\magnification1200\footline{\ifnum\pageno=1 \else\number\pageno\fi}
\def\whichexam{Exam I}
\def\examdate{February 3, 2004}
\def\numberpages{0}
\def\numberofproblems{14}
\def\multiplechoicemax{5}
\def\partialcreditmax{15}
%%
\newif\ifprelim
\newif\ifproblemspacing

\def\START#1{\ifnum#1=0{\global\prelimtrue\global\problemspacingfalse}%
\else{\ifnum#1=1{\global\prelimfalse\global\problemspacingtrue}%
\else{\global\prelimfalse\global\problemspacingfalse}\fi}%
\fi}

\START 2 %% 0 - initial run for instructors
                  %% 1 - run for the actual exam
                  %% 2 - for the webpage
%%
%%%%%  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 32415
\ll  2 52341
\ll  3 23541
\ll  4 25314
\ll  5 54312
\ll  6 12345
\ll  7 41523
\ll  8 31425
\ll  9 34251
\ll 10 41352
\ll 11 15432
}

\def\LoadPDF#1#2#3#4{
\pdfximage {#1}
\setbox0=\hbox{\pdfrefximage\pdflastximage} % get dimensions of the image
                                            % in order to include the image
                                            % at a specific resolution 
\dimen0=#2\wd0         % calculate the image width at 1200 dpi (0.06 = 72/1200)
\pdfximage              % include the image at resolution 1200 dpi
    width \dimen0 {#1}  % by setting image width to the calculated value
\vtop to#4{\vss\hbox to#3{\hbox to \dimen0{\pdfrefximage \pdflastximage}\hss}}
}


\long\def\hide #1\endhide{}

\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}

\magnification1200\footline{\ifnum\pageno=1 \hfil\else\hfil\number\pageno\hfil\fi}
\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
\newcount\pspace\pspace=0
\newcount\pc\pc=1
\newcount\breakpc\breakpc=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\breakpc<3 \newpage\else\ifprelim\else%
\ifproblemspacing\newpage\else\bigskip\fi\fi\fi\null%
\vskip10pt\centerline{{\bf#1}}\mc=0 %
\ifnum\breakpc<3\credit=\multiplechoicemax \else\credit=\partialcreditmax \fi}
\newif\ifmultichoice
\def\problemspacing{\ifprelim\vskip10pt\headline{\hfill}\else%
\ifproblemspacing
\ifmultichoice%
\ifnum\mc>0 \ifodd\breakpc\relax\newpage\null\vskip10pt\else\vfill\fi\fi%
\else%
\ifnum\mc>0 \newpage\null\vskip10pt\fi\fi%
\else \vskip10pt\headline{\hfill}\fi%
\mc=1 \fi}

\long\def\Newp#1.#2�#3a)#4\endp{
\newp#1.#2
\Xan#3
a)#4\endp}


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

\long\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.}&
\ifprelim
\global\ansmark=1 \relax%
\else
\global\ansmark=#2 \relax%
\fi%
\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 
\ifprelim
\rightappenditem{12345}\to\list
\else
\rightappenditem{#2#3#4#5#6}\to\list%
\fi%
}

\def\nomark{\pageno=1 %
\global\notmarkanswerstrue
\headline{\vbox{\rhs{3in}{Name:\ \UL{3in}{\hfil}\hss}\vskip10pt%
\rhs{3in}{Instructor:\ \UL{2.3in}{\hfill}\hss}%
}}
\answersheet}

\def\Nomark#1{\pageno=1 %
\global\notmarkanswerstrue
\headline{\vbox{\rhs{3in}{Name:\ \UL{3in}{\hfil}\hss}\vskip10pt%
\rhs{3in}{Instructor:\ \UL{2.3in}{\quad #1\hfil}\hss}%
}}
\answersheet}


\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}

\let\\=\relax
\long\def\Xan#1a)#2b)#3c)#4d)#5e)#6\endp{%
\Ans#1�
#2�
#3�
#4�
#5�
#6�
\endp}

%%%%%%%
\font\sfm=cmr8

\def\eat#1{}
\def\exam#1{\ifprelim\Exam{ }
\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}}}}

\setbox0=\vbox{\answersheet}\pageno=2
\multichoicetrue
\newchoice{Multiple Choice}
\def\kg{{\rm kg}}
\def\log{{\rm log}}
\credit=5
\null\vskip6pt

%: 1
\Newp 1.
Suppose $f(x)$ is a one-to-one function on its domain
and $f(2) = 6$, $f(6) = 3$, $f^\prime(2) = 4$, and $f^\prime(6) = 5$.  Find
$(f^{-1})'(6)$.
�5
a) 1/4 
b) 1/5 
c) 3/4 
d) 4 
e) 5 
\endp
%: 2
\Newp 2. 
Of the following functions with specified domains, which one
does NOT have an inverse function?
�2
a) $\cos x$ for $-\pi/2 \leq x  \leq \pi/2$ 
b) $\sin x$ for $-\pi/2 \leq x  \leq \pi/2$ 
c) $x^2$ for $0 \leq x < \infty$ 
d) $\ln x$ for $0 < x < \infty$ 
e) $x^3 + 2$ for $-\infty < x < \infty$
\endp

%: 3
\Newp 3.
Solve the following equation for $x$.
$$\ln \bigl(7 - e^{2x}\bigr) = 1\ . $$
�3
a) $x = \ln \sqrt{7 - e}$ 
b) $x = \ln \sqrt{7}  - \ln \sqrt{e}$ 
c) $x = \ln \sqrt{7}$
d) $x = \ln (7 - e )^{-2}$ 
e) There is no solution.
\endp

%: 4
\Newp 4.
Find $f'(x)$ for
$$ f(x) = \ln (x^2 + 2) + \arcsin (x^{-1})\  . $$
�2
a) $\frac{2x}{x^2 + 2} - \frac{1}{\sqrt{x^4 - x^2}}$ 
b) $\frac{1}{x^2 + 2} - \frac{1}{\sqrt{x^2 - x^4}}$ 
c) $\frac{2x}{x^2 + 2} - \frac{x\ln x}{\sqrt{x^2 - 1}}$
d) $\frac{2x + 2}{x^2 + 2} - \frac{1}{x^2 + 1}$  
e) $\frac{1}{x^2 + 2} - \frac{x}{\sqrt{x^2 - 1}}$
\endp
%: 5
\Newp 5. 
Find the equation of the tangent line to $f(x)$ at $x = 1$
if
$$ f(x) = e^{\sqrt{x} + 4}\ .$$
�2
a) $y = e^5 + \frac{e^5}{2} (x - 1)$ 
b) $y = e^5 - \frac{e^5}{2} (x + 1)$ 
c) $y = \frac{e^5}{2} + e^5 (x - 1)$
d) $y = e^5 + e^{5/2} (x - 1)$ 
e) $y = e^5 + \frac{1}{5} (x - 1)$
\endp
%: 6
\Newp 6.
Evaluate the following definite integral.
$$ \int_{1}^{e} \frac{2x^3 + x + 1}{x^2} dx\ .$$
�3
a) $1 + e^2 - e^{-1}$ 
b) $3 - e^2 - e^{-1}$ 
c) $1 - 2e^2 - e^{-1}$
d) $1 - e$ 
e) $3 - e^2$
\endp
%: 7
\Newp 7.
Simplify the following expression for $x$.
$$x = \log_3 81 + \log_3 \frac{1}{9}\ .$$
�3
a) $x = 2$ 
b) $x = 9$ 
c) $x = 6$
d) $x = \ln 3$
e) $x = \ln 9 - \ln 3$
\endp

%: 8
\Newp 8.
Find $f^\prime(x)$ if
$$f(x) = x^{\ln x}\ .$$
�3
a)  $2 (\ln x) x^{(\ln x) - 1} $ 
b)  $x^{(\ln x) - 1} \ln x$ 
c)  $2 (\ln x) x^{\ln x}$ 
d)  $x^{\ln x} \ln x$ 
e)  $x^{\ln x}(\ln x + 1)$
\endp

%: 9
\Newp 9.
Calculate the following definite integral.
$$\int_0^1 \frac{5^x}{5^x + 7} dx\ .$$
�5
a) $\frac{\ln(3/2)}{\ln 5}$
b) $\ln(3/10)$
c) $\frac{\ln(12/7)}{\ln 5}$
d) $\ln(3/2)$ 
e)  $\ln(15/2)$
\endp

%: 10
\Newp 10.
Calculate the following integral.
$$\int \frac{\arctan x}{1 + x^2}\ dx\ .$$
�2
a) $\frac{\bigl(\arctan x\bigr)^2}{2} + C$ 
b) $\ln \vert 1 + x^2 \vert + C$
c) $\frac{\arctan(x^2)}{2} + C$
d) $\bigl(\ln \vert 1 + x^2 \vert \bigr)\ \arctan(x) + C$
e) $\frac{\arctan(x)}{2x}\ \bigl(\>\ln \vert 1 + x^2 \vert\>\bigr) + C$
\endp
\hide
\Newp 10.
Calculate the following definite integral.
$$\int_0^1 \frac{\arctan x}{1 + x^2} dx\ .$$
�3
a) $\pi^2/32$ 
b) $\pi^2/16$
c) $-3\pi^2/32$
d) $\pi/4$
e) $-\pi/8$
\endp
\endhide

%: 11
\Newp 11.
Determine the following limit.
$$ \lim_{x \rightarrow \infty} x\ \bigl(e^{{1/x}}-1\bigr)\ .$$
�5a) 1  
b) 0 
c) $\infty$ 
d) $-\infty$ 
e) $e$
\endp

\multichoicefalse
\def\vvbv{\vbox{\hbox to 6in{\hfill Partial Credit\hfill}%
\hbox to 6in{\rm%
\hfill You must show your work on the partial credit problems 
to receive credit!\hfill}%
\vskip10pt}}
\newchoice{\vvbv}


%: 12
\newp 12.
Use logarithmic differentiation to find $f^\prime(x)$ if
%%$$f(x) = \frac{\sin^2(x + 7) \sqrt{x + e^x}}{(2x^3 + 2)^8}\ .$$
$$f(x) = \frac{x\ \sin^2(x + 7)}{(2x^3 + 2)^8}\ .$$
{\it Please show your work.}
\endp

%: 13
\newp 13. 
Evaluate the following limit.
$$ \lim_{x \rightarrow \infty} \frac{(\ln x)^2}{x}\ .$$
{\it Please show your work.}
\endp

%: 14
\newp 14.
Calculate the following indefinite integral.
$$\int x \ln x\ dx\ .$$
{\it Please show  your work.}
\endp
\xdef\numberpages{\number\pageno}
\setbox0=\hbox{\numberpages}
\vfill\eject
}


\exam{}
\marked{}
\vfill\eject
\nomark
\end


