Locate roots of \(f(x) = 0\) by considering changes of sign of \(f(x)\) in an interval of \(x\) on which \(f(x)\) is sufficiently well behaved.

Students should know that sign change is appropriate for continuous functions in a small interval.

Understand how change of sign methods can fail.

When the interval is too large sign may not change as there may be an even number of roots.

If the function is not continuous, sign may change but there may be an asymptote (not a root).

Locating roots by considering changes of sign

Solve equations approximately using simple iterative methods; be able to draw associated cobweb and staircase diagrams.

Understand that many mathematical problems cannot be solved analytically, but numerical methods permit solution to a required level of accuracy.

Use an iteration of the form \(x_{n+1} = f(x_n)\) to find a root of the equation \(x = f(x)\) and show understanding of the convergence in geometrical terms by drawing cobweb and staircase diagrams.

Iterative methods

Solve equations using the Newton-Raphson method and other recurrence relations of
the form \(x_{n+1} = g(x_n)\)

Understand how such methods can fail.

For the Newton-Raphson method, students should understand its working in geometrical terms, so that they understand its failure near to points where the gradient is small.

The Newton-Raphson method

Understand and use numerical integration of functions, including the use of the trapezium rule and estimating the approximate area under a curve and limits that it must lie between.

For example, evaluate \(\displaystyle\int_0^1{\sqrt{2x+1}}~dx\)

using the values of \(\sqrt{2x+1}\) at \(x = 0\), \(0.25\), \(0.5\), \(0.75\) and \(1\) and use a sketch on a given graph to determine whether the trapezium rule gives an over-estimate or an under-estimate.

The trapezium rule

Use numerical methods to solve problems in context.

Iterations may be suggested for the solution of equations not soluble by analytic means.

Solve problems with numerical methods