Some populations initally grow exponentially but eventually level off. Equations of the form P ( t ) = M 1 + A e − k t Where M , A , and k are positive constants, are called logistic equations and are often used to model such populations. (We will investigate these in detail in Chapter 9.) Here M is called the carrying capacity and represents the maximum population size that can be supported, and A = M − P 0 P 0 , where P 0 is the initial population. (a) Compute lim t → ∞ P ( t ) . Explain why your answer is to be expected. (b) Compute lim M → ∞ P ( t ) . (Note that A is defined in terms of M. ) What kind of function is your result?
Some populations initally grow exponentially but eventually level off. Equations of the form P ( t ) = M 1 + A e − k t Where M , A , and k are positive constants, are called logistic equations and are often used to model such populations. (We will investigate these in detail in Chapter 9.) Here M is called the carrying capacity and represents the maximum population size that can be supported, and A = M − P 0 P 0 , where P 0 is the initial population. (a) Compute lim t → ∞ P ( t ) . Explain why your answer is to be expected. (b) Compute lim M → ∞ P ( t ) . (Note that A is defined in terms of M. ) What kind of function is your result?
Some populations initally grow exponentially but eventually level off. Equations of the form
P
(
t
)
=
M
1
+
A
e
−
k
t
Where M, A, and k are positive constants, are called logistic equations and are often used to model such populations. (We will investigate these in detail in Chapter 9.) Here M is called the carrying capacity and represents the maximum population size that can be supported, and
A
=
M
−
P
0
P
0
, where P0 is the initial population.
(a) Compute
lim
t
→
∞
P
(
t
)
. Explain why your answer is to be expected.
(b) Compute
lim
M
→
∞
P
(
t
)
. (Note that A is defined in terms of M.) What kind of function is your result?
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