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Exercises 8.5 Homework

1.

For each of the following functions, find all of its fixed points. Classify each fixed point as attracting, repelling, or neutral.

  1. \(\displaystyle f(x) = x^5\)
  2. \(\displaystyle g(x) = 16/x^3\)
  3. \(\displaystyle h(x) = x^3 - x/2\)

No programming is involved in this problem. Ignore the solutions in complex numbers like \(x=i\text{;}\) we only need real numbers here.

2.

Write a script that solves the equation \(e^x + \log x = n\) using the fixed point iteration. Here \(n\) is the number formed by the first two digits of your SUID.

In order to use the fixed point method, you need to rewrite the given equation as \(g(x) = x\text{.}\) There are several ways to do this, try at least two different ones, or as many as are needed until a root is found. Starting from some value of \(x\text{,}\) such as \(x=10\text{,}\) below, run fixed point iteration until either it converges to a root, or the number of iterations becomes extremely large. The script should display either “Root found at x = ... after ... steps” or “Failed to converge”. Add a comment with the different functions \(g\) you tried, and what the outcome was.

Here, as in in Matlab and in most mathematical texts, \(\log\) means the natural logarithm, base \(e\text{.}\)