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\noindent
{\Large {\bf Mathematics 1210  \hspace{5ex} PRACTICE EXAM III 
\hspace{6ex}Fall 2004}}\\ 
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%\begin{center}
%{\bf Mathematics 1210, Fall 2000}\\
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%{\bf K. M. Golden}\\
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%{\bf PRACTICE EXAM II}\\
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%\vspace{1ex}



\begin{enumerate}

\item[1.] Calculate the following:

\begin{enumerate}

\item[a.] $\ds \lim_{x \rightarrow 0} \;\frac{\tan x -
\sin x}{x^2 \tan x} $
\hspace{4ex}
%\item
b. $\ds \lim_{x \rightarrow 0}\; \frac{x^{10/3}}{\sin x -x}$
\item[c.] $\ds \lim_{x \rightarrow \pi/2}\; \frac{\cos x}{x - \pi/2}$
%\item  
\hspace{9ex}
d. $\ds \lim_{x \rightarrow 0} 
                 \frac{\ds \int_0^x{(1-\cos{t})\;dt}}{x^3}$
\item[e.] $\ds \lim_{n \rightarrow \infty}\; \frac{1}{n}
           \sum_{i=1}^{n}{\sin{\frac{i}{n}}}$

\end{enumerate}

\item[2.] Find the general solutions to the following differential
          equations.
\begin{enumerate}
\item[a.] $\ds \frac{dy}{dx} = \sqrt[3]{\frac{x}{y}} \hspace{3ex}$ 
b. $\ds \frac{d^2 x}{dt^2} = - \omega^2 x $
\item[c.] $\ds \frac{d^2x}{dt^2} = - g $
\end{enumerate}

\item[3.] section 5.2, \#22,35,36

\item[4.] For each function below, determine whether or not
         it is Riemann integrable on the interval (0,1],
         and {\bf fully} explain your result: (a) 
         $f(x)=\sin{(1/x)},\;$ (b) $f(x)=1/\sqrt{x^2\sin{x}},\;\;$
         (c) $f(x)=x$ for $x$ rational and $=-x$ for 
         $x$ irrational.

\item[5.] Calculate $\ds \int_1^2{(3x^2-2)\;dx}$ from the definition
          of the integral, that is, using Riemann sums.
\item[6.] Calculate the following integrals:

\begin{tabular}{lll}
(a) $\quad \ds \int^4_0 \sqrt{x} \;dx \;\;\;\;$ (b) $\ds \int^{\pi/2}_0
\sin x \;dx \;\;\;\; $(c) $\ds \quad \int^3_1 \frac{1 - 3x^3}{x^2} \;dx$\\
\vspace{0ex}\\
(d) $\ds \quad \int^\pi_0 \sin^2 x\;dx \;\;\;\; $ (e)$\ds \quad \int^\pi_0 x \cos
(x^2 + \pi)\; dx \;\;\;\;$ (f) $\ds \quad \int^3_{-3} x^3 \;dx$
\end{tabular}

\item[7.] Find $\ds \frac{d}{dx} \int^{x^2}_0 \tan \theta \;d\theta$.
\end{enumerate}

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