Scientists, engineers and economists, working on a wide variety of problems, nowadays find it useful to set up mathematical models of the systems which they are investigating. To do this, they give a simplified description of the problem which allows equations to be set up and then solved to make predictions.

The following sample of the problems treated in this book shows the wide-ranging areas in which the modelling process is currently being applied.

• Mechanics. We model the tension in a rope passing over pulleys and then use the model to explain how it is possible for a worker pulling on a rope to raise a heavy load many times his own weight.

• Genetics. In a genetics problem we describe a model which allows us to predict how many generations it takes for a recessive gene resulting in defective offspring to effectively disappear.

• Thermal physics. By modelling the loss of heat through the insulation surrounding a hot water pipe, we derive the paradoxical result that, in certain circumstances, it is possible for more heat to be lost from the pipe with insulation than without it.

• Engineering. We study the phenomenon of resonance, which was responsible for a famous bridge collapsing due only to wind fluctuations.

• Fisheries management. We study a model of interacting populations of certain types offish and sharks and use it to explain how increasing the amount of fishing can actually increase the fish population.