   Chapter 4.4, Problem 81E ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

Chapter
Section ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

# 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?

(a)

To determine

To compute: The value of limit limtP(t) and explain the reason why the answer is to be expected.

Explanation

Given:

The logistic equation of growth of population is, P(t)=M1+Aekt , where M is carrying capacity and A=MP0P0 and k is a positive constant.

Calculation:

Consider, limtP(t)=limtM1+Aekt

(b)

To determine

To compute: The value of limit limMP(t) and identify the type of the function of answer.

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