CC MATHEMATICS   MATHEMATICAL EXPLORATIONSSYLLABUS

Homework 2: Due Monday 9/12 at 4PM

Try to complete as much as possible on this homework by Friday, as you will have a quiz on Monday afternoon also.

Turn in individual solutions. You may work together in solving the problems, or get help from others, but the final write-up should be your own. For homework, show full work for full credit. Do not turn in anything you cannot explain completely.

Part I: From Schaum's Calculus  Problem Further instructions
Chapter 7 16c,f,i,l  
Chapter 8 8a,b,e  
Chapter 9 18a, c, g Use the definition of derivative IN EACH PROBLEM. 
    Use Good Notation. Show work clearly.

 Part II: Chaos Problems
Number Problem
1 With the tent map using s =1.6:  Find the first 12 iterates of x=.5 and x=.49.  Does the dynamical system for the tent map with s=1.6 exhibit sensitive dependence on initial conditions?
2 Use the graph of this tent map to explain why the attractor for this s value is what the bifurcation diagram on p 155 shows it to be.  Use the graph of the tent map and its equation to find numerical values (decimals okay) for the upper and lower bounds of the attractor for this s. 
3 Show that the tent map has a fixed point at x = s/(1 + s). 
4 At what point does the "eye" in the bifurcation diagram on page 155 close off? Hint: In figure 4.36 on page 151, the second diagram shows two distinct regions or bands between which the tent map iterates are bound to oscillate.  Show that the s value at which these two bands merge is a root of s^3 - s^2 - 2s + 2 = 0.