Rates of change in other applied contexts (non-motion problems)
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Marginal cost & differential calculus
Let's say I run some type of a factory, and I've studied my operations. And I'm able to figure out how my cost varies as a function of quantity over a week, on a weekly period. And so to visualize that, let me draw it. I could draw this cost function. So this is my cost axis. This right over here could be my quantity axis. So that's quantity, or q, let me just call that q. That's my q-axis. And my function might look something like this. It seems reasonable to me. Even if I produce nothing, I still have fixed costs. I have to pay rent on the factory. I have to probably pay people even if we produce nothing. And so let's say that fixed costs in the week is $1,000. And then as my quantity increases, so do my costs. So if I produce 100 units right over here, then my cost goes up to $1,300. If I produce more than that, you see my costs increase and they increase at an ever faster rate. Now, I go into a lot more depth on things like cost functions in the Economics playlist, but what I want to think about in the calculus context is what would the derivative of this represent? What would the derivative of c with respect to q, which could also written as c prime of q, what does that represent? Well, if we think about it visually, we know that we can think about the derivative as the slope of the tangent line. So, for example, that's the tangent line when q is equal to 100. So the slope of that tangent line you could view as c prime, or it is c prime of 100. But what is that slope telling us? Well, the slope is the change in our cost divided by the change in our quantity. And it's the slope of the tangent line. This is what we first learned in calculus. As we get to smaller and smaller and smaller changes in quantity, we essentially take the limit as our change in quantity approaches 0. That's how we get that instantaneous change. So one way to think about it is this is the instantaneous. This is the rate right on the margin at which is our cost is changing with respect to quantity. So if I were to produce just another drop, another atom of whatever I'm producing, at what rate is my cost going to increase? And the reason why I'm saying right on the margin is we see that it's not constant. If our cost function were aligned, we would have a constant slope. The tangent line would essentially be the cost function. But we see it changes right over here. The incremental atom to produce here costs less than the incremental atom right over here. The slope has gone up. And it might make sense. Maybe I'm using some raw material out there in the world. And as I use more and more of it, it becomes more and more scarce. And so the market price of it goes up and up and up. But you might say, well, why do I even care about the rate at which my costs are increasing on the margin? Which is why this is called marginal cost. Well, the reason why you care about it is you might be trying to figure out when do I stop producing? Let's say this is orange juice. If I know that next gallon is going to cost me $5 to produce and I can sell it for $6, then I'm going to do it. But if that next gallon, if I'm up here, and I've already produced a lot, and I'm taking all the oranges off the market, and now I have to transport oranges from the other side of the planet or whatever it might be, and now if that incremental gallon of oranges or gallon of orange juice costs me $10 to produce, and I'm not going to be able to sell it for more than $6, it doesn't make sense for me to produce it anymore. So in a calculus context, or you can say in an economics context, if you can model your cost as a function of quantity, the derivative of that is the marginal cost. It's the rate at which costs are increasing for that incremental unit. And there's other similar ideas. If we modeled our profit as a function of quantity, if we took the derivative, that would be our marginal profit. If we modeled revenue, that would be our marginal revenue. How much does a function increase as we increase our input, as we increase our quantity on the margin?
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