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Numerical methods: A Level Maths knowledge organiser

Everything to know about numerical methods on one page: key definitions, the formulas, a worked example, the mistakes to avoid and a checklist of what you should be able to do.

Download the A4 sheet (PDF)

Key definitions

Change of sign
If a continuous f(x) changes sign between a and b, there is a root between them.
Iteration
Using xₙ₊₁ = g(xₙ) again and again to home in on a root.
Newton–Raphson
xₙ₊₁ = xₙ − f(xₙ)/f′(xₙ); it fails if f′(xₙ) is zero or close to it.
Trapezium rule
Estimates an area with trapezia; it overestimates for a convex curve.

Key formulas

Formulas marked with a label are given in the exam (we only say so where our formula sheet confirms it). Learn the rest.

Trapezium rule, \(h=\tfrac{b-a}n\)In the formula booklet: Edexcel, AQA, OCR\(\int_a^by\,dx\) \({}\approx\tfrac12h\big\{(y_0+y_n)\) \({}+2(y_1+\cdots+y_{n-1})\big\}\)
Newton–RaphsonIn the formula booklet: Edexcel, AQA, OCR\(x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}\)
Sign change: continuous \(f\) with \(f(a)f(b)<0\) has a root in \((a,b)\). Iteration \(x_{n+1}=g(x_n)\): staircase/cobweb

Worked example

f(x) = x³ − 2x − 5. Starting at x₀ = 2, use Newton–Raphson twice. Give x₂ to 4 decimal places.

  1. f′(x) = 3x² − 2; f(2) = −1 and f′(2) = 10, so x₁ = 2 + 0.1 = 2.1
  2. f(2.1) = 0.061 and f′(2.1) = 11.23, so x₂ = 2.1 − 0.061/11.23

Answer: x₁ = 2.1 and x₂ = 2.0946 (4 d.p.)

Common mistakes

  • Change of sign: incomplete conclusion
  • Change of sign across an asymptote
  • Iterating with rounded values
  • Newton–Raphson failing at a stationary point not explained

More on what examiners see students get wrong: Examiner Insights.

You should be able to…

  • Locate roots of f(x) = 0 by finding a change of sign over an interval, and explain when this method can fail, such as at asymptotes.
  • Use iterations of the form xₙ₊₁ = g(xₙ) to approximate roots, and draw staircase and cobweb diagrams to show whether an iteration converges.
  • Apply the Newton–Raphson method xₙ₊₁ = xₙ − f(xₙ)/f′(xₙ) to find roots, and explain why it can fail near a stationary point.
  • Use change of sign, iteration and Newton–Raphson to solve equations that arise from models, judging the accuracy of the answers in context.
  • Use the trapezium rule to estimate the value of a definite integral, and decide whether the estimate is an overestimate or underestimate from the curve's shape.

The printable sheet

Numerical methods knowledge organiser for A Level Maths: one A4 page of key definitions, formulas, a worked example and common mistakes
Numerical methods knowledge organiser (A Level Maths), A4. Download the PDF.

Revise it next

Other A Level Maths topics: Proof · Algebra and functions · Coordinate geometry · Sequences and series · Trigonometry · Exponentials and logarithms · Differentiation · Integration · Vectors · Sampling and data · Probability · Binomial and normal distributions · Hypothesis testing · Kinematics · Forces and Newton's laws · Moments · All A Level Maths organisers