 Hello and welcome to the screencast today about estimating distance traveled with some data. So if you remember back at the beginning of the semester, we gave you some distance values, gave you some time intervals, and then we asked you to calculate the velocity. Well now we're going to be coming full circle. We're going to be giving you the velocity and then the time intervals, but then we're going to figure out the distance. Okay, so in this particular chart I gave you some cars velocities and the time is in hours and the velocity is in miles per hour. So the units all match so we don't have to do any conversions there. Then today we're going to work on using these given values to estimate the distance the car traveled during the two hour time interval from t equals zero to t equals two. Okay, and again the key word here is estimate. We're not going to be able to find anything perfectly, but it'll be pretty darn close. Okay, look at the values that we're given and let's look at the units again. So we've got miles per hour for our velocity and then we're also given units of hours for our time. So what can we do with these two units in order to get a unit of distance out? So in this case that would be miles. Well if we multiply these two, those hours are going to cancel basically and we'll end up with our miles. And if you think back to the last screencast, that's exactly what we did. So here we made rectangles and the base of that rectangle is going to be our time and the height of our rectangle is going to be the velocity. Okay, so exactly what we did in the last one, except on this particular screencast, we don't know a lot of details. Like we can't necessarily connect these dots. So what I did was I went ahead and drew out what this would look like because I'm a very visual person and I need to see things. So I basically just went through in GeoGebra and I plotted these various points. So in the last screencast as you guys saw, you can make two rectangles, you can make one rectangle, you can make 20 rectangles. Well in our case we really don't have as many choices unfortunately. And I want to do as many rectangles as we can in order to make this make sense. So to me if we're going to go from zero to two, I think making four rectangles make sense. Okay, because we know things at a half, we know at one and we know at one and a half. So I'm going to go ahead and circle those values. And then we also know two and then we also know zero as it turns out. Okay. Alright, so let's start with some left-hand endpoints. And again our number of rectangles, our N, is going to be four. If I start at the left endpoint though, this one's kind of awkward. Okay, so this one's going to have a height of zero. There's not much rectangle here. But then something go ahead and make our next rectangle. So I'm going to pop up here and hit this value at C and pop over. So there's my second rectangle technically. Then here my value at one is up here at this height of D. And I'm going to go ahead and hop over here. So there's my third rectangle. And then my value at 1.5 is up here at E. So I'm going to pop over and that's going to make my four rectangles. So my left-hand endpoints. And would you guys say this is an underestimate or an overestimate? Well, I'm going way under on this one, right? And I didn't even touch these two numbers up here. So yes, this is definitely an underestimate. And that's pretty much because these values are increasing. Okay, so not totally, but yeah, that's a good reason why. Okay, so I know that each of my rectangles is going to be a half. So I'm going to go ahead and factor that half out. And then my height of my first one is going to be zero. The height of my second one is 30. And if you forget, again, I've got this data written down up here for us. So I can leave that up there too. The height of my third rectangle is at 42. Oopsie, I don't want that there yet. I said four rectangles, I only did three. Okay, then the height of my fourth rectangle is at 50. So again, we're not using all the data points that I gave you, but we're using a few of them. And we're using the ones that are really going to give us the best estimate here. Okay, so that gives us a grand total of 61. And this would be in miles as we talked about earlier. Alright, I'm going to switch colors. So let me go to a different color here and let's do our right end points. And again, let's stick with n equals four. So for me, I usually like to start then at my right end point. So I'm going to start here at two and I'm going to work my way backwards. Okay, so my right end point at two is up here at g. So my first rectangle is going to be larger than this one was. Okay, then here at 1.5. And again, this red rectangle goes all the way down. I'm just going to make a big mess of my graph. Okay, my second rectangle here then is at 1.5. So that's going to go backwards. I'm going to go backwards here from e. So that's going to be my second rectangle. Then my third rectangle is going to be here at d because that's my value at one. So that's going to pop backwards here this way. And then my fourth rectangle is going to start here at c and is going to pop back this way. Okay, so hopefully you can see then this is an overestimate because I went over my data. Okay, not by a ton, but certainly by some. Okay, I think honestly this overestimate may be a little bit better than my underestimate, but I don't know, it's kind of hard to tell for some of these. Okay, so again, the width of all my rectangles is a half. And then my different heights were, let's see, my g height. And again, I'm going to move right to left. If you want to move left or right, you're going to get the same answers. But we're going to end up with, let's see, let me bring those heights down here again. Let's see, my height at g was 64. So that was my height at 2. And then my height at e was 1.5 times. So that would be 50. And then I had 42. And then I had 30. Okay, when I tally that up, that's going to give me a grand total of 93 miles. Okay, so I've got a left-hand endpoint. I've got a right-hand endpoint. One's an overestimate, one's an underestimate. So what's kind of the natural thing then to do to get a really close estimate? Well, you know, take the middle point of those two, right? Take the average of these two values. So for a good estimate, and I'm going to put good in quotes here, we can actually go ahead. And this is only because we don't have more data information, okay? So for this one, a good estimate is going to be 61 plus 93 and divide that by 2. So that gives us a grand total of 77 miles. All right, thank you for watching.