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.
population growth rate
If [latex]P(t)[/latex] is the number of entities present in a population, then the population growth rate of [latex]P(t)[/latex] is defined to be [latex]P^{\prime}(t)[/latex].
The population of a city is tripling every [latex]5[/latex] years. If its current population is [latex]10,000[/latex], what will be its approximate population [latex]2[/latex] years from now?
Let [latex]P(t)[/latex] be the population (in thousands) [latex]t[/latex] years from now. Thus, we know that [latex]P(0)=10[/latex] and based on the information, we anticipate [latex]P(5)=30[/latex]. Now estimate [latex]P^{\prime}(0)[/latex], the current growth rate, using
thus, in [latex]2[/latex] years the population will be approximately [latex]18,000[/latex].
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