A-Level Maths OCR B (MEI) H640

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.

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.

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*

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

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.

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*

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

Use numerical methods to solve problems in context.