PhD. in Mathematics
Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.
The marginal cost function is the derivative of the total cost function, C(x). To find the marginal cost, derive the total cost function to find C'(x). This can also be written as dC/dx -- this form allows you to see that the units of cost per item more clearly. So, marginal cost is the cost of producing a certain numbered item.
Let's do a problem that involves marginal cost. I specifically want to find out how marginal cost actually compares to the cost of producing one more item. Let's take a look at our skateboard example. Suppose C(x) is the total cost of producing x skateboards. This is our cost function; C(x) is 1800 plus 10x plus 0.02x². Of course the cost is going to be in dollars.
We'll do three things. We'll find the marginal cost function, that's just C'(x). B; we'll find the c'(500) and give the units. In part c, we'll find the actual cost of producing the 501st skateboard, and compare that with our answer top part b.
We want to see really how good of an approximation the marginal cost is for producing that 501st skateboard. So first part a; find the marginal cost function. Most important thing to remember about marginal cost is it's just the derivative of cot. So the marginal cost is going to be C'(x). That's going to be well the derivative of 1800 is 0, the derivative of 10x is 10 plus, the derivative of 0.02x² is 2 times 0.02, 0.04x. That's pretty easy. So this is my marginal cost function.
Part b; find the marginal cost at 500, and give units. So I'm just going to plug 500 into this function. C'(500) is 10 plus 0.04 times 500. Now 0.04 times 500, whenever I'm multiplying by decimals, I can think of this as multiplying by 4 then dividing by 100. Multiplying by 4 gives me 2,000. Dividing by a 100 gives me 20. 20 and 10 is 30. So that's 30, and what are the units?
Let's remember that C'(500) is actually the same as dc/dx. So I can write c'(x) this way. When you write the derivative this form, it's much easier to see what the units would be. Units of the cost function divided by units of x. The cost function has units of dollars. X is just numbers of skateboards, so this would be dollars per skateboard, and that's what we have here; dollars per skateboard. So that's a nice way to get the units for a derivative was to look at it form.
In part c, we want to find the actual cost of the 501st. Let me just sketch out what I'm going to do here. The actual cost is going to be C(501) minus c(500). Let's see that this is a much more complicated computation than what we just did, but it will give us the actual cost of the 501st skateboard. So let's take this calculation over here to the right.
So I need C(501) minus C(500). Let me calculate each of those separately. First C(501). This is my cost function. It's 1800 plus 10 times 501 plus 0.02 501². So that's 1800 plus 10 times 501 is 5,010 plus 0.02 times 501² is 251,001. Then I've got to multiply this by 0.02. That's the same as multiplying by 2, and dividing by a 100. Multiplying by 2 would give me 502,002. Dividing by a 100 would give me that. So plus 5,010 plus 1800. Now adding all this together, I notice I have 10,000, 30, and 2 cents. Plus 1800 is 11,832, and 2 cents. That's my cost at 501 skateboards.
What's my cost at 500? I have to use this function again 1,800 plus 10 times 500 plus 0.02 times 500². That's just 1,800 plus 5,000 plus 500² is 250,000 times 0.02 again multiply by 2,500,000, and divide by a 100 means I put a decimal point right there. So this is 5,000 plus another 5,000 plus 1800. This is going to give me 11,800.
Now the difference C(501) minus C(500) is going to be $30, and 2 cents. Now this is the actual cost of producing the 501st skateboard. Look at all the work I just did just to find that the actual cost is $30, and 2 cents. That's the actual cost of that 501st skateboard.
My approximation using marginal cost over here was $30 per skateboard. This was a lot easier to calculate too. So this is the value of marginal cost. Take the derivative, plug in 500, and you get a very accurate approximation of the cost of one more skateboard, versus this calculation did over here which took me half of the board. So marginal cost is a really valuable concept. It gets you a very quick estimate too of the cost of producing one more skateboard.