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Math 1210\\
Fall 2004\\
K. M. Golden
\vspace{5ex}

{\bf EXAM III}\\
Friday, November 19, 2004
\vspace {.5in}

\begin{tabular}{|l|l|l|l|l|l|l|}
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%\multicolumn{2}{|c|}{Exam Scores}\\
\hline
Problem &Points & Score~~~~~~\\
\hline
1.~~ & 20 & \\
\hline
2.~~ & 20 & \\
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3.~~ & 10 &\\
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4.~~ & 10 &\\
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5.~~ & 20 &\\
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6.~~ & 20 &\\
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&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)] $\ds \lim_{x \rightarrow 0} 
                 \frac{\sin{x}-x}{x^3}$
\vspace{40ex}
\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}
\eject

\item[(20 points) ~~2.] 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{40ex}
\item[(b)] Solve this differential equation, that is, find the 
        particular solution which incorporates both conditions.
\vspace{40ex}
\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[(10 points) ~~3.] 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[(20 points)~~ 5.]  Using the fact that 
     $\ds \sum^n_{i = 1} \; i^2 = \frac{n(n + 1)(2n+1)}{6}\;$, calculate 
     $\ds\int^2_0
      x^2\;dx$ directly from the {\bf definition} of
      the integral, i.e., using Riemann sums. {\it Check} your result
      using the {\bf fundamental theorem of calculus}.
\eject
\item[(20 points)~~ 6.]  Calculate the following integrals:
\begin{enumerate}
\item[(a)] $\ds \int^3_1 \, (4x^3-2x^2+x-5) \; dx$
\vspace{30ex}
\item[(b)]  $\ds \int \tan^2{x}\; \sec^2{x}\; dx$
\vspace{30ex}
\item[(c)]  $\ds \int^1_{0} \; \frac{u(u+1)^2}{\sqrt{u}} \; du$
\vspace{30ex}
\item[(d)]  $\ds \int^{\pi}_{0} \; \sin^2{x} \; dx$
\end{enumerate}
\eject


\end{enumerate}

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