\documentclass{article}

\input{../preamble.tex}
\begin{document}

\begin{flushright}
\large
\textbf{Name:}$\underline{\hspace{3in}}$\\
\bigskip
\bigskip
\textbf{Pledged:}$\underline{\hspace{3in}}$\\
\bigskip
Check here if you are handing in\\the Class Survey Form:$\underline{\hspace{.5in}}$
\end{flushright}

\vspace{0.5in}

\vspace{0.5in}

\begin{center}
Rhodes College
\par
Math 115: Applied Calculus
\par
Final Exam
\par
Spring, 2008
\end{center}

\vspace{0.5in}


\begin{center}
\setlength{\extrarowheight}{8pt}
\begin{tabular}{| >{\LARGE}c | >{\LARGE}c | >{\LARGE}c |}
\hline \ \ Problem\ \  & \ \ \ Points\ \ \  & \ \ \ Score\ \ \ \\
\hline 1 & 20 & \\
\hline 2 & 15 & \\
\hline 3 & 15 & \\
\hline 4 & 20 & \\
\hline 5 & 10 & \\
\hline 6 & 10 & \\
\hline 7 & 10 & \\
\hline Total & 100 & \\
\hline
\end{tabular}
\end{center}

\vspace{1in}


\noindent Note: \textbf{SHOW ALL WORK.} Answers with no support will
receive no credit, even if the answer is correct.  May the force be
with you.

\bigskip

Have you taken a calculus class in the past? \underline{\hspace{1in}}

\bigskip

If you have, what class, where, and when?

\bigskip
\bigskip
\bigskip



\begin{enumerate}

\newpage
%XXXXXXXXXXXXXXXXXXXXXXXXXXXXX      PROBLEM 1



\item (20 pts)
Find the derivative of each of the following functions.

{\LARGE

\noindent {\bf a.} (5 pts) $f(t) = \frac{2t^3 + 3}{t^2 + 2t}$} (Use the quotient rule or equivalent).

{\LARGE

\vspace{2in}

\noindent {\bf b.} (5 pts) $F(x) = (\ln x - x^5)(3\sqrt{x} + x)$

\vspace{2in}

\noindent {\bf c.} (5 pts) $S(x) = e^{\sin x + \cos x}$

\vspace{2in}

\noindent {\bf d.} (5 pts) $T(x) =  \frac{1}{\sqrt{x}} + 3\sqrt[3]{x}$


}


\newpage
%XXXXXXXXXXXXXXXXXXXXXXXXXXXXX      PROBLEM 2

\item
(15 pts) Let
{\LARGE
\[
    f(x, y) = 3x^2 + 5y^2 - 12x - 5y
\]
}

{\bf \noindent a.} (5 pts) Find the partial derivative of $f(x, y)$
with respect to $x$.

\vspace{2.3in}

{\bf \noindent b.} (5 pts) Find the partial derivative of $f(x, y)$
with respect to $y$.

\vspace{2.3in}

{\bf \noindent c.} (5 pts) Locate all critical points in the domain
of $f(x, y)$.


\newpage
%XXXXXXXXXXXXXXXXXXXXXXXXXXXXX      PROBLEM 3

\item (15 pts)
Consider the function
{\LARGE
$f(x) = 2x^3 e^x$.
}

\noindent {\bf a.} (4 pts) What is the derivative of $f(x)$?

\vspace{2in}

\noindent {\bf b.} (4 pts) For what values of $x$ is $f(x)$
increasing?

\vspace{2.5in}

\noindent {\bf c.} (7 pts) Find all local maxima and minima of
$f(x)$.



\newpage
%XXXXXXXXXXXXXXXXXXXXXXXXXXXXX      PROBLEM 4

\item (20 pts)
Compute each definite or indefinite integral.

{\LARGE
\noindent {\bf a.} (5 pts) $\int\limits_0^\pi \cos \left(\frac{x}{4}
\right) \, dx$

\vspace{2in}

\noindent {\bf b.} (5 pts) $\int \frac{3}{x} \, dx$

\vspace{2in}

\noindent {\bf c.} (5 pts) $\int e^{2x} + 4x \, dx$

\vspace{2in}

\noindent {\bf d.} (5 pts) $\int\limits_0^3 x^2 + x^5 \, dx$

\vspace{2in}
}



\newpage
%XXXXXXXXXXXXXXXXXXXXXXXXXXXXX      PROBLEM 5

\item (10 pts)
Determine the derivative of {\LARGE $f(x) = 2x^2 - 1$ } directly from the
(limit) definition of the derivative (i.e. the ``3-Step Method").

\newpage
%XXXXXXXXXXXXXXXXXXXXXXXXXXXXX      PROBLEM 6

\item (10 pts)
Suppose the derivative of a function $F(x)$ is given by {\LARGE $f(x) =
2^x + 7$.}

\noindent {\bf a.} (5 pts) Use a Right Riemann Sum with $n = 4$ to
approximate how much $F(x)$ increases between $x = 0$ and $x = 4$.

\vspace{4.5in}

\noindent {\bf b.} (5 pts) Use a Midpoint Riemann Sum with $n = 4$
to approximate how much $F(x)$ increases between $x = 0$ and $x = 4$


\newpage
%XXXXXXXXXXXXXXXXXXXXXXXXXXXXX      PROBLEM 7


\item (10 pts)
Compute the double integral.
\[
    {\LARGE  \int\limits_2^4 \left( \int\limits_0^2 x^2y + y^2x \, dy \right) dx}
\]


\end{enumerate}
\end{document}
