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\begin{document}
\begin{center}

\textbf{College of the Holy Cross, Fall 2016} \\
\textbf{Math 136 -- Makeup Midterm Exam 3} \\
\textbf{December 5}
\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/ 20\  \hfil \\
\hline
II  & \hfil/ 30\  \hfil \\
\hline
III  & \hfil/ 20\  \hfil \\
\hline
IV  & \hfil/ 10\  \hfil \\
\hline
V  & \hfil/ 20\  \hfil \\
\hline\hline
Total   & \hfil/100\hfil \\
\hline
\end{tabular}
\end{center}

\vfil\eject

\noindent
I.  [20~points] Integrate with the partial fraction method:  
${\displaystyle \int \frac{2x + 3}{x^3 + 4x}\ dx}$

\vskip 7.5truein
\MyAnswer{Answer:}{$\phantom{A very very very very very very very long answer here}$}%
\vfill\eject
\noindent
II.  For each integral, say why the integral is \emph{improper}, then set up and 
evaluate the appropriate limits to determine whether the integral converges.  If so, find its value; if not, say ``does not converge.''  (Full credit will be given only for the correct limit calculation.)
\begin{enumerate}
\item[A.]  [15~points] $\displaystyle \int_0^1 \frac{1}{\sqrt{1 - x^2}}\ dx$.  \\
\MyAnswer{Why improper:}{$\phantom{A very very very long answer here}$}%
\vskip 6.5truein
\MyAnswer{Answer:}{$\phantom{A very long answer here}$}%
\vfill\eject
\item[B.] [15~points] ${\displaystyle \int_4^\infty \frac{1}{(x-2)(x-3)} \ dx}$\\
\MyAnswer{Why improper:}{$\phantom{A very very very long answer here}$}%
\vskip 7.0truein
\MyAnswer{Answer:}{$\phantom{A very long answer here}$}%
\end{enumerate}
\vfill\eject
\noindent
III.  Both parts of this problem deal with the differential equation 
$\frac{dy}{dx} + 2y = e^{-3x}$.
\begin{enumerate}
\item[A.] [15~points] Find the general solution of the equation.
\vskip 3.5truein
\MyAnswer{}{$\phantom{A very long answer here}$}%
\item[B.] [5~points] Find the particular solution $y(x)$ satisfying the initial 
condition $y(0) = 4$.  
\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}
\vfill\eject
\noindent
IV. [10~points]  A population $y$ satisfies the logistic growth equation
$$\frac{dy}{dt} = (.2)y\left(1 - \frac{y}{50}\right)$$
and the initial condition $y(0) = 1$.  
What is the population level when the population is \emph{growing the fastest}?
\vskip 5.0truein
\MyTallAnswer{Population level for fastest growth: }{$\phantom{A very very long answer here}$}%

\vfill\eject
\noindent
V.  
\begin{enumerate}
\item[A.] [10~points] A function $y = f(x)$ is plotted below.
\begin{center}
\includegraphics[width=1.5truein,height=1.5truein]{1363V.png} 
\end{center}
\noindent
Check the appropriate boxes for statements about $\int_0^3 f(x)\ dx$:
\vskip 10pt
\item[] The trapezoidal rule approximation is an overestimate $\Box$/underestimate $\Box$.
\item[] The midpoint rule approximation is an overestimate $\Box$/underestimate $\Box$.
\item[B.] [10~points]  Suppose you know that the function $f(x)$ plotted
here is $g''(x)$ for some other function $g(x)$.  How big would you 
need to take $N$ to get a trapezoidal rule approximation to $\int_0^3 g(x)\ dx$
with error $< 10^{-3}$?
\vglue 3.0truein
\MyTallAnswer{$N \ge $}{$\phantom{A very very long answer here}$}%
\end{enumerate}
\end{document}

