%Taylor
\def\whichexam{Final}
\def\examdate{May 7, 2001}

\newcount\n\newcount\na\newcount\nb

%% List of problem number at top of each page with last entry = number of problems +1
%% Begin with the problem number at the top of the second page since problem 1 is
%% always at the top of the first page of problems.
\gdef\pagebreaks{\\{3}\\{5}\\{7}\\{9}\\{11}\\{13}\\{15}\\{17}\\{19}%
\\{21}\\{23}\\{25}\\{26}}

%% These lines are bold on the marked answer sheet but not the
%% regular student answer sheet.
\gdef\boldlines{\\{34}}
\gdef\boldlines{\\{1}\\{7}\\{12}\\{18}\\{34}}


\newif\ifprelim
\prelimtrue
\prelimfalse

\newif\ifprofver
\profvertrue
\profverfalse
%%
\def\Sum#1#2{\displaystyle\sum_{#1=#2}^\infty}
\def\ellipse#1#2{%
\setbox0=\hbox{$99$}%
\dimen0=\wd0 %
\frac{x^{\hbox to0pt{$2$\hss}}}{\hbox to\wd0{\hfil$#1$\hfil}}+%
\frac{y^{\hbox to0pt{$2$\hss}}}{\hbox to\wd0{\hfil$#2$\hfil}}=1}

\def\dd#1#2{{{\textstyle d\> #1}\over{\textstyle d#2}}}
 \def\frac#1#2{{{\textstyle#1}\over{\textstyle#2}}}
 \def\smfrac#1#2{{{#1}\over{#2}}}

\def\Lim{\displaystyle\lim}
\def\Int{\displaystyle\int}
\def\brace#1#2{%
\def\xbx{#1}%
\ifx\xbx\empty\relax(#2)\else%
\csname #1l\endcsname(#2\csname #1r\endcsname)%
\fi}
%%
\magnification1200\footline{\ifnum\pageno=1 \hfil\else\hfil\number\pageno\hfil\fi}

\def\setdata#1{\count0=1 \def\\##1{\advance\count0 by1 }#1 %
\edef\numberpages{\number\count0}%
\def\\##1{\count0=##1 }#1 \advance\count0 by-1 %
\edef\numberofproblems{\number\count0 }}%
\setdata\pagebreaks

%%
\long\def\hide#1\endhide{}

  \input boxedeps.tex 
  \input boxedeps.cfg 
\SetTexturesEPSFSpecial
%%
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\newtoks\ta\newtoks\tb
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\def\list{}
\newcount\pcc
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\def\OR{%
\bigskip
\noindent{\bf OR}\bigskip
\expandafter\leftinsertitem\moose\to\list%
}

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\ifprelim\vskip8pt\else%
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(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%
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\vskip\Ls\noindent(e)\quad\box\Ae}

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\setbox\ac=\box#2\or\setbox\ad=\box#2\or\setbox\ae=\box#2\fi}

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\resetboxes\ifcase\Ax \or\praone\or\pratwo\or\prathree\or \or\prafive\fi\vfill}




\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\rhs#1#2{\par\noindent\hbox to\hsize{\hfill\hbox to#1{#2\hfill}}}
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\def\UL#1#2{\hbox to0pt{\qquad #2\hss}\ul{#1}}
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\def\+{\tabalign}
\long\gdef\answersheet{
\null\vskip10pt
\centerline{\bf Math 126, \whichexam}
\centerline{\examdate}
\vskip10pt

\itemitem{$\bullet$}
The Honor Code is in effect for this examination.
All work is to be your own.
\itemitem{$\bullet$} Be sure that you have all \numberpages\ pages of the test. 
\itemitem{$\bullet$} No calculators are to be used.
\itemitem{$\bullet$} The exam lasts for two hours.
\itemitem{$\bullet$} You are to hand in just the front page.
\vskip5pt
\centerline{Good Luck!}
\vskip10pt
\def\leaderfill{\leaders\hbox to3pt{\hss.\hss}\hfill}
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\def\HH{\hskip.42in}
\settabs\+9.\qquad &\HH&\HH&\HH&\HH&\HH&\cr
\centerline{Please mark your answers with an {\bf X}!\qquad Do NOT circle them!}
\par\centerline{\small The dotted lines in the answer box indicate page breaks.}
\hskip-45pt\boxit{%
\noindent\hbox to0pt{%
\multiplechoicebox
\ll 1 15324
\ll 2 35124
\ll 3 12435
\ll 4 14523
\ll 5 45213
\ll 6 24315
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\ll 13 53142
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\mcbbreak
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\ll 16 45132
\ll 17 21345
\ll 18 52314
\ll 19 32154
\ll 20 41235
\ll 21 13254
\ll 22 24315
\ll 23 32541
\ll 24 45123
\ll 25 53412
\vskip6pt
Score 1: \ul{1cm}\hskip1cm Score 4: \ul{1cm}
\vskip6pt
Score 2: \ul{1cm}
\vskip6pt
Score 3: \ul{1cm}\hskip1cm Total: \ul{1cm}
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\long\def\Exam{%
\coversheet{ }%

\vfill\eject\headline{\hfil}
\five
\newp 1. 1@
Let $f(x)= 2x^3+1$ and
let $f^{-1}$ denote the inverse function.
Then $(f^{-1})^\prime(17)=$
\ans
$\frac1{24}$€
$\frac1{17}$€
$\frac 1{7}$€
$\frac1{3}$€
$\frac{24}{17}$€
\endp 1.

\five
\newp 2. 1@
$\Int_e^{e^2}{1\over x(\ln x)^2}\>dx =$
\ans
$\frac12$€
$\frac1{2e^2}-\frac1{e}$€
$2$€
$2e^2-e$€
Diverges€
\endp 2.

\five
\newp 3. 1@
If $y^\prime(t) = t\cdot\cos(t^2)$ and $y(0)=1$, then $y\bigl(\sqrt{\smfrac\pi2\, }\ \bigr)=$
%-sin(t^2)/2 +1
\ans
$\frac32$€
$\sqrt{\frac\pi2\, }$€
$\sqrt{\frac\pi2\, }-1$€
$-2$€
$\sqrt{\frac\pi2\, }\cdot\frac32$€
\endp 3.

\two
\newp 4. 1@
The solution to the initial value problem
$$x\frac{dy}{dx}+x^2 y+x^2 = 0\qquad y(0) = 0$$
is
\def\xs{-{{\vrule width 0pt height 0pt depth 2pt\,\scriptstyle x^{\hbox to 0pt{$\scriptscriptstyle2$\hss}}}%
\over{\vrule width 0pt height 6pt depth 0pt\scriptstyle2}}}
\ans
$y=e^{^{\xs}}-1$€
$y=e-e^{^{\xs\,+\,1}}$€
$y=e^{-x}-1$€
$y=1-e^{-x}$€
$y=xe^x$€
\endp 4.

\five
\newp 5.  1@
Assuming uniform density $\delta$, the moment about the $y${--}axis of 
the plane region bounded by the axes and the line $y=6-3x$ is
\ans
$4\delta$€
$4.5\delta$€
$3\delta$€
$3\smfrac13\delta$€
$3.5\delta$€
\endp 5.

\three
\newp 6. 1@
The solution to the initial value problem
$$y^\prime=\frac{\sin x}{2y+1}\hskip 1in y(0)=2$$
satisfies the implicit equation
\ans
$y^2+y=7-\cos x$€
$y^2+y=6\cos x$€
$2y+1=5e^{-\cos x}$€
$2y+1=6-e^{-\cos x}$€
$e^{2y+1}=e^5+\arcsin x$€
\endp 6.


\five
\newp 7. 2@
$\Lim_{x\to 0^+}\ \brace{}{1+\cot x}^{\smfrac 1x}=$
\medskip

\ans
$\infty$€
$0$€
$e^{-1}$€
$1$€
Does not exist€
\endp 7.


\four
\newp 8.  2@
$\Lim_{x\to\infty}\ \frac{(\ln x)^{2.5}}{x^{0.01}}=$
\noindent
\ans
$0$€
$\infty$€
$e^{0.01}$€
$\ln(2.5)$€
Does not exist€
\endp 8.

\five
\newp 9. 2@
$\Int^{\smfrac{\sqrt2}{2}}_0\frac{dx}{\sqrt{1-x^2\>}}=$ 
\ans
$\frac\pi4$€
Diverges€
$\ln(\sqrt{2\,}-1)$€
$\frac\pi{\sqrt{2\ }}$€
$\frac\pi4-1$€
\endp 9.

\five
\newp 10. 2@
$\Int_0^{\pi/2}x\cos(x)dx=$ 
\ans
$\frac\pi2 - 1$€
$1-\frac\pi2$€
$0$€
$\frac{\sqrt{3\,}}{2}-\frac{\sqrt{2\,}}{2}$€
Diverges€
\endp 10.

\four
\newp 11. 2@
$\Lim_{t\to \infty}{\tanh\, t}=$
\ans
$1$€
$0$€
$\infty$€
$-\infty$€
Does not exist€
\endp 11.

\three
\newp 12. 2@
$\frac{x^2+x+2}{(x-1)(x^2+1)}=$
\ans
$\frac{2}{x-1}-\frac{x}{x^2+1}$€
$\frac{2}{x-1}-\frac{1}{x^2+1}$€
$\frac{2}{x-1}+\frac{3}{x^2+1}$€
$\frac{2}{(x-1)^2}-\frac{1}{x+1}$€
$\frac{2}{x-1}+\frac{1}{x^2+1}$€
\endp 12.

\five
\newp 13.  3@
Find $\Int_{0}^{1} \frac{x\ dx}{x^2-1}$.
\ans
Diverges€
$0$€
$1$€
$2$€
$ -1$€
\endp 13.

\five
\newp 14.  3@
Find 
$\Sum n1 \frac{2^{2n}}{5^{n-1}}$ 
\ans
$20$€
$\frac{4}{5}$€
$\frac{5}{4}$€
$4$€
$5$€
\endp 14.

\one
\newp 15. 3@
Which of the following series converge absolutely?

\noindent\null\hfil(1) $\Sum n0 \frac{\sin(2n)}{n!}$\hfil(2) $\Sum n2 \frac{n}{(\ln n)^2}$\hfil
(3) $\Sum n1(-1)^n\frac{n^2}{n^{3}+1}$\hfil\null
\ans
(1) converges absolutely, (2) and (3) do not€
(1) and (2) converge absolutely, (3) does not€
(1) and (3) converge absolutely, (2) does not€
(2) and (3) converge absolutely, (1) does not€
(3) converges absolutely, (1) and (2) do not€
\endp 15.

\four
\newp 16. 3@
Find $\Lim_{n \rightarrow \infty} n\cdot \sin\brace{bigg}{\smfrac 1n}$
\ans
$1$€
$\infty$€
$0$€
$e^{-1}$€
Does not exist€
\endp 16.

\one
\newp 17.  3@
Test the following series for
absolute convergence, conditional convergence or divergence:

\noindent\null\hfil
(1) $\Sum n1 \frac{(-1)^n}{\sqrt{n\,}}$\hfil
(2) $\Sum n0 \frac{(-1)^n}{(1.2)^n}$\hfil
(3) $\Sum n0 \frac{(-1)^n}{n^{1.2}}$\hfil\null
\ans
(1) converges conditionally, (2) and (3) converge absolutely€
(1) and (2) converge conditionally, (3) converges absolutely€
(1) and (2) converge absolutely, (3) converges conditionally€
(1) and (3) converge absolutely, (2) converges conditionally€
(1) converges absolutely, (2) and (3) converge conditionally.€
\endp 17.

\five
\newp 18.  3@
Find the interval of convergence for 
$$\Sum n1 \frac{(-1)^nx^n}{\sqrt{n^2+2}}$$
\par\noindent{\bf Remark:} 
$\frac{1}{\sqrt{n^2+2}}$ is decreasing for $n>0$.
\ans
$-1<x\leq 1$€
$-1<x < 1$€
$-1\leq x < 1$€
$-1\leq x\leq 1$€
all $x$€
\endp 18.

\three
\newp 19.  4@
Which series below is the MacLaurin series (Taylor series centered at $0$)
 for $x\,\sin(x^2)$?
\ans
%$\Sum n0 (-1)^n\frac{x^{2n+3}}{(2n+1)!}$€
${x^3}-\frac{x^7}{3!}+\frac{x^{11}}{5!}-\cdots$€
%$\Sum n0  x^{2n+1}$€
$x+x^3+x^5+\cdots$€
%$\Sum n0 \frac{x^{2n}}{n}$€
$x^2+\frac{x^4}{2}+\frac{x^6}{3}+\cdots$€
%$\Sum n0 \frac{(-1)^nx^{2n+1}}{n!}$€
$x-\frac{x^3}{3!}+\frac{x^5}{5!}-\cdots$€
%$\Sum n0 (-1)^nx^{2n}$€
$x-x^2+x^4-\cdots$€
\endp 19.

\three
\newp 20.  4@
Find the order 2  MacLaurin polynomial (Taylor polynomial centered at $0$)
for the solution to the initial value problem
$$y^\prime + 2y = 2x\qquad\qquad y(0)=1$$
\ans
$1-2x+3x^2$€
$1+x+x^2$€
$1-\smfrac1{2!} x+\smfrac1{3!} x^2$€
$1+x-\smfrac1{2!} x^2$€
$1-2x+\smfrac 29 x^2$€
\endp 20.

\two
\newp 21. 4@
Which MacLaurin series (Taylor series centered at $0$) 
represents the function
$\Int_0^x\cos\sqrt{t\,}\>dt$?
\ans
%$\Sum n0 \frac{(-1)^n\,x^{n+1}}{(n+1)(2n)!}$€
$x-\frac{x^2}{4}+\frac{x^3}{3\cdot 4!}-\cdots$€
%$\Sum n0 \frac{(-1)^n\,x^{n+1}}{(n+1)(2n+1)!}$€
$\frac{x}{2}-\frac{x^2}{2\cdot 3!}+\frac{x^3}{3\cdot 5!}-\cdots$€
%$\Sum n0 \frac{(-1)^n\,x^{2n+1}}{(n+1)(2n)!}$€
$\frac{x}{2}-\frac{x^3}{2\cdot 2!}+\frac{x^5}{3\cdot 4!}-\cdots$€
%$\Sum n0 \frac{(-1)^n\,x^{n+1}}{(n+1)!}$€
$x-\frac{x^2}{2!}+\frac{x^3}{3!}-\cdots$€
The given function has no MacLaurin series\hss€
\endp 21.

\four
\newp 22. 4@
The point $\brace{bigg}{2,\frac{7\pi}3}$ in polar coordinates corresponds to 
which point below in Cartesian coordinates?
\ans
$(1,\sqrt 3\>)$€
$(\sqrt 3, 1)$€
$(-1,\sqrt 3\>)$€
$(-\sqrt 3, 1)$€
Since $\frac{7\pi}3>2\pi$, there is no such point\hss€
\endp 22.

\four
\newp 23. 4@
$\Lim_{x\to 0}\ \frac{\ln(1+x^2)-x^2}{x^4}=$
\vskip10pt\noindent{\bf Hint:}
Without MacLaurin series this may be a hard problem.
\ans
$-\frac12$€
$\infty$€
$0$€
$-\infty$€
Does not exist€
\endp 23.

\five
\newp 24. 4@
Find the area inside the cardioid $r=2+2\cos\theta$.
\HideDisplacementBoxes
\par\pictboxdim=120pt
\pictboxstrut\ForceWidth{\pictboxdim}\vskip-100pt\BoxedEPSF{cardioid.eps}
\par
\ans
$6\pi$€
$8\pi$€
$6$€
$8$€
$3\pi+\ln 4$€
\endp 24.

\two
\newp 25. 4@
Which graph below is the graph of the polar curve $r=2-3\sin(\theta)$?
\HideDisplacementBoxes
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\ans
\pictboxstrut\ForceWidth{\pictboxdim}\BoxedEPSF{graphA.eps}€
\pictboxstrut\ForceWidth{\pictboxdim}\BoxedEPSF{graphB.eps}€
\pictboxstrut\ForceWidth{\pictboxdim}\BoxedEPSF{graphC.eps}€
\pictboxstrut\ForceWidth{\pictboxdim}\BoxedEPSF{graphD.eps}€
\pictboxstrut\ForceWidth{\pictboxdim}\BoxedEPSF{graphE.eps}€
\endp 25.


}

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Exam 1 = \number\eA\quad
Exam 2 = \number\eB\quad
Exam 3 = \number\eC\quad
Exam 4 = \number\eD
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