The main concept studied in this chapter is that of the steady state, or equilibrium, of a dynamic system. We explore several changing systems and the corresponding mathematical models. In some cases the system approaches equilibrium, in others it does not.

Investigating whether a system approaches equilibrium or not is one of the best motivations for the notions of limit and convergence. Computation of sequences and observation of their behaviour will often make a subsequent rigorous discussion of limits more natural.

Our treatment differs from the usual one in two ways. First, we deal with sequences arising in real problems, not ad hoc exercises. Secondly, we are not restricted to sequences defined by explicit elementary formulas.

Limits and the calculator are discussed in section 6.2. We point out how the very limitations of the calculator lead to a need for a mathematical theory. In section 6.3 we develop a more realistic model of the struggle for life. This leads to a study of quadratic equations and difference equations. In this case, as also in the sections on heat conduction (6.1) and chemical reactions (6.4), the static problem of finding the equilibrium state involves simpler mathematics than the dynamic problem of whether and how the system approaches equilibrium. By using difference equations, we can present many of these problems at an elementary level; for instance, we can discuss some aspects of heat conduction even as early as grades 5-6.

In many phenomena we encounter a resistance to disturbance of the equilibrium. With heat conduction there is a restoring velocity proportional to the deviation from equilibrium.