MA005: Calculus I

There are applications which feed the output from a function machine back into the same machine as the new input. Each time through the machine is called an iteration of the function.

Example 4: Suppose $f(x) = \dfrac{5/x + x}{2}$, and we start with the input $x = 4$ and repeatedly feed the output from $f$ back into $f$ (Fig. 9). What happens?

Solution:

Once we have obtained the output 2.236067977, we will just keep getting the same output. You might recognize this output value as $\sqrt5$. This algorithm always finds $±\sqrt 5$ . If we start with any positive input, the values will eventually get as close to $\sqrt5$ as we want. Starting with any negative value for the input will eventually get us to $–\sqrt 5$ . We cannot start with $x = 0$, since 5/0 is undefined.

Practice 6: What happens if we start with the input value $x = 1$ and iterate the function $f(x) =\dfrac{9/x + x}{2}$ several times? Do you recognize the resulting number? What do you think will happen to the iterates of $g(x) = \dfrac{A/x + x}{2}$ ? (Try several positive values of $A$).