Be able to locate the roots of \(f(x) = 0\) by considering changes of sign of \(f(x)\) in an interval of \(x\) in which \(f(x)\) is sufficiently well-behaved.

Finding an interval in which a root lies. This is often used as a preliminary step to find a starting value for the methods in 10.3 and 10.4.

Locating roots by considering changes of sign

Be aware of circumstances under which change of sign methods may fail.

e.g. when the curve of \(y = f(x)\) touches the x-axis.
e.g. when the curve of \(y = f(x)\) has a vertical asymptote.
e.g. there may be several roots in the interval.

Locating roots by considering changes of sign

Be able to carry out a fixed point iteration after rearranging an equation into the form x = g(x) and be able to draw associated staircase and cobweb diagrams.

e.g. write \(x^3 - x - 4 = 0\) as
\(x = \sqrt[3]{x+4}\) and use the iteration
\(x_{n+1} = \sqrt[3]{x_n+4}\) with an appropriate starting value.

Includes use of \(\boxed{ANS}\) key on calculator.

Notation: iteration, iterate

Iterative methods

Be able to use the Newton-Raphson method to find a root of an equation and represent the process on a graph.

The Newton-Raphson method

Understand that not all iterations converge to a particular root of an equation.

Know how Newton-Raphson and fixed point iteration can fail and be able to show this graphically.

Iterative methods The Newton-Raphson method

Be able to find an approximate value of a definite integral using the trapezium rule, and decide whether it is an over- or an under-estimate.

In an interval where the curve is either concave upwards or concave downwards.

Notation: Number of strips

The trapezium rule

Use the sum of a series of rectangles to find an upper and/or lower bound on the area under a curve.

The trapezium rule

Use numerical methods to solve problems in context.

Solve problems with numerical methods