\def\whichexam{Exam III - 126 S2000}
\def\examdate{April 25, 2000}
\def\numberpages{11}
\def\numberofproblems{15}
%%
\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 51423
\ll  2 41235
\ll  3 42513
\ll  4 32415
\ll  5 35421
\ll  6 25341
\ll  7 14325
\ll  8 15234
\ll  9 14352
\ll 10 54132
}

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

\long\def\LANp\item#1\item#2\item#3\item#4\item#5\end#6{%
\ANp(a)#1(b)#2(c)#3(d)#4(e)#5\Pra%
}

\def\dxd{\displaystyle}

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\def\iterate{\body\let\next=\iterate\else\let\next=\relax\fi\next}
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\hfuzz=20pt

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\def\ANp#1(a)#2(b)#3 (c)#4 (d)#5 (e)#6\Pra{%
\AN{#2}{#3}{#4}{#5}{#6}\Pra%
}

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(c)\quad\box\Ac(d)\quad\box\Ad%
(e)\quad\box\Ae}

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(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{\pC}{\Ac}\switchbox{\pD}{\Ad}\switchbox{\pE}{\Ae}%
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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}}}}

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\newcount\pc
\def\pr{\vskip10pt\noindent\number\pc.\global\advance\pc by 1}
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}

\def\marked{\pageno=1 %
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\headline{\vbox{\rhs{3in}{Name:\ \ul{3in}\hss}\vskip10pt%
\rhs{3in}{Instructo:\ \UL{2.3in}{Bullwinkle}\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 %
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}}
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\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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\tabalign\hfill\number\numberpartialcredit.&\quad\ul{1in}\cr\global\advance\numberpartialcredit by 1
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}}{10pt}{1in}\vss}}}}
\answersheet
\multichoicetrue
\newchoice{Multiple Choice}


\three
\newp 1.
Find the foci of the ellipse 
$\displaystyle 
{x^2\over 4} + {y^2\over 13} = 1.$
\LANp
\item
$(0,3),(0,-3)$
\item
$(3,0),(-3,0)$
\item
$(\sqrt{17},0),(-\sqrt{17},0)$
\item
$(0,\sqrt{17}),(0,-\sqrt{17})$
\item
$(3,\sqrt{17}),(-3,\sqrt{17})$
\end{enumerate}


\three
\newp 2.
Find the equation of the hyperbola having foci $(6,0),(-6,0)$ and 
eccentricity $e = {6\over 5}$.
\LANp
\item
$\dxd{x^2\over 25}-{y^2\over 11}=1$
\item
$\dxd{x^2\over 11}-{y^2\over 25}=1$
\item
$\dxd{x^2\over 25}-{y^2\over 36}=1$
\item
$\dxd{x^2\over 36}-{y^2\over 25}=1$
\item
$\dxd{x^2\over 11}-{y^2\over 36}=1$
\end{enumerate}

\five
\newp 3.
Find the interval of convergence for the power series
$$\sum_{n=1}^\infty {{(x-5)^n}\over{ 3^n n}}$$

\noindent The radius of convergence is $3$.
\LANp
\item
$[2,8)$
\item
$(2,8]$
\item
$[2,8]$
\item
$(2,8)$
\item
$(-\infty, \infty)$
\end{enumerate}

\two
\newp 4.
Find the first 4 nonzero terms of the Maclaurin series for 
$${{\cos x}\over {(1-x})}.$$

\noindent{\bf Hint:}
Multiply a power series for $\cos x$ by a power series for $\frac1{1-x}$.
\LANp
\item
$1+x+{{1} \over {2}}x^2+{{1} \over {2}}x^3$
\item
$1+x+x^2+x^3$
\item
$1-{{1} \over {2!}}x^2+{{1} \over{ 4!}}x^4-{{1} \over {6!}}x^6$
\item
$x+{{1} \over{ 2!}}x^3+{{1} \over {4!}}x^5-{{1} \over {6!}}x^6$
\item
$1-x+{{1} \over {2!}}x^2+{{1} \over {3!}}x^3$
\end{enumerate}

\two
\newp 5.
Find the third Taylor polynomial $P_3(x)$ for $f(x)={1 \over x}$ at $a=2$.
\LANp
\item
${1\over 2}-{1\over 4}(x-2)+{1\over 8}(x-2)^2-{3\over 48}(x-2)^3$
\item
$1+x+x^2+x^3$
\item
${1\over 2}+{1\over 4}(x-2)+{1\over 4}(x-2)^2+{3\over 8}(x-2)^3$
\item
$(x-2)+{1\over 2!}(x-2)^3+{1\over 4!}(x-2)^5-{1\over 6!}(x-2)^6$
\item
$1-x+{1\over 2!}x^2+{1\over 3!}x^3$
\end{enumerate}

\three
\newp 6.
Which one of the following series converges conditionally?
\LANp
\item
$\sum_{n=1}^\infty {(-1)^n \over n^{1 \over 3} + 1}$
\item
$\sum_{n=1}^\infty {1 \over n^{1 \over 3}}$
\item
$\sum_{n=1}^\infty {(-1)^{n+1} \over n^5}$
\item
$\sum_{n=1}^\infty {1 \over n^5}$
\item
$\sum_{n=1}^\infty n$
\end{enumerate}

\newp 7.
Find the first three terms of the binomial series 
$${1 \over (1+x)^{1 \over 2}} = (1+x)^{-{1 \over 2}}$$
\LANp
\item
$1-{x \over 2} + {3 x^2\over 8}$
\item
$1+{x \over 2} + {3 x^2\over 8}$
\item
$1+x+x^2$
\item
$1-x+x^2$
\item
$1-{x^2 \over 2} + {x^4\over 4!}$
\end{enumerate}


\newp 8.
Which of the following series converge?
$$(1) \sum_{n=0}^\infty {n \over (n^7+1)^{1 \over 3}}$$
$$(2) \sum_{n=0}^\infty {n 2^{3n} \over 3^{2n}}$$
$$(3) \sum_{n=0}^\infty {n! \over 2^n}$$
\LANp
\item
(1) and (2)
\item
(1), (2) and (3)
\item
(1) only
\item
(2) only
\item
none of the above
\end{enumerate}

\two
\newp 9.
Find Taylor's remainder $R_3(2)$ for the function $f(x)=e^{2x}$ at $a=0$. 
\LANp
\item
$\dxd{{16 e^c} \over {4!}} (2)^4$ for some $0 < c < 2$
\item
$\dxd{{16 e^c} \over {4!}} (2)^4$ for some $2 < c $
\item
$\dxd{{8 e^c} \over {3!}} (2)^3$ for some $0 < c < 2$
\item
$\dxd{{8 e^c} \over {3!}} (2)^3$ for some $c > 2$
\item
$\dxd{{16 e^2} \over 4!} (c)^4$ for some $0 < c < 2$
\end{enumerate}

\five
\newp 10.
Find the radius of convergence for the power series 
$$\sum_{n=0}^\infty {(x-2)^n \over 12^n}$$ 
\LANp
\item
$12$
\item
$6$
\item
$24$
\item
$\infty$
\item
$0$
\end{enumerate}


\multichoicefalse
\newchoice{Partial Credit}

\newp 11.
\item{a.}Show that the series 
$$\sum_{n=2}^{\infty} {(-1)^n \over n \ln n}$$ converges
by applying the Alternating series test.
To receive credit you should explicitly verify the hypotheses of
this test.

\item{b.}Use the Integral Test to show that this series does not absolutely 
converge. Again you must check the necessary hypotheses.

\newp 12.
Write down the first four non{--}zero terms of the MacLaurin series
solution to the differential equation with initial value
$$y^\prime - 2 y = 0\ ,
\quad y(0)=1\ .$$
\newp 13.
Find the interval of convergence for the power series
$$\sum_{n=1}^{\infty} {(-1)^n(x-3)^{2n} \over 2^n}$$

\newp 14.
Find the Maclaurin series for each of the following functions:
\item{(a)}$\dxd{1 \over 1-2x}$
\item{(b)}$\dxd{2 \over (1-2x)^2}$\qquad (hint: differentiate your answer to (a))
\item{(c)}$\dxd{2x \over (1-2x)^2}$

\newp 15.
Given the function $f(x)=\sum_{n=1}^{\infty} (-1)^n {n x^{n-1} \over 11^n}$, 
let $F(x)=\dxd\int_0^x f(t) dt$. 
Use what you know about alternating series to estimate how closely the order
$3$ MacLaurin polynomial approximates $F(x)$ for any $x$ in the interval $[0,1]$.

\vfill\eject}

\exam{}
\marked{}
\end


