Mathematics is applied a lot in science, commerce and industry to solve problems. This means that scientific research, the development of new technology, a production process, and so on, pose questions stemming from an existing situation that need te be answered. The answer often comes in the form of a mathematical model.
A mathematical model is a simplification of reality based on reasonable assumptions. In a good model all important factors are still present, only the insignificant ones are left out. Most of the time the model consists of a set of formulas describing the behaviour of the important variables.
Mathematical theories are then let loose on these formulas ...

You can find more about this in the chapter "Problem approach" on this website.

Optimization is about finding solutions using mathematical models to find the maximum or minimum value. Often this is the maximum or minimum of a function. You can find this using differentiation or your graphing calculator.