OCR B (MEI) H640

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#10.1

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.

#10.2

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.

#10.3

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

#10.4

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

#10.5

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.

#10.6

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

#10.7

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

#10.8

Use numerical methods to solve problems in context.