Calculus > Find the maximum profit in calculus

Profit maximization is one of the topics that are likely to be tested in the short-answer section of the **AP Calculus** exam. Profit is equal to a business’s revenue minus the costs incurred in producing that revenue. Profit maximization is an important concern because businesses are run in order to earn the highest profits possible. Calculus can be used to calculate the profit-maximizing number of units produced.

## Find the maximum profit in calculus: Steps

**Sample question: **Find the **profit equation** of a business with a revenue equation of 2000x – 10x^{2} and a cost equation of 2000 + 500x.

Step 1: Set profit to equal revenue minus cost. For example, the revenue equation 2000x – 10x^{2} and the cost equation 2000 + 500x can be combined as profit = 2000x – 10x^{2} – (2000 + 500x) or profit = -10x^{2} + 1500x – 2000.

Step 2: Find the first derivative of the profit equation. For example, the profit equation -10x^{2} + 1500x – 2000 becomes -20x + 1500.

Step 3: Set the equation **equal to zero**:

-20x + 1500 = 0

Step 4: Use **algebra** to find how many units are produced from the equation you wrote in Step 3.

20x = 1500

x = 75.

Step 5: Calculate the **maximum profit** using the number of units produced calculated in the previous step. In this example, inserting x = 75 into the profit equation -10x^{2} + 1500x – 2000 produces -10(75)^{2} + 1500(75) – 2000 or 54,250 in profit.

That’s how to find the maximum profit in calculus!

Tip:

If answering this type of question on the **AP calculus exam**, be sure to record this figure using the unit of measurement presented in the short-answer problem. For profit maximization short-answer problems on the AP Calculus exam, this unit of measurement is almost certainly US dollars or $. Some equations might present more than one possible answer. Some of these answers can be picked out and discarded using common sense but most often cannot be treated the same. In these cases, insert all possible answers into the profit equation to calculate their profits and then select the answer that produces the highest profit as the profit maximizing number of units produced.

Sources:

http://earthmath.kennesaw.edu/main_site/review_topics/economics.htm Retrieved July 12, 2015.

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