Newton's Method

The Increment Theorem for derivatives shows that when f'(c) exists and x ≈ c, f(x) is infinitely close to the tangent line f(c) + f'(c)(x - c) even compared to x - c. Thus intuitively, when x is real and close to c, f(x) is closely approximated by the tangent line f(c) + f'(c)(x - c). Newton's method uses the tangent line to approximate a zero of f(x). It is an iterative method that does not always work but usually gives a very good approximation.

Consider a real function f that crosses the x-axis as in Figure 5.9.1. From the graph we make a first rough approximation x₁ to the zero of f(x). To get a better approximation, we take the tangent line at x₁ and compute the point x₂ where the tangent line intersects the x-axis. At x₂, the curve f(x) is very close to zero, so we take x₂ as our new approximation. The tangent line has the equation

y = f(x₁) + f'(x₁) (x - x_l).

We get a formula for x₂ by setting y = 0 and x = x₂ and then solving for x₂.

Figure 5.9.1

0 = f(x₁) + f'(x₁)(x₂ - x₁)

We may then repeat the procedure starting from x₂ to get a still better approximation x₃ as in Figure 5.9.2,

Figure 5.9.2