1
Solutions
Section 1.1
1.
The rate of change in the population
P
(
t
)
is the derivative
P
0
(
t
)
. The
Malthusian Growth Law states that the rate of change in the population is
proportional to
P
(
t
)
. Thus
P
0
(
t
) =
kP
(
t
)
, where
k
is the proportionality
constant. Without reference to the
t
variable, the differential equation
becomes
P
0
=
kP
2.
a. This statement mathematically is
b
(
t
) =
b
0
P
(
t
)
where we have used
b
0
to represent the proportionality constant.
b. This statement translates as
d
(
t
) =
d
0
P
2
(
t
)
where we have used
d
0
to
represent the proportionality constant.
c. The overall growth rate is
P
0
(
t
)
. Thus the Logistic Growth Law is
P
0
(
t
) =
b
(
t
)
d
(
t
)
=
b
0
P
(
t
)
d
0
P
2
(
t
)
= (
b
0
d
0
P
(
t
))
P
(
t
)
:
3.
Torricelli’s law states that the change in height,
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 Fall '08
 BELL,D
 Exponential Function, Derivative, highest order derivative, Logistic Growth Law

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