%% MATH 132-01 Final Exam, Spring 2008
\documentclass[12pt]{article}
\usepackage{latexsym,amsmath,graphicx}
\newcommand{\ds}{\displaystyle}
\newcommand{\vp}{$\vphantom{\big|}$}

\newcommand{\R}{\textbf{R}}
\newcommand{\Item}{\item{$\Box$}\quad}

\newcommand{\Answer}[1]{$\phantom{.}$\hfill
        #1\quad\fboxsep 20pt \fbox{$\phantom{\Big|Answer}$}} 

\newcommand{\MyTallAnswer}[2]{$\phantom{.}$\hfill
        #1\quad\fboxsep 20pt\fbox{$\vphantom{\Big|}#2$}}

\newcommand{\MyAnswer}[2]{$\phantom{.}$\hfill
        #1\quad\fboxsep 5pt\fbox{$\vphantom{\Big|}#2$}}

\newcommand{\soln}{\paragraph{Solution}}
\newcommand{\sep}{{\medskip\hrule\bigskip}}
\newcommand{\blank}{\underline{$\phantom{A blank}$}}

\setlength{\textwidth}{6.5in}
\setlength{\topmargin}{-0.5in}
\setlength{\oddsidemargin}{0in}
\setlength{\textheight}{9.0in}

\begin{document}

\begin{center}

\textbf{College of the Holy Cross, Spring Semester, 2008} \\
\textbf{MATH 132, Section 01, Final Exam} \\
\textbf{Thursday, May~8, 2:30~PM} \\
Professor Little
\end{center}


\vfil

\noindent\textbf{Your Name}: $\underline{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}$
\vskip 0pt plus .3fil

\vskip 30pt
\noindent\textbf{Instructions}:  For full credit, you must show 
{\it all work} on the test pages and place your final answer
in the box provided for the problem.  Use the back of the
preceding page if you need more space for scratch work.  The numbers
next to each part of the questions are their point values.

\vfil

\noindent\textbf{Please do not write in the space below}

\begin{center}
\begin{tabular}{|l||r|}
\hline
Problem & Points/Poss \\
\hline\hline
I  & \hfil/ 15\  \hfil \\
\hline
II  & \hfil/ 55\  \hfil \\
\hline
III  & \hfil/ 15\  \hfil \\
\hline
IV  & \hfil/ 15\  \hfil \\
\hline
V  & \hfil/ 30\  \hfil \\
\hline
VI & \hfil/ 10\  \hfil \\
\hline
VII  & \hfil/ 20\  \hfil \\
\hline
VIII & \hfil/ 15\  \hfil \\
\hline
IX & \hfil/ 25\  \hfil \\
\hline\hline
Total   & \hfil/200\hfil \\
\hline
\end{tabular}
\end{center}

\vfil

\[
\emph{Have a safe, productive, and enjoyable summer!}
\]

\eject
\noindent
I.  
\begin{enumerate}
\item[(A)] (5) Compute the derivative of the function $\ds g(x) = \int_0^{3x} \frac{\sin(t)}{t}\, dt$.
\vskip 1.0truein
\MyTallAnswer{$g'(x)$ = }{$\phantom{A very very very very long answer here}$}% 
\item[(B)] (4) Let 
$\ds
f(x) = \begin{cases} 1 & \text{ if } 0 \le x \le 2\\
                    x-1 & \text{ if } 2 < x \le 4.\\
                   11-2x & \text{ if } 4 < x \le 6
\end{cases}$  
Sketch the graph $y = f(x)$ on the axes provided below.

\begin{center}
\includegraphics[width=2.5truein,height=2.5truein]{132FI.ps}
\end{center}
\item[(C)] (6) Let $F(x) = \int_0^x f(t)\, dt$, where $f(t)$ is the function from part (B).  
Complete the following table of values for $F(x)$:
\vskip 1.0truein
\[
\begin{array}{|c||r|r|r|r|r|r|r|}
\hline
x          & 0 & 1 & 2 & 3 & 4 & 5 & 6 \strut \\
\hline
F(x)       & \quad 0 & \qquad  & \qquad & \qquad & \qquad & \qquad & \qquad \strut \\
\hline
\end{array}
\]
 
\end{enumerate}
\vfill\eject

\vskip 10pt
\noindent
II.  Compute the following integrals.  Some of these may be forms covered by entries in the
table of integrals.  Half credit will be given for using a table entry; full credit only for showing
all work leading to the final answer.
\begin{enumerate}
\item[(A)] (5) $\ds \int \frac{x^4 - 3x^3 + \sqrt{x}}{x^{2/3}}\, dx$
\vskip 3.0truein
\MyAnswer{Answer}{$\phantom{A very very very very very very long answer here}$}% 
\item[(B)] (5) $\ds \int x \cos(x^2)\, dx$
\vskip 3.0truein
\MyAnswer{Answer}{$\phantom{A very very very very very very long answer here}$}%
\vfill\eject
\item[(C)] (10) $\ds \int \frac{\sec^2(5x)\, dx}{(\tan(5x) + 7)^5}$
\vskip 3.0truein
\MyAnswer{Answer}{$\phantom{A very very very very very very long answer here}$}%
\item[(D)] (10) $\ds \int_1^e x^7 \ln(x)\, dx$.  
\vskip 3.5truein
\MyAnswer{Answer}{$\phantom{A very very very very very very long answer here}$}%
\vfill\eject
\item[(E)] (12.5) $\ds \int \frac{x^2}{\sqrt{25 - x^2}}\, dx$
\vskip 7.5truein
\MyTallAnswer{Answer}{$\phantom{A very very very very very very long answer here}$}%
\vfill\eject
\item[(F)] (12.5) $\ds \int \frac{x}{(x+1)(x^2 + 1)}\, dx$
\vskip 7.5truein
\MyTallAnswer{Answer}{$\phantom{A very very very very very very long answer here}$}%
\end{enumerate}
\vfill\eject
\noindent
III.  
\begin{enumerate}
\item[(A)] (5)  Use a midpoint Riemann sum with $n = 4$ to approximate 
$\ds \int_0^2 e^{x^2}\ dx$.
\vskip 2.5truein
\MyAnswer{Answer}{$\phantom{A very very very very very very long answer here}$}%
\item[(B)] (5) Use the Trapezoidal Rule with $n = 4$ to approximate
$\ds \int_0^2 e^{x^2}\ dx$.
\vskip 2.5truein
\MyAnswer{Answer}{$\phantom{A very very very very very very long answer here}$}%
\item[(C)] (5) Check the appropriate boxes:
\item[]The midpoint approximation is a overestimate $\Box$/underestimate $\Box$, and 
\item[]the trapezoidal rule approximation is a overestimate $\Box$/underestimate $\Box$, because
\item[]$e^{x^2}$ is concave up $\Box$/concave down $\Box$ on $[0,2]$.  
\end{enumerate}
\vfill\eject
\noindent
\emph{Note:}  From this point on in the exam, if an entry from the table of integrals
applies, you may use it for full credit if you state which entry you are using and indicate
what $u$ and what constant values $a$, $b$, etc. are involved.  
\vskip 10pt
\noindent
IV.  For each of the following improper integrals, set up and evaluate the appropriate
limits to determine whether the integral converges.  If so, find its value; if not,
say ``does not converge.''
\begin{enumerate}
\item[(A)] (6) $\ds \int_0^2 \frac{1}{{\root 3\of {x - 1}}}\, dx.$
\vskip 2.0truein
\MyAnswer{Answer}{$\phantom{A very very very very very very long answer here}$}%
\vskip 10pt
\item[(B)] (9) $\ds \int_0^\infty \frac{1}{(x^2 + 4x + 5)^{3/2}}\, dx$.
\vskip 3.0truein
\MyAnswer{Answer}{$\phantom{A very very very very very very long answer here}$}%
\end{enumerate}
\vfill\eject
\noindent
V.  A region $R$ in the plane is bounded by the graphs $y = 9 - x^2$, $y = 2x$, $x = 0$ and $x = 1$.
\begin{enumerate}
\item[(A)] (5) Sketch the region on the following axes:
\begin{center}
\includegraphics[width=2.0truein,height=2.0truein]{132FV.ps}
\end{center}
\item[(B)] (5)  Compute the area of the region $R$.
\vskip 2.0truein
\MyAnswer{Area $=$}{$\phantom{A very very very long answer here}$}%
\item[(C)] (10) Compute the volume of the solid obtained by rotating $R$ about the $x$-axis.
\vskip 2.5truein
\MyAnswer{Volume $=$}{$\phantom{A very very very long answer here}$}%
\item[(D)] (10) Set up the integral(s) to compute the volume of the solid
obtained by rotating $R$ about the $y$-axis.  \emph{You do not need to compute the value.}
\vskip 3.0truein
\MyTallAnswer{Volume $=$}{$\phantom{A very very very very very very very long answer here}$}%
\end{enumerate}
\vskip 10pt
\noindent
VI. (10) The daily solar radiation $x$ per square meter (in hundreds of calories) in Florida
in October has a probability density function $f(x) = \frac{3}{32}(x - 2)(6 - x)$ if 
$2 \le x \le 6$, and zero otherwise.  Find the \emph{mean} daily solar radiation.
\vskip 3.0truein
\MyAnswer{Mean $=$}{$\phantom{A very very very long answer here}$}%
\vfill\eject
\noindent
VII.  Does each of the following infinite series converge or diverge? For full
credit, you must justify your answer completely by showing how the indicated test
applies and leads to your stated conclusion.
\begin{enumerate}
\item[(A)] (10)  $\ds \sum_{n=1}^\infty \frac{n}{e^{2n}}$ -- Integral Test
\vskip 3.0truein
\MyAnswer{Answer}{$\phantom{A very very very long answer here}$}%
\item[(B)] (10) $\ds \sum_{n=1}^\infty \frac{3^n}{n^2 e^n}$ -- any applicable method.
\vskip 3.0truein
\MyAnswer{Answer}{$\phantom{A very very very long answer here}$}%
\end{enumerate}  
\vfill\eject
\noindent
VIII.  
\begin{enumerate}
\item[(A)] (7.5) Using the Ratio Test and testing the endpoints, 
determine the interval of convergence for the power series $\ds \sum_{n=1}^\infty \frac{2^n x^n}{n}$.
\vskip 3.0truein
\MyAnswer{Interval of convergence}{$\phantom{A very very very long answer here}$}%
\item[(B)] (7.5)  Find the Taylor series for the function $f(x) = \sin(x/2)$.  Express 
your answer in summation (``sigma'') notation.
\vskip 3.0truein 
\MyTallAnswer{Taylor series $=$}{$\phantom{A very very very very very long answer here}$}%
\end{enumerate}
\vfill\eject
\noindent
IX.
\begin{enumerate}
\item[(A)] (5)  A drug is administered to a patient intravenously at a constant
rate of $10$mg per hour.  The patient's body breaks down the drug and removes
it from the bloodstream at a rate proportional to the amount present.  Write 
a differential equation for the function $Q(t) = $ amount of the drug present 
(in mg) in the bloodstream at time $t$ (in hours) that describes this situation.  
You \emph{do not need to solve} the equation.
\vskip 1.0truein
\MyTallAnswer{Answer}{$\phantom{A very very very long answer here}$}%
\item[(B)] (10) Find the general solution of
the differential equation $y' = y\sqrt{e^{2x} + 1}$.
\vskip 4.5truein
\MyTallAnswer{General solution $=$}{$\phantom{A very very very very very long answer here}$}%
\vfill\eject
\item[(C)] (10) A population $P$ of tree frogs 
is undergoing logistic growth following the differential equation
$P' = (.1)P(1-\frac{P}{100})$, $t$ in years.  
If the initial population is $P(0) = 10$, how long does it
take for the population to reach $45$?
\vskip 7.5truein
\MyAnswer{Time to reach $45 = $}{$\phantom{A very very very long answer here}$}%
\end{enumerate}
\end{document}
