(a) Draw a logistic curve. Label the carrying capacity L and...

(a) Draw a logistic curve. Label the carrying capacity $L$ and the point of diminishing returns $t_{0}$ (b) Draw the derivative of the logistic curve. Mark the point $t_{0}$ on the horizontal axis. (c) A company keeps track of the rate of sales (for example, sales per week) rather than total sales. Explain how the company can tell on a graph of rate of sales when the point of diminishing returns is reached.

(a) Draw a logistic curve. Label the carrying capacity $L$ and the point of diminishing returns $t_{0}$
(b) Draw the derivative of the logistic curve. Mark the point $t_{0}$ on the horizontal axis.
(c) A company keeps track of the rate of sales (for example, sales per week) rather than total sales. Explain how the company can tell on a graph of rate of sales when the point of diminishing returns is reached.

Key Concepts

Inflection Point (Point of Diminishing Returns)

The inflection point, or point of diminishing returns, is the point on the logistic curve where the growth rate reaches its maximum before beginning to decline. At this point, increases in the independent variable result in less-than-proportional increases in the dependent variable, indicating that further input yields diminishing benefits.

Derivative of the Logistic Curve

The derivative of the logistic curve represents the rate of change or instantaneous growth rate of the process. It is typically a bell-shaped curve that peaks at the inflection point. This peak clearly identifies the moment when the system experiences the highest rate of change, which corresponds to the point of diminishing returns, helping decision makers understand when additional efforts produce reduced incremental improvements.

Logistic Growth Curve

A logistic growth curve is an S-shaped function that models growth processes limited by a maximum capacity. Initially, the growth is almost exponential, but as the process progresses, resource limitations slow the growth until it eventually levels off at the carrying capacity.

Carrying Capacity

The carrying capacity is the upper limit of the system, representing the maximum sustainable level that the system can support. In the logistic curve, it is depicted as the horizontal asymptote, and it restricts the growth as the curve approaches this limit over time.

Transcript

00:01 Okay, in this question we're asked to draw a logistic curve and label the carrying capacity and the point of diminishing returns, which we're going to call t0.

00:09 So from the work in this chapter, we should know the shape of a logistic curve, and we should know the carrying capacity is this l at the top here.

00:21 Okay, and we should know how this relates to our graph for a logistic function.

00:27 So this is a horizontal asymptote that our graph is going to approach in 10 wards.

00:31 It's the maximum capacity that our population can sustain.

00:39 So we're going to mark that up first.

00:42 And then we're just going to draw our logistic curve.

00:46 We're going to give it an initial value.

00:47 We can call this p0.

00:49 We don't have to know what it is.

00:51 So the population is going to start off growing fairly slowly.

00:54 It's going to increase.

00:56 It's going to reach the point of diminishing return.

00:59 So i'll put a dot right there.

01:01 And then the population.

01:03 Rate is going to slow down as it approaches this carrying capacity and it kind of plateaus off along there.

01:12 Okay.

01:12 So this is our graph of p and this point that i've marked here is the point of diminishing returns.

01:20 So this is the point where all the returns after this are no longer as good as they were before...

Please give Ace some feedback