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Functions and Sequences of Iterates

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Let f be a function defined on [a,b] and suppose that z is a point in (a,b) such that f(z) = z. Further suppose that there is a number alpha < 1 such that f ' (x) < alpha for all x contained in (a,b) and that 0 < f ' (x) for all x contained in (a, b).

a.) Prove that if z<x, then z<f(x) and that f(x) - z < alpha(x-z).

b.) Let x0 > z and set xn = f ( x n-1) for every natural number n > or = 1. Prove that 0 < xn - z < alphan (x0- z). You may use the result of part a.

c.) Short answer - this does not require a formal proof - just an indication of why it may be so. How could the result of part b be used to argue that if x0 > z, then the sequence of iterates xn = f ( x n-1) tends towards z.

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Functions and sequences of iterates are investigated. The expert proves functions for natural numbers.

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Question #3
Let f be a function defined on [a,b] and suppose that z is a point in (a,b) such that f(z) = z. Further suppose that there is a number alpha < 1 such that f ' (x) < alpha for all x contained in (a,b) and that 0 < f ' (x) for all x contained in (a, b).

a.) Prove that if z<x, then z<f(x) and that f(x) - z < alpha(x-z).
Proof. As for all , we know that y=f(x) is increasing in ...

Solution provided by:
Changping Wang, MA
Education
  • BSc , Wuhan Univ. China
  • MA, Shandong Univ.
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  • "Your solution, looks excellent. I recognize things from previous chapters. I have seen the standard deviation formula you used to get 5.154. I do understand the Central Limit Theorem needs the sample size (n) to be greater than 30, we have 100. I do understand the sample mean(s) of the population will follow a normal distribution, and that CLT states the sample mean of population is the population (mean), we have 143.74. But when and WHY do we use the standard deviation formula where you got 5.154. WHEN & Why use standard deviation of the sample mean. I don't understand, why don't we simply use the "100" I understand that standard deviation is the square root of variance. I do understand that the variance is the square of the differences of each sample data value minus the mean. But somehow, why not use 100, why use standard deviation of sample mean? Please help explain."
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