Numerical methods (Newton-Raphson, trapezium)
Newton-Raphson: xn+1 = xn − f(xn)/f'(xn). Trapezium rule for definite integrals.
Y13. When equations cannot be solved algebraically, iterate. Trapezium rule estimates a definite integral with straight-line tops on strips.
Worked examples
Newton-Raphson: solve x³ − x − 1 = 0 starting x0 = 1.5: x1 = 1.348..., converges to ~1.3247.
Trapezium rule: ∫ab f dx ≈ h/2 [y0 + 2(y1+...+yn-1) + yn] where h = (b-a)/n.
Improvement: Simpson's rule (Y13 in some boards).
Frequently asked questions
Why does Newton-Raphson work?
It uses the tangent at xn to predict the root. Quadratic convergence near a root.
When does it fail?
When f'(x) is small (tangent nearly horizontal) or near a stationary point.