WEBVTT 00:00:01.000 --> 00:00:06.300 Have you ever gone into a store to buy something, something that you really wanted, and you really 00:00:06.302 --> 00:00:15.398 didn't know what the price was going to be? Let's say for example, you wanted to buy this 10 DVD set 00:00:15.400 --> 00:00:22.198 of Golden Oldies. Suppose you went in and you thought, you know I really like these movies, and I'd pay 00:00:22.200 --> 00:00:29.600 a lot for this. So maybe you think, I'd pay oh, 300 dollars for this, and you get there and you discover 00:00:29.602 --> 00:00:37.299 to your amazement that it's 200 dollars. That 100 dollars that you didn't spend, that you would have 00:00:37.301 --> 00:00:45.701 been willing to spend, is called consumers' surplus. Alright, let's look at another example. 00:00:45.703 --> 00:00:54.000 Suppose you are manufacturing hand towels. You're the producer of the hand towels, and you think these 00:00:54.002 --> 00:01:00.300 are going to turn out really nice, and so you're going to sell them for 3 dollars apiece. Well it turns 00:01:00.302 --> 00:01:07.701 out that they look really nice, and so it is determined after careful study that you could probably sell 00:01:07.703 --> 00:01:17.800 these for 5 dollars apiece. That 2 dollars per towel is producers' surplus. This is money that producers 00:01:17.802 --> 00:01:25.901 gain because they would have been willing to sell at a lower price but didn't have to. We need to create 00:01:25.903 --> 00:01:35.000 a model for this and see if we can use calculus to find some answers to what that amount would be over 00:01:35.002 --> 00:01:45.100 the entire range of the availability of the product. Alright, let's develop a model. We start with the relationship 00:01:45.102 --> 00:02:05.501 between the quantity sold, x, and the price per item, p. By carefully studying sales records, usually both 00:02:05.503 --> 00:02:14.300 a demand and a supply function can be produced. A demand function is a decreasing function in most cases, and 00:02:14.302 --> 00:02:22.100 it could be linear, a straight line, or it could be a parabola or a piece of a parabola that opens downward, 00:02:22.102 --> 00:02:34.299 concave down, or a piece that's concave up. I'm going to just draw this demand function like this. 00:02:34.301 --> 00:02:45.100 I'm going to label it demand as a function of x. Now, the supply function. Let's think about that. 00:02:45.102 --> 00:02:54.501 Well, it's going to be an increasing function, and so I'm going to just draw that function, and here we're getting 00:02:54.503 --> 00:03:05.000 a model, so we're not doing anything specific. I'm going to label this function S of x for supply. 00:03:05.002 --> 00:03:16.600 Where these two functions cross is called the market equilibrium point. It has a value of x and a 00:03:16.602 --> 00:03:26.700 value of p as its ordered pair. We can call it (x, p). There are lots of different notations for this. 00:03:26.702 --> 00:03:35.600 Because it's something specific, we're going to call it x-sub-m and p-sub-m. But again, it depends on 00:03:35.602 --> 00:03:49.398 who's doing this what notation they use. Notice that if you drew a horizontal line parallel to the x-axis 00:03:49.400 --> 00:03:57.001 and right through the market equilibrium point, it divides the area between these two curves 00:03:57.003 --> 00:04:14.101 from 0 to x-sub-m, into two areas. This area actually represents the consumers' surplus, the amount 00:04:14.103 --> 00:04:23.398 of money that consumers didn't spend but they would have been willing to spend. We'll label that CS. 00:04:23.400 --> 00:04:31.401 This area represents the producers' surplus. That's the amount of money producers gained because they 00:04:31.403 --> 00:04:39.401 would have been willing to supply the product for less. You can see how we're going to use integration to 00:04:39.403 --> 00:04:48.000 find consumers' surplus and producers' surplus. We're going to integrate, depending on which is the upper 00:04:48.002 --> 00:04:55.803 and which is the lower, from 0 to the market equilibrium point or the value of x here. Now we don't have 00:04:55.805 --> 00:05:03.300 to find the consumer surplus all the way to this point. It is possible that you could want to evaluate 00:05:03.302 --> 00:05:10.700 either one of these at some other value of x, and if so, you would have a boundary at say some particular 00:05:10.702 --> 00:05:20.800 production level that might be something like this where this is just some other value of x. In the same 00:05:20.802 --> 00:05:28.901 way, you would integrate from 0 to this value of x, and just be careful because when you're looking for 00:05:28.903 --> 00:05:38.201 consumers' surplus it's the demand function minus the equation of this line, which is going to be y equals p. 00:05:38.203 --> 00:05:44.796 It will either be, it'll be the market equilibrium p if you're evaluating at the market equilibrium point, 00:05:44.798 --> 00:05:52.000 and it will be the p that goes with this x if it's this point. Then if you want producers' surplus, the 00:05:52.002 --> 00:06:01.800 upper function would be the line y equals either this p or this one and then minus the supply function. 00:06:01.802 --> 00:06:11.801 Let's take a look at this on a slide. The consumers' surplus is the integral from 0 to x of the demand 00:06:11.803 --> 00:06:21.500 function minus the price at that x production level. That's this area. The producers' surplus is the 00:06:21.502 --> 00:06:28.798 integral from 0 to x, so the same side boundaries, but the upper function is the price at that 00:06:28.800 --> 00:06:35.401 production level minus the supply function. Alright, let's go over the terms, the demand function, 00:06:35.403 --> 00:06:43.700 the supply function. x is the number of units produced, and p is the price per unit. 00:06:43.702 --> 00:06:56.200 Let's take a look at an example. At a current sales level of 100 sets, analysts for BTR, LLC, believe that 00:06:56.202 --> 00:07:09.000 the demand for their new 10-pack DVD set of Golden Oldies would be D of x equals 840 minus .06 x squared. The 00:07:09.002 --> 00:07:15.601 question is to find the total amount of money consumers would save because they would have been willing 00:07:15.603 --> 00:07:24.601 to pay a higher price for the DVD set. Now we have a production level of 100. We don't have the price yet. 00:07:24.603 --> 00:07:33.800 Let's first take a look at the graph of this demand function. This is going to be the shape of a parabola that's 00:07:33.802 --> 00:07:45.100 concave down. It has an intercept at (0, 840). We'll label this 840, and we know it's the demand function 00:07:45.102 --> 00:07:52.301 so we don't really need to get an exactly accurate graph. We just want to know that it's a decreasing 00:07:52.303 --> 00:08:02.600 function, and that's the general shape. Now this is going to be at a production level of 100. We need to 00:08:02.602 --> 00:08:19.000 know what the price is at that production level. So D of 100 is 840 dollars minus .06 times 100 squared. 00:08:19.002 --> 00:08:37.000 840 minus 100 squared times .06 is 600, and so the price for this DVD set at a production level of 100 is 00:08:37.002 --> 00:08:51.600 240 dollars, which is considerably lower than 840 dollars. So this point is (100, 240). That means this 00:08:51.602 --> 00:09:02.400 line, the bottom of the consumers' surplus area, has the equation y equals 240. So we are ready to find 00:09:02.402 --> 00:09:23.102 the consumers' surplus. It is the integral from 0 to 100 of the upper function over the interval, the 00:09:23.104 --> 00:09:54.200 upper minus the lower dx. Let's simplify first. 840 minus 240 is 600, minus .06 x squared dx. 00:09:54.200 --> 00:10:11.700 Let's get the antiderivative. It would be 600x minus .06 x cubed over 3, and then we're going to evaluate 00:10:11.702 --> 00:10:38.400 this from 0 to 100. Subbing in the 100, 600 times 100. Now this is .02 so let's do that right there, 00:10:38.402 --> 00:10:56.500 minus .02 times 100 [cubed]. That's F of b. This should be cubed. Let's check this. Yes, minus 600 and 00:10:56.502 --> 00:11:10.200 we're going to have 0 here and 0 here, so that's convenient. What do we have? We have 60,000 minus, 00:11:10.202 --> 00:11:18.100 now 100 cubed. A fast way to do this is to multiply one of the hundreds times the .02, and then that 00:11:18.102 --> 00:11:26.700 gives you 2. Then you have left 2 times the 100 squared, which is a 2 with four zeroes. Of course you 00:11:26.702 --> 00:11:34.900 can use a calculator, but want to hear that again? That's one of the hundreds times .02 is 2, 00:11:34.902 --> 00:11:48.701 then 2 times 100 squared. So we have 60,000 minus 20,000 is 40,000 dollars. It's a substantial sum. 00:11:48.703 --> 00:11:55.600 That is the consumers' surplus. That is the money that consumers would have been willing to spend on 00:11:55.602 --> 00:12:04.800 this but didn't have to. At the production level of 100, the price they paid was 240 dollars. 00:12:04.802 --> 00:12:10.601 This is a great application of the definite integral and the Fundamental Theorem of Calculus. 00:12:10.603 --> 00:12:18.400 Let's do another one. Analysts for Kokomo Textiles have studied the market and determined that the 00:12:18.402 --> 00:12:26.399 supply and demand functions for their new line of pineapple hand towels are, these are the two functions, 00:12:26.401 --> 00:12:36.600 demand function 30 minus x, supply function square root of x. Find the amount of surplus money 00:12:36.602 --> 00:12:46.096 that they, that's Kokomo textiles, will earn from the sale of these towels. The money they gained, and 00:12:46.098 --> 00:12:50.400 this is an explanation, the money they gained because they would have been willing to sell the towels for 00:12:50.402 --> 00:12:59.701 a lower price. Now what is that called? That's called producers' surplus. So let's draw a model. 00:12:59.703 --> 00:13:16.601 The demand function is a linear function. It has a y-intercept at (0, 30), and it has a slope of negative 1. 00:13:16.603 --> 00:13:26.598 That's really all we need. The supply function is the square root of x, which looks something like that. 00:13:26.600 --> 00:13:36.201 This is the market equilibrium point. You can tell in the problem that there is no quantity here mentioned 00:13:36.203 --> 00:13:42.001 in terms of what is being produced, and so we assume that what we're going to do here is we're going to find 00:13:42.003 --> 00:13:57.200 all of the producers' surplus, and so that will be the area bounded by this horizontal line and the 00:13:57.202 --> 00:14:04.701 square root function. We need this point. We need the x-value for the side boundary of the interval, and 00:14:04.703 --> 00:14:16.100 we need the y-value for the equation of the upper function. So we take the two functions and we find the 00:14:16.102 --> 00:14:22.600 the intersection by setting them equal to each other. We're going to need to square both sides here, so 00:14:22.602 --> 00:14:42.100 we're going to get 900 minus 60x plus x squared equals x. What did I do? I squared both sides. Now I have 00:14:42.102 --> 00:15:11.001 x squared minus 61x plus 900 equals 0. Let's see if we can factor this. This may take a few minutes. 00:15:11.003 --> 00:15:18.800 Well, the first thing that comes to mind would be 30 times 30, but of course that's going to give us 60. 00:15:18.802 --> 00:15:26.099 So we're going to have to adjust this a little bit. What if we tried something like, I'm just going to pick 00:15:26.101 --> 00:15:41.501 random numbers here. Instead of 30 and 30, maybe we'll drop down to say 25. 25 plus what is 61? 36. 00:15:41.503 --> 00:15:48.500 Ok, Well maybe you're not always going to be that lucky the first time, but I think we hit on it. Let's check it. 00:15:48.502 --> 00:16:04.300 25 times 36 is 900. 25 plus 36 is 61. But this is peculiar. We got two answers, and they're both positive. 00:16:04.302 --> 00:16:12.400 How can that be? These two graphs meet obviously here. Look at what we did. We squared both sides, and when 00:16:12.402 --> 00:16:20.401 you do that you allow for the possibility of an extraneous root. Let's see. If I put 36 into the original 00:16:20.403 --> 00:16:28.300 problem, the one without the square, I have 30 minus 36 which is negative 6, and that cannot possibly be equal 00:16:28.302 --> 00:16:38.201 to the square root of 36 which is positive 6. So this root is extraneous. So I have now 25 is the value 00:16:38.203 --> 00:16:46.901 of x I want here. I now need the price. Well, where do I go? I can go to either one of these. Let's see. 00:16:46.903 --> 00:16:56.401 If I put 25 into this one I'd have 30 minus 25 is 5. Here's a good way to check. Let's put 25 into this one. 00:16:56.403 --> 00:17:03.901 We'd better get the same thing. The square root of 25 is 5. So what we discover here is where x is 25, 00:17:03.903 --> 00:17:13.801 the price is, and let's put dollars just to help us remember that this is 5. So we have 30 minus 25 is 5, and 00:17:13.803 --> 00:17:26.800 the square root of 25 is 5. So this line has the equation y equals 5. We're looking for producers' surplus. 00:17:26.802 --> 00:17:37.000 The producers' surplus is the integral from 0 to 25 of the upper function, which is y equals 5, 00:17:37.002 --> 00:17:50.301 minus the lower function. Now, which one of these is the lower function? Mmhm. It's the square root of x. 00:17:50.303 --> 00:18:03.701 So we integrate. We get 5x minus, this is x to the 1/2, so the antiderivative would be x to the 3/2 00:18:03.703 --> 00:18:28.100 divided by 3/2, which is times 2/3, evaluated from 0 to 25. Let's find F of 25. This would be 5 times 25 00:18:28.102 --> 00:18:47.800 minus 2/3 times the square root of 25 cubed. That's x to the 3/2, so it's square root of x cubed. Then 00:18:47.802 --> 00:19:14.601 F of 0 is 0. 125. This would be the square root of 25, so that's 5. 5 cubed is 125. So that's 125 minus 00:19:14.603 --> 00:19:36.000 250/3. This would be 375/3 minus 250/3 which is 125/3. Now if all we were doing was integrating and doing 00:19:36.002 --> 00:19:42.601 skill practice that would be fine, but we want to make this look like money. So let's divide 3 into 12 00:19:42.603 --> 00:19:57.900 and get 4. 3 into 5 and get 1. And then we have 2/3 left. Well, let's write 2/3 as 2/3 of a dollar, 00:19:57.902 --> 00:20:12.400 and that is the producers' surplus. There are many, many, many applications using the Fundamental Theorem 00:20:12.402 --> 00:20:20.800 of Calculus. You now have the tools you need to apply what you've learned about calculus to applications 00:20:20.802 --> 00:20:31.801 in your major field of study. Remember three things. Stay organized. Work hard. And most of all, 00:20:31.803 --> 00:20:40.000 have confidence in your own abilities. Good luck.