\def\wsn{20}
\input worksheet.tex

\item{1.}  Suppose for some function $h(x)$, we know that $h(0)=2$ and
that $h'(x)=(1+x^3)^{1/3}$.  Draw the graph of $h(x)$.

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\item{2.}  The mathematicians at Los Alamos Laboratory developed the
following equation to describe the change over time in the number of
people infected with HIV.
$$
{dI\over dt}=\alpha I(t)\left[1-{I(t)\over N}\right]
$$
where
$$
\eqalign{
I(t)=&\hbox{ number of people inffected at time $t$}\cr
N=&\hbox{ size of the population}\cr
\alpha=&\hbox{ rate at which an infected person passes on the virus
per unit time.}\cr}
$$
Assume $t$ is measured in days, $N=100,000$ and $\alpha=0.01$.

\medskip
\itemitem{a)}  Is the number of infected people increasing or decreasing?

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\itemitem{b)}  How many people have to be infected for the number of
people infected to stop increasing?

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\itemitem{c)}  How many people are being infected per day when there
are 100 people infected?

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\itemitem{d)}  Find $\displaystyle{d^2I\over dt^2}$ using implicit
diffferentiation then write it as a function of $I(t)$.

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\itemitem{e)}  Is the number of infected people accelerating, or
increasing at a slower rate and slower rate?

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\itemitem{f)}  How many people have to be infected for the number of
people infected to stop accelerating?

\bigskip
\item{3.}  {\bf Snell's Law}  Fermat's Principle in optics says that
light travels from point A to point B along the path that requires the
least amount of time.  Suppose that light travels in one medium at
velocity $c_1$ and in a second medium at velocity $c_2$.  If A is in
medium 1 and B is in medium 2, and the $x$-axis separates the two
media, as in the figure below, show that
$\displaystyle{{\sin\theta_1\over c_1}={\sin\theta_2\over
c_2}}{\atop.}$

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\centerline{\epsfxsize=3in\epsfbox{/home/oehrtman/m210/Snells_Law.eps}}

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\item{4.}  Suppose you know that
$\displaystyle{f(x)={2\over1+x^2}-{3\over4+x^2}{\atop.}}$  Is
$f(0)&0\quad\hbox{for }x>0,\cr
\hbox{and }f''(x)<&0\quad\hbox{for }x<0.\cr}
$$
The number of critical points, number of local maxima, number of local
minima, and number of roots of $f$ are all tabulated.  Give all
possible such tabulations.



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