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\begin{document}
\begin{center}

\textbf{College of the Holy Cross} \\
\textbf{MATH 134, Calculus With Fundamentals 2}\\
\textbf{Final Examination -- Thursday, May 10}
\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.  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 total points on this exam.

\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/ 30\  \hfil \\
\hline
IV  & \hfil/ 40\  \hfil \\
\hline
V  & \hfil/ 30\  \hfil \\
\hline
VI  & \hfil/ 20\  \hfil \\
\hline
VII  & \hfil/ 30\  \hfil \\
\hline\hline
Total   & \hfil/ 200\hfil \\
\hline
\end{tabular}
\end{center}

\begin{center}
$$\mathcal{HAVE\ AN\ ENJOYABLE\ AND\ PRODUCTIVE\ SUMMER!}$$
\end{center}

\vfil\eject
\noindent
I.  Integration and the Fundamental Theorem of Calculus.  
\begin{enumerate}
\item[(A)]  (10) Let 
$\ds
f(x) = \begin{cases} 2 - x & \text{ if } 0 \le x \le 3\\
                     3x - 10 & \text{ if } 3 < x \le 4.
\end{cases}$  
whose graph is shown here:
\begin{center}
\includegraphics[width=2.5truein,height=2.5truein]{FQIP.png}
\end{center}
Let $F(x) = \int_0^x f(t)\, dt$.  Complete the following table of values for $F(x)$
and $F'(x)$.  (Assume the graph continues to the left and the right so 
$F'(0)$ and $F'(4)$ make sense.)
\vskip 2.0truein
\[
\begin{array}{|c||c|c|c|c|c|c|c|}
\hline
x          & 0 & 1 & 2 & 3 & 4 \strut \\
\hline
F(x)       & \qquad  & \qquad  & \qquad & \qquad & \qquad \strut \\
\hline
F'(x)      & \qquad  & \qquad  & \qquad & \qquad & \qquad \strut \\
\hline
\end{array}
\]
\item[(B)] (10) Compute $G'(x)$ if $G(x) = \displaystyle{\int_0^{x^3} e^{-x^2}\ dx}$.  
\vskip 1.0truein
\MyAnswer{$G'(x) =$}{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}%
\end{enumerate}
\vfill\eject
\noindent
II.  Compute the following integrals using basic rules, $u$-substitution, or integration
by parts, as appropriate.
\begin{enumerate}
\item[(A)] (10) $\displaystyle \int x^2 \cos(x^3)\ dx$
\vskip 1.0truein
\MyAnswer{Integral $=$}{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}%
\item[(B)] (10) $\displaystyle \int x^2 e^{5x}\ dx$
\vskip 2.5truein
\MyAnswer{Integral $=$}{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}%
\item[(C)] (10) $\displaystyle \int x^2(1 - 4x + 3x^2)\ dx$
\vskip 2.0truein
\MyAnswer{Integral $=$}{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}%
\end{enumerate}
\vfill\eject
\noindent
III.  Trigonometric Substitutions.
\begin{enumerate}
\item[A.]  (10) What trigonometric substitution would you use to evaluate
$$\int \frac{x^2}{\sqrt{49 - x^2}}\ dx?$$
Make the substitution and simplify so there is no square root in the resulting trigonometric
integral.  
\vskip 3.0truein
\MyAnswer{Integral $=$}{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}%
\item[B.]  (10) Find $\int \sin^2(\theta) \cos^2(\theta)$ using the trigonometric reduction formulas.
\vskip 3.0truein 
\MyAnswer{Integral $=$}{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}%
\item[C.]  (10) Suppose you had used the trig substitution $x = 3\tan(\theta)$ and you integrated
the resulting trigonometric integral to get
$$\theta + \sec(\theta) + C$$
What is the final answer, expressed as a function of $x$?  
\vskip 2.0truein
\MyAnswer{Integral $=$}{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}%
\end{enumerate}
\vskip 10pt
IV.  Partial Fractions.
\begin{enumerate}
\item[A.] (5)  How do you recognize the situation where you need to start by using polynomial division? 
Describe briefly.
\vglue 1.0truein
\item[B.] (10) To decompose $\displaystyle\frac{x^2 + 3}{(x+1)(x+2)^2(x^2 + 4)}$ 
into partial fractions, what would the form of the fractions be (leave coefficients as undetermined; 
\emph{do not try to solve for them}).  
\end{enumerate}
\vfill\eject

\begin{enumerate}
\item[C.]  (10)  Determine the values $A,B,C$ making 
$$\frac{3x + 1}{x(x + 3)(x - 1)} = \frac{A}{x} + \frac{B}{x + 3} + \frac{C}{x - 1}$$
\vglue 3.0truein
\MyAnswer{$A =$}{\qquad\qquad\qquad\qquad} \MyAnswer{$B =$}{\qquad\qquad\qquad\qquad}
\MyAnswer{$C =$}{\qquad\qquad\qquad\qquad} %
\item[D.]  (15)  You have decomposed a rational function $f(x)$ into partial fractions as
$$f(x) = 3x^3 + 4 + \frac{6}{x} + \frac{2}{x^2} + \frac{2x - 3}{x^2 + 4}.$$
What is $\int f(x)\ dx$?
\vglue 2.0truein
\MyTallAnswer{Integral $ = $}{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}%
\end{enumerate}
\vfill\eject
V.  All parts of this problem refer to the region $R$ bounded by $y = x$, $y = 4 - x^2$, $x = 0$ and $x = 1$.  
\begin{enumerate}
\item[A.] (10)  Sketch the region $R$.
\vskip 3.0truein


\item[B.] (10) Compute the area of the region $R$.   
\vskip 3.0truein
\MyAnswer{Area $=$}{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}%
\end{enumerate}

\vfil\eject
\begin{enumerate}
\item[C.] (10) Compute the volume of the solid of revolution obtained by rotating the region $R$ about 
the $x$-axis.  
\vskip 3.0truein
\MyAnswer{Volume $=$}{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}%
\end{enumerate}
VI.  Other Applications of Integrals.
\begin{enumerate}
\item[(A)] (10)  Find the general solution of the differential equation 
$\displaystyle \frac{dy}{dx} = x e^y$.
\vskip 3.0truein
\MyAnswer{General solution $=$}{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}%
\end{enumerate}
\vfill\eject
\begin{enumerate}
\item[(B)] (10) Suppose that a random variable $T$ has pdf $f(t) = 20t^3(1 - t)$ for $0 \le t \le 1$ (and
$0$ otherwise).  Find $P\left(0 \le T \le \frac{1}{3}\right)$.
\vskip 3.0truein
\MyAnswer{Probability $=$}{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}%
\end{enumerate}
\vskip 10pt
\noindent
VII.  Miscellany -- \emph{answer any 3 of the following 5 questions}.  
If you answer more than 3, all points received will be counted, 
giving a possibility of up to 20 points Extra Credit.  
\begin{enumerate}
\item[(A)] (10) The standard normal pdf is the function $f(x) = \frac{1}{\sqrt{2\pi}} e^{-x^2/2}$.
Use a left-hand Riemann sum with $N = 5$ to approximate $\int_0^{0.5} f(x)\ dx$.  
\vglue 3.0truein
\MyAnswer{$L_5 = $}{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}%
\item[(B)] (10) If $Z$ has a standard normal distribution, what is $P(0 \le Z \le 0.50)$ from our
table?  Why is this number smaller than the answer in part A?  Explain briefly.
\vglue 1.0truein
\MyAnswer{$P(0 \le Z \le 0.50) = $}{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}%
\item[(C)]  (10)  What is the future value of a series of monthly payments of $\$300$ at $5\%$ interest, 
over a period of 6 years?  If you were making these payments to pay off a car loan for $\$18000$ would
you have overpaid or still owe the loan company?
\vglue 4.0truein
\MyAnswer{FV of series of payments $ = $}{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}%  
\vskip 1pt
\MyAnswer{Circle the correct option: Overpaid/Owe by \$}{\qquad\qquad\qquad\qquad\qquad}%
\vfill\eject
\item[(D)]  (10)  According to the \emph{multiplier effect} in economics,
when there is an injection of money to consumers in an economy, the consumers
spend a certain proportion of it, then that amount recirculates through the economy and
adds additional income, which comes back to the consumers and they
spend the same percentage, etc.  The process repeats \emph{indefinitely} circulating
additional money through the economy.  Suppose the government cuts taxes by 
\$ 50 billion, thereby giving 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?
\vglue 3.0truein
\MyAnswer{Amount $=$}{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}%
\item[(E)]  (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 $45$?
\vglue 2.0truein
\MyAnswer{Time to reach $y = 5$ is:}{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}%
\end{enumerate}
\end{document}





