Let's try this last problem, problem seven here on this page. Now, you already know that the profit function is equal to total revenue minus total costs. So we're going to skip a step here, and just go right to the good stuff. Go ahead, and you try it on your own. Pause the video, and then we'll come back and do it together. So my profit is equal to 1,400Q minus 6Q squared, minus 3,000, plus 80Q. Now, let's take this negative sign and multiply it through over here. So we're left with 1,400Q minus 6Q squared, minus 3,000, minus 80Q. Now we're going to take the derivative of my profit with respect to quantity d pi dQ. Over here again, we've got differences, so we're using a difference rule. So 14 times Q to the power of one, bring the one down in front of the 14, we're left with 1,400. Q to the power of one minus one gives me a Q to the power of 0 which is equal to one, so we're just left with the 1,400 there. This is six times Q to the power of two, bring the two in front. Two times six is 12, and Q to the power of two minus one just gives me a Q right there. Negative 3,000, the derivative of that with respect to Q is zero, and then we have negative 80 times Q to the power of one, which gives me negative 80 times Q to the one minus one, which is Q to the power zero which is one, right here. So we set this whole thing equal to zero, and we say, this equals 1,400 minus 12Q minus 80. So I take my 12Q and bring it to this other side. Here's my 12Q right there. I have my 1,400 minus my 80, still gives me 1,320, divide both sides by 12, and I'm left with a 110. Now, you might say, wait a minute, that's the exact same answer that I got from question six above, why is that? I have these two different functions. Wow, truth is that they're two different functions. My cost function is just 1,500 higher in problem seven than it is in problem six. It means that overall, I'll be making less profit at that quantity, but that quantity is still the profit maximizing quantity. Profit maximization or optimization of profit is only one of the ways that we use calculus, but it's probably the most frequent way that you'll be using calculus in a business setting. Practice the rules, sum rule, product rule, difference rule, make sure you understand the power rule, make sure you understand how to calculate these simple derivatives. They'll be very useful in your studies. For this video, I want to talk about how good decisions are made on the margin. This might be something that you've heard before. A lot of the practice tests that you might be taking for GRE, or some of the math that you might be needing as you're going into a field of study, in business, economics, finance, accounting, supposes that you understand that decisions are made on the margin and the best decisions are made on the margin.We just look at the overall idea of marginal behavior, and what that really means. Recall that we might have a profit function that looks something like this. I'm going to just do it very generic profit function. Profit is equal to total revenue, which is a function of quantity, minus your total costs, which is also a function of quantity. Now, recall that, I'll draw this diagram again, recall that if we have a profit function, and it looks something like this, here's my profit, here's quantity on this axis, and here's dollars on this axis, at some point, my profit function reaches its maximum. At this point at the top, the derivative with respect to my profit function is going to be equal to zero. That is, the slope of the top of this function is equal to zero. D pi dQ is equal to zero. So what use is that? What can it tell us? Well, let's look at this profit function; profit is equal to total revenue, as a function of quantity, and also cost which is also a function of quantity. Just very generically, what if we took the derivative of that, and tried to solve it for this profit maximizing quantity? Say like, what does this mean, d pi dQ? Well, the derivative of total revenue with respect to quantity, I can write it like this, dTR, dQ. Here, the derivative of my total cost with respect to Q, I could just write like this, dTC, dQ, and I set that whole thing equal to zero. So when I set this equal to zero, I end up with this, dTR, dQ, is equal to derivative total cost, dQ. What's usefulness about this? Remember, the derivative of my total revenue with respect to quantity can be called marginal revenue. My derivative of my costs with respect to quantity is equal to marginal cost. This is a very important point. What this says is that, if I am trying to profit maximize, I can find the profit maximizing quantity by identifying the place where my marginal revenue is equal to my marginal cost. This gives me really good decision-making behavior, profit-maximizing behavior. Here's how this works in reality. Suppose that you are the manager of a clothing store, or a boutique clothing store at the mall, and you have several workers working for you. Now, you notice that it's a very, very slow day. There's not that many people coming in and buying clothes. So it might benefit you to understand, well, for the next two or three hours, what are the costs of me keeping each one of my workers here at the store? Those would be a marginal cost. What is the marginal revenue that each worker would bring in, in terms of selling clothes? So you look at your worker, and you look at, say someone named Dave, and you say, "Dave, it's going to cost me $10 an hour to keep you here for the next four hours, are you likely to sell $40 worth of clothes? If you are, then it's beneficial for me to have you stay here. If I'm not, and if we're not going to sell $40 with the clothes, then the marginal cost, that is the $40 of keeping you here for the next four hours outweighs the marginal revenue, the number of clothing sales that you'll make." So if you pay attention, you'll actually notice that if it's really slow, a manager might send one of their sales persons home. To say, "Dave, we don't need you anymore today. You can go home now." You might also notice the opposite happen. Like on a Friday night you go to a little ice cream shop or something like this, and you'll notice that they have an extra person working the cash register, or an extra person scooping ice cream, what have you. So you're like, geez, it cost them an additional $10 an hour to have that person working there. Sure, but if that $10 an hour person can actually bring in more than $10 of additional revenue, that is their marginal cost is less than their marginal revenue, that it makes sense to bring that person in and have them work. So businesses they're wise, are making these decisions on the margin. Asking the question, is the next worker going to cost me more than the revenue they're going to bring in or is the revenue that they're going to bring in greater than what that person is costing me for the next hour, for the next day, for the next year, what have you? This is not an identity. Marginal revenue doesn't equal marginal cost everywhere. This is a condition that says, "If we can find the place where marginal revenue equals marginal cost, that will reveal the quantity where profit is maximized." All of this is given to us by understanding calculus.