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\begin{document}


\title{On-line Math 21}


\email{david.johnson@lehigh.edu}


\urladdr{http://www.lehigh.edu/\textasciitilde{}dlj0}

\maketitle

\subsection{Modeling}

Modeling is the way we adapt a {}``real{}'' situation to mathematics. Much
of the time, a particular problem you will have to solve, no matter where it
comes from, has a number of particular details. Some of these details are irrelevancies,
others matter only in a limited way, and still others are fundamental to the
general way in which situations like that behave. The essence of modeling (or
\emph{modelling}) is the translation of that situation to its mathematical essence.
Although certainly {}``reducing{}'' the problem to mathematics loses the particulars
of the problem, which is often what is most interesting, it also separates out
what is relevant or irrelevant to answering the question.

The best example of this is the question of population growth:

Though undoubtedly important to those involved, the details of how a species
procreates does not affect the mathematical underpinning of how the population
changes with time. 

The rate of gestation, the frequency of, shall we say, contact, and the life
expectancy are relevant parameters that go into the particulars of the modeling
of the population of a particular species, but are not central to the model
itself. 

What is central to the model of population is the simple fact that, the more
members of a particular species there are, the more new members can be created,
and the more that die. That central fact about populations is translated to
mathematics by saying that, if \( P(t) \) is the population at time \( t \),
the rate of change of the population at time \( t \), \( P'(t) \), is proportional
to the population itself, or \[
P'(t)=kP(t).\]
That \emph{differential equation} models the dynamics of \emph{any} population
of creatures. One then applies the mathematics of differential equations (we
will be studying this somewhat in this course) to see that, necessarily, the
population has a very specific form, \[
P(t)=A\, e^{kt}.\]
 the {}``raw{}'' calculus that goes into these problems is in that step, starting
with the model and drawing from it the mathematical conclusion. 

Other factors which affect population growth, in general, such as crowding and
limitations of the food supply, can refine the model to improve its accuracy.
But that model of population growth can then be applied to any population problem
and give fairly accurate analysis of a wide variety of situations. That same
model can also be applied in analogous situations, which share the feature of
population problems that their rate of change depends upon the amount present,
such as problems of radioactive decay, cooling, and the spread of epidemics.
\end{document}
