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\title{Lab Week 4: Biorhythms}
\author{Names Here}
\date{\today}

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\begin{enumerate}

\item Some preliminary questions:
\begin{enumerate}
\item Assume that $t$ is measured in days since some fixed date in the past
(for example, your birthday).  Write the three sine functions
that would represent the ``Physical'', ``Emotional'', and ``Intellectual'' cycles
for you since you were born.

\bigskip

{\bf Hint:}  The function $A\sin(\omega (t-\delta))$ has amplitude $A$, period $2\pi/\omega$, and
has been shifted forward by $\delta$ units (this is the phase shift).


\item How many days have you been alive?  (Use Maple to compute it)
\item Compute the date it will be when you will have been alive for 20,000 days.
\end{enumerate}

\item Write Maple code that will plot your biorhythms for a two week period
centered around today's date (See the help file for \verb+TodaysDate+).   

For the $x-$axis, label it as "Date", and put some
dates on the axis (Hint:  Use the "tickmarks" example from the Maple sheet).


Don't use too many dates-  Six or so is probably plenty.


\item All three cycles start at zero when you are born.  How long
does it take to return to that initial condition?  How old were you,
or will you be, on that date?  

HINT:  If you had three functions that were periodic with periods 1, 2, and 3,
and they all started at the same point, how long would it take for them all to
return to that state?  Think about ``Least Common Multiple".


\item Is it possible for all three cycles to reach their maximum or
minimum at exactly the same time?

HINT:  This problem is different than the previous one, because the maxima
(or minima) do not START at the same time.  Think about the times each
curve reaches its maximum (or minimum- Same reasoning applies).

\item We should have found that we will never have the ``perfect day'', when
all three cycles are 100.  However, let's define the ``nearly perfect day''
to be that day when we get a maximum of the sum of the three cycles.

Find the date of your nearly perfect day.

HINT:  The maximum is difficult for Maple to find, and cannot be found
exactly.  Therefore, we have to try a numerical method.
 The date is between $t=16900$ and $t=17100$.  See if you can use the 
\verb+Maximize+ function to have Maple give you the value (go to the help file).
You'll need to include:  \verb+with(Optimization):+ for it to work.

\end{enumerate}


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