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Some populations initially grow exponentially but eventually level off. Equations of the form$$ P(t) = \frac{M}{1 + Ae^{-kt}} $$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 = \frac{M - P_0}{P_0} $, where $ P_0 $ is the initial population.(a) Compute $ lim_{t\to \infty} P(t) $. Explain why your answer is to be expected.(b) Compute $ lim_{M\to \infty} P(t) $. (Note that $ A $ is defined in terms of $ M $.) What kind of function is your result?

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02:23

Wen Zheng

01:11

Amrita Bhasin

Calculus 1 / AB

Calculus 2 / BC

Chapter 4

Applications of Differentiation

Section 4

Indeterminate Forms and l'Hospital's Rule

Derivatives

Differentiation

Volume

Oregon State University

Harvey Mudd College

Boston College

Lectures

04:35

In mathematics, the volume of a solid object is the amount of three-dimensional space enclosed by the boundaries of the object. The volume of a solid of revolution (such as a sphere or cylinder) is calculated by multiplying the area of the base by the height of the solid.

06:14

A review is a form of evaluation, analysis, and judgment of a body of work, such as a book, movie, album, play, software application, video game, or scientific research. Reviews may be used to assess the value of a resource, or to provide a summary of the content of the resource, or to judge the importance of the resource.

01:47

Some populations initally …

03:56

It can be shown that solut…

01:52

Populations grow exponenti…

Yeah. For this problem here, uh we have key approaching infinity in the logistic equation. So we have em over one plus E to the negative Katie. When t goes to infinity, this thing goes to zero, meaning the numerator goes to one. And then we're just left with Emma's results. And then for part B we have um to calculate it differently. M is going to infinity now. So we're going to treat the variable. We're gonna treat amazon a variable. However, when we send it to infinity, we get the indeterminant form. So we have to use local cows role, but we get it into this form right here. Uh, peanut class mm am minus P not I M E to the negative Katie. So when we evaluate this and we take the derivative, what we're left with is P not times E to the K T. As the final answer.

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