Posted by Elizabeth on Friday, July 20, 2012 at 10:56pm.
Use newtons method and the function f(x)=x^2a to derive the mechanics rule.
Mechanics Rule for approaching sqrt(a), a>0, is
Xn+1=1/2(Xn+a/Xn), n=1,2,3,...

Calculus  Count Iblis, Friday, July 20, 2012 at 11:16pm
Newton's method for the root of f(x) is:
x_{n+1} = x_{n}  f(x_{n})/f'(x_{n})
= x_{n}  (x_{n}^2  a)/(2 x_{n}) =
1/2 (x_{n} + a/(x_{n}))
This rule is not so convenient, because you need to divide by x_{n} at each step (division is slower than multiplication). A rule without division can be obtained by writing the square root as the root of the function:
f(x) = x^(2)  1/a
Newton's method then yields:
x_{n+1} = x_{n} 
( x_{n}^(2)  1/a)/(2x_{n}^(3)) =
1/2 x_{n} (3  x_{n}^2/a)
So, in each step you now only have a division by a, so you only have to compute 1/a once.
Answer This Question
Related Questions
 Math (Calculus)  Hello, Could somebody please help me with the following ...
 Calculus  Am I right so far? G(x)=e^x sqrt(1+x^2) =e^x(1+x^2)^1/2 =e^x(1/2)(1+x...
 Calculus  We just learned derivatives and now on the Chain Rule. y = r/[(sqrt)(...
 Calculus  Can anyone explain how to use the chain rule and power rule together ...
 Calculus  So, I have a homework question on The Quotient Rule (taking ...
 precal  Find the roots of the function f(x)= x^2+2x+2 Determine f(x) a) (x+1i...
 Calculus  Match the rule with the title: ____ 3. d/dx [f(x)/g(x) ]=(g(x) f^' (x...
 limit calc question  2 questions!!! 1. Limit X approaching A (X^1/3a^1/3)/ xa...
 Calculus  Use the General Power Rule or the Shifting and Scaling Rule to find ...
 calculus  Find the derivative of the function below in two ways. F(x) = (x  7 ...
More Related Questions