The Newton-Raphson Method

This lesson covers the Newton-Raphson method for finding roots of equations — as required by the Edexcel A-Level Mathematics specification (9MA0). You need to know the formula, understand how it works, recognise when it fails, and compare it with other numerical methods.

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The Formula

The Newton-Raphson method uses the following iteration formula:

x_{n+1} = x_n - f(x_n) / f'(x_n)

Starting from an initial estimate x₀, this generates a sequence x₁, x₂, x₃, ... that (usually) converges rapidly to a root of f(x) = 0.

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How It Works — The Geometric Idea

At each step, the method:

1. Evaluates f(x_n) and f'(x_n) at the current estimate.
2. Draws the tangent line to y = f(x) at the point (x_n, f(x_n)).
3. Finds where this tangent crosses the x-axis — this gives x_{n+1}.

The tangent at (x_n, f(x_n)) has equation: y - f(x_n) = f'(x_n)(x - x_n)

Setting y = 0 (x-axis crossing): -f(x_n) = f'(x_n)(x - x_n) x = x_n - f(x_n)/f'(x_n)

This is the Newton-Raphson formula.