Pure 2 numerical methods
Find approximate solutions and explain why you can trust them.
Eight lessons take you from locating roots to fixed-point iteration, Newton–Raphson, accuracy checks and modelling. Explore why methods converge or fail, then use them to locate stationary and inflection points.
- Locating roots by a change of signLocate roots using continuity and a change of sign. Learn what endpoint values prove, why repeated roots and asymptotes need care, and how to establish uniqueness. Includes interactive graphs and original worked practice.
- Root accuracy and error boundsProve a numerical root correct to decimal places using rounding boundaries and sign brackets. Explore bisection, midpoint error bounds, absolute and percentage error, and why rounded iterates or small residuals are not proofs.
- Fixed-point iterationLearn fixed-point iteration xₙ₊₁ = g(xₙ), rearrange equations, calculate labelled iterates and retain working precision. Explore real cube roots, square-root branches and logarithm domains with original practice.
- Convergence and divergence of iterationUnderstand convergent, divergent and oscillating fixed-point iterations using learner-controlled staircase and cobweb diagrams. Compare rearrangements, starting values and the local derivative test, with original worked practice.
- Newton–Raphson methodDerive and use the Newton–Raphson formula with learner-controlled tangent steps. Work through polynomial, exponential, logarithmic and trigonometric equations, retain working precision and verify root accuracy.
- Why Newton–Raphson can failUnderstand Newton–Raphson failure: zero derivatives, domain escape, two-cycles, divergence and unsuitable starts. Explore contrasting cases, explain tangent behaviour and recognise when an exact root has already been found.
- Numerical methods in modellingApply numerical methods to original volume and temperature models. Form a target equation, choose a physical interval, compare Newton and fixed-point estimates, verify rounding and interpret answers with units and model limitations.
- Finding stationary and inflection points numericallyUse numerical methods on first and second derivatives to find stationary and inflection points. Choose the correct Newton denominator, restore original coordinates, verify accuracy and classify with sign changes.