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Math 1210\\
Fall 2006\\
K. M. Golden
\vspace{5ex}

{\bf EXAM III}\\
Friday, November 17, 2006
\vspace {.5in}

\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.~~ & 15 & \\
\hline
2.~~ & 10 & \\
\hline
3.~~ & 20 &\\
\hline
4.~~ & 15 &\\
\hline
5.~~ & 15 &\\
\hline
6.~~ & 15 &\\
\hline
7.~~ & 10 &\\
\hline

&TOTAL&\\
\hline
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\eject

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\begin{enumerate}
\item[(15 points) ~~1.]  Calculate the following.  Be sure to
show all of your work.

\begin{enumerate}
\item[(a)] $\ds \lim_{x \rightarrow 0} 
                 \frac{\sin{x}-x}{x^3}$
\vspace{35ex}
\item[(b)] $\ds \lim_{x \rightarrow 1} 
                 \frac{x^3-1}{x-1}$
%\vspace{40ex}
%\item[(c)] $\ds \lim_{x \rightarrow 0} 
%                 \frac{\ds \int_0^x{\sin{(t^2)}\;dt}}{x^3}$
\end{enumerate}

\vspace{30ex}

\item[(10 points) ~~2.] 
Solve the differential equation 
$$\ds \frac{dx}{dt}=-32t+48$$
for the function $x(t)$
with the initial condition $x(0)=0$.



\eject

\item[(20 points) ~~3.] A population of frogs in a swamp is found
                        to grow at a rate proportional to the 
              square root of the population size. The initial
population $P$ is 400 frogs, and 5 years later there are 900
of them.
\begin{enumerate}
\item[(a)] Write the differential equation for the frog population
$P(t)$ with the two corresponding conditions.
\vspace{30ex}
\item[(b)] Solve this differential equation, that is, find the 
        particular solution which incorporates both conditions.
\vspace{50ex}
\item[(c)] How long does it take for the frog population to
      quadruple (reach 1600) from its intitial value of 400?

\end{enumerate}

%\vspace{30ex}

\eject

\item[(15 points) ~~4.] 
Consider $$\ds f(x)=x + \frac{1}{x} \;.$$ Sketch the graph of
$f(x)$, indicating any symmetry and asymptotes. Find
where the function is increasing and decreasing, and where
it is concave up and concave down. Indicate all local and
global extrema, and any inflection points. Be sure to
show all your work.

\eject


\item[(15 points) ~~5.]  
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 
with the second derivative test.





%\begin{enumerate}
%\item[(a)]  Find where $f$ is increasing and decreasing, and all local
%            maxima and minima. Does $f(x)$ have a global maximum
%            or global minimum?
%            Be sure to show all your work and justify your answers.
%\vspace{2.3in}
%\item[(b)]  Find where $f$ is 
%            concave up and where it is concave down.
%            Does $f(x)$ have any inflection points? If so, find them 
%            all, being sure to show your work.
%\vspace{2.3in}
%\item[(c)]  Sketch the graph of $f(x)$, 
%      incorporating all
%      your information from (a) and (b). 
%\end{enumerate}


%Determine whether or not
%        $\ds f(x)=\frac{1}{\sqrt{\sin{x}}}\;\;$
%         is Riemann integrable on the interval (0,1],
%         and {\bf fully} explain your result (you may
%        want to do an
%         explicit calculation to explain your result).
%\vspace{60ex}
%\item[(10 points) ~~4.] 
%Suppose the velocity $v(t)$ of a mass at the end of a spring
%is given by $v(t)=\cos{t}$. If the position $x(t)$ of the mass and
%its velocity are related by $\ds v(t)=\frac{dx}{dt}$, find
%$x(t)$, given that its initial position is $x(0)=1$.


%A ball is thrown upward from the ground with an initial
%            velocity of 64 $f/s$. Use the fact that the total distance 
%           traveled
%            from $t=a$ to $t=b$ is given by $\ds \int_a^b{|v(t)|\;dt}$,
%            to find how far the ball traveled during the first
%            3 seconds of its journey (hint: $x(t)=-16t^2+64t\;$).




\eject

\item[(15 points)~~ 6.]  
      Calculate $ \;\; \ds\int^1_0 x\;dx \;\;$ from the {\bf definition} of
      the integral $-$ using a Riemann sum. 
     Be sure to show all your work. 
     Hint: use the fact that 
     $$\ds \sum^n_{i = 1} \; i = \frac{n(n + 1)}{2}\;.$$
    Check your result
      using the {\bf fundamental theorem of calculus}.

\vspace{70ex}

\item[(10 points)~~ 7.]  Calculate the following integrals:
\begin{enumerate}
\item[(a)] $\ds \int^2_1 \, (3x^2+x+1) \; dx$
\vspace{20ex}
\item[(b)]  $\ds \int \sin^2{x}\; \cos{x}\; dx$
%\vspace{40ex}
%\item[(c)]  $\ds \int_0^1 \; \frac{u(u+1)^2}{\sqrt{u}} \; du$
%\vspace{30ex}
%\item[(d)]  $\ds \int^{\pi}_{0} \; \sin^2{x} \; dx$
\end{enumerate}
\eject


\end{enumerate}

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