Posted by kwack on Tuesday, September 7, 2010 at 3:45am.
Use Newton's method to approximate a root of the equation 3sin(x)=x as follows.
Let x1=1 be the initial approximation.
The second approximation x2 is
and the third approximation x3 is

calculus  drwls, Tuesday, September 7, 2010 at 6:09am
f(x) = 3 sinx  x
f'(x) = 3 cosx 1
if xo = 1 is the first approximation, the second approximation is
x1 = xo  f(1)/f'(1)
= 1  1.524/0.621 = 1.454
x2 = x1  f(x1)/f'(x1)
= 1.454 1.525/1.804 = 2.299
x3 = x2  f(x2)/f'(x2)
= 2.299 0.0599/(1.997)
= 2.269 
calculus  drwls, Tuesday, September 7, 2010 at 6:32am
I tried to follow the format at
http://en.wikipedia.org/wiki/Newton's_method
My xo may be your x1, etc. 
calculus  kwack, Tuesday, September 7, 2010 at 5:45pm
I put 1.454 for x2 but when I put 2.299 for x3 I got it wrong