
\gdef\localbox#1{\epsfscale=800\vtop to1.5in{\null\vskip
-1in\epsfbox{:graphs:Ans#1.eps}\vss}}

\gdef\localbox#1{\epsfscale=900\vtop to1.5in{\null\vskip
-1in\epsfbox{:graphs:Ans#1.eps}\vss}}

\gdef\Localbox#1{\epsfscale=2400\vtop to3.25in{\null\vskip
-1in\epsfbox{:graphs:#1.eps}\vss}}


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

\magnification1200\footline{\ifnum\pageno=1 \hfil\else\hfil\number\pageno\hfil\fi}
\def\whichexam{Exam I}
\def\examdate{September 24, 2002}
\def\numberpages{0}
\def\numberofproblems{15}
\def\multiplechoicemax{5}
\def\partialcreditmax{10}
%%
\newif\ifprelim
\prelimfalse
%\prelimtrue
%%
\newif\ifproblemspacing
\problemspacingtrue
\problemspacingfalse
\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 15423
\ll 2 24531
\ll 3 43251
\ll 4 21453
\ll 5 14235
\ll 6 53142
\ll 7 12345
\ll 8 42315
\ll 9 25413
\ll 10 35241
}

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

\long\def\xan#1a.#2b.#3c.#4d.#5e.#6\endp{%
\Ans#1€
#2€
#3€
#4€
#5€
#6€
\endp}
\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
\Newp 1.
If $f(2)=5$, $f(3)=2$, $f(4)=5$, $g(2)=6$, $g(3)=2$ and $g(4)=0$,
find $(f g)(2)+(f\circ g)(3)$.
€5
a)$35$\\
b)$30$\\
c)$25$\\
d)$20$\\
e)$15$\\
\endp

\Newp 2.
Let a function $f(x)$ be given by
$f(x)=\displaystyle\cases{2x+2&$x>0$\cr -x+c&$x\leq0$\cr}$.
For what value of $c$ is $f$ continuous?
€3
a)$c=2$\\
b)$c=0$\\
c)$c=1$\\
d)$c=3$\\
e)There is no such $c$.\hss\\
\endp

\Newp 3.
Compute $\Lim_{x \to 5} \frac{x-6}{x-5}$.
€2
a)Does not exist and is not $\infty$ or $-\infty$.\hss\\
b)$+\infty$\\
c)$-\infty$\\
d)$0$\\
e)$1.1$\
\endp

\Newp 4.
 Compute $\Lim_{x \to 5^+} \frac{x^2-25}{x-5}$.
€3
a)$10$\\
b)$+\infty$\\
c)$0$\\
d)$5$\\
e)Does not exist and is not $+\infty$.\hss\\
\endp

\Newp 5.
Find $f^\prime(4)$ if $f(x)=4\sqrt x-\frac{16}{\sqrt{x}}$.
€5
a)$2$\\
b)$3$\\
c)$4$\\
d)$0$\\
e)$-1$\\
\endp

\Newp 6.
Find $\frac{d\> f}{dx}$ if $f(x)=(x^2+1)(x^3+1)$.\\
€2
a)$2x(x^3+1)+(x^2+1)(3x^2)$\\
b)$6x^3$\\
c)$5x^4+8x+1$\\
d)$(x^2+1)(x^3+1)+(2x)(3x^2)$\\
e)$3x^4(x^3+1)+(x^2+1)(4x^5)$\\
\endp

\Newp 7.
Find $f^\prime(3)$ if $f(x)=\frac{x^2-x}{x+2}$.\\
€5  
a)$\frac{19}{25}$\\
b)$\frac{41}{25}$\\
c)$\frac{25}{19}$\\
d)$\frac{25}{41}$\\
e)$\frac{29}{25}$\\
\endp

\Newp 8.
Which line below is the tangent line to the graph of
$y=(x^2-3x+1)(x-3)$ when $x=3$.
€2 
a)$y=x-3$\\
b)$y=2x-3$\\
c)$y=\bigl((2x-3)(x-3)+(x^2-3x+1)\bigr)x-3$\\
d)The tangent line at $x=3$ is vertical.\hss\\
e)$y=x+3$\\
\endp

\Newp 9.
Given that $f$ and $g$ are differentiable at $x=3$ and
that $f(3)=-1$, $g(3)=2$, $f^\prime(3)=3$ and $g^\prime(3)=-4$,
what is $\Bigl(\frac gf\Bigr)^\prime(3)$?
€5
a)$-2$\\
b)$2$\\
c)$\frac 12$\\
d)$-\frac 12$\\
e)$-\frac{2}{9}$\\
\endp

\vskip1.5in\Newp 10.
For which graph below is $y=2x+1$ a tangent line?
\vskip60pt\null
€3
a)\localbox{A}\\
b)\localbox{B}\\
c)\localbox{C}\\
d)\localbox{D}\\
e)\localbox{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}

\newp 11.
Let $f(x)=\frac{1}{x + 1}$.
\par\noindent 
(a)Using the limit definition of the derivative, find $f^\prime$.
\par\noindent (b) Determine the domain of $f^\prime$.
\endp

\newp 12.
 Let $$f(x)=\frac{x}{1 + x^2}\ .$$
\par\noindent (a) Compute $f^\prime (x)$.
\par\noindent (b) Determine where $f^\prime (x) = 0$.
\endp

\newp 13. 
How and why can the Intermediate Value Theorem 
be used to show that 
$$y=f(x)=\frac{x^3 - x - 2}{x + 1}$$
has a root between $x=1$ and $x=2$?
\endp

\newp 14.
 Draw the graph of a continuous function $y=f(x)$
with $f(0)=3,$ $f^\prime (0)=-1,$ $f^\prime (2)=0,$ and
$f^\prime(-2) = 2$.
\vskip 2true in
\Localbox{BlankGraph}
\endp

\newp 15.
A ball thrown in the air at 4 m/sec on planet
X has height $s(t)=4t - 3t^2 - t^3$ meters above the surface $t$
seconds after it is thrown.
\par\noindent (a) Find the velocity at time $t$.
\par\noindent (b) Find the velocity when the ball hits the ground.
\endp

\xdef\numberpages{\number\pageno}\vfill\eject
}


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