Practice question · Multiple choice
Newton's method finds a root by repeatedly stepping along the tangent line: x − f(x)/f'(x). Why does following a straight line converge on the root of a curve?
Hints
- The tangent is the linear approximation. What does solving the approximation give you?
- Ask why the second step is better than the first.
Show the answer
A. Because near the root the tangent closely tracks the curve.
Why
Each step replaces the curve by its tangent and solves the easy problem of where a line crosses zero, not the true root, but closer, and the next tangent approximates better still. The number of correct digits roughly doubles per step. It is not unconditional: a nearly horizontal tangent throws the guess far away, and a poor start can cycle or find a different root.
Practise Curve Sketching
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