91Ó°ÊÓ

In the secant method, prove that if \(x_{n} \rightarrow q\) as \(n \rightarrow \infty\), and if \(f^{\prime}(q) \neq 0\), then \(q\) is a zero of \(f\).

If Newton's method is used with \(f(x)=x^{2}-1\) and \(x_{0}=10^{10}\), how many steps are required to obtain the root with accuracy \(10^{-8} ?\) (Solve analytically, not experimentally.)