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\flushright Name
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\flushright Student ID \#
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\flushright Class (circle one) ~~~~~~~~~~~~~~ 9:40 ~~~~~~ 10:45\\
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%\flushright Discussion Instructor
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\begin{center}

Math 1210\\
Fall 2009\\
K. M. Golden
\vspace{5ex}

{\bf EXAM II}\\
Friday, October 23, 2009
\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.~~ & 30 & \\
\hline
2.~~ & 25 & \\
\hline
3.~~ & 20 &\\
\hline
4.~~ & 15 &\\
\hline
5.~~ & 10 &\\
%\hline
%6.~~ & 10 &\\
%\hline
%7.~~ & 10 &\\
  \hline

&TOTAL&\\
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\begin{enumerate}
\item[(30 points) ~~1.]  Calculate the following limits, and be sure to
      show all of your work.  
  If a particular limit does not exist, state this clearly and tell why.


\begin{enumerate}
\item[(a)] $\ds \lim_{x \rightarrow 2} 
                \frac{x^2-x-2}{x-2}$

%\item[(a)] $\ds \lim_{x \rightarrow 0^+} 
%                x\sin{\left(\frac{1}{x}\right)}$
\vfill
\item[(b)] $\ds \lim_{x \rightarrow +\infty} 
    \frac{7 x^5+\pi x^4-10^{20}x^3-x+1}{(x+1)^2(x+2)^3}$
\vfill
\item[(c)] $\ds \lim_{x \rightarrow 0} \frac{1-\cos{x}}{x}$
%      \left({\frac{1}{x+h}}-{\frac{1}{x}}\right)
%       \left(\frac{1}{h}\right)$
\vfill

\item[(d)]
$\ds \lim_{x \rightarrow 0} f(x)$, ~~~ 
where $f (x) = \left \{
\begin{array}{ll}
\sin{x}, & x < 0 \\
\cos{x}, & x \geq 0 
\end{array}
\right.$
\vfill
\item[(e)] $\ds \lim_{x \rightarrow 0}\frac{\sin{(\pi x)}}{3x}$
\vfill
\item[(f)]
$\ds \lim_{x \rightarrow 3} f(x)$, ~~~ 
where $f (x) = \left \{
\begin{array}{ll}
1, & x \;\; \mbox{rational} \\
0, & x \;\; \mbox{irrational} 
\end{array}
\right.$


\end{enumerate}

\eject





\item[(25 points) ~~2.]  Find the following.  Be sure to
show all of your work.

\begin{enumerate}
\item[(a)] $\ds \frac{d}{dx}
           \left( x \sin{x} \right)$
\vfill
\item[(b)] $\ds \frac{d}{dx}
           \left(\frac{x}{1+x^2} \right)$
\vfill
\item[(c)] $\ds \frac{d^2x}{dt^2}\;\;$ where $\;\;
          x(t) = \sin({ \sqrt{2}\;t})$
\vfill
\item[(d)] $\ds \frac{dm}{dv}\;\;$  % $\ds \frac{dm}{dv}\;,\;\;\;\;
            where $\;\; \ds 
       m(v)=m_0 \left( 1-\frac{v^2}{c^2}\right)^{-1/2}$
   %\frac{m_0}{\sqrt{1-\frac{v^2}{c^2}}}$
%\vfill
%\item[(e)] $\ds \lim_{x \rightarrow \infty} 
%             \frac{\sin{x}}{x}$
\vfill
\item[(e)] $\ds \frac{dy}{dx}\;\;$  % $\ds \frac{dm}{dv}\;,\;\;\;\;
            where $\; \ds x^2 + y^2 =1$
\end{enumerate}
\eject
\item[(20 points) ~~3.]  
Consider 
$f (x) = \left \{
\begin{array}{ll}
|x+1|, & x \leq 0 \\
5-(x-2)^2, & x > 0 
\end{array}
\right.,   \;\;$ 
defined on all of $\mathbb{R}$.

\begin{enumerate}
\item[(a)]  Sketch the graph of $f(x)$.
\vspace{25ex}
\item[(b)]  Sketch the graph of the derivative of 
      $f(x)$. 
\vspace{25ex}
\item[(c)]  Where is $f(x)$ continuous?
\vspace{20ex}
\item[(d)]  Where is $f(x)$ differentiable?
\vspace{20ex}
\item[(e)]  Use the derivative to find the maximum value of $f(x)$ on the interval $[0,\infty)$. 
           Where does the maximum occur? Justify your results.
      
\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. 








%\item[(10 points) ~~4.] 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*}
%Find its velocity $\ds v(t)=\frac{dx}{dt}\;$.

%\vspace{20ex}




\item[(15 points) ~~4.]  
A primordial lightning bolt ignites a flame in the 
top of a 100 foot tall Cretaceous tree.
A 20 foot tall {\it Tyrannosaurus rex}
runs away from the flame
at a speed of 10 $f/s$. How fast is the tip of
his shadow moving when he is 150 feet away from the 
base of the tree? Be sure to show all your work.

%\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.

\vspace{88ex}


\item[(10 points) ~~5.]  Use linear approximation 
                 (the differential) to estimate $\sqrt{4.25}\;$.

%\vspace{40ex}
%\item[(10 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}

\end{document}
