This module can be used twice. It provides an excellent example when you first study Euler's method. The simplest version of Newton's Model of Cooling

in which the ambient temperature is constant is an easy example to do "by hand." Then they can continue using either Maple or Mathematica to investigate the more difficult situation

obtained by replacing the constant ambient temperature A in the first model by a fluctuating ambient temperature.

We want to examine the effects of two things.

• The size and insulation of the object being studied. This is represented mathematically by the constant k. If this constant is large than heat is transfered to and from the object very quickly. This is what happens if the object is small and poorly insulated. If this constant is small then the heat transfer is slow. This is what happens if the object is large and well insulated.

• The frequency with which the ambient temperature oscillates. This is represented by the greek letter omega .

Later in the course after using symbolic or algebraic methods to study this same differential equation you can return to this module to look at more general models of the form

where the function A(t) describing the ambient temperature is more complicated. This makes two nice points--first, numerical methods can be used even when algebraic methods fail and, second, the effects that we observed when we varied the constants k and omega were not consequences of algebraic manipulations and trigonometric identities but more fundamental physical realities.