%% MATH 132-01 Final Exam, Spring 2005
\documentclass[12pt]{article}
\usepackage{latexsym,amsmath,amsfonts,mathrsfs,psfig}

\setlength{\textwidth}{6.5in}
\setlength{\oddsidemargin}{0in}
%% \setlength{\topmargin}{-.5in}
\setlength{\textheight}{8.25in}

\newcommand{\ds}{\displaystyle}
\newcommand{\ts}{\textstyle}
\DeclareMathOperator{\lcm}{lcm}

\newcommand{\Neg}{\phantom{-}}
\newcommand{\vp}{$\vphantom{\big|}$}
\newcommand{\Vp}{$\vphantom{\Big|}$}

\newcommand{\by}{\times}

\newcommand{\Number}[1]{\mathbf{#1}}
\newcommand{\R}{\Number{R}}

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

\newcommand{\be}{\begin{itemize}}
\newcommand{\ee}{\end{itemize}}

\newcommand{\bd}{\begin{description}}
\newcommand{\ed}{\end{description}}

\newcommand{\bi}{\begin{itemize}}
\newcommand{\ei}{\end{itemize}}

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

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

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

%\usepackage{math131}

\pagestyle{myheadings}
\markright{MATH 132-01, Final Exam}

\setlength{\oddsidemargin}{0in}
\setlength{\textheight}{8.5in}

\def\Answer#1{$\phantom{.}$\hfill
        #1\quad\fbox{$\phantom{\Big|Put longggg answer here}$}}
\def\ShortAnswer#1{$\phantom{.}$\hfill
        #1\quad\fbox{$\phantom{\Big|answer}$}}

\def\Name{$\underline{\phantom{Please write your name in this
        space}}$} 
\def\Strut{$\vphantom{\Big|}$}

\begin{document}

\begin{center}

\textbf{Holy Cross College, Spring Semester, 2005} \\
\textbf{MATH 132, Section 01, Final Exam} \\
\textbf{Tuesday, May 10, 2:30~PM} \\
Professor Little
\end{center}

\vfil

\noindent{\bf Your Name}: \Name
\vskip 0pt plus .3fil

\noindent
{\it Instructions.} Clearly mark your answers, and show work on
the test itself. Use the back of the preceding page if you need more
space for scratch work. {\it You must show all work for full
credit}, but please place answers in the boxes provided where appropriate.

\vfil

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

\begin{center}
\begin{tabular}{|l||r|}
\hline
Problem & Points/Poss    \Strut \\
\hline\hline
1       & \hfil/\  20\hfil \Strut \\
\hline
2       & \hfil/\  45\hfil \Strut \\
\hline
3       & \hfil/\  45\hfil \Strut \\
\hline
4       & \hfil/\  10\hfil \Strut \\
\hline
5       & \hfil/\  15\hfil \Strut \\
\hline
6       & \hfil/\  10\hfil \Strut \\
\hline
7       & \hfil/\  30\hfil \Strut \\
\hline
8       & \hfil/\  25\hfil \Strut \\
\hline\hline
Total   & \hfil/200\hfil \Strut \\
\hline
\end{tabular}
\end{center}

\vfil

\begin{center}
{\it Have an enjoyable and productive summer!}
\end{center}
\eject


\noindent
1. The following graph shows the function $y = f(x)$.

\centerline{\psfig{figure=1321a.ps,height=2.0in,width=2.4in,angle=-90}}

\noindent
a. [10~points] Let $F(x)$ be the antiderivative of $f(x)$ with $F(0) = 0$.
Compute the entries in the following table of values:
\[
\begin{array}{|c||r|r|r|r|r|}
\hline
x          & 0 & 1 & 2& 3& 4\Strut \\
\hline
F(x)       & \quad 0 & \qquad  & \qquad & \qquad & \qquad \Strut \\
\hline
\end{array}
\]

\vglue 2.0truein
\sep
\noindent
b. [10~points] Sketch the graph $y = F(x)$ on the interval $0 \le x \le 4$:

\vskip 10pt
\centerline{\psfig{figure=1321b.ps,height=2.0in,width=2.0in,angle=-90}}

\vfil\eject
\noindent 
2. Integrate each of the following.  You must show all work
and cite any table entries you use for full credit.

\vskip 10pt
a. [5~points]  $\ds \int \frac{x^2 + 3x + 5}{x^{3/2}}\ dx$
\vglue 3.0truein

\Answer{Answer:}

\sep
\noindent
b. [10~points]  $\ds \int \frac{e^t + 1}{(e^t + t)^2}\ dt$ 
\vglue 3.0truein
\Answer{Answer:}
\eject
\noindent
c. [10~points]  $\ds \int \frac{dx}{(9 - x^2)^{3/2}}$
\vglue 3.0truein  
\Answer{Answer:}

\sep
\noindent
d. [10~points] $\ds\int_0^1 x^2 e^{2x}\ dx$
\vglue 3.0truein
\Answer{Answer:}
\eject
\noindent
e. [10~points]  $\ds \int \frac{(x - 1)\ dx}{x^3 + 25 x}$
\vglue 6.0truein
\Answer{Answer:}

\eject
\noindent
3.  All parts of this problem refer to the region $R$ in the plane
bounded by $y = 2 + \cos(x)$, $y = 0$, $x = 0$, and $x = \pi$,
shown below:

\centerline{\psfig{figure=1323.ps,height=2.0in,width=2.0in,angle=-90}}

\vskip 10pt
\noindent
a. [10~points] Find the volume of the solid with base $R$ whose cross-sections by
planes perpendicular to the $x$-axis are squares.
\vglue 3.0truein
\Answer{Answer:}

\sep
\noindent
b. [10~points] True or False:  The solid obtained by rotating $R$ about the $x$-axis
has volume $\pi$ times the volume from part a.  (For full credit,
give the integral that would compute the volume of this solid of
revolution, and answer the question.)

\vglue 1.0truein

\Answer{Answer (T/F)}

\eject

\noindent
c. [15~points] A thin metal plate has the shape of the region $R$ (units in meters) and constant
mass density $1 kg/m^2$.  Find the $x$-coordinate of its center of mass.

\vglue 4.5truein

\Answer{Answer:}

\sep
\noindent
d. [10~points] Set up, {\it but do not evaluate} the integral giving the 
arclength of the top edge of $R$.
\vglue 1.0truein

\Answer{Answer:}

\eject
\noindent
4. [5~points each] The following graph shows a probability density function (pdf)
for a quantity $x$:

\centerline{\psfig{figure=1325.ps,height=2.0in,width=2.0in,angle=-90}}

\vskip 10pt
\noindent
a.  Which of the following two plots shows the corresponding cumulative
distribution for $x$:

\vskip 10pt
\centerline{\psfig{figure=1325a1.ps,height=2.0in,width=2.0in,angle=-90},
\psfig{figure=1325a2.ps,height=2.0in,width=2.0in,angle=-90}}

\vglue 1.0 truein
\Answer{Answer (A/B):}

\sep
\noindent
b.  Using the appropriate graph, estimate the {\it median} of $x$.

\vglue 1.0truein
\Answer{Answer:}

\eject

\noindent
5. [5~points each] Does each of the following series converge?  For full credit, you must
justify your answer completely by showing how the indicated test leads
to your stated conclusion.
\vskip 10pt

\noindent
a.  $\ds \sum_{n=1}^\infty \frac{(-1)^n}{n^{1/3}}$ -- alternating series test.

\vglue 2.5truein

\Answer{Answer (converges/diverges):}
\sep
\noindent
b.  $\ds \sum_{n=1}^\infty \frac{1}{n^{9/10}}$ -- integral test

\vglue 2.5truein

\Answer{Answer (converges/diverges):}
\eject
\noindent
c.  $\ds \sum_{n=1}^\infty \frac{1}{n e^n}$ -- any applicable method
\vglue 2.5truein
\Answer{Answer (converges/diverges):}
\sep
\noindent
6. 
\noindent
[10~points]  Using the ratio test,
determine the radius of convergence of the following power series
$$\sum_{n=0}^\infty \frac{(x+2)^n}{n^2 4^n}$$
\vglue 3.0truein
\Answer{Answer:}

\eject
\noindent
7.  All parts of this problem refer to $f(x) = \cos\left(\frac{x}{2}\right)$.

\noindent
a. [15~points]  Using the definition of Taylor polynomials, compute the Taylor polynomial of 
degree $n = 6$ for $f(x)$ at $a = 0$.

\vglue 2.5truein
\Answer{Answer:}
\sep
\noindent
b. [5~points] Use your answer from part a to approximate $\cos(1.2)$.
\vglue 1.5truein
\Answer{Answer:}
\sep
\noindent
c. [10~points] Using the error bound for Taylor approximations, estimate how big 
$n$ must be to guarantee 4 decimal place accuracy in approximations to 
$f(x)$ for all $x$ between $0$ and $2$.  
\vglue 1.5truein
\Answer{Answer:}
\eject
\noindent
8.  When a course ends, it's an unfortunate fact of life that students
start to forget the material they have learned.  One mathematical model for this
process states that {\it the rate at which a student forgets material is
proportional to the difference between the material currently remembered
and some positive constant $a < 1$}.  Let $y$ be the fraction of the
original material remembered $t$ weeks after the course has ended. 
Then  $y(0) = 1$, and the model assumes $y(t) > a$ for all $t$.    
\vskip 10pt
\noindent
a. [5~points] Which of the following differential equations is the 
translation of the model described above:
$$I:  \frac{dy}{dt} = -k (y - a) \qquad II: \frac{dy}{dt} = \frac{-k}{y-a}$$
\vskip 10pt
\Answer{Answer (I/II):}
\sep
\noindent
b. [10~points] Solve the equation you selected in part a by separation
of variables.
\vglue 3.5truein
\Answer{Answer:}
\eject
\noindent
c. [10~points] Suppose $a = .2$, and after 3 weeks the student only 
remembers $80\%$ of the material from the course (that is, $y(3) = .8$). 
How long will it be before the student only remembers one half of the 
material?
\vglue 2.5truein
\Answer{Answer:}
\end{document}