To measure the temperature in your house, you might place a
thermometer at a variety of positions, and record the temperature.
Although you would not know the temperature at *every* point in
the house (did you crawl under the sink?), you could
probably get a good idea of how the temperature varies from room to
room.

One of the roles of science and mathematics is to *model* and
*predict* natural phenomenon. A scientist will use knowledge
of your house (*e.g.*, the location of heat sources, the amount
of insulation in your walls and attic, the temperature outside) to
build a mathematical model of the heat distribution in your house. We
say that the model is a "good model" if it can predict actual
temperature measurements.

Scientists and engineers use *functions* to model phenomenon
such as temperature. To each point in your house, we can associate a
temperature. Thus a function is a process that takes an input
(*e.g.*, a point in the room) to an output (*e.g.*, the
temperature at that point). We call the set of inputs to a function
the *domain* of that function. The set of all outputs we call
the *range* of the function. In our example, the domain of the
temperature function is a set of points, the range of our function is
the set of temperatures reached by some point in your house.
Sometime visualizing functions becomes easier if we ask the question:
for a given value of the output, what is the subset of the domain that
are associated to that value? In our temperature example, this is the
question "what is the set of points in the house which are exactly 22
degrees Celsius (72 degrees Fahrenheit)?" The set of points which
satisfy this condition are called an *isotherm*. (We see
isotherms daily on weather maps.) We say that the isotherm is defined
*implicitly* since it is the set of points that satisfies a
condition.

Scientists can learn a lot by analyzing functions. For example, if
your oven is on, the temperature in your kitchen changes rapidly over
a short distance between the refrigerator and the center of the
oven. Mathematicians quantify this behavior by saying that the
temperature function is changing rapidly. The change in a function can
be measured by looking at the so-called *derivative function*:
When the temperature is not changing we say it has zero derivative;
when the temperature is changing rapidly, we say that it has a large
derivative.

Comments to: pisces@geom.umn.edu

Last modified: Sun Nov 26 15:57:19 1995

Copyright © 1995 by The Geometry Center, all rights reserved.