Reading: Examples of Instantaneous Rates of Change
So far we have emphasized the derivative as the slope of the line tangent to a graph. That interpretation is very visual and useful when examining the graph of a function, and we will continue to use it. Derivatives, however, are used in a wide variety of fields and applications, and some of these fields use other interpretations. The following are a few interpretations of the derivative that are commonly used.General
Rate of Change
f′(x) is the rate of change of the function at x. If the units for x are years and the units for f(x) are people, then the units for [latex] \frac{df}{dx} [/latex] are [latex] \frac{\text {people}}{\text {year}} [/latex], a rate of change in population.Graphical
Slope
f′(x) is the slope of the line tangent to the graph of f at the point (x, f(x) ).Physical
Velocity
If f(x) is the position of an object at time x, then f′(x) is the velocity of the object at time x. If the units for x are hours and f(x) is distance measured in miles, then the units for [latex] f\prime (x) = \frac {df}{dx} [/latex] are [latex] \frac {\text {miles}}{\text {hour}} [/latex] (miles per hour), which is a measure of velocity.Acceleration
If f(x) is the velocity of an object at time x, then f′(x) is the acceleration of the object at time If the units are for x are hours and f(x) has the units [latex] \frac {\text {miles}}{\text {hour}} [/latex], then the units for the acceleration [latex] f\prime (x) = \frac {df}{dx} [/latex] are [latex] \frac {\text {miles/hour}}{\text {hour}} = \frac{\text {miles}}{\text {hour}^2} [/latex] (miles per hour).Business
Marginal Cost
If f(x) is the total cost of x objects, then f′(x) is the marginal cost, at a production level of x. This marginal cost is approximately the additional cost of making one more object once we have already made x objects. If the units for x are bicycles and the units for f(x) are dollars, then the units for [latex] f\prime (x) = \frac {df}{dx} [/latex] are [latex] \frac {\text {dollars}}{\text {bicycle}} [/latex], the cost per bicycle.Marginal Profit
If f(x) is the total profit from producing and selling x objects, then f′(x) is the marginal profit, the profit to be made from producing and selling one more object. If the units for x are bicycles and the units for f(x) are dollars, then the units [latex] f\prime (x) = \frac {df}{dx} [/latex] are [latex] \frac {\text {dollars}}{\text {bicycle}} [/latex], dollars per bicycle, which is the profit per bicycle. In business contexts, the word "marginal" usually means the derivative or rate of change of some quantity. One of the strengths of calculus is that it provides a unity and economy of ideas among diverse applications. The vocabulary and problems may be different, but the ideas and even the notations of calculus are still useful.Licenses & Attributions
CC licensed content, Shared previously
- Business Calculus. Provided by: Washington State Colleges Authored by: Dale Hoffman and Shana Calaway. Located at: https://docs.google.com/file/d/0B1lkHWwO61QEM0gwOFhES2N5Tlk/edit. License: CC BY: Attribution.
