#1.09a
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.
#1.09b
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.
#1.09c
Be able to solve equations approximately using simple iterative methods, and be able to draw associated cobweb and staircase diagrams.
#1.09d
Be able to solve equations using the Newton-Raphson method and other recurrence relations of the form \(x_{n+1} = g(x_n)\).
#1.09e
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.
#1.09f
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]
#1.09g
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.