\documentclass[12pt,fleqn]{article}
\pagestyle{empty}


\usepackage{amssymb}
\usepackage{amsmath}


\newcommand{\adj}{\mathrm{Adj}}
\newcommand{\mbf}{\mathbf}
\newcommand{\R}{\mathbb{R}}
\newcommand{\ra}{\rightarrow}
\newcommand{\proj}{\mathrm{proj}}
\newcommand{\dotprod}{\bullet}

\begin{document}

\begin{center}
{\Large Final

Math 202}
\end{center}

\begin{itemize}
\item  Do your work on separate paper.  Make sure your work is neat!
\item  You can use a calculator and book, but no notes!
\item  Your answers need to be justified by your work!  (Saying you found a similar problem in the book is not enough!)
\item  There are 6 problems on the test (number~\ref{extra} is extra credit) for a total of 64 points.
\end{itemize}



\begin{enumerate}


\item  (15 points)  Calculate the following integrals
	\begin{enumerate}
	\item  \[ \int_{\pi/6}^{\pi/2} \frac{d\theta}{\tan \theta} \]
	\item  \[  \int \frac{x^2}{x^2+1} \; dx \]
	\item  \[  \int_1^\infty \frac{\ln x}{x^2} \; dx \]
	\end{enumerate}

\item  (10 points)  Find the radius of convergence for the following power series.
	\begin{enumerate}
	\item  \[ \sum_{n=1}^\infty \frac{2^n x^n}{\sqrt{n}} \]
	\item  \[ \sum_{n=1}^\infty n! x^n \]
	\end{enumerate}

\item  (12 points)  Test the following series for absolute and conditional convergence
	\begin{enumerate}
	\item  \[ \sum_{n=1}^\infty \frac{e^{1/n}}{n^2} \]
	\item  \[ \sum_{n=1}^\infty (-1)^{n+1} \left(\frac{\ln n}{\sqrt{n}} \right) \]
	\item  \[ \sum_{n=1}^\infty \frac{n^2+1}{3n^2-4n} \]
	\item  \[ \sum_{n=1}^\infty \left( \frac{3n}{1+9n} \right)^n \]
	\end{enumerate}



\item  (12 points)  
	\begin{enumerate}
	\item  Find the Taylor series for $f(t) = te^t$ centered at $t=0$.
	\item  Using your power series, find a Taylor series, centered at $x=0$, for the function
		\[
		F(x) = \int_0^x te^t \;dt
		\]
	\item  Find the exact value for the sum.  (Use the previous parts!)
		\[
		\sum_{n=0}^\infty \frac{1}{(n+2)n!}
		\]

	\end{enumerate}

\newpage

\item  (15 points)  \label{polar}
	\begin{enumerate}
	\item  Graph the polar curve. (Graph all values of $\theta$, but below we will only be interested in $0 \leq \theta \leq \pi/2$.)
		\[
		r = 4 \cos^2(\theta)
		\]
	\item  Set up (but do not solve) an integral representing the length of this curve for $0 \leq \theta \leq \pi/2$.
	\item  Set up (but do not solve) an integral representing the area between this curve and the $x$-axis for $0 \leq \theta \leq \pi/2$.
	\item  Rotate this curve (for $0 \leq \theta \leq \pi/2$) around the $x$-axis.
		Set up (but do not solve) an integral representing the volume of this solid.
	\item  Rotate this curve (for $0 \leq \theta \leq \pi/2$) around the $x$-axis.
		Set up (but do not solve) an integral representing the surface area of this surface.
	\end{enumerate}


\item  \label{extra}  (6 points)  Extra Credit - Compute the integrals in number~\ref{polar}. \\
	Hint: don't bother with the length integral.

\end{enumerate}




\end{document}