A mathematical model is a mathematical description (often by means of a function or an
equation) of a real-world phenomenon such as the size of a population, the demand for a
product, the speed of a falling object, the concentration of a product in a chemical reaction,
the life expectancy of a person at birth, or the cost of emission reductions. The purpose
of the model is to understand the phenomenon and perhaps to make predictions about
future behavior.
Given a real-world problem,our first task is to formulate a mathematical model by identifying
and naming the independent and dependent variables and making assumptions that simplify
the phenomenon enough to make it mathematically tractable. We use our knowledge of the
physical situation and our mathematical skills to obtain equations that relate the variables.
In situations where there is no physical law to guide us, we may need to collect data (either from a
library or the Internet or by conducting our own experiments) and examine the data in the
form of a table in order to discern patterns. From this numerical representation of a function
we may wish to obtain a graphical representation by plotting the data. The graph
might even suggest a suitable algebraic formula in some cases.
The second stage is to apply the mathematics that we know (such as the calculus that
will be developed throughout this book) to the mathematical model that we have formulated
in order to derive mathematical conclusions. Then, in the third stage, we take those
mathematical conclusions and interpret them as information about the original real-world
phenomenon by way of offering explanations or making predictions. The final step is to
test our predictions by checking against new real data. If the predictions don’t compare
well with reality, we need to refine our model or to formulate a new model and start the
cycle again.
A mathematical model is never a completely accurate representation of a physical situation—
it is an idealization. A good model simplifies reality enough to permit mathematical
calculations but is accurate enough to provide valuable conclusions. It is important to
realize the limitations of the model. In the end, Mother Nature has the final say.
There are many different types of functions that can be used to model relationships
observed in the real world. In what follows, we discuss the behavior and graphs of these
functions and give examples of situations appropriately modeled by such functions.