Some populations initially grow exponentially but eventually level off. Equations of the formP (t) = M/(1 + Ae-kt)where M, A, and k are positive constants, are called logistic equations andare often used to model such populations. Here M is called the carryingcapacity and represents the maximum population size that can be supported,and A =(M−P0 )/Powhere P0 is the initial population.(a) Compute limt→∞P (t).Explain why your answer is to be expected.(b)Compute limM→∞P (t). (Note that A is defined in terms of M.) What kindof function is your result? Some populations initally grow exponentially but eventually level off. Equa-tions of the formMP (t)1+ Ae-ktwhere M, A, and k are positive constants, are called logistic equations andare often used to model such populations. Here M is called the carryingcapacity and represents the maximum population size that can be supported,and A = M-P where Po is the initial population.Po(a) Compute lim P (t). Explain why your answer is to be expected.(b) Compute lim P(t). (Note that A is defined in terms of M.) What kindof function is your result ?

Answered: Some populations initially grow…

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Some populations initially grow exponentially but eventually level off. Equations of the form
P (t) = M/(1 + Ae^-kt)
where M, A, and k are positive constants, are called logistic equations and
are often used to model such populations. Here M is called the carrying
capacity and represents the maximum population size that can be supported,
and A =(M−P₀ )/P_o
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?

Some populations initally grow exponentially but eventually level off. Equa-
tions of the form
M
P (t)
1+ Ae-kt
where M, A, and k are positive constants, are called logistic equations and
are often used to model such populations. Here M is called the carrying
capacity and represents the maximum population size that can be supported,
and A = M-P where Po is the initial population.
Po
(a) Compute lim P (t). Explain why your answer is to be expected.
(b) Compute lim P(t). (Note that A is defined in terms of M.) What kind
of function is your result ?

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