478
Chapter 4 | Applications of Derivatives
Figure 4.81 The approximations continue to alternate between 0and1 and never approach the root of f .
For f ( x ) = x 3 −2 x +2, let x
4.47
0 =−1.5 and find x 1 and x 2 .
From Example 4.48 , we see that Newton’s method does not always work. However, when it does work, the sequence of approximations approaches the root very quickly. Discussions of how quickly the sequence of approximations approach a root found using Newton’s method are included in texts on numerical analysis. Other Iterative Processes As mentioned earlier, Newton’s method is a type of iterative process. We now look at an example of a different type of iterative process. Consider a function F and an initial number x 0 . Define the subsequent numbers x n by the formula x n = F ( x n −1 ). This process is an iterative process that creates a list of numbers x 0 , x 1 , x 2 ,…, x n ,…. This list of numbers may approach a finite number x * as n gets larger, or it may not. In Example 4.49 , we see an example of a function F and an initial guess x 0 such that the resulting list of numbers approaches a finite value. Example 4.49 Finding a Limit for an Iterative Process Let F ( x ) = 1 2 x +4 and let x 0 =0. For all n ≥1, let x n = F ( x n −1 ). Find the values x 1 , x 2 , x 3 , x 4 , x 5 . Make a conjecture about what happens to this list of numbers x 1 , x 2 , x 3 …, x n ,… as n →∞. If the list of numbers x 1 , x 2 , x 3 ,… approaches a finite number x *, then x * satisfies x * = F ( x *), and x * is called a fixed point of F .
Solution If x 0 =0, then
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