Exam 1: MAP 4484
*
February 6, 2009
Name
:
Student ID
:
This is a
closed book
exam and the use of calculators is
not
allowed.
1. A population model is described by the following equation:
x
n
+1
=
2
1 +
x
n
•
Calculate all fixed points and all period 2 points (if any).
Fixed points are solutions of:
x
=
2
1 +
x
,
or equivalently of:
x
2
+
x

2 = 0
.
This quadratic equation has two solutions:
x
1
,
2
=

1
±
3
2
= 1
,

2
We discard the negative solution, and thus there is a single fixed point:
x
*
= 1
.
Period two points must satisfy:
x
=
2
1 +
2
1+
x
,
which reduces to
x
2
+
x

2 = 0
,
the same quadratic equation we solved before. Thus, there are no period two points, as
the only solution of the above quadratic equation only yields the fixed point determined
earlier.
•
What happens to the solution sequences as
n
→ ∞
? (
Hint
: Use a global result from
our cobwebbing notes)
Since the function
f
(
x
) = 2
/
(1 +
x
) is decreasing and bounded, and since there are
no period two points, we conclude that all solutions converge to the unique fixed point
x
*
= 1.
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 Summer '06
 DeLeenheer
 Quadratic equation, Elementary algebra, Stability theory

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