Homework

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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.

$\displaystyle f(x) = x^5$

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$\displaystyle g(x) = 16/x^3$

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$\displaystyle h(x) = x^3 - x/2$

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

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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.

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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.

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

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