Marginal Analysis as an Instantaneous Rate of Change
Primary tabs
SAMPLE LESSON
Sign in for easy to read lessons narrowly focused on a specific subject. All lessons are linked to each other allowing you to easily see the connections between different Calculus concepts. Our unique Calcmap will give you a bird's-eye view of all our lessons and help you see how important concepts are related to each other. All of the formulas are available to our subscribers, but here is a sample of the lesson you will see when you log in:
In business it is important to know how much money it costs to sell a product and how much money you expect to make in selling that product. Furthermore, these two quantities determine the amount of profit you can make in creating and selling a product. Let's denote $n$ as the number of units of our product we want to sell, $C(n)$ as the cost to make $n$ items, and $R(n)$ as the revenue gained in selling those $n$ products. Our profit in making and selling $n$ products is just the revenue minus the cost: <Sign in to see all the formulas>.
It is important to determine how much more cost or revenue will be gained in making just one more unit. For example, how much more money will it cost to increase our production from 1,000 units to 1,001 units? How will our revenue increase if we increase our sells from 1,000 units to 1,001 units? And most importantly, how will our profit increase as we increase our sells from 1,000 units to 1,001 units?
Let's focus on cost for the moment to answer these questions. Let's also assume that we are given a function $C(n)$ that tells us our cost to make $n$ units of our product. The cost to make one more unit is <Sign in to see all the formulas>. Now we know that the additional amount of cost to produce one more unit is <Sign in to see all the formulas>. But we also know that we can approximate this amount by taking the derivative of $C'(n)$ and multiplying it by 1 unit. The figure below illustrates this point.
The quantity $C'(n)$ is referred to as the marginal cost of production and allows us to easily approximate how much more it would cost to produce one more unit of a product. Similarly, we can define the marginal revenue as $R'(n)$ and the marginal profit as $P'(n)$. The marginal revenue is the additional revenue gained in selling just one more unit. The marginal profit is the additional profit gained in producing and then selling just one more unit.
Marginal analysis is not limited to cost, revenue, and profit. We can apply marginal analysis to many functions in business and economics. For example, rental property, advertising, investments, and demand functions are all subjects where marginal analysis has been applied.
