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

\magnification1200\footline{\ifnum\pageno=1 \hfil\else\hfil\number\pageno\hfil\fi}
\def\whichexam{Exam II}
\def\examdate{October 30, 2001}
\def\numberpages{10}
\def\numberofproblems{14}
\def\multiplechoicemax{6}
\def\partialcreditmax{10}
%%
\newif\ifprelim
\prelimfalse
%\prelimtrue
%%
\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 24315
\ll 2 15342
\ll 3 32415
\ll 4 43125
\ll 5 31425
\ll 6 13425
\ll 7 45321
\ll 8 54312
\ll 9 45123
\ll 10 25341
}

\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\credit=6
\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%
\ifmultichoice%
\ifnum\mc>0 \ifodd\breakpc\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{\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}

\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.}&
\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}



%%%%%%%
\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}

\def\kg{{\rm kg}}
\def\log{{\rm log}}
\newp 1.
If $xy^2+x^2y=\sin(x+y)$ what is $y^\prime$ at the point $(-1,1)$?
\Ans5
$-1$€
$-2$€
$0$€
$1$€
$2$€
\endp
\newp 2.
Find $\frac{d\ (\tan x\ \sec x)}{dx}$.
\Ans3
$\sec^3 x+\sec x \tan^2 x$€
$\sec^2 x+\sec x \tan^3 x$€
$\sin^3 x+\sin x \cos^2 x$€
$\sec^3 x-\sec x \tan^2 x$€
$\sin^3 x-\sin x \cos^2 x$€
\endp
\newp 3.
If $f^\prime(x)=x^2(x^2-1)(x-2)^3$ find the local minima of $f$.
Note that you are given $f^\prime$, {\bf NOT} $f$.
\Ans4
$-1$ and $2$€
$-1$, $0$, $1$ and $2$€
$0$ and $1$€
$0$€
Can't tell from the given information.\hss€
\endp
\newp 4.
If $f^{\prime\prime}(x)=x^2(x^2-1)(x-2)^3$ find the points of inflection of $f$.
Note that you are given $f^{\prime\prime}$, {\bf NOT} $f$.

\Ans4
$-1$, $1$ and $2$€
$-1$, $0$, $1$ and $2$€
$0$ and $1$€
$0$€
Can't tell from the given information.\hss€
\endp
\newp 5.
Find $\frac{d \cos\biggl(\frac{x^2}{x^2+1}\biggr)}{dx}$.
\Ans2
$-\frac{2x}{(x^2+1)^2}\cdot\sin\biggl(\frac{x^2}{x^2+1}\biggr)$€
$-\sin\biggl(\frac{x^2}{x^2+1}\biggr)$€
$-\frac{2x}{(x^2+1)^2}\cdot\cos\biggl(\frac{x^2}{x^2+1}\biggr)$€
$-\sin\biggl(\frac{2x}{(x^2+1)^2}\biggr)$€
$\cos\biggl(\frac{2x}{(x^2+1)^2}\biggr)$€
\endp
\newp 6.
Find $\displaystyle\lim_{u\to\infty}\frac{(u^2-1)^2}{4u^4-3u^3+2u^2-u}$.
\Ans5
$\frac{1}{4}$€
$+\infty$€
$-\infty$€
$-\frac13$€
Does not exist.€
\endp
\newp 7.
If $y=\frac{\sin x}{x}$ find the differential $dy$.
\Ans3
$\frac{x\cos x-\sin x}{x^2} dx$€
$\cos x dx$€
$\frac{\cos x}{x} dx$€
$\frac{x\sin x-\cos x}{x^2} dx$€
$\frac{\cos x}{x^2}  dx$€
\endp
\newp 8.
If $y=x^3-3x^2+4x+1$, find $y^{\prime\prime}$.
\Ans3
$6x-6$€
$3x^2-6x+4$€
$6$€
$4x+1$€
$7x^2-3x+2$€
\endp
\newp 9.
Find $\frac{d^{401} \sin x}{dx^{401}}$.
\Ans5
$\cos x$€
$-\cos x$€
$\sin x$€
$-\sin x$€
$\cos x+\sin x$€
\endp
\newp 10.
Which number below occurs if you use linear approxiamtion to
estimate the relative error in the area of a circle and if you have
made a relative error of $5\%$ in measuring the diameter?
\Ans5
$10\%$€
$1\%$€
$0$€
$20\%$€
$150\%$€
\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.
Find an equation for the tangent line to the curve
$x^2+y=3\sin(x+y)$ at the point $(1,-1)$.
Does the curve lie above or below the tangent line in a neighborhood of the point?
Why?
\endp
\newp 12.
You are watching an ant hill grow and you want to know how much material
the ants are excavating without disturbing them.
You observe that the hill is shaped like a cone and
you remember that the volume of a cone is $V=\frac{\pi}{3}r^2h$.
\gdef\cm{{\rm cm}}
\gdef\rt{{\rm cm}/{\rm hr}}
At a particular moment you observe that the height and the radius are each $2\cm$.
Moreover the radius is increasing at a rate of $4\rt$ and the height is increasing at a 
rate of $2\rt$.
How fast is the volume increasing at this same moment?
\endp
\newp 13.
Show that $\bigl\vert \sin(2.674)-\sin(2.670)\bigr\vert\leq 0.004$.
Also explain why it is that $\sin(2.674)-\sin(2.670)<0$?
(Hint: You may use that $\frac{\pi}2< 2.670<2.674<\pi$.)
\endp
\newp 14.
Sketch the curve $y=\sqrt{x^2+1}$ on the axes below.
You may use that $y^\prime=\frac{x}{\sqrt{x^2+1}}$ and that
$y^{\prime\prime}=\frac{1}{(x^2+1)^{3/2}}$.
Indicate the intervals on which $f$ is increasing/decreasing.
If there are no such intervals, say so.
Indicate the intervals on which $f$ is concave up/down.
If there are no such intervals, say so.
Find all the vertical asymptotes.
If there are none, say so.
Check that the line $y=x$ is a slant asymptote as $x$ goes to $+\infty$.
Check that the line $y=-x$ is a slant asymptote as $x$ goes to $-\infty$.
\vskip30pt
\null\hskip1in\epsfxsize=2.7in \epsfbox{p14.eps}\endp

\vfill\eject}


\exam{}\marked{}
\end