Newton–Raphson method
Objective. Use tangent iteration and recognise failure cases.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
The Newton–Raphson method approximates a root of f(x) = 0 using tangent lines, with the iteration xₙ₊₁ = xₙ − f(xₙ)/f′(xₙ), starting from an estimate x₀ close to the root. Each step follows the tangent at xₙ down to where it crosses the x-axis. For f(x) = x² − 2, f′(x) = 2x, so starting at x₀ = 1 gives x₁ = 1 − (1−2)/2 = 1.5, closer to √2. The method can fail if f′(xₙ) is zero or if x₀ is chosen near a turning point, causing divergence.
Worked example
Use one iteration of the Newton–Raphson method with f(x) = x³ − 5 and x₀ = 2 to find x₁.
- 1Differentiate: f′(x) = 3x²
- 2Evaluate f(2) = 8 − 5 = 3 and f′(2) = 12
- 3Apply x₁ = x₀ − f(x₀)/f′(x₀) = 2 − 3/12
- 4Simplify to x₁ = 1.75
Answer: x₁ = 1.75
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Practice
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Start practisingCommon mistake. Students often forget to recompute f′(xₙ) at each new xₙ, reusing the original derivative value throughout. Always evaluate both f and f′ freshly at the current approximation before each iteration.
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