Find the square root of 612 with an error less than 10-4 by newton-rapson method.
We have to find x such that
x2=612
The derivative of
f(x) = x2-612
is f’(x) = 2x
With an initial guess of x0=10, the sequence given by the method is
X1= x0 -(f(x0)/f'(x0)) = 10-(102-612)/(2*10)=35.6
x2 = x1 - (f(x1)/f'(x1)) = 26.3955056
Similarly,
x3 = 24.7906355
x4 = 24.7386883
x5 = 24.7386338
Where the correct digits are underlined.
f(24.7386338) = (24.7386338)2-612 = 0.0000022905 <10-4
The required square root is 24.7386338.