2017 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 10

Problem

Newton’s method applied to the equation $f(x) = x^3-x$ takes the form of the iteration $x_{n+1} =x_n-\frac{(x_n)^3-x_n}{3{x_n}^2-1} , n = 0,1,2,\ldots$

(a) What are the roots of $f(x) = 0$ ?

(b) Study the behavior of the iteration when $x_0 > \frac{1}{\sqrt{3}}$ to conclude that the sequence $\{x_0,x_1,\ldots\}$ approaches the same root as long as you choose $x_0 > \frac{1}{\sqrt{3}}$ . It may be helpful to start with the case $x_0>1$ .

(c) Assume $-\alpha < x_0 < \alpha$ . For what number $\alpha$ does the sequence always approach $0$ ?

(d) For $x_0 \in(\alpha,\frac{1}{\sqrt{3}})$ the sequence may approach either of the roots $\pm x^{*}$ . Can you ﬁnd an (implicit) expression that can be used to determine limits $a_i$ and $a_{i+1}$ such that if $x0 \in (a_i,a_{i+1})$ then the sequence approaches $(-1)^i{x^{*}}$ . Hint: $a_1 = \frac{1}{\sqrt{3}}, a_i > a_{i+1}$ and $a_i$ approaches $\frac{1}{\sqrt{5}}$ when $i$ becomes large.

2017 UNM-PNM Contest II (Problems • Answer Key • Resources)
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2017 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 10

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