A Level Maths: Numerical Methods Topic Summary and Resources
Video Lessons
Watch alongside the worksheet for the full lesson experience, then test your understanding with the lesson questions.
Revision Notes
Handwritten notes summarising the key ideas for each lesson. Ideal for quick review before a test.
Exam Questions
Past-paper-style questions organised by topic, with full mark schemes.
- Trapezium Rule (OCR)
- Trapezium Rule (Edexcel)
- Bounds on Integrals Using Rectangles (OCR)
- Fixed Point Iteration (OCR)
- Fixed Point Iteration (Edexcel)
- The Newton-Raphson Method (OCR)
- The Newton-Raphson Method (Edexcel)
Drawn from OCR and Edexcel past papers but designed to be useful for students of all UK exam boards — including AQA and OCR MEI — unless a sheet is explicitly board-specific.
Before You Start This Topic
It will help if you are confident with the following:
- Differentiation — needed for Newton-Raphson
- Integration — needed for understanding what is being approximated
- Concave and Convex Curves and Points of Inflection — needed for over/under estimate analysis
A Level Maths numerical methods is the practical side of calculus, giving you tools for problems that cannot be solved exactly. You locate roots of equations using sign changes, solve equations iteratively using fixed-point iteration and the Newton-Raphson method, and estimate integrals using the trapezium rule. These are the techniques real scientists and engineers use every day.
You locate roots by spotting where $f(x)$ changes sign on a continuous function — if $f(a)$ and $f(b)$ have opposite signs and $f$ is continuous, there is a root between them. You solve equations iteratively using $x_{n+1} = g(x_n)$ (fixed-point iteration), with cobweb and staircase diagrams to visualise convergence. The Newton-Raphson method uses the iteration $x_{n+1} = x_n – \frac{f(x_n)}{f'(x_n)}$, converging quickly near simple roots but failing near stationary points or in chaotic regions. You estimate integrals using the trapezium rule, determining whether the estimate is an over- or under-estimate by sketching. You also use rectangle approximations to bound integrals above and below.
Numerical methods is part of the Pure Maths strand of A Level Maths for AQA, Edexcel, OCR, and OCR MEI students.
Watch out for…
A few things to be careful with: a sign change implies a root only if the function is CONTINUOUS — discontinuities (like asymptotes) can cause spurious sign changes without roots; for the Newton-Raphson method, if $f'(x_n) = 0$ the iteration FAILS — you cannot divide by zero; the trapezium rule UNDERESTIMATES area for convex curves and OVERESTIMATES for concave curves; and iteration formulas only converge if $|g'(\text{root})| < 1$ — questions often ask you to verify this.