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.
Rate of Change: [latex]f'(x)[/latex] is the rate of change of the function at [latex]x.[/latex] If the units for x are years and the units for [latex]f(x)[/latex] are people, then the units for [latex]\frac[/latex] are [latex]\frac[/latex], a rate of change in population.
Slope: [latex]f'(x)[/latex] is the slope of the line tangent to the graph of f at the point [latex]( x, f(x) ).[/latex]
Velocity: If [latex]f(x)[/latex] is the position of an object at time x, then [latex]f '(x)[/latex] is the velocity of the object at time x. If the units for x are hours and [latex]f(x)[/latex] is distance measured in miles, then the units for [latex]f '(x) = \frac\;are \frac,[/latex] miles per hour, which is a measure of velocity.
Acceleration: If [latex]f(x)[/latex] is the velocity of an object at time x, then [latex]f '(x)[/latex] is the acceleration of the object at time x. If the units for x are hours and [latex]f(x)[/latex] has the units [latex]\frac,[/latex] then the units for the acceleration $$f ‘(x) = \frac\;are \frac<\frac> = \frac, \;\text.$$
Marginal Cost, Marginal Revenue, and Marginal Profit: We’ll explore these terms in more depth later in the section. Basically, the 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 [latex]f(x)[/latex] are dollars, then the units for [latex]f '(x) = \frac \;are \frac ,[/latex] the cost per bicycle.
In business contexts, the word “marginal” usually means the derivative or rate of change of some quantity. Thus when we are interested in a marginal function such as a marginal profit function, this will be the derivative of the profit function, and the marginal cost function will be the derivative of the cost function.
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.
Suppose you are producing and selling some item. The profit you make is the amount of money you take in minus what you have to pay to produce the items. Both of these quantities depend on how many you make and sell. (So we have functions here.) Here is a list of definitions for some of the terminology, together with their meaning in algebraic terms and in graphical terms.
Your cost is the money you have to spend to produce your items.
The Fixed Cost (FC) is the amount of money you have to spend regardless of how many items you produce. FC can include things like rent, purchase costs of machinery, and salaries for office staff. You have to pay the fixed costs even if you don’t produce anything.
The Total Variable Cost (TVC) for q items is the amount of money you spend to actually produce them. TVC includes things like the materials you use, the electricity to run the machinery, gasoline for your delivery vans, maybe the wages of your production workers. These costs will vary according to how many items you produce.
The Total Cost (TC, or sometimes just C) for q items is the total cost of producing them. It’s the sum of the fixed cost and the total variable cost for producing q items.
The Average Cost (AC) for q items is the total cost divided by q, or [latex]\frac.[/latex] You can also talk about the average fixed cost, [latex]\frac,[/latex] or the average variable cost, [latex]\frac.[/latex]
The Marginal Cost (MC) at q items is the cost of producing the next item. Really, it’s
$$MC(q) = TC(q + 1) – TC(q).$$
In many cases, though, it’s easier to approximate this difference using calculus (see Example below). And some sources define the marginal cost directly as the derivative,
In this course, we will use both of these definitions as if they were interchangeable.
The units on marginal cost is cost per item.
Why is it OK that are there two definitions for Marginal Cost (and Marginal Revenue, and Marginal Profit)?
We have been using slopes of secant lines over tiny intervals to approximate derivatives. In this example, we’ll turn that around – we’ll use the derivative to approximate the slope of the secant line.
Notice that the “cost of the next item” definition is actually the slope of a secant line, over an interval of 1 unit:
So this is approximately the same as the derivative of the cost function at q:
In practice, these two numbers are so close that there’s no practical reason to make a distinction. For our purposes, the marginal cost is the derivative is the cost of the next item.
The table shows the total cost [latex](TC)[/latex] of producing [latex]q[/latex] items.