2017 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 10
Problem
Newton’s method applied to the equation takes the form of the iteration
(a) What are the roots of ?
(b) Study the behavior of the iteration when to conclude that the sequence approaches the same root as long as you choose . It may be helpful to start with the case .
(c) Assume . For what number does the sequence always approach ?
(d) For the sequence may approach either of the roots . Can you ﬁnd an (implicit) expression that can be used to determine limits and such that if then the sequence approaches . Hint: and approaches when becomes large.
Solution
(a) The roots are .
(b) First consider . Let and with . The iteration gives . Next consider . As the signs of the numerator and denominator in the rational part of the iteration does not change on the interval under consideration we ﬁnd that . Finally, produces .
To answer (c), rewrite the iteration as, and note that for the next iterate will be non-positive. Insisting that , so that will be closer to zero than gives the limiting case , or , which has the solution .
Finally the implicit recurrence in (d) is obtained by running Newton backwards = .
See also
2017 UNM-PNM Contest II (Problems • Answer Key • Resources) | ||
Preceded by Problem 9 |
Followed by Last Question | |
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All UNM-PNM Problems and Solutions |