Let's try some profit maximizing problems. Remember, we're focusing on the calculus, so there might be some elements that come up in profit maximization that you might not be very aware of, and that's okay. What we're going to do is I might reveal some elements of economics and finance that are involved in this profit maximization problems, but I'll show why else are these come about a little bit later. Let's look at this, the first problem here. Suppose the demand curve is price is 18- Â½Q? And also recall that total revenue is equal to price times quantity. Part a asks us to find the total revenue. How do we do that? You can pause the video and try, and we'll come back, and we'll do it together. All right, so let's look at our function here, We've got price is equal to 18- Â½Q. Now if total revenue is price times quantity, on the left-hand side I have a price, I can multiply by a quantity right there. Well, if I'm going to multiply that by quantity, I have to multiply everything on the right hand side by quantity as well. So now this becomes my total revenue, and then I multiply both of my elements inside my parentheses by my quantity. Here's an 18Q- Â½Q squared, there it is. Now, that's my total revenue function. Let's find the marginal revenue. Well, the marginal revenue is the derivative of my total revenue with respect to my quantity. Go ahead, try to calculate that and then we'll come back and do it together. All right, so let's take the derivative of this with respect to quantity. This is dTR, dQ, now, I've got some power rules and I also have a difference. So I take the exponent of my 18Q to the power of 1, bring it down in front of my 18, that leaves me with an 18. And quantity to the power of 1- 1 is quantity to the power of 0 which is just a 1. So at least leaves you with 18 minus I think there's 2 and bring it down in front. So I have 2 times one-half quantity to the power of 2- 1 gives me 1, these guys cancel and I'm left with 18- Q, that's the derivative of total revenue with respect to quantity. That is my marginal revenue. Let's try number 5 here, suppose a firm has the following profit function profit = -Q squared + 11Q- 24. Determine the amount of output this firm should produce to maximize its profit, remember that our profit, Our profit curve, our function looks something like this. And right here, With the slope of that, d pi, dQ, when that is equal to 0, that identifies the profit maximizing quantity, so we will call it Q*. So I can take the derivative of this thing, solve it, set it equal to 0 and solve for Q and that will tell me what my quantity profit max is. Go ahead, you try it and then we'll come back and do it together. All right, so my profit function is equal to -Q squared + 11Q- 24. The derivative of my profit function with respect quantity, we're using the product rule and the summation rule here. So 2 comes down in front of that -1, I end up with -2, Q to the power of 2- 1 gives me a Q. Here's an 11, 11Q to the power of 1, so it's going to be 11 to the Q to the power of 1- 1, which basically gives me a Q. And then there is no derivative, derivative of 24 with respect to Q is 0 and now I'm left with this, I'm going to set this whole thing equal to 0. So now I've got- 2Q + 11 = 0. I'm going to move the Q to the other side so I'm left with this, 11 = 2Q and then I will divide each side by 2 and say Q = 5.5. So then this becomes my profit maximizing quantity. There's no other quantity that's going to give me a higher profit. Now you might look at this and say, well, that doesn't make any sense, am I going to produce 5.5 of these units? This could be in thousands, this could be in millions, this could be in billions of units. So producing 5.5 million units or 5.5 thousand units, 5,500 units, that's okay, right? We've got to make sure that we understand that our answer doesn't always have to look pretty, let's try another one. Suppose the firm has the following total revenue function and cost function. Total revenue equals 1,400Q- 6Q squared and total cost equals 1,500 + 80Q. Determine the amount of output this firm should produce to maximize its profits. This is a little bit more complex but still it's going to use calculus. Go ahead, give it a try and we'll come back and do it together. All right, recall that our profit is equal to our total revenue minus our total cost. So now I can substitute these functions in for each one of these elements. So my total revenue function is 1400Q- 6Q squared. And my total cost function is 1,500 + 80Q. Now you'll notice I wrote the negative sign and then I put everything in the parentheses. In a second, I'm going to use my algebra and multiply that -1 times each of those elements. So let's write it out long, 1,400Q- 6Q squared,- 1,500- 80Q. So now, we're going to use the calculus, right? So I'm going to take the derivative of pi with respect to quantity and looks like lot of differences, so we're going to end up using the difference rule. All right, d pi, dQ, So this is a 1,400Q to the power of 1. I bring 1 in front of 1,400 and I'm left with 1400. Q to the power of 1- 1, it gives me Q to the power of 0, which means 1, so I'm left with the 1,400 -. I bring the 2 down in front of the 6, that's a 12 then, and I multiply that by Q to the power of 2- 1, means Q to the power of 1, I'm left with the Q right there. - 1,500 the derivative of that with respect to quantity is 0. So that becomes a 0. And over here I have this -80 times Q to the power of 1. So I have this -80 times 1 gives me -80. Q to the power of 1- 1 gives me Q to the power of 0, which means that's 1 and I'm left with this. I set all of this equal to 0. So I say 0 = 1,400- 12Q- 80. So what I'm going to do is I'm going to add a 12Q each side. Here's my 12Q. Let's simplify this, gives me 1,320. I divide both sides by 12 here. I'm left with Q = 1320 divided by 12, which gives me 110. This becomes my profit maximizing quantity.