Many calculators compute reciprocals using the approximation1/a xn+1, wherexn+1 = xn(2
Chapter 4, Problem 26(choose chapter or problem)
Many calculators compute reciprocals using the approximation \(1 / a \approx x_{n+1}\), where
\(x_{n+1}=x_{n}\left(2-a x_{n}\right), n=1,2,3, \ldots\)
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
\(f(x)=\frac{1}{x}-a\)
to derive this approximation.
(b) Use the formula to approximate \(\frac{1}{17}\).
Equation Transcription:
Text Transcription:
sqrt a approximately equal to x_n+1
x_n+1 = x_n(2-ax_n), n = 1,2,3,...
x_1
1/a
f(x) = 1/x-a
1/17
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