Be able to 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.

Includes verifying the level of accuracy of an approximation by considering upper and lower bounds.

Locating roots by considering changes of sign

Understand how change of sign methods can fail.

e.g. when the curve \(y = f(x)\) touches the x-axis or has a vertical asymptote.

Locating roots by considering changes of sign

Be able to solve equations approximately using simple iterative methods, and be able to draw associated cobweb and staircase diagrams.

Iterative methods

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

The Newton-Raphson method

Understand and be able to show how such methods can fail.

In particular, learners should know that:
1. the iteration \(x_{n+1} = g(x_n)\) converges to a root at \(x = a\) if \(|g'(a)| < 1\), and if \(x_1\) is sufficiently close to \(a\);
2. the Newton-Raphson method will fail if the initial value coincides with a stationary point.

The Newton-Raphson method

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

Learners will be expected to use the trapezium rule to estimate the area under a curve and to determine whether the trapezium rule gives an under- or overestimate of the area under a curve.

Learners will also be expected to use rectangles to estimate the area under a curve and to establish upper and lower bounds for a given integral. See also 1.08g.

[Simpson’s rule is excluded]

The trapezium rule

Be able to use numerical methods to solve problems in context.

i.e. for solving problems in context which lead to equations which learners cannot solve analytically.

Solve problems with numerical methods