Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2024 / Paper 2 / 5

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- Solution 5

Solution

The Newton root-finding iteration replaces the graph by its tangent line at

x_{0}

and uses that line's zero:

0 = f (x_{0}) + f^{'} (x_{0}) (x_{1} - x_{0}), x_{1} = x_{0} - \frac{f ( x _{0} )}{f ^{'} ( x _{0} )} .

Comparing with the printed reciprocal-ratio form gives

α = - 1, β = - 1 .

This step requires

f^{'} (x_{0}) \neq = 0

. If

f (x_{0}) = 0

, the iteration has already reached a zero of a function and the printed reciprocal-ratio expression need not be evaluated.

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