%% Denzler & Hind 
\magnification1200\footline{\ifnum\pageno=1 \hfil\else\hfil\number\pageno\hfil\fi}
\def\whichexam{Exam III}
\def\examdate{November 28, 2000}
\def\numberpages{10}
\def\numberofproblems{13}
\def\multiplechoicemax{5}
\def\partialcreditmax{12}
%%
\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 12435
\ll 2 31425
\ll 3 52413
\ll 4 23415
\ll 5 24351
\ll 6 43125
\ll 7 41532
\ll 8 35214
}

\def\rhb#1{\noindent\hbox to\hsize{\hfil#1}}

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\def\arctan{{\rm arctan\>}}

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\AN{#2}{#3}{#4}{#5}{#6}\Pra}

\def\dxd{\displaystyle}

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\vskip10pt\centerline{{\bf#1}}\mc=0 %
\ifnum\pc<3\credit=\multiplechoicemax \else\credit=\partialcreditmax \fi}
\newif\ifmultichoice
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\hfuzz=20pt

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\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{%
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\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%
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\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}}}
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\hskip\AA\cr\global\advance\numbermultiplechoice by1 
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\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}
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\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

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DO NOT WRITE IN THIS BOX!
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\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}


\newp 1.
Evaluate $$\int y^2\cos \frac{y^3}{3}\,dy\;.$$
The answer is:
\Ans4
$\sin\frac{y^3}{3} + C$€
$\frac{y^3}{3}\sin\frac{y^3}{3} + C$€
$y^2\sin\frac{y^3}{3} + C$€
$3\sin\frac{y^3}{3} + C$€
$y^2\sin\frac{y^4}{12} + C$€
\endp

\newp 2.
Given the following values, what is the value of
$\dxd\int_1^2 f(x)\>dx$?\par
$\dxd\int_1^{10} f(x)\>dx=10$; $\dxd\int_4^{10} f(x)\>dx=5$;
$\dxd\int_2^4 f(x)\>dx=6$.

\Ans4
$-1$€
$1$€
$0$€
$21$€
The answer can not be determined from the data.\hss€
\endp

\newp 3.
Let $f(x) = \Int_0^{1+x^2} \frac{1}{t^3+1}\,dt$. 
Evaluate $f^\prime(2)$.
\Ans5
$\frac{2}{63}$€
$\frac{1}{126}$€
$\frac{1}{9}$€
$\frac{2}{9}$€
$\frac{5}{9}$€
\endp

\newp 4.
The equation $x^4+x-1=0$ has exactly one positive solution, which is
near~$1$. Which answer below is the result of one iteration of
Newton's method applied to this equation with $1$ as the starting point?
\Ans5
$\frac45$€
$1$€
$\frac54$€
$\frac34$€
$\frac43$€

\endp

\newp 5.
Which of the following is a correct solution of the initial value problem
$$
\frac{dy}{dx} = \frac{1}{3+\sin x}
\;,\quad
y(1)=2
$$
\Ans2
$\dxd y(x)=  2+\int_1^x \frac{1}{3+\sin t}\,dt$€
$\dxd y(x)=  1+\int_2^x \frac{1}{3+\sin t}\,dt$€
$\dxd y(x)=  2+\int_1^{\smfrac{1}{3+\sin x}}t \>dt$€
$\dxd y(x)=  1+\int _1^{x^2}\frac{1}{3+\sin t}\,dt$€
$\dxd y(x)=  1+\int_1^x \frac{1}{3+\sin t}\,dt$€
\endp

\newp 6.
Evaluate
$$
\sum_{k=1}^{100} k^{-1/2} - \sum_{k=1}^{99} k^{-1/2}
$$
\Ans2
$\frac{1}{10}$€
$\int_1^{100} x^{-1/2}\,dx - \int_1^{99} x^{-1/2}\,dx $€
$99^{-1/2}$€
$100^{-1/2} - 99^{-1/2}$€
\hbox to0pt{Cannot be determined without knowledge of the value of~$k$.\hss}€
\endp

\newp 7.
$$\int_0^{\pi/2} (1-\sin^2x)\,\cos x\,dx = ?$$
\Ans5
$\frac{2}{3}$€
$\frac{\pi}{2}-\frac{\pi^3}{24}$€
$\frac{4}{3}$€
$\frac{\pi}{2}+\frac{\pi^3}{24}$€
$1$€
\endp

\newp 8.
Let $f(x)=1/\cos(x)=\sec(x)$ and consider the definite integral $\Int_0^1f(x)\,dx$?
Divide the interval of integration into $5$ equal pieces.
Which sum below is the Riemann sum for this partition where the point in each interval
is a point at which $f(x)$ obtains its minimum in that interval.

%\epsfscale=500
\noindent\rhb{\epsfbox{p8.eps}}%
\Ans1
$\frac{1}{5} \Bigl( \f(0) + \f(\smmfrac{1}{5}) + \f(\smmfrac{2}{5}) + 
\f(\smmfrac{3}{5}) + \f(\smmfrac{4}{5}) \Bigr)$€
$\frac{1}{5} \Bigl( \f(\smmfrac{1}{5}) + \f(\smmfrac{2}{5}) + \f(\smmfrac{3}{5}) + \f(\smmfrac{4}{5}) + \f(1)\Bigr)$€
$\frac{1}{2} \Bigl( \f(0) + \f(1) \Bigr)$€
$\frac{1}{10} \Bigl( \f(0) + 2 \f(\smmfrac{1}{5}) + 2 \f(\smmfrac{2}{5}) + 
2 \f(\smmfrac{3}{5}) + 2 \f(\smmfrac{4}{5}) + \f(1) \Bigr)$€
$\f(1)$€
\endp

\multichoicefalse
\newchoice{Partial Credit}

\newp 9.
Write down the formula for Simpson's Rule applied to the integral
$$
\int_0^6 \frac{x^2+1}{x^4+1}\,dx
$$
where you have divided the interval into $6$ pieces.

\noindent{\bf No credit will be given for simplifying your answer, but points may
be deducted for blatant arithmetical errors if you attempt to simplify.}
(The {\sl style\/} of the answers for problem 8 is what is wanted here.)
\endp

\newp 10.
The curves $y=\sin x$ and $y=\cos x$ enclose an area as given in the figure. 
Set up a definite integral which calculates the area of this region.

\noindent{\bf No credit will be given for evaluating your integral but points may be deducted for especially
``creative'' attempts.}

\epsfscale=800 \rhb{\epsfbox{p10.eps}}%

\endp

\newp 11.
Remember that an error estimate for the trapezoidal rule is given by
$$
\Bigg\vert\int_a^b f(x)\,dx - T\>\Bigg\vert \le \frac{b-a}{12} h^2 
\Bigl(\>\max_{x\in[a,b\,]}\>\big\vert f^{\prime\prime}(x)\big\vert\Bigr)\qquad;\qquad
h=\frac{b-a}{n}
\leqno(\ast)$$
where $T$ is the formula you are expected to know from the Trapezoid Rule and $n$ is the
number of subintervals. 

\bigskip
Assume you want to approximate $\int_0^2\sin(x^2)\,dx$ using the trapezoidal
rule and will tolerate an error of at most 0.1. 
\medskip
\item{(a)}
   Calculate the second derivative (you should know of which expression) and
   show that its absolute value is $\le 18$.
\item{(b)}
   Use the formula $(\ast)$ above with 
$\max_{x\in[a,b\,]}\>\big\vert f^{\prime\prime}(x)\big\vert\leq18$
   from (a) to find the smallest number $n$ so that the resulting estimate of the error is  less than $0.1$.

\endp

\newp 12.
Out of a solid ball of radius~$R$, a cylindrical hole of radius~$r$ ($r<R$) has
been drilled centrally (i.e., the axis of the cylinder passes through the
center of the ball). 
Set up a definite integral, in terms of $R$ and $r$, for the volume of
the remaining body.

\noindent{\bf No credit will be given for evaluating your integral but points may be deducted for especially
``creative'' attempts.}

\noindent\rhb{\epsfbox{p12.eps}}%
\medskip
\epsfscale=500\noindent\rhb{\epsfbox{jdfig.eps}}%

\endp

\newp 13.
The triangle whose corners have coordinates $(0,0)$, $(1,3)$, $(3,0)$ is
rotated around the $x$-axis to produce a solid of revolution. 
\itemitem{(a)} Set up a definite integral which gives the volume of this solid.
\itemitem{(b)} Evaluate your integral.

\noindent{\bf Hint:} The line $x=\frac y3$ goes through $(0,0)$ and $(1,3)$ while
the line $x=-\frac{2y}3+ 3$ goes through $(3,0)$ and $(1,3)$

\epsfscale=1600 \rhb{\epsfbox{p13.eps}}%

\endp

\vfill\eject}


\exam{}\marked{}
\end
