Chapter 4.7, Problem 26ES

Many calculators compute reciprocals using the approximation $1/a\approx x_{n+1},$ where $x_{n+1}=x_n\left(2-a{x}_n\right),n=1,2,3,\dots$ and $x_1$ is an initial approximation to $1/a$ . This formula makes it possible to perform divisions using multiplications and subtractions, which is a faster procedure than dividing directly.

(a) Apply Newton’s Method to $f\left(x\right)=\frac{1}{x}-a$ to derive this approximation.

(b) Use the formula to approximate $\frac{1}{17}$ .