A-Level Maths: Numerical Methods

Numerical methods is the A2 topic where understanding failure matters as much as executing success: every method on the specification comes with named failure cases, and exam questions probe both sides. This quiz does the same.

The sign-change questions establish the foundation: locating a root of x³ − 5x + 2 by evaluating endpoints and finding the interval with genuinely opposite signs (a question rebuilt during verification so exactly one listed interval qualifies), why opposite signs guarantee a root for continuous functions, and — the flip side — the three standard ways the method fails: discontinuities that fake a sign change, an even number of roots hiding between same-sign endpoints, and repeated roots that touch the axis without crossing. All three failure modes are keyed, a correction made during verification when the even-roots case proved just as valid as the others.

Newton-Raphson gets computational and conceptual treatment: applying the formula to x³ − 3x + 1 from x₀ = 0 (where dividing by the negative derivative flips the sign — the key itself was corrected during verification after independent recomputation showed x₁ = +1/3), building the symbolic iterate for eˣ − 3 from x₀ = 1, and the failure-case question covering zero derivatives, poor starting points near stationary values, and oscillation.

Fixed-point iteration is tested through convergence: the |g'(x)| < 1 criterion, cobweb behaviour when the derivative is negative (for the iteration √(5 − x)), and a direct convergence verdict on the linear map 0.5x + 2. The trapezium rule completes the set with its defining features: linear interpolation between ordinates and error shrinking as strips increase.

Every explanation states both the computation and the criterion at work, which is precisely the pairing that "explain why the method fails here" questions demand.

• Locate roots by sign change and identify when the method is guaranteed to work
• Name the failure modes: discontinuities, paired roots and repeated roots
• Apply the Newton-Raphson formula numerically and symbolically
• Judge fixed-point iteration convergence via |g'(x)| < 1, including cobweb behaviour
• Describe the trapezium rule's construction and error behaviour

Topic scope follows section I (Numerical methods, A level only) of the DfE's prescribed AS and A level mathematics subject content — 100% common across all exam boards: sign-change root location, fixed-point iteration, the Newton-Raphson method with failure cases, and the trapezium rule.