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\begin{center}

\textbf{College of the Holy Cross, Spring 2014} \\
\textbf{Math 136 -- Midterm Exam 3} \\
\textbf{May 2}
\end{center}

\vfil

\noindent\textbf{Your Name}: $\underline{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}$

\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/ 25\  \hfil \\
\hline
II  & \hfil/ 20\  \hfil \\
\hline
III  & \hfil/ 10\  \hfil \\
\hline
IV  & \hfil/ 20\  \hfil \\
\hline
V  & \hfil/ 25\  \hfil \\
\hline\hline
Total   & \hfil/100\hfil \\
\hline
\end{tabular}
\end{center}

\vfil\eject

\noindent
I.  Let $R$ be the region bounded by $y = \sin(x)$, the $x$-axis, and 
$0 \le x \le \pi/3$.  
\begin{enumerate}
\item[A.] [10~points] Write down (but {\it do not try to evaluate}) the integral
that would compute the arc-length of the top edge of $R$.
\vskip 2.0truein
\MyTallAnswer{Arclength integral:}{$\phantom{A very very very long answer here}$}%
\item[B.] [15~points] A thin metal plate of constant density has 
the shape of $R$.  Find the $x$-coordinate of the center of mass of 
the plate.
\vskip 4.0truein
\MyAnswer{$\overline{x}=$}{$\phantom{A very very very long answer here}$}%
\end{enumerate}

\vfill\eject
\noindent
II.  Both parts of this problem deal with the differential equation $\frac{dy}{dx} = xy$.
\begin{enumerate}
\item[A.] [15~points] Find the general solution $y(x)$ of the equation 
by separating variables and integrating.
\vskip 3.5truein
\MyAnswer{}{$\phantom{A very long answer here}$}%
\item[B.] [5~points] Find the particular solution $y(x)$ satisfying the initial 
condition $y(0) = 4$ and compute the exact value of $y(2)$.  
\vskip 2.5truein
\MyAnswer{Particular solution: }{$\phantom{A very very very long answer here}$}%

\MyAnswer{$y(2) = $}{$\phantom{A very long answer here}$}%
\end{enumerate}
\vskip 10pt
\noindent
III. [10~points]  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, with some
proportionality constant $k$.  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.  
Note: You \emph{do not need to solve} the equation.
\vskip 5.0truein
\MyTallAnswer{Differential Equation: }{$\phantom{A very very very very very long answer here}$}%
\vfill\eject
%\noindent
%IV. All parts of this question deal with the infinite series 
%\[
%\frac{1}{2} + \frac{2}{3} + \frac{3}{4} + \frac{4}{5} +  \cdots\ .
%\]
%\begin{enumerate}
%\item[A.]  [5~points] Find a general formula for the $n$th term of the series as a function
%of $n$ and write the series in summation (``sigma'') notation.
%\vskip 1.0truein
%\MyTallAnswer{Series $=$}{$\phantom{A very very very very very long answer here}$}%
%\item[B.]  [5~points] Call the $n$th term in the series $a_n$.  
%What is ${\displaystyle \lim_{n\to\infty} a_n}$?
%\vskip 1.5truein
%\MyAnswer{Limit $=$}{$\phantom{A very long answer here}$}%
%\item[C.]  [5~points] Add up the first 3 \emph{partial sums} of the series.
%
%\MyAnswer{$s_1 = $}{$\phantom{A very long answer here}$}%
%
%\MyAnswer{$s_2 = $}{$\phantom{A very very long answer here}$}%
%
%\MyAnswer{$s_3 = $}{$\phantom{A very very very long answer here}$}%
%\item[D.]  [5~points] Does this series converge or diverge? Explain your answer.
%\vskip 1.0truein
%\MyAnswer{Answer:}{$\phantom{A very long answer here}$}%
%\end{enumerate}
%\vfill\eject
\noindent
IV. 
\begin{enumerate}
\item[A.]  [10~points] Does the series 
$\ds \sum_{n=0}^\infty 4\cdot \frac{2^n}{\pi^n}$ converge or diverge?  
If it is convergent, say why and find the 
sum; if it is not convergent say why not.
\vskip 2.0truein
\MyAnswer{Answer: }{$\phantom{A very very very very long answer here}$}%
\item[B.] [10~points] Explain why the Integral Test can be applied to the series
$\ds \sum_{n=1}^\infty \frac{1}{n^2 + 4}$ and use it to determine if the series
converges or diverges.
\vskip 4.0truein
\MyAnswer{Answer: }{$\phantom{A very very very long answer here}$}% 
\end{enumerate}
\vfill\eject
\noindent
V.  All parts of this question refer to the power series 
\[
\sum_{n=1}^\infty \frac{(x - 1)^n}{n 3^n}.
\]
\begin{enumerate}
\item[A.] [15~points] Use the Ratio Test to determine the radius of convergence.
\vskip 3.0truein
\MyAnswer{Radius of convergence: }{$\phantom{A very very very long answer here}$}% 
\item[B.] [10~points] Test convergence at the endpoints of the interval from part A to determine 
the interval of convergence.  Explain your conclusions.

\vskip 3.0truein
\MyAnswer{Interval of convergence: }{$\phantom{A very very very long answer here}$}% 
\end{enumerate}

\end{document}

