SI Calculus I
Weber State University
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This 2 page Class Notes was uploaded by Rosa Farrell on Wednesday October 28, 2015. The Class Notes belongs to MATH 1210 at Weber State University taught by Staff in Fall. Since its upload, it has received 15 views. For similar materials see /class/230806/math-1210-weber-state-university in Math at Weber State University.
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Date Created: 10/28/15
ADDITIONAL NOTES AND EXERCISES FOR 63 Newton7s method does not always work The following result gives suf cient conditions for when it does Theorem 1 Newton Let f ab A R be continuous Suppose that 0 for some c in ab Suppose that f exists and is bounded on ab39 that is there exists M gt 0 such that fI S M for every x E a b Suppose that f is bounded away from zero that is there exists 6 gt 0 such that f 2 e for every x 6 ab Then there is a closed interval I I d e C ab containing c such that the sequence given recursively by f1n71 In 7 Inil f 1n71 converges to c Moreover xn is in I for each n For this to make sense it helps if we have a notion of convergence of sequences We will do this in second semester calculus For now we will just use the intuitive statement as n gets large xn gets arbitrarily close to c77 and write xn c Let us now analyze the various conditions in the statement Bounding f away from zero should seem reasonable from the de nition of xn but it is not necessary The following exercise show that Newton7s method may work even if f is not bounded away from zero Exercise 2 Let x2 Adopt the notation of Newton7s theorem Show that if xn1 0 then xn 2 1 Suppose that x0 1 What is x1 How about x2 How about xn Argue that x7 converges to zero as n goes to 00 when x0 l In the above example any x0 except zero will do The next exercise illustrates what can go wrong if we do not bound f away from zero Exercise 3 Let x474x711 Then lt 0 lt so by the intermediate value theorem there exists a root between x 0 and x 3 What happens if we apply Newton7s method when we start at x0 1 Sketch the graph of f on the interval 0 3 We also require a bound on the second derivative The following exercise shows what can go wrong Exercise 4 Let xi Observe that x 0 is a root of Compute f x and explain why f is unbounded near x 0 Attempt to use Newtonls method with x0 1 What happens Illustrate your conclusion with a diagram 1 2 ADDITIONAL NOTES AND EXERCISES FOR 63 Newton7s method when it works is extremely fast The number of decimal places of accuracy approximately doubles with each successive iterationi The statement of Newton7s theorem does not tell us how to how to nd the closed interval 1 The proof gives some indication but is beyond the scope of the course For our purposes we can use a graphing calculator to give us an idea of where to look for the roots of a given function and then apply Newton7s method to get a good decimal approximation Exercise 5 Use Newton7s method to nd the coordinates of the in ection point 3 of y 772 sinz correct to six decimal places Exercise 6 Let a be a nonzero numberi Suppose we want to compute a decimal expansion of This could be dif cult if a is big or has a large number of decimal places Newton7s method gives us a way of computing a without divisioni et 1 7 7 a Notice that f has a root at z 7 Using the notation of Newton7s theorem show that 2 1 In 21ml 7 amnili
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