Past exam of the year 1 of the computer science course of the University of Oxford / 2014 / Discrete mathematics, continuous mathematics and probability / 5 / b / i / Solution

For $f(x)=x^2-a$, its derivative is $f^{\prime }(x)=2x$. Substitution into Newton's method givesEvery positive input gives another positive input, so a starting value $x_0>0$ keeps the denominator nonzero at every step.