3.4 Derivatives as rates of change  (Page 3/15)

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.

Changes in cost and revenue

In addition to analyzing motion along a line and population growth, derivatives are useful in analyzing changes in cost, revenue, and profit. The concept of a marginal function is common in the fields of business and economics and implies the use of derivatives. The marginal cost is the derivative of the cost function. The marginal revenue is the derivative of the revenue function. The marginal profit is the derivative of the profit function, which is based on the cost function and the revenue function.

We can roughly approximate

by choosing an appropriate value for $h.$ Since x represents objects, a reasonable and small value for $h$ is 1. Thus, by substituting $h=1,$ we get the approximation $MC\left(x\right)=C^{\prime }\left(x\right)\approx C\left(x+1\right)-C\left(x\right).$ Consequently, $C^{\prime }\left(x\right)$ for a given value of $x$ can be thought of as the change in cost associated with producing one additional item. In a similar way, $MR\left(x\right)=R^{\prime }\left(x\right)$ approximates the revenue obtained by selling one additional item, and $MP\left(x\right)=P^{\prime }\left(x\right)$ approximates the profit obtained by producing and selling one additional item.