## OCR A H240

### Are you studying this syllabus?

#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.