Example 2.  Use Mathematica's built in subroutine "FixedPointList" and experiment finding the fixed point(s) for the function   .

Solution 2.

Investigate iteration near the "attractive fixed point."

How could we generate this list ?  Execute the following cells one cell at a time.
It will start with a list of one number and "Append" numbers to the list.

Thus we have constructed the essential ideas for Mathematica's  "FixedPointList" subroutine.

Aside. It has become "pop math" to see all those crazy figures about "chaos", etc. They all depend on drawing line segments between points. This graphical excursion is not the usual academic treatment of numerical analysis.  It is just for fun!

Consider the following graph consisting of line segments joining the points given above.

Show this graph and the one in Example 1 on the same plot.

(c) John H. Mathews 2004