The Mechanics Rule for approximating is where is an approximation of (a) Use Newtons

Chapter 3, Problem 19

(choose chapter or problem)

Mechanic's Rule The Mechanic's Rule for approximating \(\sqrt{a}\), a > 0, is

\(x_{n+1}=\frac{1}{2}\left(x_{n}+\frac{a}{x_{n}}\right), \quad n=1,2,3 \ldots\)

where \(x_{1}\) is an approximation of \(\sqrt{a}\).

(a) Use Newton's Method and the function \(f(x)=x^{2}-a\) to derive the Mechanic's Rule.

(b) Use the Mechanic's Rule to approximate \(\sqrt{5}\) and \(\sqrt{7}\) to three decimal places.

Text Transcription:

sqrt a

x_n+1=1/2(x_n+a/x_n), n=1,2,3…

x_1

f(x)=x^2-a

sqrt 5

sqrt 7