Population Change
In addition to analyzing velocity, speed, acceleration, and position, we can use derivatives to analyze various types of populations, including those as diverse as bacteria colonies and cities. We can use a current population, together with a growth rate, to estimate the size of a population in the future. The population growth rate is the rate of change of a population and consequently can be represented by the derivative of the size of the population.
Definition
If \(P \left(\right. t \left.\right)\) is the number of entities present in a population, then the population growth rate of \(P \left(\right. t \left.\right)\) is defined to be \(P^{'} \left(\right. t \left.\right) .\)
Example 3.37
Estimating a Population
The population of a city is tripling every 5 years. If its current population is 10,000, what will be its approximate population 2 years from now?
Solution
Let \(P \left(\right. t \left.\right)\) be the population (in thousands) \(t\) years from now. Thus, we know that \(P \left(\right. 0 \left.\right) = 10\) and based on the information, we anticipate \(P \left(\right. 5 \left.\right) = 30 .\) Now estimate \(P^{'} \left(\right. 0 \left.\right) ,\) the current growth rate, using
By applying Equation 3.10 to \(P \left(\right. t \left.\right) ,\) we can estimate the population 2 years from now by writing
thus, in 2 years the population will be 18,000.
Checkpoint 3.23
The current population of a mosquito colony is known to be 3,000; that is, \(P \left(\right. 0 \left.\right) = 3,000 .\) If \(P^{'} \left(\right. 0 \left.\right) = 100 ,\) estimate the size of the population in 3 days, where \(t\) is measured in days.
This lesson is part of:
Derivatives