1. Suppose you deposit $1000 each week into a special savings account, but the bank takes
5% of the total in fees. A discretetime system modelling your investment is
x
n
+1
=0
.
95
x
n
+ 1000
.
(a) At the end of the ±rst week, there is
x
1
=0
.
95(1500) + 1000 = $2425
in the account. At the end of the second week, there is
x
2
=0
.
95(2425) + 1000 = $3303
.
75
in the account. At the end of the third week, there is
x
3
=0
.
95(3303
.
75) + 1000 = $4138
.
56
in the account.
(b) The updating function is
f
(
x
)=0
.
95
x
+ 1000
(c) The equilbrium points satisfy
p
=0
.
95
p
+ 1000
0
.
05
p
= 1000
p
=20
,
000
.
(d) The solution is
x
n

20
,
000 = 0
.
95
n
(
x
0

20
,
000)
x
n
=

18
,
500
×
0
.
95
n
+20
,
000
(e) See Figure 1.
(f) See Figure 2.
(g) The equilibrium point is stable.
2. A disease is spreading through campus. Each day, the number of people infected depends
on how many were infected the day before, according to the formula
y
n
+1
=
6
.
2976
y
n
1+0
.
0112
y
n
.
1
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 Fall '08
 DUMITRISCU
 Savings account, Equilibrium point, Stability theory

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