Math 404/504 - Assignment 3
Due:
Tuesday, January 31. Nothing accepted after Thursday, February 2. 2 points off for being late. Please work
by yourself. See me if you need help.
You are studying the net worth of seventh graders. Let xn be the net worth of a certain seventh grader at 12 noon on
Sunday n weeks from now. A negative value means the individual is in debt to their parents for this amount. Suppose
the following difference equation xn+1 = (xn)4 - 2 (xn)2 describes how xn varies from one week to the next.
1.
(2 points) This equation has the form xn+1 = f(xn). What is f(x)? Make the graph showing both the curve y = f(x)
and the line y = x. What sort of symmetry does the graph of f(x) have?
2.
(2 points) What are the equilibrium values?
3.
(2 points) Classify each equilibrium values as to being a sink, source or neither. Explain why you classify each
equilibrium value the way you do.
4.
(1 point) Suppose xo >
5.
(1 point) Suppose xo < -
6.
(1 point) In general, what is the relationship between the two solutions where the first has initial condition xo and
the second has initial condition - xo.
7.
(1 point) Suppose
8.
(1 point) Suppose. 0 < xo <
9.
(1 point) Suppose -1 < xo <
1+ 5
. (Recall xo is the net worth of a seventh grader at time n = 0). Without actually
2
solving the difference equation sketch the graph of the solution xn. The graph should have n on the horizontal axis
and xn on the vertical axis. What happens to xn as n   in this case?
1+ 5
. Without actually solving the difference equation sketch the graph of the solution
2
xn. What happens to xn as n   in this case? How does it compare to one of the solutions in part 4?
1- 5
< xo < 0. Without actually solving the difference equation sketch the graph of the
2
solution xn. What happens to xn as n   in this case?
5-1
. Without actually solving the difference equation sketch the graph of the
2
solution xn. What happens to xn as n   in this case? How does it compare to one of the solutions in part 7? To
1- 5
5-1
summarize parts 7 and 8, suppose
< xo <
. What happens to xn as n   in this case?
2
2
1- 5
. Without actually solving the difference equation sketch the graph of the
2
solution xn. What happens to xn as n   in this case?
10. (1 point) Find the four solutions of f(x) =
1- 5
. Suppose when arranged in order they are a1 < a2 < a3 < a4.
2
11. (1 point) Suppose a1 < x < a2 or a3 < x <a4. Without actually solving the difference equation sketch the graph of
the solution xn. What happens to xn as n   in this case?
1+ 5
. What are the three possible behaviors of xn as
2
1+ 5
n  . The answer depends on xo. Show how to divide the interval a4 < x <
into sub-intervals in such a
2
way that for each xo in the interval the corresponding xn has the same behavior as n  . Also, what happens if xo
is an endpoint of one of the subintervals? There are an infinite number of intervals. Describing them all is a little
tricky. See me if you need help.
12 (for students in Math 504 only) (4 points) Suppose a4 < x <