%% MATH 136-01 Final Exam, Spring 2014
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\begin{document}

\begin{center}

\textbf{College of the Holy Cross, Spring Semester, 2014} \\
\textbf{MATH 136, Section 01, Final Exam} \\
\textbf{Wednesday, May~14, 11:30~AM} \\
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.  There
are 200 regular points and 10 possible Extra Credit points (see problem IV).

\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/ 50\  \hfil \\
\hline
III  & \hfil/ 20\  \hfil \\
\hline
IV  & \hfil/ 40\  \hfil \\
\hline
V & \hfil/ 20\  \hfil \\
\hline
VI  & \hfil/ 25\  \hfil \\
\hline
VII & \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)] (10) Let 
$\ds
f(x) = \begin{cases} 1 & \text{ if } 0 \le x \le 2\\
                    x-1 & \text{ if } 2 < x \le 4.\\
                   11-2x & \text{ if } 4 < x \le 6
\end{cases}$  
whose graph is shown here:
\begin{center}
\includegraphics[width=2.5truein,height=2.5truein]{132FIS.ps}
\end{center}
Let $F(x) = \int_0^x f(t)\, dt$.  Complete the following table of values for $F(x)$:
\vskip 1.0truein
\[
\begin{array}{|c||r|r|r|r|r|r|r|}
\hline
x          & 0 & 1 & 2 & 3 & 4 & 5 & 6 \strut \\
\hline
F(x)       & \quad 0 & \qquad  & \qquad & \qquad & \qquad & \qquad & \qquad \strut \\
\hline
\end{array}
\]
\item[(B)] (10) Compute the derivative of the function $\ds g(x) = \int_0^{2x} \frac{\cos(t)}{t^2}\, dt$.
\vskip 1.0truein
\MyTallAnswer{$g'(x)$ = }{$\phantom{A very very very very long answer here}$}% 
\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.  If you use a table entry, state which one.
\begin{enumerate}
\item[(A)] (5) $\ds \int \frac{x^4 - 3\pi^3 + \sqrt{x}}{x^{2/3}}\, dx$
\vskip 3.0truein
\MyAnswer{Answer}{$\phantom{A very very very very very very long answer here}$}% 
\item[(B)] (5) $\ds \int x e^{x^2}\, dx$
\vskip 3.0truein
\MyAnswer{Answer}{$\phantom{A very very very very very very long answer here}$}%
\vfill\eject
\item[(C)] (10) $\ds \int \frac{\csc^2(5x)\, dx}{\cot(5x) + 7}$
\vskip 3.0truein
\MyAnswer{Answer}{$\phantom{A very very very very very very long answer here}$}%
\item[(D)] (10) $\ds \int_1^e x^5 \ln(x)\, dx$.  
\vskip 3.5truein
\MyAnswer{Answer}{$\phantom{A very very very very very very long answer here}$}%
\vfill\eject
\item[(E)] (10) $\ds \int \frac{1}{\sqrt{16 + x^2}}\, dx$
\vskip 7.5truein
\MyTallAnswer{Answer}{$\phantom{A very very very very very very long answer here}$}%
\vfill\eject
\item[(F)] (10) $\ds \int \frac{1}{x^3+x^2+x+1}\, dx$
\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)  Set up an integral to compute the length of the curve
$y = x^3$ from $x = 1$ to $x = 3$.
\vskip 2.5truein
\MyAnswer{Answer}{$\phantom{A very very very very very very long answer here}$}%
\item[(B)] (10) Use a midpoint Riemann sum with $n = 4$ to approximate your integral
from part (A).
\vskip 2.5truein
\MyAnswer{Answer}{$\phantom{A very very very very very very long answer here}$}%
\item[(C)] (5) Given:  The graph of the function in the arclength integral
is concave up on the interval $[1,3]$.  Check the appropriate box:
\item[]The midpoint approximation is a overestimate $\Box$/underestimate $\Box$.  
\end{enumerate}
\vfill\eject
%\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
IV.  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)] (20)  Compute the area of the region $R$.
\vskip 2.0truein
\MyAnswer{Area $=$}{$\phantom{A very very very long answer here}$}%
\item[(B)] (20) 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[(C)] {\it Extra Credit (10)} Set up 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 1.0truein
\MyTallAnswer{Volume Integral(s) $=$}{$\phantom{A very very very very very  long answer here}$}%
\end{enumerate}
\vfill\eject
\noindent
V.  (20) The daily solar radiation $x$ per square meter (in hundreds of calories) in Florida
in October has a probability density function $f(x) = c (x - 2)(6 - x)$ if 
$2 \le x \le 6$, and zero otherwise.  Find the value of $c$, 
and then compute the probability that the daily solar
radiation is $> 4$ hundred calories per square meter.
\vskip 6.0truein
\MyAnswer{$c=$}{$\phantom{A very very very long answer here}$}%
\vskip 5pt
\MyAnswer{Probability $=$}{$\phantom{A very very very long answer here}$}%
\vfill\eject
\noindent
VI.  An avian flu epidemic has broken out in Birdsburgh, a large city 
with total population 10 million.  Let $N$ be the number, in millions,
of people who have been infected, as a function of time $t$ in weeks.  The Birdsburgh
Public Health department determines that the rate of change of $N$ is 
proportional to the product of the number of infected people ($N$) and 
the number of people not yet infected.   
\begin{enumerate}
\item[(A)] (10)  Write the statement above as a differential equation, calling
the constant of proportionality $k$.
\vskip 1.0truein
\MyTallAnswer{Differential Equation: }{$\phantom{A very very very very very very long answer here}$}%
\item[(B)] (5)  The function $N(t) = 10/(1 + 9999 e^{-t})$ should be a solution of your 
differential equation from part A.   What is the value of $k$? 
\vskip 4.0truein
\MyAnswer{$k = $}{$\phantom{A very very long answer here}$}%
\vfill\eject
\item[C)]  (10)  If the epidemic proceeds according the function given in part B,
how many weeks will pass before the number of infected people reaches 1 million?
\end{enumerate}
\vskip 7.5truein
\MyAnswer{Time to reach 1 million $=$}{$\phantom{A very very very long answer here}$}%
\vfill\eject
\noindent
VII.  (25) Using the Ratio Test and testing the endpoints, 
determine the interval of convergence for the power series 
$\ds \sum_{n=1}^\infty \frac{4^n x^n}{n^2}$.  Explain.
\vskip 6.0truein
\MyAnswer{Interval of convergence}{$\phantom{A very very very long answer here}$}%
\end{document}
