Mathematical Modeling

Mathematical modeling is commonly used to predict the

behavior of phenomena in the environment. Basically,

it involves analyzing a set of points from given data

by plotting them, finding a line of "best fit" through

these points, and then using the resulting graph to

evaluate any given point. Models are useful in

hypothesizing the future behavior of populations,

investments, businesses, and many other things that

are characterized by fluctuations. A mathematical

model usually describes a system by a set of variables

and equations which form the basis of the

relationships between the variables.

The variables represent independent and dependent

properties of the system. Models are classified in a

variety of ways. One of these ways is "linear versus

nonlinear." A linear model is any system whose

behavior can be explained or predicted using a linear

equation or an entire set of linear equations. On the

other hand, a nonlinear model uses at least one

nonlinear equation to describe its behavior. Models

may also be classed as either deterministic or

probabilistic. A deterministic model always performs

the same way under a given set of initially occurring

conditions, while a probabilistic model is

characterized by randomness. Another way of evaluating

models is to determine whether it is static or

dynamic. Static models do not account for time, while

dynamic models do take this element into

consideration. In calculus, dynamic models are often

represented using differential equations. Finally,

models can have lumped parameters or distributed

parameters. A model with lumped parameters has a

consistent state throughout the system and are said to

be homogenous. A model with distributed parameters has

a changing state throughout the system. It is said to

be heterogeneous.

Another way of understanding models is to figure out

if the given model is a "white box" or a "black box"

model. White box models are constructed with all

necessary information on hand. Black box models are

constructed with pieces missing. There is no "a

priori" information available. White box models are

easier for the average pre-calculus student since one

is allowed to work with all of the needed variable

relationships: "if you have used the information

correctly, then the model will behave correctly" in a

white box model (Wikipedia). Many models have some

white box, and some black box characteristics. When

one is given a "black box" problem, estimation of

relationships between variables is used. Black box

models are, therefore, much less accurate. As the

number of parameters for the model increases, so does

the difficulty level of solving the problem.

When creating three-dimensional models which can be

used to visually explain concepts or behavior,

engineers