NEWTON'S METHOD

Method

1. Given a function f(x), approximate a root r.
2. Find the tangent line at (r, f(r)).
3. Find the x-intercept (x,0) for this tangent line.
4. Using x as a new approximation, repeat steps 2 through 4 until desired accuracy is reached.

Proof

• The equation of the tangent line containing (x₁, f(x₁)) is  y - f(x₁) = f '(x₁)(x - x₁)
• The x-intercept is (x₂,0) so setting x = x₂ and y = 0 produces
• -f(x₁) = f '(x₁)(x₂ - x₁)
• -f(x₁)  =  x₂ - x₁
f '(x₁)
• x₂ = x₁ -   f(x₁)
                f '(x₁)

Newton's Method May Fail when

1. a critical point lies between x₁ and the true root
2. endpoints are used

Sufficient Condition for Newton's Method to Work