
Mathematical Modeling II
with Calculus BC
Mathematical Modeling
One of the primary uses of mathematics is in making
predictions or inferences about future or past events based upon data collected
from natural processes or experiments. In order to do this effectively, the
mathematician or scientist must determine a mathematical "model" which does a good
job of simulating the results of these processes or experiments. Such a model
may take the form of an equation or an inequality, a system of equations or
inequalities, geometric patterns, or
computer programs that emulate numerical or geometric patterns in the data, or any
combination of these. We often
say that we want our mathematical models to "capture" the trend of the data so
that we may be more confident in predictions made from the models. In truth, we
rarely, if ever, obtain a mathematical model which perfectly captures the data we
are studying. In fact, a mathematical model should not be viewed as a fixed final
construction, but as a work in progress, subject to evaluation and revision if it
fails to account for new results in the system being studied.
The two major types of mathematical models are discrete models and continuous
models. The MMII/BC course at Polytechnic School gives almost equal weight to
each type. The first half of the course concentrates on discrete dynamical systems,
which in many cases may be modeled using difference equations, the discrete
counterpart of the differential equations studied in calculus. The second half of the
course is devoted to the continuous models of the calculus. Until the last third
of the twentieth century, discrete mathematics remained in the shadow of the calculus
of Isaac Newton and Gottried Leibniz, which is the foundation upon
which modern physics is built. The advent of the digital computer in the 1950s and
60s ushered in a new era in mathematical modeling using discrete mathematics.
Most of the discoveries in chaos theory and
fractal geometry would have remained beyond the reach of humans without the
assistance of computers to simulate discrete dynamical systems.
Throughout the MMII/BC course, we will stress the interplay between discrete
and continuous models, noting that often the choice between using a discrete
model and a continuous model is a matter of convenience. Nevertheless, our
study of the continuous logistic differential equation and its discrete
counterpart, the logistic difference equation, will warn us that sometimes
the model we choose can significantly effect what we see in the system we are studying.
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