\documentclass[11pt]{article}

\setlength{\oddsidemargin}{0.25in}
\setlength{\evensidemargin}{0.25in}
\setlength{\textwidth}{6in}
\setlength{\topmargin}{-.5in}
\setlength{\textheight}{9in}

\pagestyle{empty}

\def\ds{\displaystyle}

\begin{document}


\noindent
{\Large {\bf Mathematics 1210  \hspace{5ex} PRACTICE EXAM II 
\hspace{6ex}Fall 2004}}
%\vspace{2ex}


%\begin{center}
%{\bf Mathematics 1210, Fall 2000}\\
%\vspace{1ex}
%{\bf K. M. Golden}\\
%\vspace{3ex}
%{\bf PRACTICE EXAM II}\\
%\end{center}
%\vspace{1ex}



\begin{enumerate}

\item[1.] Calculate the following:

\begin{enumerate}

\item[a.] $\ds \frac{d^2x}{dt^2}\;, \;\;\; x(t) = A\sin({\omega
t-\phi})$
\hspace{4ex}
%\item
b. $\ds \frac{df}{dx}\;,\;\;
                f(x)=\left( \frac{x-2}{x-\pi} \right)^{3}$
\item[c.] $\ds \frac{dy}{dx}\;,\;\;\; \cos{xy}=y^2 + 2x$
%\item  
\hspace{9ex}
d. $\ds \lim_{x \rightarrow 0}
                \frac{\sin{x}\tan{x}}{1-\cos{x}}\;$
\item[e.] $\ds \frac{df}{dx}\;,\;\;
                f(x)=\sin{\sqrt{ \frac{\tan{x}}{1+x^2} }}$
\hspace{7ex}
%\item
f. $\ds \frac{df}{dx}\;,\;\;
                f(x)=x^2\sin^2{(x^3)}$


\end{enumerate}

\item[2.] Find all solutions to the differential equation for a mass on
a spring, $$\ds \frac{d^2x}{dt^2}=-\omega^2 x(t),$$ where
$x(t)$ is the position as a function of time $t$,
$\ds \omega=\sqrt{k/m}\;$, $k$ is the spring constant and $m$ is
the mass. (Hint: see problem 1(a) above.)
Choose a set of parameters characterizing a particular
solution, and graph it.
Also find in this case for which times the position is maximized,
and for which times the velocity is maximized.

\item[3.] 
A circular oil slick spreads so that its radius 
increases at the rate of 1.5 feet/second.
How fast is the area of the enclosed oil increasing
at the end of two hours?

%A metal disk expands during heating. If its radius increases
%          at the rate of 0.02 inch per second, how fast is the area
%          of one of its faces increasing when its radius is 8.1 inches?

\item[4.] The hands on a clock are of length 5 inches (minute hand)
          and 4 inches (hour hand). How fast is the distance between 
          the tips of the hands changing at 3:00? 


\item[5.] Approximate $\ds \sqrt{66}$ and
          $\ds \sin\left({\frac{\pi}{100}}\right)$ using
          linear approximation (i.e., the differential).
          Also do problem \#35 on page 155, section 3.10.

\item[6.] Use the differential to approximate
the increase in volume of a spherical bubble as its radius
increases from 3 to 3.025 inches.

\item[7.] Consider $f (x) = \left \{
\begin{array}{ll}
|x|, x < 0, \\
\sin{x},  x \geq 0.
\end{array}
\right.$ Find all local maxima and minima of $f$, where $f$ is increasing
and decreasing, where $f$ is concave up and concave down, and 
all inflection points.
Does $f$
have a global maximum or a global minimum? Sketch the graph of $f(x)$. Do the same for
$f(x)=x^3-12x+1$ and $\ds f(x)=\frac{1}{1+x^2}\;$.
% p. 188 #19


\item[8.] Consider $f(x)=2\sin{(x-\frac{\pi}{4})}$ on the interval
$[\frac{\pi}{2},\frac{5\pi}{4}]$. Find where $f$ is increasing and decreasing. Find the maximum and minimum values of $f$ on the interval.


\item[9.] A rectangle has two corners on the $x-$axis and the
other two on the parabola $y=12-x^2$, with $y\geq0$. What are the
dimensions of the rectangle of this type with maximum area?

\item[10.] Problem \#29 on Snell's Law, page 186, section 4.4.

\end{enumerate}

\end{document}

