
\documentclass[12pt]{article}
\usepackage{latexsym,amsmath,graphicx}
\newcommand{\ds}{\displaystyle}
\newcommand{\vp}{$\vphantom{\big|}$}

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

\newcommand{\Answer}[1]{$\phantom{.}$\hfill
        #1\quad\fboxsep 20pt \fbox{$\phantom{\Big|Answer}$}} 

\newcommand{\MyTallAnswer}[2]{$\phantom{.}$\hfill
        #1\quad\fboxsep 20pt\fbox{$\vphantom{\Big|}#2$}}

\newcommand{\MyAnswer}[2]{$\phantom{.}$\hfill
        #1\quad\fboxsep 5pt\fbox{$\vphantom{\Big|}#2$}}

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

\setlength{\textwidth}{6.5in}
\setlength{\topmargin}{-0.5in}
\setlength{\oddsidemargin}{0in}
\setlength{\textheight}{9.0in}

\begin{document}

\begin{center}

\textbf{College of the Holy Cross, Fall 2016} \\
\textbf{MATH 136, Section 02, Final Exam} \\
\textbf{Tuesday, December 13, 11:30am} \\
\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 necessary to justify your answers} 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 Extra Credit points.

\vfil

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

\begin{center}
\begin{tabular}{|l||r|}
\hline
Problem & Points/Possible \\
\hline\hline
I  & \hfil/ 20\  \hfil \\
\hline
II  & \hfil/ 20\  \hfil \\
\hline
III  & \hfil/ 60\  \hfil \\
\hline
IV  & \hfil/ 20\  \hfil \\
\hline
V  & \hfil/ 30\  \hfil \\
\hline
VI & \hfil/ 30\  \hfil \\
\hline
VII  & \hfil/ 20\  \hfil \\
\hline\hline
Total   & \hfil/200\hfil \\
\hline
Extra Credit & \hfil/ 10\  \hfil \\
\hline
Exam I Subscore & \hfil/ 60\ \hfil\\
\hline
\end{tabular}
\end{center}

\vfil

\[
\emph{Have a peaceful and joyous holiday season!}
\]

\eject
\noindent
I.  
\begin{enumerate}
\item[(A)] (5) Compute the derivative of the function $\ds g(x) = \int_0^{3x} \frac{\sin(t)}{t}\, dt$.
\vskip 10pt
\MyTallAnswer{$g'(x)$ = }{$\phantom{A very very very very long answer here}$}% 
\item[(B)] (10) Let 
$\ds
f(x) = \begin{cases} 6x+2 & \text{ if } 0 \le x \le 1\\
                    -x+9 & \text{ if } 1 < x \le 4.\\
                   5 & \text{ if } 4 < x \le 5
\end{cases}$  
which is plotted in the graph above.  
\begin{figure}
\begin{center}
\includegraphics[width=2.5truein,height=2.5truein]{136FI.png}
\caption{Figure for problem I, parts B and C.}
\end{center}
\end{figure}
Let $F(x) = \int_0^x f(t)\, dt$, where $f(t)$ is the function above.  
Complete the following table of values for $F(x)$:
\vskip 1.0truein
\[
\begin{array}{|c||r|r|r|r|r|r|}
\hline
x          & 0 & 1 & 2 & 3 & 4 & 5 \strut \\
\hline
F(x)       & \quad 0 & \qquad  & \qquad & \qquad & \qquad & \qquad \strut \\
\hline
\end{array}
\]
\item[(C)]  (5)  On which interval(s) contained in $(0,5)$ is the graph 
$y = F(x)$ from part B concave down?
\vskip 10pt
\MyAnswer{Concave down on: }{$\phantom{A very very very very long answer here}$}% 
\end{enumerate}
\vfill\eject
\noindent
II.  
\begin{enumerate}
\item[(A)] (5)  Use a left-hand Riemann sum with $n = 4$ to approximate 
$\ds \int_0^1 e^{-x^2/2}\ dx$.
\vskip 2.5truein
\MyAnswer{Answer}{$\phantom{A very very very very very very long answer here}$}%
\item[(B)] (5) Use a midpoint Riemann sum with $n = 4$ to approximate
$\ds \int_0^1 e^{-x^2/2}\ dx$.
\vskip 2.5truein
\MyAnswer{Answer}{$\phantom{A very very very very very very long answer here}$}%
\item[(C)] (10) Check the appropriate boxes and fill in the blank:
\item[] The left-hand sum is an overestimate $\Box$/underestimate $\Box$,
\item[] because: $\underline{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}$
\item[] The midpoint approximation is a overestimate $\Box$/underestimate $\Box$,  
because 
\item[]$e^{-x^2/2}$ is concave up $\Box$/concave down $\Box$ on $[0,1]$.  
\end{enumerate}
\vfill\eject
\noindent
III.  Compute the following integrals; full credit only for showing
all work leading to the final answer.
\begin{enumerate}
\item[(A)] (5) $\ds \int \frac{x^{2/3} - x^4 + \sqrt{x}}{x^{2/3}}\, dx$
\vskip 3.0truein
\MyAnswer{Answer}{$\phantom{A very very very very very very long answer here}$}% 
\item[(B)] (5) $\ds \int x \cos(x^2)\, dx$
\vskip 3.0truein
\MyAnswer{Answer}{$\phantom{A very very very very very very long answer here}$}%
\vfill\eject
\item[(C)] (10) $\ds \int \frac{\sec^2(5x)\, dx}{(\tan(5x) + 7)^5}$
\vskip 3.0truein
\MyAnswer{Answer}{$\phantom{A very very very very very very long answer here}$}%
\item[(D)] (10) $\ds \int_1^{e^2} x^3 \ln(x)\, dx$.  
\vskip 3.5truein
\MyAnswer{Answer}{$\phantom{A very very very very very very long answer here}$}%
\vfill\eject
\item[(E)] (15) $\ds \int \frac{x^2}{\sqrt{16 - x^2}}\, dx$
\vskip 7.5truein
\MyTallAnswer{Answer}{$\phantom{A very very very very very very long answer here}$}%
\vfill\eject
\item[(F)] (15) $\ds \int \frac{x + 1}{(x - 3)(x^2 + 1)}\, dx$
\vskip 7.5truein
\MyTallAnswer{Answer}{$\phantom{A very very very very very very long answer here}$}%
\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)] (10) $\ds \int_1^3 \frac{1}{{\root 3\of {x - 1}}}\, dx.$
\vskip 2.0truein
\MyAnswer{Answer}{$\phantom{A very very very very very very long answer here}$}%
\vskip 10pt
\item[(B)] (10) $\ds \int_0^\infty \frac{1}{x^2 + 4x + 5}\, dx$.  (Hint: There are 
two ways to do this.  You can compute the integral if you complete
the square.  Or you can use $\frac{1}{x^2 + 4x + 5} < \frac{1}{x^2}$ for all $x > 0$.)
\vskip 3.0truein
\MyAnswer{Answer}{$\phantom{A very very very very very very long answer here}$}%
\end{enumerate}
\vfill\eject
\noindent
V.  A region $R$ in the plane is bounded by the graphs $y = x^2$, $y = x + 6$.
See Figure 2. 
\begin{figure}
\begin{center}
\includegraphics[width=1.6truein,height=1.6truein]{136FV.png}
\caption{Figure for problem V.}
\end{center}
\end{figure}
\begin{enumerate}
\item[(A)] (10)  Compute the area of the region $R$.
\vskip 2.0truein
\MyAnswer{Area $=$}{$\phantom{A very very very long answer here}$}%
\item[(B)] (10) 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)] (10) Set up the integral(s) to compute the volume of the solid
obtained by rotating $R$ about the line $y = 12$.  \emph{You do not need to compute the value.}
\vskip 3.0truein
\MyTallAnswer{Volume integral $=$}{$\phantom{A very very very very very very very long answer here}$}%
\end{enumerate}
\vfill\eject
VI.
\begin{enumerate}
\item[(A)] (10)  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.  Write 
a differential equation for the function $y(t) = $ amount of the drug present 
(in mg) in the bloodstream at time $t$ (in hours) that describes this situation.  
You \emph{do not need to solve} the equation.
\vskip 1.0truein
\MyTallAnswer{Answer}{$\phantom{A very very very long answer here}$}%
\item[(B)] (10) Find the general solution of
the differential equation $\frac{dy}{dx} = xy\sqrt{x^2 + 1}$.
\vskip 4.5truein
\MyTallAnswer{General solution $=$}{$\phantom{A very very very very very long answer here}$}%
\vfill\eject
\item[(C)] (10) Let $y(t)$ represent the population of a colony of tree frogs 
that is undergoing logistic growth following the differential equation
$\frac{dy}{dt} = (.1)y(1-\frac{y}{100})$, $t$ in years.  
If the initial population is $y(0) = 10$, how long does it
take for the population to reach $45$?
\vskip 3.0truein
\MyAnswer{Time to reach $45 = $}{$\phantom{A very very very long answer here}$}%
\end{enumerate}
\noindent
VII. For full credit, you must justify your answer completely by showing how 
the indicated test applies and leads to your stated conclusion.
\begin{enumerate}
\item[(A)] (5)  Use the Integral Test to determine if
$\ds \sum_{n=1}^\infty \frac{n}{e^{2n}}$ converges.
\vskip 3.0truein
\MyAnswer{Answer}{$\phantom{A very very very long answer here}$}%
\item[(B)] (15) Using the Ratio Test and testing convergence at the endpoints, 
determine the interval of convergence for the power series 
$$\sum_{n=1}^\infty \frac{x^n}{n 3^n}.$$
\vskip 6.0truein
\MyAnswer{Interval of convergence}{$\phantom{A very very very long answer here}$}%
\end{enumerate}
\vfill\eject
\noindent
Extra Credit. (10) In economics, the \emph{multiplier effect} refers to the fact
that when there is an injection of money to consumers in an economy, the consumers
spend a certain proportion of it. That amount recirculates through the enconomy and
adds additional income, which comes back to the consumers and they
spend the same percentage.  The process repeats \emph{indefinitely} circulating
additional money through the economy.  Suppose that in order to stimulate 
the economy, the government cuts taxes by \$ 50 billion, thereby injecting
that much money back to consumers.  If consumers 
save 10\% of the money they get and spend the other 90\%, what is the total
additional spending circulated through the economy by the tax cut?
\end{document}
