1. Marginal Functions in EconomicsBy
Dr. Julia Arnold and Ms. Karen Overman
using Tan’s 5th edition Applied Calculus for the managerial , life, and social sciences text

2. Cost Function
Suppose the total cost of manufacturing refrigerators per week was
Now imagine you wanted to know the actual cost of manufacturing the 251st refrigerator.
Since the equation above gives you the cost of making x refrigerators, to do this you would need to find the total cost of making 251 refrigerators and subtract from it the cost of making 250 refrigerators, or C(251)-C(250).

3. Find C(251)-C(250).
So we have found the actual cost to manufacture the 251st
refrigerator was $99.80

4. The actual cost of making the 251st refrigerator is $99.80.
Notice however, that
is the slope formula of a secant line between the x values of 250
and 251 or the average cost over the interval [250, 251].
Let’s compare the actual cost with C’(250)
The difference between the actual cost, $99.80 and C’(250)
is only 20 cents. Which was easier to compute?
I hope you are saying the C’ way.

5. You may be asking yourself: What is C’(250)?
We can say C’(250) represents the approximate cost of making one
more unit.
The actual cost incurred in producing an additional unit of a certain commodity given that a plant is already at a certain level of operation is called the marginal cost.
Economists have defined the marginal cost function to be the derivative of the corresponding total cost function. In other words, if C is a total cost function, then the marginal cost function is defined to be its derivative C’. Thus, the adjective marginal is synonymous with derivative of.

6. Average Cost Functions
Suppose C(x) is a total cost function. Then the average cost function, denoted by (read C bar of x) is denoted by:
This function represents the average cost of producing x units of the commodity.
The derivative of is called the marginal average cost function and measures the rate of change of the average cost function with respect to the number of units produced.

7. Revenue Function
The Revenue Function is R(x) = px where x is the number of units produced and p is the selling price. The selling price p is related to the quantity x of the commodity demanded.
This relationship p= f(x) is called a demand equation.
Thus R(x) = xf(x)
The derivative R’(x) is called the marginal revenue function, and measures the rate of change of the revenue function with respect to the number of units produced.

8. Profit Function
The Profit Function is P(x) = R(x) - C(x) , where x is the number of units produced, C(x) is the cost function and R(x) is the revenue function.
The derivative P’(x) is called the marginal profit function, and measures the rate of change of the profit function with respect to the number of units produced and provides us with a good approximation of the actual profit or loss realized from the sale of the (x + 1)st unit, assuming the xth unit has been sold.