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\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}

\noindent
\begin{enumerate}
\item The following graph shows the function $y = f(x)$.

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

\noindent
\begin{enumerate}
\item[(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:
\[
\begin{array}{|c||r|r|r|r|r|}
\hline
x          & 0 & 1 & 2& 3& 4\Strut \\
\hline
F(x)       & \quad 0 & \qquad  & \qquad & \qquad & \qquad \Strut \\
\hline
\end{array}
\]

\item[(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}}
\end{enumerate}

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

\begin{enumerate}
\item[(a)] [5~points]  $\ds \int \frac{x^2 + 3x + 5}{x^{3/2}}\ dx$

\item[(b)] [10~points]  $\ds \int \frac{e^t + 1}{(e^t + t)^2}\ dt$ 

\item[(c)] [10~points]  $\ds \int \frac{dx}{(9 - x^2)^{3/2}}$ (use a trigonometric substitution)

\item[(d)] [10~points] $\ds\int_0^1 x^2 e^{2x}\ dx$

\item[(e)] [10~points]  $\ds \int \frac{(x - 1)\ dx}{x^3 + 25 x}$
\end{enumerate}

\item 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}}

\begin{enumerate}
\item[(a)] [10~points] Find the volume of the solid with base $R$ whose cross-sections by
planes perpendicular to the $x$-axis are squares.

\item[(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.)

\item[(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.


\item[(d)] [10~points] Set up, {\it but do not evaluate} the integral giving the 
arclength of the top edge of $R$.
\end{enumerate}

\item 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}}

\begin{enumerate}
\item[(a)] [5~points] 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}}

\item[(b)] [5~points] Using the appropriate graph, estimate the {\it median} of $x$.
\end{enumerate}

\item
 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.

\begin{enumerate}
\item[(a)] [5~points] $\ds \sum_{n=1}^\infty \frac{(-1)^n}{n^{1/3}}$ -- alternating series test.

\item[(b)] [5~points] $\ds \sum_{n=1}^\infty \frac{1}{n^{9/10}}$ -- integral test

\item[(c)] [5~points] $\ds \sum_{n=1}^\infty \frac{3^n}{\pi^n}$ -- any applicable method
\end{enumerate}

\item  Using the ratio test,
determine the radius of convergence of the following power series
$$\sum_{n=0}^\infty \frac{(x+2)^n}{n^2 4^n}$$
([15~points]).

\item All parts of this problem refer to $f(x) = \cos\left(\frac{x}{2}\right)$.
\begin{enumerate}
\item[(a)] [15~points]  Using the definition of Taylor polynomials, compute the Taylor polynomial of 
degree $n = 6$ for $f(x)$ at $a = 0$.

\item[(b)] [10~points] Use your answer from part a to approximate $\cos(1.2)$.
\end{enumerate}

\item  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 $t$.    

\begin{enumerate}
\item[(a)] [5~points] Which of the following differential equations is the 
correct translation of the model described above:
\begin{eqnarray*}
I:  \frac{dy}{dt} = y - a && II: \frac{dy}{dt} = \frac{-k}{y-a}\\
III: \frac{dy}{dt} = k(y-a) && IV: \frac{dy}{dt} = t(y-a).
\end{eqnarray*}

\item[(b)] [10~points] Solve the equation you selected in part a by separation
of variables.

\item[(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?
\end{enumerate}
\end{enumerate}
\end{document}
