Population models The population of a species is given by the function P(t) = Kt²/(t² + b) , where t ≥ 0 is measured in years and K and b are positive real numbers.


c. For arbitrary positive values of K and b, when does the maximum growth rate occur (in terms of K and b)?

Verified step by step guidance

1. To find when the maximum growth rate occurs, we need to determine the rate of change of the population, which is the derivative of the function P(t) with respect to time t. Start by finding the derivative P'(t) using the quotient rule.
2. The quotient rule states that if you have a function in the form of a quotient, u(t)/v(t), its derivative is given by (u'(t)v(t) - u(t)v'(t))/(v(t))^2. Here, u(t) = Kt² and v(t) = t² + b.
3. Calculate the derivatives: u'(t) = d/dt[Kt²] = 2Kt and v'(t) = d/dt[t² + b] = 2t.
4. Substitute these derivatives into the quotient rule formula to find P'(t): P'(t) = [(2Kt)(t² + b) - (Kt²)(2t)] / (t² + b)².
5. Simplify the expression for P'(t) and set it equal to zero to find the critical points. Solve for t to determine when the maximum growth rate occurs in terms of K and b.

Key Concepts

Population Growth Rate

The population growth rate refers to the change in population size over time, often expressed as a derivative of the population function with respect to time. In this context, it is essential to find the derivative of the function P(t) to determine when the growth rate is maximized, which involves analyzing critical points where the derivative equals zero.

Critical Points

Critical points are values of the independent variable (in this case, time t) where the derivative of a function is either zero or undefined. These points are crucial for identifying local maxima and minima in the population growth function, as they indicate where the growth rate changes direction, helping to determine when the maximum growth occurs.

Second Derivative Test

The second derivative test is a method used to classify critical points as local maxima, minima, or points of inflection. By evaluating the second derivative of the population function at the critical points found, one can confirm whether the growth rate is at a maximum, providing insight into the behavior of the population over time.