




This method is used to solve an equation, which we write in the form f(x)=0, when f is differentiable.
Step 1
Start at some point x_{1}
Find the point x_{2} where the tangent line to f at x_{1} meets the xaxis.
Calculation of x_{2}:
The equation of the tangent line at x_{1} is
y = f(x_{1}) + f '(x_{1})(x  x_{1})
x_{2} solves
0 = f(x_{1}) + f '(x_{1})(xx_{1})
Step 2
Repeat this step with x_{2} and so until f(x_{k}) = 0
This method can be used to solve f(x) = h(x) (Just apply Newton's method to solve g(x) = f(x)  h(x) = 0.)
Illustration
Find a solution of f(x) = x^{3}  3x^{2} + 1 = 0
Curve and iterations
f(x) = x^{3}  3x^{2} + 1
f '(x) = 3x^{2}  6x
hence
Step 1
We start at x_{1} = 35
Step 2
We repeat this step with x_{2} and so on...