%% Alber
\gdef\localbox#1{\epsfscale=800\vtop to1.5in{\null\vskip -1in\epsfbox{Ans#1.eps}\vss}}
\long\def\hide #1\endhide{}

\magnification1200\footline{\ifnum\pageno=1 \hfil\else\hfil\number\pageno\hfil\fi}
\def\whichexam{Exam III}
\def\examdate{December 4, 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 14523
\ll 2 32145
\ll 3 41235
\ll 4 43125
\ll 5 42153
\ll 6 53142
\ll 7 21435
\ll 8 52341
\ll 9 23514
\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}

\long\def\xan#1a.#2b.#3c.#4d.#5e.#6\endp{%
\Ans#1
#2€
#3€
#4€
#5€
#6€
\endp}

%%%%%%%
\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.
Evaluate the following integral by making an appropriate
substitution $$\int_0^1 \frac{t+t^2}{\sqrt {3t^2+2t^3+1}} dt = ?$$
\xan5
a. $\frac{\sqrt{6}-1}{3}$
b. $\frac{\sqrt{3}-1}{2}$
c. $\frac{\sqrt{3}-\sqrt{2}}{3}$
d. $\frac{5}{2}$
e. $\frac{1}{3}$
\endp

\newp 2.
If f(x) is continuous and $\int_0^3 f(x) dx = 12$ find
$$\int_1^2 x f(x^2-1) dx\ .$$
\xan5
a. $6$
b. $0$
c. $-4$
d. $10$
e. $24$
\endp

\newp 3.
Calculate the definite integral
$$\int_{-\pi}^{\pi} \sin (\sin u) du = ?$$
\xan5
a. $0$
b. $2 \pi$
c. $\frac{\pi}{2}$
d. $2$
e. $\frac{3\pi}{2}$
\endp

\newp 4.
Let $y(x)$ denote the solution of the equation
$$\frac{dy}{dx} = x^2 (x-1)$$
such that $y(1)=2$.
Then $y(0)=?$.
\xan5
a. $\frac{25}{12}$
b. $\frac{16}{15}$
c. $\frac{1}{4}$
d. $\frac{20}{3}$
e. $\frac{21}{15}$
\endp

\newp 5.
Calculate the following definite integral
$$\int_1^4 \frac{2u -1}{\sqrt{u}} du = ?$$
\xan5
a. $\frac{22}{3}$
b. $3$
c. $\frac{30}{3}$
d. $0$
e. $\frac{21}{2}$
\endp

\newp 6.
A rocket is fired vertically into the air. Its velocity at $t$
seconds after lift-off is $v(t)= 8t + 1$ meter per second. Before
launch, the top of the rocket is 10 meters above the launch pad.
Find the height of the rocket (measured from the top of the rocket
to the launch pad) at $3$ seconds.
\xan5
a. $49$ meters
b. $39$ meters
c. $25$ meters
d. $35$ meters
e. $42$ meters
\endp

\newp 7.
Evaluate the following integral
$$\int_0^2 |x^2-1| dx = ?$$
\xan5
a. $2$
b. $0$
c. $\frac{1}{3}$
d. $\frac{5}{3}$
e. $\frac{2}{3}$
\endp

\newp 8.
Calculate the definite integral
$$\int_{0}^{\pi} ((\sin \theta)^2 +1) \cos \theta d \theta = ?$$
\xan5
a. $0$
b. $\pi$
c. $\frac{\pi}{3}$
d. $1$
e. $\frac{3\pi}{2}$
\endp

\newp 9.
Find the definite integral
$$\int_{-\pi}^{\pi} \bigl(\cos (x+\pi) - \sin (x - \pi)\bigr) dx.$$
\xan5
a. $0$
b. $\pi$
c. $1$
d. $2 \pi$
e. $2$
\endp

\newp 10.
Find the area of the domain bounded by the graph of $y=x^3 -1$,
the $x$-axis and the two vertical lines $x=1$ and $x=4$.
\hide wrong answers
a. $61$
b. $-24$
c. $10$
d. $4$
e. $-5$\endhide
\xan5
a. $60\> \smfrac34$
b. $-(23\>\smfrac14)$
c. $11\>\smfrac12$
d. $4\>\smfrac13$
e. $-(5\>\smfrac14)$
\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.
Write down the formula given by Newton method for solving the
equation $$ x^3 + x^2 + 1 =0\ .$$
Tell why there is a solution to this equation in the interval $[-2,-1]$.
\endp

\newp 12.
A rectangular garden of area 80 square feet is to be surrounded on
three sides by a brick wall costing \$20 per foot and on one side
by a fence costing \$10 per foot. Find the dimensions of the
garden such that the cost of materials is minimized. Describe your
solution in detail.
\endp

\newp 13.
Find all points on the parabola $y=x^2$ that are closest to the
point (0,1). Justify your answer in detail.
\endp
\newp 14.

Use Riemann sums to calculate the following integral:
$$ \int_0^1 (3x^2 +2) dx\ .$$
Pick $x_i^*$ to be the right-hand endpoint of each subinterval.
Justify your solution in detail.

Hint: You can use the following formulae:

$$ \sum_{i=1}^n i = \frac{n(n+1)}{2}\ ,\qquad \sum_{i=1}^n i^2 =
\frac{n(n+1)(2n+1)}{6}\ ,\qquad \sum_{i=1}^n i^3 = \frac{n^2 (n+1)^2}{4}\
.$$

\endp


\vfill\eject}


\exam{}\marked{}
\end
