Chapter 3, Explorations 57 and 58

Minimize the left sidebar so you can see both graphs. The left figure always shows a graph of $$y = f_a(x)$$. The right figure shows a graph of $$y = f_a^n(x)$$ where $$n$$ can also be set using a slider.

Exploration 57 - For each of the $$a$$-values given in your text, the logistic equation has an attracting periodic orbit. Select the appropriate value of $$n$$ and determine its period. How many periodic points are there of this period? How many prime periodic orbits are there of this period?

Exploration 58 - For each parameter value use the slider to change $$n$$ so that it agrees with the period of the attracting periodic orbit you found in Exploration 57. For example, if you found an attracting period 2 orbit in Exploration 57, make $$n$$ equal to 2 using the slider. What can you say about the derivative of the right graph at the attracting periodic points? What about the repelling ones?

a:

n: