 The first homework assignment was officially due half an hour ago, but it got extended until Saturday morning or early. You have two more days. So you already did it. Remember that you get extra credit. If you do it two days before it's due. So you probably do something so far before you get the credit. In fact, any that you do this afternoon also weren't extra credit. There were a couple of people, but there were some people that had problems with the homework assignment. I want to cover a couple more things that you should know already, but it seems there are several people that don't. So where we left off last time, I think the question is a lot of people are happy with it. So where we left off last time, remember you don't find the area under some terms between say here and here and say there. It says chop this up into a bunch of pieces, say n pieces. So I want to go from a to b, y equals f of x. And then in this piece, I choose some point, let's say the point on the right. I call this point x1, I call this point x2, x3. The last point is xn. And then we just make the rectangle of that height, whenever the appropriate height is, these heights are negative, add them up. So that would be written as the width of each rectangle, this big, and the height of each rectangle is the function value there, and I just add them up. So for those of you that forgot, so this is the width, location needs, I mean factor out the d minus a. This gives me the approximation of the area. Now I take many, many, many, many, many, many rectangles, with the widths going to zero, lots of really skinny rectangles, and this is the area. So we have to prove that this is a given. I'm not proving this to supposedly give this in your last class. So this is defined to be the integral from a to b, that's limit. So I'll do this little job, add up all of these areas. These are choices. You're going to tell me which one's right, and then I'll tell you whether you're right. Is there an answer in? People like to answer one. That's good because they answer one with garbage. And very few people like to answer d, which is also good because that's garbage. 35% of people think it's b. So about half of you think this is the answer, and about a third of you think this is the answer, and the rest, like a and d. So let's, I guess, do this problem. These two are very similar, except this one has a one here, and this one doesn't, but we take the limit as n goes from infinity. It's infinity, but if we put the y's in infinity, then we're adding up. We have to have the width shrink and height vary. So that's why we have to go to n and stop. But then we take the limit as the number of slices we take goes from infinity. So in terms of the picture, it's kind of like this. As I take n bigger and bigger and bigger, these vertices can be skinny. So my picture looks like this. I'm going to 1 to 3. This has a 2 over n. Stuff the region from 1 to 3 into a bunch of little pieces. So how wide is any given piece? 2 over n. So the width here of each rectangle. If I move over, so for the first one, I move over one space. How far am I gone? When I start at 1, I move over one step. 2 over n. So x1 is 1 plus 2 over n. I start here at 1, and I take a step of the size 2 over n. There I am. And x2 is wherever x1 was plus 1 over rectangle. Which is 1 plus 2 over n plus 2 over n more. Which is 1 plus 2 times 2 over n. x3 is 1 plus 3 times 2 over n. And x22 is 1 plus 22 times 2 over n. So in general, xI is 1 plus I times 2 over n. So if I go over here to xI, this is sitting here at 1 plus I times 2 over n, and the height is f with that value. So that means that I want f of 1 plus 2 over n times 5. That was the law. And then I want to take the limit. Crazy looking notation is just a way for us to capture this item there. And we just have to be a little careful about chopping it up. The problem on the homework that I got about, yeah, it was a number between 1 and n. So here I is 1, and then it's here, and then it's 3, and then it's 4. So it's representing which rectangle? Here I is 1, I is 2, here I is 3. Since I don't know how many rectangles I have, I just have n. Maybe n is 100, maybe n is a million, maybe n is 5. I want to write the general rectangle for one of those rectangles. So I have to have it end on time. Now these limits are a pain in the neck to calculate, so we're not going to do that. It's also possible to go the other way. You see a limit like this, and you say what area does this represent. That question doesn't actually have a single answer. It's equal on the other way around. You already know that. But it also equals the integral from, say, 0 to 2. Here it's shifting my picture by 2. In this picture I'm thinking, okay, here's the log of x, and I'm going from here to here, and I want that area. In this guy, instead I have a different function. It's moved over by 1. Starting at 0 and going out to 2. Those are the same area. I've just shifted it over a little bit. My notion of what x is has changed, and I can write infinitely many different variations on this piece, but they're all the same thing. That's like saying, write the name of a student in this class. There are lots of different names in this class. There are lots of different integrals that this represents. They all have the same value. I'm trying to spend a lot more time on this. This is supposedly stuck in notes, so I want to move on. As I said, so any questions on this? Viewing these limits explicitly is tedious and nasty, so we don't usually do it just like we could just buy the limit. We earn some rules, and then we use those rules in the way we go, but to get those rules sometimes we have to be joined back to the definition. So those of you, well, I guess all of you, have probably already seen something that turns this into, that means the derivative of G integral from A to B of B of B, the absolute derivative that we can. So we know that this should not be used to anybody, I hope. So this is actually half of something called the fundamental theorem of calculus. We'll write it in a minute. The reason it's a fundamental theorem, or the fundamental theorem, is it interrelates two very big things that you do in calculus. It says that to do integrals you have to know about derivatives. You don't have to. Knowing about derivatives includes you know about integrals. Now if you remember, when I said last time, in calculus, is if we understand microscopic detail, then we understand macroscopic or full-size behavior. This is, again, a version of that because the derivative here is microscopic information. It says how is the function changing on a very small scale? We can relate that to how the function changes on a big scale. So one thing that we do in this class is we prove stuff. So we've got things in math. You can give a solid explanation to prove something. You can give a solid explanation of why it's true. I'm going to give a mostly solid explanation to give a solid view I'm going to gloss over a number of the details that are mathematically subtle. But why would this be true? Well we can do it in a fuzzy way. Does anyone have a clue why this is true? Well that's what the theorem says. Integrals and derivatives are opposite. But why? So, I mean, yes, it's true because integrals and derivatives are opposite. But it's like asking why is the sky blue? Well, because it's just many answerings. I mean, I don't even know how many answerings So, the sky is blue because the light that comes from the sky is blue. That's not an answer. I mean, it's an answer. But it doesn't tell you why. It just tells you it is because it is. So, okay. So has anyone seen a proof of this? Okay, you just forgot. Okay. One more person in this room can see a proof of this in their life. Two people will give a clue. Alright, so why is this true? Well, let's think about what's going on with this integral. It's my function to chop it up into a bunch of rectangles. But let me not take the limit. Let me just do it at n. So we know that. So g, this is b. And this is n. This is g of xn. And I'm going to write g of x1 way over here. g of x, the one before. And that's true. So nothing changed. And then I'm going to do it again. But I'm going to do it again. I still didn't leave enough room. So I'll put it down here. And then I'm going to get down to g of x2. g of x2. g of x1 plus g of x1. So nothing changed here. Just did that. Those are certainly equal. Called the mean value theorem. You might be confusing it with the mean value theorem. Which you went over in your conference class because it's important. But it seems stupid. Says that if I have a differentiable function, let's call it. So if g is a differentiable function, and it grows on some interval, then if I take two points from here to here, my two points which usually you call them a and b, but they're youth a and b. And the slope of the line through there is some point in between. At least one place. Just driving your car in the New Jersey Turnpike. They hand you a ticket when you get off. You go 75 miles and an hour later they hand you a ticket when you get off. And they say you were speeding because you traveled 75 miles in one hour. You must have been going at least 75 miles an hour sometimes. The speed is the same as the average speeds here. We don't know when it happened but we know it happened sometime. He only speeded once, but he speeded. Okay, so that's what that says. Now why is that useful here? G prime, let me call it because it sits between them. So that means that G prime is at distance, right? This is the slope of the line and there's the derivative. The slope of the line there is the same as the average of the two names. Times G prime of C i is this difference. But I can just replace the derivative. This guy, since I'm going to take the limit times some number h, actually, I don't need to replace the derivative. I've confused myself. Anyway, I get a bunch of different scores. I can't do this one. I can't do this one. I'm subtracting. So this is what I started with and this is what I started with. Oh yeah, thank you. Okay, so now what? I've almost stopped what I want because now I have a bunch of derivatives sitting there. I'm going to put this into an image. So that means that is going to be that first thing over times G prime and the second thing is h times G prime of C n minus 1. And the next one h G prime of C n minus 1. Well, okay. Now I'm going to take the limit and so this is the same as n going to infinity. Well, what is this? This is exactly something that looks like is the width of the rectangle. And G prime of the number of infinity. And this is the integral. Each of these guys with the slope of some places is equal to 0 so I'm adding up more and more and more and more slopes. So this tells me that I can just do that instead of just having definite areas I want to think of functions defined by things. Start somewhere here. I want here I get an area. I get more area. So this is actually a function. I can define underneath the curve. I go a certain distance t and then I stop. So an example of this position is velocity. The derivative of velocity acceleration you can think of this as being graph of the velocity and I want to know the position. If I drive my car at 60 miles an hour for one hour I know the position is 60 miles away. If I drive my car at 60 miles an hour for one hour and 10 minutes it's 66 miles away. And so I want to define a new function as the integral of the old one. So this is the integral from where I start not to t t of my old functions. To define a function the variable here is not where I stop. Not what I did in between. Fundamental theorem tells me well anywhere t is not enough. In terms of a picture this is telling me that the rate of change of the area is changed. The area here here is my area. And I took south area and now I move the end a little bit how much did this move so just the height of the function that's how much I would say width of this is just the height of the function are related. So but it's also extremely important let's how many percent of you got the right answer also this is not right so if you take the derivative and put it in that's almost the same thing 73 percent of you got it right for all stupid answers that's a stupid answer we're down to these two is the same as g and q squared we're going to do the j more so h prime of t g prime of g squared times the derivative of the way we go examples like this let me move on to find the examples the depth of recitation is important so let me point out one last things that you were supposed to know before you came to this class and that's how to do easy integrals either as definite integrals suppose I want to do the integral from 1 to 3 say that I just need to find a function whose derivative is x squared so a function whose derivative is x squared while the power here is 2 when I take the derivative of the power decreases by 1 so that I need a third power but when I take the derivative of the third power I get an extra factor of 3 that I don't like here there's no 3 signal so I just divide everything by which would be oops, what did I do wrong? this is x cubed over 3 and I plug in the values from 1 to 3 so this is anybody confuse by that at all? okay I have something that looks slightly different from 0 to 2 plus 1 squared the function's derivative is x plus 1 squared but it looks a lot like x cubed here because I say I wish this x plus 1 were really by the same amount and so that means that this becomes so in terms of u but x is 2 it was 3 because u is x plus 1 in terms of u this is u equals 1 so I'll say more about this next time it's important to write at least u these are akin to using the units inside dx is like meters now here I've just shifted by 1 so dx will vary