#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.