Chapter 5, Exploration 30

Explore by making adjustments to the slider for the tool that zoomed in on the graph of \(y=f_a^2(x)\). As the slider moves, the fixed point moves and the graph itself looks like it is rotating about the fixed point. Focus on the rotation. Do you see how the graph of \(y=f_a^2(x)\) "rotates" through the line \(y=x\) as \(a\) passes through the bifurcation value of \(a=3\)?

  1. Explain why this property is important for the creation of three fixed points of \(y=f_a^2(x)\) as \(a\) passes through the bifurcation value.
  2. What can you say about \(\frac{\partial}{\partial x}\left(f_a^2\left(x\right)\right) \) for \(x<3\)? For \(x=3\)? For \(x>3\)?
  3. What does this exploration imply about the value of \( \frac{\partial}{\partial a}\left( \frac{\partial}{\partial x}\left(f_a^2\left(x\right)\right)\right) = \frac{\partial^2 }{\partial a\partial x}\left(f_a^2\left(x\right)\right) \) evaluated at \(a=3, \, x= 2/3\)?