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Math · Calculus
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As you have just seen with the function f cosx x, Newtons method is frequently an efficient way of finding a root to an equation. But Newtons method doesnt always work. Sometimes it gives you an answer you dont want, sometimes it cant be used, and sometimes it takes you further and further away from the root you are looking for! 1. Consider f (x) 3 +2r x2. Use algebra to find the positive root of f. 2. Now lets say that some people wanted to try Newtons method out on the above problem. What would happen if they started with x 0? What would happen if they started with xi l? 3. Suppose f x) was a very complicated function with a lot of roots in a very small interval. What problems could arise in trying to use Newtons method to find all the roots? 4. Now we consider an even worse scenario: one where Newtons method takes you further away from the only root of a function, rather than closer. In order to understand the circumstances which cause this to happen, consider the following: Suppose you are at a point x where the function is positive but the derivative is negative. Will the next iteration give you a number larger or smaller than x? 5. Let f What is the only root of f? 1 +x 6. Graph f on the interval [0, 10) using technology. Notice that it is first increasing, and then decreasing, What is the global maximum of f? 7. What happens if we use Newtons method starting at x 2? Can you explain why Newtons method thinks it is approaching a root?

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