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\magnification1200\footline{\ifnum\pageno=1 \hfil\else\hfil\number\pageno\hfil\fi}
\def\whichexam{Exam III}
\def\examdate{April 23, 2002}
\def\numberpages{10}
\def\numberofproblems{15}
\def\multiplechoicemax{6}
\def\partialcreditmax{10}
%%
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\prelimfalse
\prelimtrue
%%
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\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 25431
\ll 2 32145
\ll 3 32541
\ll 4 23145
\ll 5 42531
\ll 6 12435
\ll 7 13524
\ll 8 53124
\ll 9 45231
\ll 10 53142
\ll 11 23145
\ll 12 45321
}

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

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

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\else%
\ifnum\mc>0 \newpage\null\vskip10pt\fi\fi%
\mc=1 \fi}

\long\def\Newp#1.#2�#3a)#4\endp{
\newp#1.#2
\Xan#3
a)#4\endp}


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\global\advance\breakpc by1 %
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\newbox\aa\newbox\ab\newbox\ac
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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}

\long\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%
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\newcount\pcc
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\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%
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\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
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\newcount\pc
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\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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\ifnum\ansmark=2 \relax($\bullet$)\else(b)\fi&
\ifnum\ansmark=3 \relax($\bullet$)\else(c)\fi&
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\hskip\AA\cr\global\advance\numbermultiplechoice by1 
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}

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}}
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\Ans#1�
#2�
#3�
#4�
#5�
#6�
\endp}
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\Ans#1�
#2�
#3�
#4�
#5�
#6�
\endp}

%%%%%%%
\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 Cao please indicate 10:40 or 11:45}%
}}
\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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}}{10pt}{1in}\vss}}}}
\answersheet
\multichoicetrue
\newchoice{Multiple Choice}

\def\kg{{\rm kg}}
\def\log{{\rm log}}
\credit=5

\Newp 1.
Let $x=\sin(9t)$ and $y=\cos(9t)$.
Then $\frac{dy}{dx}=?$
�5
a)$-\tan(9t)$\\
b)$\cot(9t)$\\
c)$\tan(9t)$\\
d)$9\tan(t)$\\
e)$81\sec^2(9t)$\\
\endp

\Newp 2.
Find the points on the curve $x=t^2$, $y=t^2-2t$
where the tangent line is horizontal.
�3
a)$(x,y)=(1,-1)$\\
b)$(x,y)=(1,1)$\\
c)$(x,y)=(2,1)$\\
d)$(x,y)=(1,2)$\\
e)$(x,y)=(2,-1)$\\
\endp

\Newp 3.
Find the length of the curve $x=4e^{^{\smfrac{t}{2}}}$, $y=e^t-1$,
$0\leq t\leq1$.
�2
a)$\displaystyle\int_0^1 \sqrt{e^{2t}+4e^t\ }\ dt$\\
b)$\displaystyle\int_0^1 \sqrt{1+e^{2t}\ }\ dt$\\
c)$\displaystyle\int_0^1 \sqrt{1+4e^t\ }\ dt$\\
d)$\displaystyle\int_0^1 \sqrt{e^{2t}+14e^t+1\ }\ dt$\\
e)$\displaystyle\int_0^1 \sqrt{e^{2t}-14e^t+1\ }\ dt$\\
\endp

\Newp 4.
Find the area of the surface obtained by rotating the curve $x=3t^2$, $y=t^3-3t$, $0\leq t\leq 1$
around the $y${--}axis.
�4
a)$\frac{48}{5}\pi$\\
b)$18\pi$\\
c)$3$\\
d)$1$\\
e)$0$\\
\endp

\Newp 5.
Find $\displaystyle\sum_{n=0}^{\infty}\ 3\biggl(\frac{-2}{5}\biggr)^n$.
 �5
a)$\frac{15}{7}$\\
b)$\frac{6}{5}$\\
c)$\frac{17}{5}$\\
d)$\frac{5}{3}$\\
e)$\frac{3}{5}$\\
\endp

\Newp 6.
Consider the parabola with vertex $(0,0)$, axis the $x${--}axis 
and passing through $(1,-4)$. Which point below is the focus of this parabola?
�5
a)$(4,0)$\\
b)$(-4,0)$\\
c)$(2,0)$\\
d)$(-2,0)$\\
e)$(1,-2)$\\
\endp 

\Newp 7.
Given the sequence $s_n=\frac{9n+1}{10n+9}$, compute $\Lim_{n\to\infty}\ s_n$. 
�5
a)$\frac{9}{10}$\\
b)$\frac{10}{9}$\\
c)$\frac{1}{9}$\\
d)$0$\\
e)$+\infty$\\
\endp

\Newp 8.
Evaluate the sum
$\displaystyle\sum_{n=1}^{\infty}\biggl(\frac{1}{n^{100}}-\frac{1}{(n+1)^{100}}\biggr)$
�4
a)$1$\\
b)$0$\\
c)$\frac{1}{100}$\\
d)$+\infty$\\
e)Divergent but bounded\hss\\
\endp

\Newp 9.
Let $s=\displaystyle\sum_{n=1}^{\infty}\ \frac{1}{n^3}$,
$s_n=\displaystyle\sum_{i=1}^{n}\ \frac{1}{i^3}$ and
$R_n=s-s_n$.
Use the integral test to estimate $R_n$.
�2
a)$\frac{1}{2}\frac{1}{(n+1)^2}\leq R_n\leq \frac{1}{2}\frac{1}{n^2}$\\
b)$\frac{1}{(n+1)^2}\leq R_n\leq \frac{1}{n^2}$\\
c)$\frac{1}{3}\frac{1}{(n+1)^3}\leq R_n\leq \frac{1}{3}\frac{1}{n^3}$\\
d)$\frac{1}{(n+1)^3}\leq R_n\leq \frac{1}{n^3}$\\
e)$\frac{1}{2}\frac{1}{(n+1)^3}\leq R_n\leq \frac{1}{2}\frac{1}{n^3}$\\
\endp

\Newp 10.
Use Comparison Tests to determine which {\bf one} of the following series is divergent.
�3
a)$\displaystyle\sum_{n=1}^{\infty}\ \frac{n^2-1}{n^3+100}$\\
b)$\displaystyle\sum_{n=1}^{\infty}\ 7\biggl(\frac 56\biggr)^n$\\
c)$\displaystyle\sum_{n=1}^{\infty}\ \frac{1}{n^2+8}$\\
d)$\displaystyle\sum_{n=1}^{\infty}\ \frac{1}{n^{\smfrac32}+1}$\\
e)$\displaystyle\sum_{n=1}^{\infty}\ \frac{n}{n+1}\biggl(\frac12\biggr)^n$\\
\endp

\Newp 11.
The series $\displaystyle\sum_{n=1}^\infty \frac{(-1)^{n-1}}{n^3}$ is an alternating
series which satisfies the conditions of the alternating series test.
Find the smallest number $k$ on the list below so that the $k$-th partial sum
is within $\frac{1}{1,000}$ of the actual sum.
�4
a)$10$\\
b)$15$\\
c)$20$\\
d)$25$\\
e)$50$\\
\endp

\Newp 12.
Given that $\displaystyle \sum_{n=1}^{\infty}\ \frac{1}{n^2}=\frac{\pi^2}{6}$ it follows that
$\displaystyle \sum_{n=1}^{\infty}\ \frac{1}{(2n)^2}=
\frac14\biggl( \sum_{n=1}^{\infty}\ \frac{1}{n^2}\biggr)=
\frac{\pi^2}{24}$.
Then $\displaystyle \sum_{n=0}^{\infty}\ \frac{1}{(2n+1)^2}=1+\frac{1}{9}+\frac{1}{25}+\cdots=\ ?$
�5
a)$\frac{\pi^2}{8}$\\
b)$\frac{\pi^2}{9}$\\
c)$\frac{2\pi^2}{9}$\\
d)$\frac{3\pi^2}{8}$\\
e)$\ln2$\\
\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}
\credit=13
\newp 13.
Determine whether the series
$$\sum_{n=1}^{\infty}\ (-1)^n\ \frac{n^2+1}{n^3-9}$$
is absolutely convergent, conditionally convergent or divergent.
The correct answer with no justification is worth 2 points.
\endp

\newp 14.
Find the radius of convergence and interval of convergence of the power series
$$\sum_{n=1}^{\infty}\ \frac{(-1)^n}{\sqrt n}\ (x-3)^n$$
The correct answer with no justification is worth 2 points.
\endp

\credit=14
\newp 15.
Find the power series centered at the origin for $f(x)=\frac{1}{(2-x)^2}$.
\par\noindent{\bf Hint:}$f(x)$ is related to the derivative of a series you should know.
\par\noindent The correct answer with no justification is worth 2 points.
\endp

\vfill\eject}


\exam{}\marked{}
\end
