A-Level Maths OCR A H240

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.

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.

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

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

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.*

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*

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.