Suppose that the number of individuals at time t in a...

Suppose that the number of individuals at time $t$ in a certain wildlife population is given by
$$N(t)=\frac{340}{1+9(0.77)^{t}}, \quad t \geq 0$$
where $t$ is in years. Use a graphing utility to estimate the time at which the size of the population is increasing most rapidly.

Key Concepts

Logistic Model

A logistic function is used to describe population growth where there is an initial exponential increase that later tapers off as the population approaches a carrying capacity. Its S-shaped curve reflects the slowing of growth due to resource limitations, making it a common model for evaluating biological systems over time.

Inflection Point

The inflection point of a function is where the curve changes concavity, and for logistic growth, it is of particular interest because it represents the time at which the population growth rate is at its maximum. This point is critical in understanding when a population is experiencing its fastest increase.

Rate of Change (Derivative)

The derivative of a function measures its rate of change and is used to determine how fast the population is increasing at any given time. In the context of a logistic growth model, finding the maximum value of the derivative helps identify the time at which the population experiences its most rapid growth.

Graphical Analysis

Graphing utilities are essential tools for visualizing functions and their derivatives. They allow for the estimation of key characteristics such as the inflection point and the maximum growth rate in logistic models by providing a clear visual representation of the function's behavior over time.

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