Root Finding Methods! Newtons! Bisectional! Algebra!?

Question

Newton's Root Finding Method:
x1=x0-(f(x0)/f'(x0)
x2=x1-(f(x0)/f'(x0)
xn=x{n-1}-(f(x{x-1})/f'(x{x-1})
The "1" "2" "n" "n-1" are subscripts.

1. Use newton's method to appproximate the value of square root of 6. Perform 4 iterations to find your approximate answer.

I tried to do that, but im not sure what # to start x0 at! Should it be higher or lower?

2.wat is approximate error between actual value and approximated value at x4?

3. Use newtons method to approximate the root of the function for 4 itterations:
f(x)=e^(3x+2)-x^2
x0=0
Whats f(x4)=?

Not sure how to do this, i ran into problems with alegebra. I cannot do x2 for some reason.

5. Which method is more effective? Bisection method or Newtons' root finding method?

I have no clue which is bettter, maybe you guys can tell why with facts.

Answer 1

I found the Wikipedia's article on Newton's method to be very good:

http://en.wikipedia.org/wiki/Newton's_method

Answer 2

g(x) = x^2 - 2 g'(x) = 2x x_(n+a million) = x_(n) - g(x)/g'(x) x_(n+a million) = x_(n) - (x^2 - 2)/(2x) if x_(0) = a million x_(a million) = a million - (a million^2 - 2)/(2 * a million) x_(a million) = a million.5 x_(2) = a million.5 - ((a million.5)^2 - 2)/(2 * a million.5) x_(2) = a million.4166667