\magnification1200\footline{\ifnum\pageno=1 
\hfil\else\hfil\number\pageno\hfil\fi}
\def\whichexam{Final Exam }
\def\examdate{December 14, 2001}
\def\numberpages{14}
\def\numberofproblems{25}
\def\multiplechoicemax{6}
\def\partialcreditmax{12}

\gdef\pagebreaks{%
\\{1}\\{3}\\{5}\\{7}\\{9}\\{11}\\{13}\\{15}\\{17}\\{19}\\{21}\\{23}\\{25}\\{ 
25}\\{26}%
}
\gdef\answersheetbreaks{\\{1}\\{13}\\{100}}

\def\checkPBN#1@{%
\edef\yy{\number#1}\def\\##1{ \edef\yyy{\number##1}\ifx\yyy\yy 
\gdef\PBN{1}\fi}%
\xxpagebreaks}
\global\let\xxpagebreaks=\pagebreaks

\def\epsfbox#1{}
\input epsf.def

\def\anspic#1#2{}

\def\answersheetbreakcount{}
\def\pagebreakcount{}
\def\leaderfill{\leaders\hbox to3pt{\hss.\hss}\hfill}
\def\pskip{\vskip -8pt\hskip20pt\hbox to 2 true in{\leaderfill}\vskip -8pt}
%%
\newif\ifprelim
\prelimfalse
%\prelimtrue
%%
\def\un#1{\>\hbox{#1}}
\def\fs{\un{ft/sec}}

\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 31245
\ll 2 23415
\ll 3 42135
\ll 4 53214
\ll 5 25134
\ll 6 25134
\ll 7 45123
\ll 8 35142
\ll 9 32154
\ll 10 15243
\ll 11 53142
\ll 12 12354
\ll 13 14352
\ll 14 53421
\ll 15 12345
\ll 16 32451
\ll 17 51243
\ll 18 14532
\ll 19 32415
\ll 20 41532
\ll 21 45321
\ll 22 52413
\ll 23 21453
\ll 24 41253
\ll 25 21543
}

\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{}

\def\beginmcbox{\vtop\bgroup\hsize=1 true in}
\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{
\ifprelim\vskip8pt\else%
\gdef\PBN{0}\checkPBN\pc@%
\def\PBNx{1}%
\ifnum\pc=1 \vskip10pt\else
\ifx\PBN\PBNx\vfill\eject\null\vskip10pt\else\vfill\fi%
\fi\fi%
%\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{%
\gdef\yy{#1}%
\ifx\yy\answersheetbreakcount \ifnum#1>1 \egroup%
\beginmcbox\fi%
\leftremove\Pagebreaks\to\pagebreakcount%
\leftremove\Answersheetbreaks\to\answersheetbreakcount%
\fi%
\ifx\yy\pagebreakcount\ifnum#1>1 \pskip\fi%
\leftremove\Pagebreaks\to\pagebreakcount%
\fi%
\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}

\font\sfm=cmr8

%%%%%%%
\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}%
\rhs{3in}{\hskip 1cm\sfm If Hind please indicate 10:40 or 1:55}}%
}%

\long\gdef\answersheet{
\global\let\Answersheetbreaks=\answersheetbreaks
\global\let\Pagebreaks=\pagebreaks
\leftremove\Answersheetbreaks\to\answersheetbreakcount
\leftremove\Pagebreaks\to\pagebreakcount
\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 two hours.
\itemitem{$\bullet$} You will only hand in this page, so be sure
you have marked the answer sheet below correctly. Dotted lines and new 
columns indicate page breaks
in the test.
\itemitem{$\bullet$} Be sure that you have all \numberpages\ pages of the test.
\vskip5pt
\centerline{Good Luck!}
\vskip4pt
\def\HH{\hskip.45in}
\settabs\+9.\qquad &\HH&\HH&\HH&\HH&\HH&\cr
%\hskip.5 true in%
\boxit{
\vbox{\hfuzz=10in\baselineskip20pt
\noindent
PLEASE MARK YOUR ANSWERS WITH AN X, not a circle!
\vskip2pt\noindent
\hbox to 6.5true in{{\beginmcbox\correctanswers\egroup}
}}}{10 true pt}{6.5 true in}
%%\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!
\vskip4pt
\tabalign {\bf Final Exam Total:}&\quad\ul{1in}\cr

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

\newp 1.
Find the limit
$\Lim_{x\to 1}\>\frac{\sin (x-1)}{x-1}$.
\Ans 5
$1$€
$0$€
$-1$€
$+\infty$€
Does not exist.€
\endp

\newp 2.
Find the limit
$\Lim_{x\to 0}\>\frac{\sqrt{x+9} - 3}{x}$.
\Ans 5
$\frac 16$€
$\frac 13$€
$3$€
$-3 $€
Does not exist€
\endp

\newp 3.
If $f$ and $g$ are two continuous functions with $f(1) = 2$\par
\hskip 30pt and $\lim_{x 
\to 1}[3f(x) + g(x)]
= 6$, find $g(1)$.
\Ans 5
$0$€
$2$€
$1$€
$-3$€
$3$€
\endp

\newp 4.
For what value of the constant $c$ is the function $f$ continuous on 
$(-\infty, \infty)$
where $f(x) =  c x + 1$ for $x \le 2$, 
$f(x) =   c x^2 -1 $ for $x> 2$?
\Ans 5
$1$€
$3$€
$-1$€
$2$€
$4$€
\endp

\newp 5.
Find an equation for the tangent line to the curve $y= \frac{1}{\sqrt{x}}$ 
at the point
$(1, 1)$.
\Ans 4
$2y= - x + 3 $€
$2y= x + 3 $€
$2y= - x - 3 $€
$2y= x - 3 $€
$2x= - y + 3 $€
\endp

\newp 6.
If $f(3) = 4$, $g(3) = 2$, $f^\prime(3) = -6 $ and $g^\prime(3) = 5$,
find $\biggl(\frac fg\biggr)^\prime(3)$.
\Ans 5
$-8$€
$\frac 18$€
$-1$€
$ 8 $€
$0$€
\endp

\newp 7.
The solution to the initial value problem
$\frac{dy}{dx}=\sec(x^2)$; $y(0)=1$ is given by
\Ans 3
$y = 1+\int^x_0\sec(t^2)\> dt$€
$y = \int^x_1\sec(t^2)\> dt$€
$y = 1+\int^{x^2}_0\sec(t)\> dt$€
$y = \int^{x^2}10\sec(t)\> dt$€
$y = \int^x_0\sec(t^2+1)\> dt$€
\endp

\newp 8.
The position function of a particle is given by $s = t^3 - 4.5 t^2 - 7 t$, 
for $t \ge 0$.
When does the particle reach a velocity of $5 \fs$ ?
\Ans 5
$4$ seconds€
$2$ seconds€
$7$ seconds€
$5$ seconds€
$0$ seconds€
\endp

\newp 9.
If $g(3) = 6$, $g^\prime(3) = 4$, $f^\prime(3) = 2$, 
$f^\prime(6) = 7$ and $F(x) = f(g(x))$, 
find
$F^\prime(3)$.
\Ans 5
$28$€
$6$€
$24$€
$14$€
$42$€
\endp

\newp 10.
Let $x^2- y^2 = 1$. Find $\frac{dy}{dx}$.
\Ans 5
$\frac xy$€
$-\frac xy$€
$\frac yx$€
$- \frac yx$€
$ 2x - 2y$€
\endp

\newp 11. Find the equation of the tangent line to the curve $y^2 = 
x^3(2-x)$ \par \hskip30pt at the point
$(1,1)$.
\Ans 5
$y = x $€
$y = -x $€
$y = x +1 $€
$y = x - 1$€
$y = \frac 1x$€
\endp

\newp 12.
Find the linearization $L(x)$ of $f(x) = x^3$ at the point $a = 1$.
\Ans 5
$3x - 2 $€
$3x + 1$€
$2x -3 $€
$x - 1 $€
$2x + 3$€
\endp

\newp 13.
Find the maximum value of $ f(x) = \frac{x}{x+2}$ on the closed interval 
$[3, 4]$.
\Ans 5
$\frac 23$€
$\frac 35$€
$ \frac 12$€
$\frac 34$€
$\frac 13$€
\endp


\newp 14. Find all numbers $c \in [ -1, 1]$ that satisfy $f^\prime(c) = 
\frac{f(1)- f(-1)}{2}$\par\hskip30pt for $f(x) = 3 x^2 + 2x - 5$.
\Ans 5
$ 0 $€
$ 1 $€
$ -1$€
$ 3 $€
$ -5 $€
\endp

\newp 15.
What are the $x${--}coordinates of all
the inflection points of \par\hskip30pt $f(x) = x^4 - 6 x^2 + 100 x + 99$.
\Ans 3
$x=-1$ and $x=1$€
$x=-6$€
$x=0$€
$x=0$ and $x=-1$€
$x=-6$ and $x=1$€
\endp

\newp 16.
Find all the slant asymptotic lines of $ y = f(x) = \frac{x^2 + x +1 }{x}$.
\Ans5
$y = x + 1$€
$y = x - 1 $€
$y = -x + 1$€
$y = x $€
$y = \frac{1}{x}$€
\endp

\newp 17.
Find {\bf all} the points on the hyperbola $y^2 - x^2 = 4$ that are closest
\par\hskip30pt to the point $(2, 0)$.
\Ans5
$(1, \pm \sqrt{5})$€
$(1, \sqrt{5})$€
$(\sqrt{5}, 1)$€
$(-1, \sqrt{5})$€
$(1, \pm 5)$€
\endp

\newp 18.
If $f(x) = \int^{5x}_{0} \cos (u^2) du $, find $f^\prime(x)$.
\Ans 4
$5 \cos (25 x^2)$€
$5 \cos (5 x^2)$€
$-5 \cos (25 x^2)$€
$-5 \cos (5 x^2)$€
$-25\cos (5 x^2)$€
\endp

\newp 19.
Evaluate the integral $\int^{\sqrt\pi}_0 x \sin(x^2) dx$.
 \Ans 5
$1$€
$2$€
$1 - \frac 1\pi$€
$\frac14$€
$\frac\pi4$€
\endp

\newp 20.
Compute the area of the region below the curve
$y = 2x $ and above the curve\par\hskip30pt $ y = x^2 - 4x$.\par
\null\hskip 1in\epsfbox{graph20.eps}
\Ans 2
$\int_0^6\bigl(2x-(x^2-4x)\bigr)\> dx$€
$\int_0^6\bigl((x^2-4x)-2x\bigr)\> dx$€
$\int_0^4\bigl(2x-(x^2-4x)\bigr)\> dx$€
$\int_0^4\bigl((x^2-4x)-2x\bigr)\> dx$€
$\int_0^4\bigl(2x-(x^2-4x)\bigr)\> dx+ \int_4^6\bigl((x^2-4x)-2x\bigr)\> dx$€
\endp

\newp 21.
The shape of a football can be obtained by rotating the curve
$y = x - x^2$ around  $x$-axis for $0 \le x \le 1$.
Compute its volume.\par
\null\hskip 1in\epsfbox{graph21.eps}
\Ans 3
$\pi \int_0^1\bigl(x-x^2\bigr)^2\> dx$€
$2\pi \int_0^1x\bigl(x-x^2\bigr)\> dx$€
$\pi \int_0^1\bigl((x-x^2)^2-1^2\bigr)\> dx$€
$2\pi \int_0^1\bigl(x-x^2\bigr)\> dx$€
$\pi \int_0^1\bigl(x^2-x^4\bigr)\> dx$€
\endp

\newp 22.
The solid is obtained by rotating
about the y-axis the region bounded by $ y = 2 + \sin x $,
$y = \sin x$, $ x = 0$ and $ x = 2 \pi$. Find its volume.
\par
\null\hskip 1in\epsfbox{graph22.eps}
\Ans 5
$8 \pi^3$€
$4 \pi^3$€
$2 \pi^3 $€
$8 \pi^2$€
$\pi^3 $€
\endp

\newp 23.
An ice cube is melting at a rate of $3{\rm cm}^3/{\rm hr}$.
Assume it remains a cube during the melting process.
How fast is the length of a side changing when this length is $2{\rm cm}$?
\def\un{\ {\rm cm}/{\rm sec}}
\Ans 5
$\frac14\un$€
$3\un$€
$\frac38\un$€
$\frac83\un$€
$2\un$€
\endp

\newp 24.
If $f(x) = \int^x_0\sec(t)\>dt$, find $f^{\prime\prime}(x)$.
\Ans 4
$\sec(x) \>\tan(x)$€
$\sec(x^2)$€
$\sec(x)$€
$\cot(x)$€
$0$€
\endp


\newp 25.
Find the average value of the function $f(x) = 4x - x^2$ on the interval
$[0, 3]$.
\Ans 5
$3$€
$4$€
$\frac 13$€
$\frac 14$€
$9$€
\endp

\vfill\eject}


\exam{}\marked{}
\end





