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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 xaxis as in Figure 5.9.1. From the graph we make a first rough approximation x_{1} to the zero of f(x). To get a better approximation, we take the tangent line at x_{1} and compute the point x_{2} where the tangent line intersects the xaxis. At x_{2}, the curve f(x) is very close to zero, so we take x_{2} as our new approximation. The tangent line has the equation y = f(x_{1}) + f'(x_{1}) (x  x_{l}). We get a formula for x_{2} by setting y = 0 and x = x_{2} and then solving for x_{2}. Figure 5.9.1 0 = f(x_{1}) + f'(x_{1})(x_{2 } x_{1}) We may then repeat the procedure starting from x_{2} to get a still better approximation x_{3} as in Figure 5.9.2, Figure 5.9.2


Home Limits, Analytic Geometry, and Approximations Newton's Method Newton's Method 