Chapter 3: Applications of Differentiation
Section 3.2: Newton's Method
Newton's method for solving the equation fx=0 is an iterative numeric technique defined by
This "formula" expresses the simple idea that the x-intercept on a tangent line drawn at xk near an x-intercept for the graph of f, is generally closer to the intercept of f than xk. Indeed, the point-slope form of the equation of the line tangent at xk is
The x-intercept on this line is the solution for x when y=0. This x becomes xk+1 by the calculation in Table 3.2.1.
Table 3.2.1 Derivation of the Newton iteration formula
In general, Newton's method converges rapidly to a root of f=0 provided the initial point is chosen "close enough" to the root. However, the method is not infallible, and there are functions for which the method will converge more slowly, will cycle back and forth between two distinct iterates, will seem to converge but not to a root, and will fail to find an actual root. One of these anomalies is the subject of Example 3.2.3.
Use Newton's method to approximate the positive zero of the function fx=sinx−x/2.
Use Newton's method to approximate the positive root of the equation 5 tan−1x=1/x.
Apply Newton's method to fx=x2−2 x−1.
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