-4}{x^5-32}$.
\ANp
(a)$\frac{1}{20}$
(b)$\frac{1}{40}$
(c)$\frac{1}{10}$
(d)$\frac{1}{5}$
(e)$\frac{2}{5}$
\Pra

\three
\newp 7.
Calculate\quad $\Lim_{x\to 2}\>x^x$.
\ANp
(a)$4$
(b)$3$
(c)$2$
(d)$1$
(e)Limit does not exist.
\Pra

\five
\newp 8.
Calculate $\Int^1_0\frac{x}{\sqrt{\vrule width0pt height 9pt depth 1pt 1-x^4\>}}\>dx$.
\ANp
(a)$\smfrac12\arcsin(1)$
(b)$\arcsec(1)$
(c)$2\arctan(1)$
(d)$\smfrac12\arcsec(1)$
(e)$\smfrac12\arctan(1)$
\Pra

\two
\newp 9.
Calculate $\frac{d\ }{dx}\>\arctan(\ln x)$.
\ANp
(a)$\frac1{x(1+(\ln x)^2)}$
(b)$\frac1{(1+(\ln x)^2)}$
(c)$\frac{x}{(1+(\ln x)^2)}$
(d)$\frac1{\ln x(1+x^2)}$
(e)$\frac{\ln x}{(1+x^2)}$
\Pra

\two
\newp 10.
Calculate $\Int^{3}_{2}\frac{dx}{(x+1)\sqrt{x^2+2x\>}}$.
\ANp
(a)$\arcsec(4)-\arcsec(3)$
(b)$\arcsec(3)-\arcsec(2)$
(c)$\arctan(2)-\arctan(1)$
(d)$\arctan(3)-\arctan(1)$
(e)$\arcsin(\smfrac13)-\arcsin(\smfrac12)$
\Pra

\multichoicefalse
\newchoice{Partial Credit}

\newp 11.
Consider the function $f(x)=\sqrt{2x^4+x^2}$.
\item{a)} Show that $f$ is one to one on the domain $(0,\infty)$.
\item{b)} Find the slope of the tangent line to the graph of
the inverse function $f^{-1}$ at the point $f^{-1}(6)=2$.

\newp 12. 
Find the derivative of the function
$$f(x)=\root{\textstyle x}\of{x}=x^{\smfrac1x}\quad .$$

\newp 13. 
The quantity of a radioactive substance decreases from $100\%$ to
$80\%$ in three hours.
Compute the half{--}life (the time until you have $50\%$ of your sample left) 
as a quotient of logs.


\newp 14. 
Determine $\Lim_{x\to\infty}\sqrt{e^x+x\>}-\sqrt{e^x+1\>}$.

\noindent{\bf Hint:}
Rewrite the expression using algebra and then
use what you know about rates of growth.

\newp 15.
Express
$\sec\bigl(\arctan(x)\bigr)$
as an algebraic function of $x$.

\vfill\eject}

\exam{}
\marked{}
\end