- Familiarity with functions and their graphs.
- Familiarity with solving equations.
- Familiarity with differentiation, and the geometric meaning of the derivative as the gradient of tangent to a curve.
Although this may seem a paradox, all exact science is dominated by the idea of approximation. When a man tells you that he knows the exact truth about anything, you are safe in inferring that he is an inexact man.
— Bertrand Russell
Sometimes we can find nice, or simple, or exact solutions to equations. Linear equations and quadratic equations, for example, can be solved exactly. But other types of equations can be much more difficult to solve exactly. In fact, sometimes it can be impossible to write down an exact expression for a solution.
Exercises in school mathematics textbooks are often deliberately designed to give nice answers. But in solving many mathematical equations deriving from real-world situations, there is no reason to expect the answer to be particularly nice. Often, the best that we can hope for is an approximate solution, to a desired degree of accuracy.
When equations are difficult to solve, we can resort to approximate numerical methods to find a solution. It is sometimes more efficient to find an approximate answer.
In this module we will examine two of the most common and useful numerical methods for finding approximate solutions to equations: the bisection method, and Newton's method. These methods are quite interesting in their own right, and lead to some beautiful pictures.