\documentclass[12pt]{article}
\usepackage[sumlimits]{amsmath}
\usepackage{amssymb}

% key.sty (Mark W. Johnson, Sept. 8, 2000)
% A LaTeX style (used with the article class) for typing
% tests.
% Use with article.cls by inserting \usepackage{test}.




%%%%% Multiple Choice Question formats
%%%%% 	Each of the following commands takes five arguments, namely
%%%%%	the five answers to a multiple choice question.  They differ
%%%%%	in how the answers are formatted, with letters automatically
%%%%%	added.  If formatting problems arise, one need only change
%%%%%	the brief command to switch to a different format.

\newcommand{\Answers}[5]{\begin{center}(a) #1\hfil(b) #2\hfil(c) #3
\hfil(d) #4\hfil(e) #5 \end{center}}
%%%%%	Simply places all five answers on one line.  (short answers)

\newcommand{\Answerm}[5]{\begin{center}(a) $#1$\hfil(b) $#2$\hfil(c) $#3$
\hfil(d) $#4$\hfil(e) $#5$ \end{center}}
%%%%%	Simply places all five answers on one line, in math mode.

\newcommand{\threetwo}[5]{\begin{center}(a) $#1$\hfil(b) $#2$\hfil(c) $#3$ \\
\vspace{10pt}(d) $#4$\hfil(e) $#5$\end{center}}
%%%%%	Three answers on the first line, with two on the second, math mode.

\newcommand{\twothree}[5]{\begin{center}(a) $#1$\hfil(b) $#2$ \\
\vspace{10pt}(c) $#3$\hfil(d) $#4$\hfil(e) $#5$\end{center}}
%%%%%	Two answers on the first line, and three on the second, math mode. 

\newcommand{\twotwoone}[5]{\begin{center}(a) $#1$\hfil(b) $#2$ \\
\vspace{10pt}(c) $#3$ \hfil(d) $#4$ \\ \vspace{10pt}(e) $#5$\end{center}}
%%%%%	Two answers on each of the first two lines, with a single
%%%%%	answer on the third, math mode.

\newcounter{lett}
\newcommand{\singles}[5]{\begin{list}{(\alph{lett})}
{\usecounter{lett}} \item#1\item#2\item#3\item#4\item#5
\end{list}}
%%%%%	A simple list with one answer on each line.  (left-aligned)

\newcounter{lette}
\newcommand{\singlem}[5]{\begin{list}{(\alph{lette})}
{\usecounter{lette}} \item$#1$\item$#2$\item$#3$\item$#4$\item$#5$
\end{list}}
%%%%%	A simple list with one answer on each line, math mode.  (left-aligned)




%%%%% Essay Question formats

\newenvironment{halfpage}{\begin{list}{\textbf{\arabic{numb}.}}
{\setlength{\itemsep}{200pt plus 10pt minus 10pt}
\setlength{\topsep}{60pt plus 10pt minus 10pt}\usecounter{numb}}}
{\end{list}}
%%%%%	Partial credit problem in a (roughly) half-page format.

\newcommand{\half}[2]{\begin{halfpage}\setcounter{numb}{#1}#2\end{halfpage}}
%%%%%	The short command for the (roughly) half-page environment.
%%%%%	The first argument
%%%%%	indicates how many questions have gone before, in order
%%%%%	to number problems correctly.

\newenvironment{thirdpage}{\begin{list}{\textbf{\arabic{numb}.}}
{\setlength{\itemsep}{155pt plus 10pt minus 10pt}
\setlength{\topsep}{155pt plus 10pt minus 10pt}\usecounter{numb}}}
{\end{list}}
%%%%%	Partial credit problem in a (roughly) third of a page format.

\newcommand{\third}[2]{\begin{thirdpage}\setcounter{numb}{#1}#2
	\end{thirdpage}}
%%%%%	Short command for the third of a page format.  The first argument
%%%%%	indicates how many questions have gone before, in order
%%%%%	to number problems correctly.





%%%%% Now some standard test formatting commands.

\newenvironment{tops}{\begin{center}\bfseries}{\end{center}\vspace{20pt}}
%%%%% 	A header environment.

\newcommand{\blank}[1]{\makebox[#1]{\hrulefill}}
%%%%%	A generic blank line, with length argument.

\newcommand{\Name}[1][2.5in]{\textbf{Name:}\blank{#1}}
\newcommand{\Instructor}[1][2.5in]{\textbf{Instructor:}\blank{#1}}
\newcommand{\Section}[1][1in]{\textbf{Section:}\blank{#1}}
%%%%%	Name, Instructor or Section blank, with optional argument
%%%%%	to change the length of the blank.





%%%%% For Mild Texperts Only
%%%%% 	This segments sets up environments for an answer grid on the cover.

%% The first commands deal with a cover-page answer grid.

\newcommand{\checks}[2][9]{\boxed{\parbox[c][#2pt]{#1pt}{\large \bfseries a}} 
\boxed{\parbox[c][#2pt]{#1pt}{\large \bfseries b}} 
\boxed{\parbox[c][#2pt]{#1pt}{\large \bfseries c}} 
\boxed{\parbox[c][#2pt]{#1pt}{\large \bfseries d}} 
\boxed{\parbox[c][#2pt]{#1pt}{\large \bfseries e}}}
%%%%% 	An intermediate command which includes choices of height
%%%%%	or width for answer boxes.  

\newcommand{\answers}[1][9]{\checks[#1]{9}}
%%%%%  [1]=box width, default is 9pt

\newcommand{\ans}{\item \answers}
%%%%%  	This command gives one line of answer boxes if typed inside any
%%%%%	list environment.

\newcounter{numb}
\newenvironment{mylist}
{\begin{list}{\textbf{\arabic{numb}.}}{\usecounter{numb}}}{\end{list}}
%%%%%	Environment is simply an enumerated list with bold-face numbers
%%%%%	using the counter numb.


\newcommand{\Mult}[3][2in]{\item {#2}\newline\parbox[r][#1][t]{300pt}{#3}}

\newcommand{\Multlist}[2]{{\begin{mylist}
	\setcounter{numb}{#1} #2 \end{mylist}}}


\newcommand{\First}[2][2.5in]{\parbox{#1}{\begin{mylist} #2 \end{mylist}}}
%%%%%	This command produces the first column of answer boxes.
%%%%%	The optional argument is the width of the column, while the
%%%%%	standard argument is expected to be a number of \ans commands.
 
\newcommand{\Second}[3][2.5in]{\parbox{#1}{\begin{mylist}
	\setcounter{numb}{#2} #3 \end{mylist}}}
%%%%%	This command produces the second column of answer boxes
%%%%%	(do NOT skip a line after the argument of \First).
%%%%%	The optional argument changes the column width, the first
%%%%%	standard argument indicates how many answers are in the first
%%%%%	column, while the second should be a number of \ans commands.

\newcommand{\Firsts}[3][2.5in]{\hspace{#3}\parbox{#1}
		{\begin{mylist} #2 \end{mylist}}}
%%%%%	A slight variation of the \First command above which allows the
%%%%%	third argument to add an \hspace before the first column
%%%%%	of answer boxes.







%%%%% Notre Dame-Specific commands
%%%%% 	These commands are essentially class specific and should be 
%%%%%	ignored by almost everyone. 

\newcommand{\points}[3][100]{\vspace{25pt} \noindent
This examination consists of #2 problems,
worth a total of #1 points on #3 pages, including this front cover.  If 
problems are missing from your copy, you must ask for a new copy right
away.  The first ten problems are multiple choice 
\textbf{with no partial credit for any reason}.  Be sure to indicate
your single answer to each question by placing an $\times$ through
that letter on the answer grid below.  Students will \textbf{NOT} be
allowed extra time to fill in the grid after the exam has ended
if they forget to do so during the exam!
\vspace{10pt}}
%%%%% 	Between \Name and answer grid comments.  The optional argument
%%%%%	indicates the total points (default is 100).  The first argument
%%%%%	indicates the number of problems, the second deals with the
%%%%%	number of pages.


\newcommand{\finalpoints}[3][150]{\vspace{25pt} \noindent
This examination consists of #2 problems,
worth a total of #1 points on #3 pages, including this front cover.  If 
problems are missing from your copy, you must ask for a new copy right
away.  All of the problems are multiple choice 
\textbf{with no partial credit for any reason}.  Be sure to indicate
your single answer to each question by placing an $\times$ through
that letter on the answer grid below.  Students will \textbf{NOT} be
allowed extra time to fill in the grid after the exam has ended
if they forget to do so during the exam!
\vspace{10pt}}
%%%%% 	Between \Name and answer grid comments for the final.  
%%%%%	This is a slight variation on the \points command above.





\newcommand{\calcs}[1]{\vspace{10pt} \noindent
The last #1 problems are partial credit problems.
Be sure to show all work legibly.  Clearly indicate your final answer,
which should be simplified whenever possible.
\textbf{Books, notes and CALCULATORS are NOT permitted.}}  
%%%%%	Below answer grid comments.  The argument indicates how many
%%%%%	partial credit problems will be on the exam. 


\newcommand{\pledge}{\vspace{20pt}
\textbf{Sign the following honor code statement:}\newline 
``On my honor, I have neither given nor received unauthorized
aid on this exam.''\newline\parbox[c][35pt][b]{3.5in}{\hrulefill}}
%%%%%	Standard Notre Dame honor code statement.

\newcommand{\shortpledge}{\textbf{Sign the following honor code statement:}
\newline``On my honor, I have neither given nor received unauthorized
aid on this exam.''\quad \parbox[c][10pt][b]{4.5in}{\hrulefill} }
%%%%%	Notre Dame honor code statement for tight spaces.




\newcommand{\limn}{\underset{\scriptscriptstyle{n \to \infty}}{\lim} \ }
\newcommand{\lims}{\limn \underset{i=1}{\overset{n}{\Sigma}}}

\newcommand{\coeff}[1]{\frac{1}{{#1}!}}
\newcommand{\probfrac}[4]{\frac{\binom{#1}{#2}}{\binom{#3}{#4}}}


\begin{document}

\begin{tops}
 Math 120 -- Final Exam \\ 
 Friday, December 15, 2000 
\end{tops}
\Name \quad \Section

\finalpoints{30}{11}
\vfill
\Firsts[1.9in]{\ans\ans\ans\ans\ans
\ans\ans\ans\ans\ans}{-.7in}
\Second[1.9in]{10}{\ans\ans\ans\ans\ans
\ans\ans\ans\ans\ans}
\Second[1.9in]{20}{\ans\ans\ans\ans\ans
\ans\ans\ans\ans\ans}
\vfill
\shortpledge

\Multlist{0}{

\Mult{Suppose $x^3 \leq f(x) \leq x^2$ for $-1 \leq x \leq 1$.  Which
	of the following must be true}
	{\singles{$f(x)$ is always positive.}
	{$0 \leq \int_{-1}^1 f(x)\ dx \leq \frac23$}
	{$\frac14 \leq \int_{-1}^1f(x)\ dx \leq \frac13$}
	{$\int_{-1}^1 f(x)\ dx$ is negative.}{$f(1)=0$}}
%1 b

\Mult{Find the area of the region enclosed by $y=x$ and $y=\sqrt{x}$.}
	{\Answers{$\frac12$}{$-1$}{$0$}{$\frac16$}{$4$}}
%2 d

\Mult{Suppose the natural length of a spring is $40$cm and a $20$N force
	is required to hold it at $50$cm.  Find the work required to stretch
	the spring from $40$cm to $50$cm.}
	{\Answers{$1$J}{$.001$J}{$1000$J}{$50$J}{$.05$J}}
%3 a

\Mult{Compute the average value of $sin(x)$ over the interval from $x=0$
	to $x=\pi$.}
	{\Answers{$0$}{$\frac{\pi}2$}{$1$}{$\frac2{\pi}$}{$2\pi$}}
%4 d

\Mult{Which of the following is equal to the derivative of 
	$f(x)= ln|sin(x)|$?}
	{\threetwo{sin^2(x)}{tan(x)}{sec(x)}{csc(x)}{cot(x)}}
%5 e

\Mult{Compute \[\int_1^2 \frac{e^{\frac1x}}{x^2}\ dx .\]}
	{\threetwo{1}{\frac{e}2-e^{\frac12}}{e-\sqrt{e}}
	{e^{\frac1x}}{\frac14}}
%6 c

\Mult{Which of the following is a Riemann Sum approximating the area under
	the curve $y=\sqrt{x+3}$ between $x=1$ and $x=3$?}
	{\threetwo{\int_1^3 \sqrt{x+3}\ dx}
	{\sum_{k=1}^4\frac12 \sqrt{\frac{k}2}}
	{\sum_{k=1}^4\frac12 \sqrt{4+\frac{k}2}}
	{1+\sqrt{2}}{\frac12(1+\sqrt{2}+\sqrt{3}+\sqrt{4})}}
%7 c

\Mult{According to the Fundamental Theorem of Calculus, what is 
	$g'(x)$ when $g(x)=\int_{x^2}^2 t-3\ dt$.}
	{\threetwo{6x-2x^3}{\frac{(x^2-3)^2}{2}}{3x^2-6x}
	{4t^3-6}{2t}}
%8 a

\Mult{Compute the volume of the solid generated by rotating the region 
	between $y=x^2$ and $y=x^3$ around the $y$-axis.}
	{\Answerm{\pi}{\frac{\pi}{10}}
	{\frac{1}{10}}{\frac{1}{20}}{2\pi}}
%9 b

\Mult{Which of the following represents the work done by splashing all
	of the water out of a rectangular tub of length $5$m, width $2$m
	and depth $3$m?}
	{\threetwo{\int_0^2 9800x^4\ dx}{\int_2^5 98000x\ dx}
	{\int_3^5 98000x^4\ dx}{\int_2^3 98000x^5\ dx}{\int_0^3 98000x\ dx}}
%10 e

\Mult{According to L'Hospital's Rule, what is the limit
	of the fraction $\frac{sin(2\pi x)}{cos(\pi x)}$ as $x$ approaches
	$\frac12$?}
	{\Answerm{0}{\pi}{2}{\frac{\pi^2}8}{4\pi}}
%11 c

\Mult{Compute $\int_0^1 xe^x\ dx$.}
	{\Answerm{e+e^2}{1+e}{1-e}{1}{e}}
%12 d

\Mult{Compute $\int_0^{\frac{\pi}4} cos^2(x)\ dx.$}
	{\Answerm{-\frac{\pi}8}{\frac{\pi}8+\frac14}{14}
	{\pi+\frac14}{\frac{\pi}4}}
%13 b

\Mult{Compute \[\int_0^{\frac{\pi}4}sec^4(x) tan(x)\ dx . \]}
	{\Answerm{\frac34}{\frac43}{\frac12}{\frac23}{\frac32}}
%14 a

\Mult{Which of the following integrals would be used to solve 
	$\int \frac{x^3}{\sqrt{1+x^2}}\ dx$ by trigonometric substitution?}
	{\singlem{\int sin(\theta) tan(\theta)\ d\theta}
	{\int cos(\theta)\ d\theta}{\int sin(\theta)\ d\theta}
	{\int tan(\theta)\ d\theta}
	{\int sec(\theta) tan^3(\theta)\ d\theta}}
%15 a

\Mult{Use logarithmic differentiation to compute the derivative of
	$ y= \frac{x^5(x+7)^4}{(x+2)^3(x+5)^2}$.}
	{\singlem{y'=\frac{5}{x}-\frac4{x+7}+\frac3{x+2}-\frac2{x+5}}
	{y'=\frac{5}{x} +\frac4{x+7}-\frac3{x+2}+\frac2{x+5}}
	{y'=y(5x+4(x+7)-3(x+2)-2(x+5))}
	{y'=y(\frac{5}{x} +\frac4{x+7})-\frac3{x+2}-\frac2{x+5}}
	{y'=y(\frac{5}{x} +\frac4{x+7}-\frac3{x+2}-\frac2{x+5})}}
%16 e

\Mult{How would you rewrite $\frac6{(x-1)(x+2)}$ in order to find its
	antiderivative by the method of partial fractions?}
	{\threetwo{\frac1{x-1}-\frac5{x+2}}{\frac{4x}{x-1}-\frac{2x+3}{x+2}}
	{\frac3{x-1}-\frac1{x+2}}{\frac2{x-1}-\frac2{x+2}}
	{\frac7{x-1}-\frac{x}{x+2}}}
%17 d

\Mult{Find the $y$-coordinate of the centroid of the region bounded by
	$y=cos(x)$, $y=0$, $x=0$ and $x=\frac{\pi}2$.  The area of this
	region is 1 square unit.}
	{\singlem{\int_0^{\frac{\pi}2} sin^2(x)\ dx}
	{\frac12 \int_0^{\frac{\pi}2} xcos(x)\ dx}
	{\frac12 \int_0^{\frac{\pi}2} cos^2(x)\ dx}
	{\frac{\pi}2 \int_0^{\frac{\pi}2} cos(x)-sin(x)\ dx}
	{\frac14 \int_0^{\frac{\pi}4} cos^4(x)\ dx}}
%18 c

\Mult{Which of the following integrals represents the length along the
	curve $y=3x^2-5$ from $x=1$ to $x=4$?}
	{\singlem{\int_1^4 \sqrt{1+36x^2}\ dx}{\int_1^4 \sqrt{1+6x}\ dx}
	{\int_1^4 2\pi(3x^2-5) \sqrt{1+36x^2}\ dx}
	{\int_1^4 2\pi(3x^2-5) \sqrt{1+6x}\ dx}{\int_1^4 \sqrt{1-6x^2}\ dx}}
%19 a

\Mult{Find the surface area generated by rotating the curve
	$y=(x-2)^{\frac32}$ between $x=2$ and $x=5$ around 
	the $x$-axis.}
	{\singlem{\int_2^5 \sqrt{1+\frac94(x-2)}\ dx}
	{\int_2^5 2\pi(x-2)^{\frac32}\sqrt{1+\frac94(x-2)}\ dx}
	{\int_2^5 2\pi(x-2)^{\frac32}\sqrt{1+(x-5)}\ dx}
	{\int_2^5 2\pi(x-2)^{\frac32}\sqrt{1+\frac32(x-2)^{\frac12}}\ dx}
	{\int_2^5 \sqrt{1+\frac32(x-5)}\ dx}}
%20 b

\Mult{If the demand function is $p=5-\frac{x}{40}$ and the selling price
	is $ \$4$, find the consumer surplus.}
	{\singlem{\int_0^{4.9} \frac1{10}-\frac{x}{40}\ dx}
	{\int_0^{40} \frac1{10}-\frac{x}{40}\ dx}
	{\int_0^{4.9} 5-\frac{x}{40}\ dx}
	{\int_0^{4.9} 1-\frac{x}{40}\ dx}
	{\int_0^{40} 1-\frac{x}{40}\ dx}}
%21 e

\Mult{If you flip a fair coin five times, what is the probability of
	getting {\bf at least} four heads?}
	{\Answerm{\frac{3}{32}}{\frac{3}{16}}{\frac{3}{8}}
	{\frac{1}{32}}{\frac{1}{8}}}
%22 b

\Mult{An unfair coin has a sixty percent chance of coming up tails
	(q=$\frac35$).  In flipping the coin five times, what is the
	chance of getting exactly four heads?}
	{\threetwo{10 (\frac25)^3(\frac35)^2}{10 (\frac25)^4(\frac35)}
	{5 (\frac25)^4(\frac35)}{6 (\frac25)^3(\frac35)}
	{5 (\frac35)^4(\frac25)}}
%23 c

\Mult{Suppose $f(0)=f'(0)=f''(0)=4$, $f^{(3)}(0)=12$ and
	$f^{(4)}(0)=72$.  Which of the following is the fourth Taylor
	polynomial for $f(x)$? }
	{\singlem{4+4x+2x^2+2x^3+3x^4}
	{4+2x-x^2+2x^3+3x^4}{4-4x+2x^2-2x^3+3x^4}
	{1+4x+x^2+x^3+x^4}{4-4x+2x^2-x^3+x^4}}
%24 a

\Mult{Suppose that the second Taylor polynomial for $f(x)$
	is $P_2(x)=3+x+5x^2$.  What approximation for $f(.1)$ does
	this give?}
	{\Answerm{5.13}{325}{1.35}{3.15}{2.513}}
%25 d

%} \end{document}

\Mult{Give an upper bound on the error in approximating $e^{-1}$
	by $P_4(-1)=\frac{9}{24}$ by applying Taylor's Theorem.}
	{\Answerm{\frac{e}{3(4!)}}{\frac{1}{3(5!)}}{\frac{e}{6!}}
	{\frac{1}{8!}}{\frac1{5!}}}
%26 e 

\Mult{What integral represents the hydrostatic force on the end of a
	rectangular tub of width $2$m and depth $5$m filled with water?}
	{\threetwo{9800\int_0^5 5x\ dx}{9800\int_0^2 5x\ dx}
	{9800\int_0^5 2x\ dx}{9800\int_0^2 2x\ dx}{9800\int_0^5 x^2\ dx}}
%27 c

\Mult{Find a Taylor Series for the derivative of 
	$sinh(x)=x+\frac{x^3}{3!}+\frac{x^5}{5!}+\dots$.}
	{\singlem{1+\frac{x^4}{4!}+\frac{x^8}{8!}+\dots}
	{1+\frac{x^2}{2!}+\frac{x^4}{4!}+\dots}
	{\frac{x^2}{2!}+\frac{x^4}{4!}+\frac{x^6}{6!}+\dots}
	{\frac{x^3}{3!}+\frac{x^5}{5!}+\frac{x^7}{7!}+\dots}
	{x+\frac{x^3}{3!}+\frac{x^5}{5!}+\dots}}
%28 b

\Mult{Give a Taylor Series expansion for $tan^{-1}(x)$ if
	the Taylor Series expansion for 
	$\frac1{1+x^2}=\sum_{k=0}^{\infty}(-1)^k x^{2k}$.}
 	{\singlem{\sum_{k=1}^{\infty}(-1)^k x^{2k+1}}
	{\sum_{k=0}^{\infty}(-1)^k x^{2k+1}}
	{\sum_{k=0}^{\infty}(-1)^{k+1}\frac{x^{2k}}{k+1}}
	{\sum_{k=0}^{\infty}(-1)^k \frac{x^{2k+1}}{2k+1}}
	{\sum_{k=0}^{\infty}(-1)^k (2k)x^{2k-1}}}
%29 d

\Mult{In class, we found a Taylor Series expansion for $\int e^{-x^2}\ dx$
	by integrating the expansion for $e^u$ with $u=-x^2$.  What is
	the Radius of Convergence of the resulting Taylor Series
	expansion?}
	{\Answerm{\infty}{10}{2}{1}{0}}
%30 a

}















\end{document}

\Multlist{0}{

\Mult	{Which of the following is equal to the binomial
	coefficient $\binom{5}{3}$?}
	{\Answers{4}{5}{6}{10}{20}}
%d

%\Mult	{Which of the following represents 
%	the best upper bound for $|R_4(1)|$ given by 
%	the remainder formula in Taylor's Theorem when the fifth
%	derivative $f^{(5)}(t)=-cos(\frac{\pi}{2}\ t)$ and $x=1$?}
%	{\Answers {$\coeff{5}$}{$\coeff{6}$}{$\coeff{4}$}
%		{$\frac{\pi}{2(5!)}$}{$\frac{1}{5}$}}
%a

\Mult	{Compute the $y$-coordinate of the center of mass of a uniform
	plate bounded by the curves $y=\sqrt{9-x}$, $y=0$, $x=0$ and
	$x=5$.  The area of this region is $A=\frac{38}{3}$ square
	units.}
	{\threetwo{\frac{3}{38}\frac{65}{4}}{\frac{3}{38}\frac{65}{2}}
	{\frac{3}{38}\frac{75}{2}}{\frac{3}{38}\frac{75}{4}}
	{\frac{3}{38}\frac{-65}{4}}}
%a


\Mult	{What integral computes the length along the curve $y=ln(x^2-1)$ 
	between $x=2$ and $x=3$?}
	{\twotwoone{\int_4^9 \sqrt{1-\frac{2x}{x^2-1}} \ dx}
	{\int_2^3 1+\frac{4x^2}{(x^2-1)^2} \ dx}
	{\int_2^3 \frac{x^2+1}{x^2-1} \ dx}
	{\int_2^3 1-\frac{4x^2}{(x^2-1)^2} \ dx}
	{\int_2^3 \sqrt{1+\frac{2x}{x^2-1}} \ dx}}
%c

\Mult	{Find an integral which computes the surface area generated by
	rotating $y=x^3$ around the $x$-axis between $x=0$ and $x=4$.}
	{\twotwoone{\int_3^4 \frac{\pi}{2} x^3 \sqrt{1+x^4} \ dx}
	{\int_3^4 \frac{\pi}{2} x^3 \sqrt{1+9x^4} \ dx}
	{\int_0^4 2\pi x^3 \sqrt{1+\frac{x^4}{4}} \ dx}
	{\int_0^4 2\pi y^3 \sqrt{1-9x^4} \ dx}
	{\int_0^4 2\pi x^3 \sqrt{1+9x^4} \ dx}}
%e

\Mult	{What is the chance of getting {\bf at least} two heads when flipping
	four fair coins?}
	{\Answers{$\frac{5}{8}$}{$\frac{11}{16}$}{$\frac{3}{4}$}
		{$\frac{2}{3}$}{$\frac{1}{4}$}}
%b

\Mult	{If the demand function is $p=4-\frac{x}{20}$, find the consumer
	surplus when the price is {\$}$3$.}
	{\Answers{$7$}{$10$}{$20$}{$0$}{$14$}}
%b

\Mult	{What is the probability of drawing a spade flush in 5-card stud
	poker?}
	{\threetwo{\probfrac{13}{5}{52}{13}}{\probfrac{52}{13}{13}{5}}
	{\probfrac{13}{5}{52}{5}}{\probfrac{13}{5}{52}{39}} 
	{\probfrac{39}{5}{52}{5}}}
%c

\Mult	{Given only the following information, find the value of
	the fourth Taylor polynomial when $x=\frac{1}{10}$,$P_4(\frac1{10})$.
	$f(0)=f'(0)=f''(0)=f^{(3)}(0)=6$ and $f^{(4)}(0)=48$}
	{\threetwo{6.6314}{6.6322}{6.6324}{6.6312}{6.6323}}
%d

\Mult	{How does one write the repeating decimal $.129129129 \dots$
	as a proper fraction?}
	{\Answers{$\frac{43}{333}$}{$\frac{129}{9999}$}{$\frac{129}{909}$}
	{$\frac{43}{3333}$}{$\frac{43}{33}$}}
%a

\Mult	{What is the $x$-coordinate of the centroid of the region 
	bounded by $y=sin(x)$, $y=0$, $x=0$ and $x=\pi$?}
	{\Answers{$\frac{4}{\pi}$}{$\frac{2}{\pi}$}{$2$}
	{$\frac{\pi}{2}$}{$\frac{\pi}{4}$}}
%d

%\Mult	{}
%	{{}{}{}{}{}}
%

}

\break

\half{10}{

\item	{\bf (13 pts.)} Suppose Hank Aaron was batting $.400$ one season and 
	in one game he had 4 at bats (where he was not walked).  
	What is the probability he ended that game with exactly two hits?  
	Express your answer as a proper fraction.
	(Hint: Batting $.400$ means $p=\frac25$.)

\item 	{\bf (13 pts.)} Find the third Taylor polynomial $P_3(x)$ for
	the function $f(x)=(4-2x)^{\frac32}$.  

\break

\item	{\bf (12 pts.)} 
	Use {\bf linear approximation} to estimate $\sqrt{17}$.  
	Be sure to give (and say a few words to justify) 
	an upper bound on the error of this estimate.

\item	{\bf (12 pts.)} A trough with triangular cross-section is filled with 
	water for a horse.  If the maximum depth is $1$ meter and 
	the maximum width is $2$ meters, what is the hydrostatic force 
	on the end of the trough.  
	(You do not need to give an explicit Riemann sum,
	because it is implicit in the integral you will solve.)

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