\def\wsn{27}
\input worksheet.tex

\item{1.}  The rate at which any radioactive material decays is
directly proportional to the amount of the material present.  That is,
$$
{dS\over dt}=-kS
$$
where $S(t)$ is the amount present at time $t$ and $k$ is a positive
constant.  Suppose $k=1$, there is initially 1 kg of radioactive
material present, and $t$ is measured in days.

\medskip
\itemitem{a)}  Over the first 12 hours, approximate the rate of decay
by a constant (even though the rate is changing).  At that rate, how
much material would be left at the end of 12 hours?

\medskip
\itemitem{b)}  Now over the next 12 hours approximate the rate of
decay by another constant.  At this rate of decay, how much material
would be left at the end of the day?

\medskip
\itemitem{c)}  Now start over at the beginning of the day, dividing it
up into 24 one hour periods.  Starting with hour one, approximate the
rate of decay over that hour and find the amount of material that
would remain at the end of that hour.  Repeat this until you have an
estimate for how much material is left at the end of the day.

\medskip
\itemitem{d)}  If you had divided the day up into minute intervals in
order to improve accuracy, would your new estimate be bigger or
smaller than that in part c? 

\medskip
\itemitem{e)}  Use your calculator to find $1/e$.

\bigskip
\itemitem{2.\hskip12pt a)}  For what values of $b$ is $f(x)=\log_bx$ an
increasing function?

\medskip
\itemitem{b)}  Sketch the functions $\log_ex$ and $\log_{1/e}x$.

\medskip
\itemitem{c)}  For what values of $b$ is $f(x)=b^x$ an increasing
function?

\medskip
\itemitem{d)}  Sketch the functions $e^x$ and $(1/e)^x$.

\bigskip
\item{3.}  As a promotional gimic, your bank offers you the option of
investing a single dollar at 100\% interest for the year, that is,
they will pay you back two dollars at the end of the year.  The bank
down the street offers to compound the interest every month.  (What does
this mean?)  A third bank offers to compund the interest hourly!

\medskip
\itemitem{a)}  How much will be paid out by the second bank at the end
of one year?

\medskip
\itemitem{b)}  How much will be paid out by the third bank at the end
of one year?

\medskip
\itemitem{c)}  What is meant by continuously compounded interest?

\bigskip
\item{4.}  Explain how the constant $e$ might appear in a model of
population growth.

\bigskip
\item{5.}  Find an estimate for the infinite sum
$$
{1\over0!}+{1\over1!}+{1\over2!}+{1\over3!}+{1\over4!}+{1\over5!}
+{1\over6!}+\cdots
$$

\bigskip
\itemitem{6.\hskip12pt a)}  Sketch the graph of a nonzero function
which is its own derivative, that is, its slope is equal to its height
at every point.

\medskip
\itemitem{b)}  Suppose that such a function has $f(0)=1$.  What is the
slope of the tangent line at $x=0$?  What is the height of this tangent
line at $x=1.1$?  Why might this be used as an approximation for
$f(1.1)$?  Repeat this process nine more times to find an estimate for
$f(1)$.  Is this an overestimate or an underestimate?





\bye