%
% Calculus II Project III
%
% Inspired by the Drug Dosage Project from
% Calculus: An Active Approach with Projects
%   and
% The Strange Case of Dr. Jekyll and Mr. Hyde
%   by Robert Louis Stevenson
%

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{\large\bf Project Based Math 112, Fall 2001\\
Project \#3 --- The Sinister Serum of Dr.~Jekyll\\
Due Last Class}\\[.25cm]
\large Part 1\\
\underline{\hspace{4.5in}}\\[.25cm]
\end{center}

\begin{flushright}
Mr.~Edward Hyde \\
London, England\\
14 November, 1885
\end{flushright}

\begin{flushleft}
\noindent Math 112 Students\\
Cornell University\\
Ithaca, NY 14853\\
\end{flushleft}

\noindent My Learned Associates, \bigskip

You are all aware of my regrettable situation; that through a series of
experiments, Dr.~Henry Jekyll fragmented his personality, refining himself
and creating me; that control has switched unpredictably between us and we
have lived two separate lives in a shared body, to what extent that is
possible.  Knowing that our well-being is inexorably unified, we have
formed an uneasy truce.  That has ended.  My alter ego has developed a
stabilizing serum which would, when properly administered, prevent future
personality changes.  He means to suppress me forever!

I became aware that he was working towards such a serum from his research
journal, which he leaves unhidden---with a tantalizing outline of his
work, but just enough mathematical detail missing that I cannot
re-construct the process.  He completed the serum today.  I would have
been lost but for a miraculous turn of fortune: as he was preparing to
inject his first dose, control was exchanged.  I have been granted an
opportunity to save myself from oblivion, but my only hope is to master
his invention.  If I am to survive I must learn to use the stabilizing
serum; I must seize control of this body.  There is more at stake than my
existence.  The brilliant Dr.~Jekyll is increasingly mad.  I have
considered taking my life to bring his demented and profane experiments to
an end, but I have been too weak, too selfish.  Soon, such drastic
measures will either not be required, or not be possible.

The good doctor's notes should permit me to reproduce the serum without
trouble, but the concentration of serum in my blood must remain within
certain bounds for it to be both safe and effective.  I am personally
unable to determine an appropriate dose size and frequency.  That is why I
have turned to you.  Your mathematical abilities far exceed my own; you
are doubtlessly equal to the challenge.  I have enclosed a copy of the
journal, I can only hope that no further information is needed.

I desperately await your response.  I will pass the next several days in a
room secured by locks with combinations known only to me; it is a crude
obstruction and will buy me but a little time.  This is my final battle
with an adversary that I have come to respect and fear and who, until now,
I believed that I could never escape.  With your help, I may yet prevail.

\bigskip \hfill\parbox{2in}{ Yours sincerely,\\[1.2cm]
Mr.~Edward Hyde}

\newpage

\begin{center}
\large Part 2\\
One month later\\
\underline{\hspace{4.5in}}\\[.25cm]
\end{center}

\begin{flushright}
Mr.~Edward Hyde \\
London, England\\
14 December, 1885
\end{flushright}

\begin{flushleft}
\noindent Math 112 Students\\
Cornell University\\
Ithaca, NY 14853\\
\end{flushleft}

\noindent My Studious Benefactors, \bigskip

I am unspeakably grateful for the help that you have given me.  Your
calculational efforts have allowed me to suppress the diabolical
Dr.~Jekyll for an entire month, but it has taken a toll.  The frequent
need to intravenously administer the stabilizing serum has left my arms
covered in needle tracks that would impress a heroin addict and has
prevented me from sleeping through the night---indeed, I live in constant
fear of missing a nocturnal dose and disappearing forever into the abyss
of Jekyll's demented mind.

A better solution may exist.  I have discovered a new portion of my alter
ego's research journals; he was planning to produce a form of the serum to
be taken orally.  His notes are complete enough that I have managed to
manufacture the pill.  Dr.~Jekyll determined that the ingested serum would
have an important advantage over the injected form; it is introduced into
the bloodstream gradually, allowing larger doses to have prolonged
effectiveness without causing an overdose.  He tantalizingly scribbled in
his notebook that the new serum needs only to be ingested three times per
day, a vast improvement over my current situation.

Once again, all necessary information seems to be provided in the doctor's
journal and, once again, my ignorance of mathematics prevents me from
calculating an appropriate dosage.  Therefore I must, once again, rely on
your skills and charity.  I have enclosed a copy of this section of his
journal.

My continued existence is in your capable hands.

\bigskip \hfill\parbox{2in}{ Yours sincerely,\\[1.2cm]
Mr.~Edward Hyde}

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