%% Test Authors
\magnification1200\footline{\ifnum\pageno=1 \hfil\else\hfil\number\pageno\hfil\fi}
\def\whichexam{Exam II}
\def\examdate{October 24, 2000}
\def\numberpages{10}
\def\numberofproblems{13}
\def\multiplechoicemax{5}
\def\partialcreditmax{12}
%%
\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 54231
\ll  2 34125
\ll  3 25314
\ll  4 13254
\ll  5 31425
\ll  6 45132
\ll  7 13425
\ll  8 23514
}

\def\longans#1{\vtop{\noindent\hsize=6.25 true in #1}}
\def\arctan{{\rm arctan\>}}

\long\def\ans#1€#2€#3€#4€#5€#6\endp{%
\AN{#1}{#2}{#3}{#4}{#5}\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
\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{\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\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\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}{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}


\two
\newp 1.
Find the derivative of $f(x)=\sin^2(x^2+1)$.
\ans
$f^\prime(x)=2\sin(x^2+1) \cdot \cos(x^2+1) \cdot 2x$ €
$f^\prime(x)=2\sin(x^2+1) \cdot \cos(x^2+1)$ € 
$f^\prime(x)=\cos^2(x^2+1)$ € 
$f^\prime(x)=\cos^2(x^2+1)\cdot 2x$ € 
$f^\prime(x)=2\cos(2x)$ € 
\endp

\one
\newp 2.
The function $\displaystyle f(x)=\frac{-1}{x^2+4}$, defined for all~$x$, 

\ans
has an absolute minimum, but no other extrema (local or absolute) € 
has an absolute minimum and an absolute maximum € 
has a local minimum, which is not an absolute one € 
has a local maximum € 
has exactly one critical point,	which is neither a maximum nor a minimum. € 
\endp

\newp 3.
Mr Smith starts the engine of his car at 7:30am and drives to work.
At 7:50am, he leaves his car at  the parking lot of his workplace, 
which is exactly 10 miles away from his home. Note that 20 minutes is
${{1}\over{3}}$ of an hour.

You may assume that the distance--time function is 
differentiable for any car. Which of the following conclusions hold?
\ans
On his trip to work, Mr Smith must have had a speed of 30mi/h at least once. € 
He was 5mi from home at 7:40am.  € 
His instantaneous speed will have been  30mi/h no more than twice. € 
The mean value theorem will tell us the acceleration at any time during his trip. € 
We cannot know whether he ever had the speed 30mi/h on this trip. € 
\endp

\newp 4.
Suppose the acceleration of an object moving along a coordinate line
is given by $a(t) = 3 t^{2}$. If at time $t=0$, the position of
the object is $s(0)=5$ and the velocity is $v(0)=1$, what is the position of
the object when $t=2$?

\five
\ans
$11$€
$10$€
$9$€
$7$€
$12$€
\endp

\one
\newp 5.
Let $$f(x) = 3 x (x-4)^{\smfrac{1}{3}}.$$ 
Then $$f^\prime(x)=4(x-3)(x-4)^{-\smfrac23}$$
and
\ans
\longans{$f$ has a local minimum at $x=3$ and another critical point $x = 4$, which is not a local extremum point.}€
\longans{$f$ has a local maximum at $x=3$ and another critical point $x = 4$, which is not a local extremum point.}€
$f$ has a local minimum at $x=3$ and a local maximum at $x = 4$.€
$x=3$ is the only critical point and $f$ has a local minimum at this point.€
$x=3$ is the only critical point and $f$ has a local maximum at this point.€
\endp

\five\newp 6.
Let $y=t^2-t-1$ and $x=t^2+t+1$. 
Which formula below is the differential $dy$ when $t=1$?
\ans
$\smfrac13\> dx$€
$dx$€
$3\> dx$€
$\smfrac12\> dx$€
$2\> dx$€
\endp

\one
\newp 7.
Let $f(x) = x^{4} - 2 x^{2}$. Then the extreme values of $f$ over
the {\bf interval} $[0, 2]$ are:
\ans
Absolute maximum value is $8$. Absolute minimum value is $-1$.€
Absolute maximum value is $8$. Absolute minimum value is $-2$.€
Absolute maximum value is $16$. Absolute minimum value is $-1$.€
Absolute maximum value is $0$. Absolute minimum value is $-1$.€
No absolute maximum value. Absolute minimum value is $-1$.€
\endp

\five
\newp 8.
Use the linearization of the function $f(x)=x^{\smfrac1{10}}$ at $x=1$ 
to estimate $(1.01)^{\smfrac{1}{10}}$.
One obtains
\ans
$1.001$€
$1.01$€
$1.0001$€
$0.99$€
$0.999$€
\endp

\multichoicefalse
\newchoice{Partial Credit}

\newp 9.
The point $(3,2)$ lies on the curve given by the equation
$$y^3+y+x^3+x=40\ .$$ 

Calculate the slope of the tangent to this curve at this point.
\endp

\newp 10.
The hour hand of a large clock at a public building is 4ft long, the minute
hand has 6ft. What is the rate of change of the angle between the two
hands? At what rate is the distance between the tips of both hands
changing, when it is 2 o'clock?

\noindent{\bf Hints:} You may use the {\it law of cosines\/}, which relates the quantities
depicted in the figure for an arbitrary triangle by the formula $c^2 = a^2 + b^2 -2ab\cos\gamma$:

\hskip1.7 true in \hbox to 0pt{\vbox{
\noindent\setbox0=\hbox{\noindent\epsfxsize=3in \epsfbox{p10.eps}}%
\picV=\ht0 %
\vbox to 0pt{\box0 \vss}\vrule width 0pt height 0pt depth\picV%
}\hss}

\noindent You also know enough to find $\gamma$ at 2 o'clock but finding
$\frac{d\gamma}{dt}$ at 2 o'clock is more difficult. It helps to recall that both the 
hour hand and the minute hand revolve at fixed rates.
\endp

\vfill\eject\newp 11.
$$ f(x) = 1-9x -6x^{2} - x^{3}.$$

\noindent Find intervals where $f$ is (a) increasing (b) decreasing (c) concave up
(d) concave down. Identify local extreme values and locate inflection points.
Then sketch the graph of $f$.
\noindent\setbox0=\hbox{\noindent\epsfxsize=3in \epsfbox{p11.eps}}%
\picV=\ht0 %
\vbox to 0pt{\box0 \vss}\vrule width 0pt height 0pt depth\picV%

\endp

\newp 12.
For the autonomous differential equation
$$ \frac{dy}{dx} = y^{2} - y $$
find all equilibrium values and judge which are stable and which 
are unstable. Construct a phase line and sketch at least one solution
curve in each region divided by equilibrium solutions.
\noindent\setbox0=\hbox{\noindent\epsfxsize=3in \epsfbox{p12.eps}}%
\picV=\ht0 %
\vbox to 0pt{\box0 \vss}\vrule width 0pt height 0pt depth\picV%
\endp

\newp 13.
\$320 are available to fence in a rectangular garden. The fencing for
the side of the garden facing the road costs \$6 per foot and the
fencing for the other three sides costs \$2 per foot. Find the dimensions
of the garden with the largest possible area.
\endp

\vfill\eject}


\exam{}\marked{}
\end
