Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2017 / Paper 1 / 19 / a / Solution

The Newton root-finding iteration follows the current tangent line to its horizontal-axis intersection. Since $f^{\prime }(x)=2x$, it givesA positive initial guess is appropriate for the positive square root; negative starting values instead target the negative root.