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Chapter 4 | Applications of Derivatives
Figure 4.79 We can use Newton’s method to find 2.
Use Newton’s method to approximate 3 by letting f ( x ) = x 2 −3 and x
4.46
0 =3. Find x 1 and x 2 .
When using Newton’s method, each approximation after the initial guess is defined in terms of the previous approximation by using the same formula. In particular, by defining the function F ( x ) = x − ⎡ ⎣ f ( x ) f ′( x ) ⎤ ⎦ , we can rewrite Equation 4.8 as x n = F ( x n −1 ). This type of process, where each x n is defined in terms of x n −1 by repeating the same function, is an example of an iterative process . Shortly, we examine other iterative processes. First, let’s look at the reasons why Newton’s method could fail to find a root. Failures of Newton’s Method Typically, Newton’s method is used to find roots fairly quickly. However, things can go wrong. Some reasons why Newton’s method might fail include the following: 1. At one of the approximations x n , the derivative f ′ is zero at x n , but f ( x n ) ≠0. As a result, the tangent line of f at x n does not intersect the x -axis. Therefore, we cannot continue the iterative process. 2. The approximations x 0 , x 1 , x 2 ,… may approach a different root. If the function f has more than one root, it is possible that our approximations do not approach the one for which we are looking, but approach a different root (see Figure 4.80 ). This event most often occurs when we do not choose the approximation x 0 close enough to the desired root. 3. The approximations may fail to approach a root entirely. In Example 4.48 , we provide an example of a function and an initial guess x 0 such that the successive approximations never approach a root because the successive approximations continue to alternate back and forth between two values.
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