Newton's Method--using it on exponentials?

Question

Ok, so now i get how to do that whole 'Xn+1=Xn-f(x)/f'(x)' part of Newton's Method...but Im having trouble with finding a 'general rule for approximating the radical' when given the other info...

x=sqrt(a) [Hint: f(x)= x^2 - a ]
-------------I came up with (Xn^2+a)/(2Xn)

but now im not sure how to use this new function to approximate the sqrt(7) to three decimal places...help anyone?

Im not quite sure that my function for x=sqrt(a) might be right either... =(

Answer 1

f(x) = x² - a
f'(x) = 2x

Xn+1 = Xn - f(Xn)/f'(Xn)
= Xn - (Xn² - a)/2Xn
= (2Xn² - Xn² + a)/2Xn
= (Xn² + a)/2Xn
= ½(Xn + a/Xn) which is the average of Xn and a divided by Xn

When a = 7 Then Xn+1 = ½(Xn + 7/Xn)
Take X1 = 3
Then X2 = ½(3 + 7/3)
= ½(5.33333333.....)
= 2.66666666........
X3 = ½(2.6666666...+ 7/2.666666666......)
= 2.645833333.....
X4 = ½(2.645833333..... + 7/2.645833333.....)
= 2.64575 ....
= 2.646 (to 3 decimal places)

Answer 2

It should be Xn+1 = Xn - (Xn^2 - 7) / (2Xn)

Then just iterate this until Xn doesn't change in three decimal places.