To capture the bigger picture of where this is going and to relate it to something we've seen, you've got to remember that the derivative, the thing they object to you sitting in Calc 1 is it provides information about the rate at which the function changes. As the input changes, how fast does the output changes. This section, what we're doing now, we have a way now to measure accumulation over time or over the inputs vary. You have this ability to measure accumulation. As a quantity changes over time, then the quantity accumulates as time passes. If you have more rectangles that are adding up, you're accumulating their total area. We use it to find areas, but you can also use it to find other things, because various mathematical tools enable this approximation of this accumulation. We saw Riemann sums or the first way to do that. Just to give an example using distances, let's talk about an example here, and you'll do an easy one and then we'll get to the more complicated one here. Let's say you're driving and you're on the interstate and you have cruise control set at 73 miles an hour, nice and fast, speeding along and try not to get ticket. On cruise control, how far do you travel? What's the distance? How far do you travel in two hours? Cruise control is 73 miles an hour to two hours. How many miles? Now this is easy. You don't need total calculus here, but obviously you just do 73 times 2 and that's 146. Don't forget your units here. We have 146 miles were traveled during this time period. In another way to think of it is you've accumulated a 146 miles over two hours. If you're to plot this, you could think of time on the x-axis and you can think of your speed in miles per hour. Here's 70 and so we just want to be a little bit north of 70. You can think of your speed as constant over two hours, and then what you've accumulated would be the total area under the curve. When your speed is drawn, you get your accumulation of distance. This is a nice way to apply the abstract, like just find an area just because to something that's actually use and you can see where this is applicable to like physics or other disciplines. The total miles traveled is the product of the height of the velocity function and the width of the time interval. This is a simple example. It should feel like the other ones we did. This provides an important illustrative example of accumulation, I think, and that's how this thing is going to work. Let's do a more complicated example where you're not sitting in cruise control and let's put more complicated graph on the board here. Let's put time on the x-axis as usual, and we'll put our speed will go in meters per second just to mix it up, let's go 10 and 10. Time there is 10 seconds. Let's put a crazy graph, have something like this. Starts off low, goes to high, has changing cavity and we'll say this is the velocity of a runner. This is what you measure, you sit there with a stopwatch, you measure the time, you know the speed, and the question is, can you determine the distance traveled? Here's the diversity of the last one in the prior example, to compute the distance was as easy. It was a constant velocity, we knew the time, and now though things are changing. But the idea that we had before allows us to compute this. We have 10 time intervals, we'll split them up into equal intervals, so one second, two seconds, three seconds, we do our thing here, will break it up. The idea is, how much did I accumulate over each seconds? Maybe we'll do two sub-intervals, just the cube so we don't get lost in their calculations. Six, eight, something like that's, so not drawn to scale. Imagine two seconds. On each of these two seconds sub-intervals, the velocity over two seconds doesn't change as much as it did over the entire 10 seconds. You can consider, you can think of it, you can approximate it closer to being constant. The total distance traveled can be approximated by multiplying the constant velocity on each interval, something like a little rectangle, over the length of time. This area under the curve will approximate our distance traveled. The idea is the smaller intervals, the closer it becomes to being constant, and that's where the rectangles come in. That's what we'll find that, just like the first example where we were in cruise control. Of course, you need more rectangles. The issue with this approach is that the runner's velocity is not actually constant on each sub-interval. We're only approximating that squiggle mark is super important. You have to make some decisions on when you're approximating. Do you want to approximate over? Do you want to approximate under? If you want the actual, you're going to have to take some limits. If depending on the graph of the curve, if you pick the left endpoints to be on the graph or the right endpoints. You're going to get over-approximation or under-approximation. Let's actually do this. We'll use five rectangles. We'll do left and right and see what we actually get. In the graph that I drew, I chose right endpoints to be on the graph, so let's do the right approximation. I have some graph, it goes up, it goes over. We'll keep it at 10 seconds. This is time in seconds, and this is my speed in meters per second, 2,4,6 and 8. Let's do the right endpoint. Try to draw this as best we can, and it gets a little flutter over here. These last two rectangles contribute very little error. In theory, if I drew a better picture, they would all be the same length. If I want to calculate the right endpoint, I need to get the graph of this function. Now I don't actually know the function equation. I have no idea what this is, but I was measuring it as I went along and I can compute these values. How fast was the running of zero? Where she running zero, that part doesn't matter. We can put some numbers in here, so we'll say f of 0 was about 0. That when we actually know f is 2, will give these values based on the graph 0.25. Just making these up here, f of 4 plus 3.25, so you're measuring the speed clock and how fast they go and get your radar gun out. F of 6 is about 3.9, and then f of 8 is about 9.9. This is an example, what I like about this example is you don't actually know the formula for the graph, but you have enough data points where you have these values. We know the base of each rectangle, which would be in our formula, b minus a over n or 10 minus 0 over 5. I want to do with five rectangles and equals 5 is going to be 10 over 5 which is 2. That makes sense? You can see from the picture that each rectangle has base two. Now to get the total approximations using the right endpoints, I have to take the base times the height. I do it for the first one. I do the base times the height, and I do this for all of them. X of f4 and I worked my way all the way. Should I write it out? I'll write it out. F of 6 and the base times the height for f of 8, and the base times the height for f of 10. Gross, nasty plug in your numbers. The base is constant. You can factor that out, so it's 2 times 25 plus 3.25 times 2, and then another 2 times 9 and then 2 times 9.9, and the outcomes in the calculator and you can work this out. Remember all of this, 64.8 is an approximation. Remember our word problem here, it's a physics problem, got meters. What we're doing is we're accumulating distance. If I use the right endpoints to approximate this thing, notice I don't use zero because I'm approximating the right side, I get 64.8 meters. If I were to do the left side, if I were to take the left approximation, and so now line up all the rectangles so that I get an underestimate, and I get rectangles whose left side actually lies on the curve. Now you can check me here, but based on the picture alone, hopefully see you do get this. If I were to estimate just picking five rectangles just because and working this out, it's the same formula. The only difference is that I include now f of 0, and I go all the way up to the base times the height. I use f of 8, I don't use f of 10. If you plug in all the numbers, you can check you get 44.8 for this example. My lower approximation, my left using left sums is 44.8 meters by right sum is the 64.8. The actual sum is somewhere in between. But I don't actually know the answer. I do is I can say that the actual ran around between 44.8 and 64.8 meters during the first 10 seconds of the sprint. If I wanted the actual area, the true distance, I would have to take a limit. We can work that out, but there's that gets a little harder if you're limited by the number of data points you have, or if you don't know the graph of the curve. But this is the idea of, if I know how the change of velocity over time, I can accumulate distance, I can find the formula for distance. We'll do some more examples with these, with and without the function as we go along. See you next time.