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

{\bf FINAL EXAM}\\
\vspace{2ex}
Thursday, December 17, 2009
\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.~~ & 20 & \\
\hline
2.~~ & 15 & \\
\hline
3.~~ & 10 &\\
\hline
4.~~ & 10 &\\
\hline
5.~~ & 15 &\\
\hline
6.~~ & 15 &\\
\hline
7.~~ & 10 &\\
\hline
8.~~ & 10 &\\
\hline
9.~~ & 10 &\\
\hline
10.~~ & 10 &\\
\hline
11.~~ & 10 &\\
\hline
12.~~ & 15 &\\
\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)] $\ds \lim_{x \rightarrow 2}
       \frac{x^3-8}{x-2}$
\vfill
\item[(b)] $\ds \lim_{x \rightarrow 0} 
                 \frac{\ds \int_0^x{{(\cos{t}\;-1)}\;dt}}{x^3}$
\vfill
\item[(c)] $\ds \lim_{x \rightarrow 0}
       \frac{\sin{(x^2)}}{\pi x}$ 

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

\end{enumerate}
\eject



\item[(15 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. Write the differential equation for the frog population
$P(t)$ with the two corresponding conditions, and find the 
        particular solution which incorporates both conditions.




%\item[(15 points) ~~2.] Find $x(t)$ satisfying 
%         $\ds \frac{d^2x}{dt^2}=2t+1$ and the conditions
%         $x(0)=1$ and $x'(0)=0$.


\vspace{85ex}


\item[(10 points) ~~3.] Use the differential to approximate
    $\sqrt[3]{28}$, showing all your work.


\eject

\item[(10 points) ~~4.] 
Suppose a ball is propelled upward from the ground $x_0=0$
with an initial velocity of 
$v_0=64$ feet/second. Its position $x(t)$ after $t$
seconds is then given by $x(t)=-16t^2+64t$. 
When does the ball reach its maximum height?
What is the maximum height reached by the ball?


\vspace{40ex}


\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.








%A bullet is fired directly upward
%from the ground with an initial velocity of $v_0$ $f/s$,
%so that its height is given by $x(t)=-16t^2 + v_0 t$.
%What must its initial velocity $v_0$ be in order for
%the bullet to reach a maximum height of 1 mile (5,280 feet)? 


\eject



\item[(15 points)~~ 6.]  Find the temperature profile 
$T(x)$ on the interval $[0,2]$ satisfying the 
steady state heat equation $$\ds \frac{d^2 T}{dx^2}=0$$
and the boundary conditions $T(0)=-10$ and $T(2)=-2$.

\vspace{70ex}



\item[(10 points)~~ 7.]  
For a given spring, the force required to keep it stretched
$x$ feet is given by $F=9x$ pounds. How much work in 
ft$\cdot$lb 
is done
in stretching the spring 1 foot from its unstretched equilibrium state?
Be sure to show all your work.


\eject

\item[(10 points)~~ 8.]  
Consider the region %$\mathcal{R}$ 
bounded by the 
graph of $y(x)=\sin{x}$ and the $x-$axis 
between $x=0$ and $x=2\pi$.
Revolve this region around the $x-$axis, and find the 
volume of the resulting solid.

\vspace{60ex}


%(b) Suppose there is a {\it positive} electrical charge at the 
%origin $r=0$ and an equal
%but opposite {\it negative} electrical charge 1 cm away at $r=1$.
%If the Coulomb force of attraction on the negative charge is given by
%$$
%F_{coul}(r)=-\frac{40}{r^2} \;\;\; \mbox{dynes},
%$$
%find how much work in dyne$\cdot$cm you must do to move the negative charge
%from 1 cm away at $r=1$ to 5 cm away at $r=5$.


%Give a sophisticated graph of 
%     $\ds f(x) = \frac{1}{1+x^2}\;$.  
%     Indicate if $f$ has any symmetry, and find 
%     where $f$ is increasing, 
%     decreasing, concave up and concave down. Identify all 
%     local maxima, local minima, and inflection points. 
%     Does $f$ have a global maximum? Does $f$ have a global
%     mimimum?





\item[(10 points)~~9.] Find the area of the finite region bounded
   by the graphs of $y=-x+2$ and $y=x^2$. Sketch the region.



\vspace{55ex}

\item[(10 points)~~10.] Find the volume of the solid generated
   by revolving about the $x-$axis the region bounded by the
   $x-$axis and the upper half of the ellipse  
   $$ \ds \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1. $$
   

\vspace{65ex}


\item[(10 points)~~11.] Consider a circle of radius 5 centered at the origin,
parameterized by $(x(t),y(t))=(5\cos{t},5\sin{t})$.
Find the circumference of this circle by calculating the arc length 
of the curve parameterized by $(x(t),y(t))$ with $t \in [0,2\pi]$.


\eject

\item[(15 points)~~12.] Use the Theorem of Pappus to find 
the volume of the doughnut, 
or {\it torus}, obtained when the region bounded by the circle
of radius 1, centered at the point $(3,0)$, is revolved around
the $y-$axis. The equation of the circle is $(x-3)^2+y^2=1$. 
(Hint: where is the centroid of the circular region?)



\end{enumerate}

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