Marginal function like marginal cost, marginal revenue, and marginal profit are all derivatives. This means that we can undo these derivatives to obtain the cost, revenue, and profit functions by taking their antiderivatives.
For example, suppose the marginal cost for a product is given by
where x is the number of units produced. Also suppose the fixed cost are $1000. The antiderivative of the marginal cost is
where K is an arbitrary constant. By requiring that the fixed cost is $1000, we know that the cost of producing no items is $1000 or C(0) = 1000. This means that
or K = 1000. The cost function matching the marginal cost and fixed cost is
Here are several more examples worked out by students.
Example 1 Find the demand function corresponding to the marginal revenue
Remember. no revenue is incurred when no items are sold.
Example 2 Find the cost function corresponding to the marginal cost function
Assume that 16 units costs $45.
Example 3 The marginal profit in dollars per pound on Brie cheese is
where x is the amount of cheese sold in hundreds of pounds. Assume the profit is -40 when no cheese is sold.
a. Find the profit function.
b. Find the profit from selling 200 pounds of cheese.