This method is used to solve an equation, which we write in the form f(x)=0, when f is differentiable.
Start at some point x1
Find the point x2 where the tangent line to f at x1 meets the x-axis.
Calculation of x2:
The equation of the tangent line at x1 is
y = f(x1) + f '(x1)(x - x1)
0 = f(x1) + f '(x1)(x-x1)
Repeat this step with x2 and so until f(xk) = 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.)
Application to tin cans
Find a solution of f(x) = x3 - 3x2 + 1 = 0
Curve and iterations
f(x) = x3 - 3x2 + 1
f '(x) = 3x2 - 6x
We start at x1 = 35
We repeat this step with x2 and so on...