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\begin{center}

Math 1210\\
Fall 2004\\
K. M. Golden
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{\bf EXAM II}\\
Friday, October 22, 2004
\vspace {15ex}

\begin{tabular}{|l|l|l|l|l|l|l|}
%\hline
%\multicolumn{2}{|c|}{Dept. Use Only}\\
%\multicolumn{2}{|c|}{Exam Scores}\\
\hline
Problem &Points & Score~~~~~~\\
\hline
1.~~ & 20 & \\
\hline
2.~~ & 20 & \\
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3.~~ & 15 &\\
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4.~~ & 20 &\\
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5.~~ & 10 &\\
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6.~~ & 15 &\\
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%  6.~~ & 10 &\\
%  \hline

&TOTAL&\\
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\begin{enumerate}
\item[(20 points) ~~1.]  Calculate the following.  Be sure to
show all of your work.

\begin{enumerate}
\item[(a)] $\ds \frac{d}{dx}
           \left( x \sin^2{x} \right)$
\vfill
\item[(b)] the differential $dm$ of $m(v)$,  % $\ds \frac{dm}{dv}\;,\;\;\;\;
            where $\ds m(v)=\frac{m_0}{\sqrt{1-v^2/c^2}}$
\vfill
\item[(c)] $\ds \frac{d}{dx}
           \left(\frac{1-x}{1+x^2} \right)$
\vfill
\item[(d)] $\ds \lim_{x \rightarrow 0} \frac{1-\cos{x}}{x}$
%\vfill
%\item[(e)] $\ds \lim_{x \rightarrow \infty} 
%             \frac{\sin{x}}{x}$
\end{enumerate}
\eject
\item[(20 points) ~~2.]  
Consider $\ds f(x)=x^2-4x+3$ on the interval $[0,3]$.
\begin{enumerate}
\item[(a)]  Find where $f$ is increasing and decreasing.
\vspace{2.75in}
\item[(b)]  Find the maximum value of $f(x)$ on $[0,3]$
             and find the minimum value of $f(x)$ on $[0,3]$. Be sure
             to justify your answers.
\vspace{2.75in}
\item[(c)]  Find where $f$ is 
            concave up and where it is concave down.
            Does $f(x)$ have any inflection points on $[0,3]$?
\eject
\item[(d)]  Sketch the graph of $f(x)$ on $[0,3]$, 
      incorporating all
      your information from (a)--(c). 
\end{enumerate}

%\eject

%\item[(10 points) ~~3.] Use implicit differentiation
%to find $\ds \frac{dy}{dx}$ for any point $(x,y)$ on the
%circle $x^2 + y^2 = 1$. Use your result to 
%find all points on the circle
%for which the tangent line is vertical, and
%then all points for which the tangent line
%is horizontal. 


\vspace{40ex}



\item[(15 points) ~~3.] Find the dimensions of the rectangle
with maximum area inscribed in a circle of unit radius.
Be sure to show all your work. 
%and verify your result 
%using the 
%$1^{st}$ or $2^{nd}$ derivative test.







%A mass connected to a spring moves 
%along the $x-$axis so that its $x-$coordinate at time $t$ is
%\begin{eqnarray*}
%x(t) = \sin{2t} + \sqrt{3}\cos{2t}.
%\end{eqnarray*}
%What is the farthest the mass gets from the origin?




\eject


\item[(20 points) ~~4.]  
A primordial lightning bolt ignites a flame in the 
top of a 100 foot tall Cretaceous tree.
A {\it tyrannosaurus rex}, who is 20 feet tall, 
runs away from the flame
at a speed of 10 $f/s$. How fast is the length of
his shadow moving when he is 250 feet away from the 
base of the tree?

\eject

%\vspace{4in}
%\item[(10 points) ~~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[(10 points) ~~5.]  Use linear approximation 
                 (the differential) to estimate $\sqrt{24}\;$.

\vspace{40ex}
\item[(15 points) ~~6.] Most of the gases in earth's atmosphere are contained
in the first 20 miles of altitude above the surface of the earth. 
The radius of the earth is about 4,000 miles. Use the differential
to estimate the volume (in cubic miles) 
of this main part of our atmsophere.

%estimate the amount of paint in cubic centimeters needed
%           to apply a coat of paint 0.04 cm thick to a hemispherical
%           dome with a diameter of 40 meters.

\end{enumerate}

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