Summary of Derivatives as Rates of Change

Using [latex]f(a+h)\approx f(a)+f^{\prime}(a)h[/latex], it is possible to estimate [latex]f(a+h)[/latex] given [latex]f^{\prime}(a)[/latex] and [latex]f(a)[/latex].
The rate of change of position is velocity, and the rate of change of velocity is acceleration. Speed is the absolute value, or magnitude, of velocity.
The population growth rate and the present population can be used to predict the size of a future population.
Marginal cost, marginal revenue, and marginal profit functions can be used to predict, respectively, the cost of producing one more item, the revenue obtained by selling one more item, and the profit obtained by producing and selling one more item.

Glossary

acceleration: is the rate of change of the velocity, that is, the derivative of velocity

amount of change: the amount of a function [latex]f(x)[/latex] over an interval [latex][x,x+h][/latex] is [latex]f(x+h)-f(x)[/latex]

average rate of change: is a function [latex]f(x)[/latex] over an interval [latex][x,x+h][/latex] is [latex]\dfrac{f(x+h)-f(a)}{b-a}[/latex]

marginal cost: is the derivative of the cost function, or the approximate cost of producing one more item

marginal revenue: is the derivative of the revenue function, or the approximate revenue obtained by selling one more item

marginal profit: is the derivative of the profit function, or the approximate profit obtained by producing and selling one more item

population growth rate: is the derivative of the population with respect to time

speed: is the absolute value of velocity, that is, [latex]|v(t)|[/latex] is the speed of an object at time [latex]t[/latex] whose velocity is given by [latex]v(t)[/latex]