Norm Prokup

**Cornell University**

PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

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The marginal cost and marginal revenue are the additional amount of cost or revenue that arise from producing one more item. If you take the derivative of the cost and revenue functions, you get approximately the marginal cost and revenue. **Marginal cost and revenue** are useful in solving calculus optimization problems involving economics.

In Economics there are 2 special terms that I need to introduce to you marginal cost and marginal revenue. I'll explain what they are in a second, but first let's imagine that c of x is the total cost of manufacturing x units of your product. For example if your company manufactures skateboards c of x would tell you what the cost is of manufacturing x skateboards. If your company produces skateboards c of 400 would be the cost of producing 400 of them. And the question is how much would it cost to produce one more skateboard so you're already producing 400 what does it cost to produce 401.

Well that's something that the marginal cost would tell you, c prime of 400 and c prime is just what it looks like. It's the derivative of the cost function evaluated at 400, so in Economics the derivative of the cost is called the marginal cost c prime of 400 is defined as the limit as h approaches 0 of c of 400+h-c of 400 over h. This is just the definition of the derivative as you've seen before. And of course a way of approximating this limit is to take a very small value of h and when you're on the order of hundreds of skateboards h=1 is a pretty small value, so this expression here is the difference quotient evaluated with h=1.

Now if you look at what the numerator is c of 401 that's the cost for producing 401 skateboards c of 400 is the cost of producing 400 the difference would be the cost of producing that 401 skateboards. So that cost of 401 skateboards is approximately equal to the derivative of the cost function at 400 and that's why the marginal cost is often interpreted as telling us the cost of that 401 skateboard the cost of producing one more. And there's a similar concept with revenue, so suppose r of x is the revenue function for your company. When you sell x units of your product, if your product is skateboards then r of 400 would be the revenue generated by selling 400 skateboards. So how much revenue would you get if you sold 1 more?

That's where marginal revenue comes in marginal revenue is the derivative of the revenue function, the marginal revenue at 400 will tell you about how much you would get from selling that 401 skateboards. So here again is the definition of the derivative and here again I've approximated this limit by plugging in h=1 and this quantity is exactly the revenue from selling 401 skateboards this is exactly the revenue from selling 400 and the difference would be the revenue you get from selling that 401 skateboards, the revenue from selling one more. That's what the marginal revenue tells us, so marginal cost and marginal revenue the thing to remember is that these are just the derivatives of the cost and revenue functions respectively.