Calculus: Three Big Theorems

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Curve Sketching
Introduction
Three Big Theorems Page [1 of 1]
There's some really great results that come out of calculus and things that are related to calculus, and three, in particular, I really want to share with you, because they're big results. And one of them, in fact, is one of my all-time favorite theorems of all calculus. So let me just show you these.
The first one is called the intermediate value theorem, which really isn't exactly a theorem about calculus, but sort of relates to it, so naturally it seems like a great opportunity to share it with you. So the intermediate value theorem basically says what values do you take on intermediate between two values? Well, the answer is, if you have a function that's continuous, if you have a function that has no rips in it, then, in fact, all intermediate values are taken on. So let me try to show this to you with a picture, because the intermediate value theorem really is a theorem that's designed to be viewed, rather than read.
So here's the idea: suppose I have a function that is continuous from a to b. So it's a continuous function, so you have to be able to draw it without lifting your pen or, if you're using a pencil, without lifting your pencil. And so, and if you had a crayon, I don't know why you're during calculus, but it's none of my business. So this is y = f(x), let's say. And now let's just pick two values. You can see that this is actually continuous everywhere. There's no rips at all, but all I care about is that it's continuous from some point, let's say a, and some other point here, let's call this - where should I pick b? Should I pick be over here, you think? I'll pick b way over here.
Now, let's take a look at the value of the function at a. So I go up here, so that's right here. That would be affectionately known as f(a), and this value right here, I think of this as f(b). Now, remember that this thing is continuous, there's no breaks. And the intermediate value theorem says the following, that if you give me any value at all in between f(a) and f(b) - so pick any value you want, I don't care, but just make sure it's between those two. So maybe this value right here. I'll call that Q, just to show you it could be any value at all, just to show you it could be any value at all, even Q. Then there has to exist at least one point between a and b, so that the value of the function at that point is Q. And actually this makes complete sense, like in the picture. If I just go like this and not go backwards, you see that I'm going to hit the curve somewhere, because it's continuous. I go from f(a) height to f(b) height, which means I've got to hit everything in between. That's why it's called the intermediate value theorem. Every intermediate value must be hit. So, in fact, the only way to draw that is for me to pass through that height. For example, think about that as being a tollbooth. It's a point in between. How can I possibly get that green curve to go from down below the tollbooth to above the tollbooth without actually hitting the toll? Well, I can't, it has to hit the toll somewhere and you can just come down and you'll find some point c, so that f(c) = Q.
So, the intermediate value theorem says if you're a continuous function, then if you have a point a and b and you look at f(a), f(b), if you pick any point Q in between f(a) and f(b), there will always be some point c in between a and b, for which f(c) = Q. You hit somewhere inside there. So that's a neat, fun theorem. I don't know if it's really calculus or not, but everyone talks about it in calculus. And when was the last time you saw this in Art History? Never!
Now another really neat theorem is a theorem called Rolle's theorem. This is really neat. Rolle's theorem is sort of fun. Let me show you what Rolle's theorem looks like.
Now here, not only do you need continuity, but you actually need differentiability. You've got to have the function not only without breaks, but very smooth, very smooth, so no jagged edges or anything like that. None of that will be allowed. So here's the function, very smooth and pretty. And so this is y = f(x).
Okay, now suppose that I can find two points, a and b, so here's a - if I run up to the function, by the way, then this would be f(a). And suppose I could find some other point, which I will call b, that has the property that, when you run up to it, look what you see. You see actually the exact same value as f(a). So, in fact, f(a) and f(b) are the same. These are at the same height. So suppose I have a and b such that f(a), the value of the function at a, actually equals f(b), the value of the function of b. So they're the same height. Then, if the function f(x) is differentiable, at least even in between a and b, so it's smooth everywhere, they what must happen? Well, this is an old expression that you might have heard, so if I start here and I'm going to end here and I wiggle somewhere in between, then what goes up must come down. So, in fact, there must be a point where I turn. So there must be a turning point, unless, of course, I'm just constant all the way through. That's a possibility, I guess, but, in this picture, you can see there must be at least one turning point. There may be many; watch this function: up and down and up and down. That actually had a whole bunch of turning points, but there has to be at least one turning point if the function moves.
So the statement of Rolle's theorem is that there has to exist a point in between a and b such that the derivative at that point must equal 0. So what does that mean visually? It means that there has to be a point in between a and b, I'm calling it c, such that, if you look at c and look at the derivative there - now remember, the derivative represents the slope of a tangent line, so if you think about the tangent line there, what you see is the tangent line would be horizontal. So that would have slope 0. And that's where our turning point would go on.
Now remember that we're assuming that it's differentiable, so I can't just have a sharp point up there. It has to be a very smooth up and then curve down, so it really has a tangent line there, and that tangent line has to be 0. So the slope = 0.
So Rolle's theorem says that, if you have a function that is both continuous and differentiable between two points a and b, and if f(a) = f(b), then there always exists at least one point in between a and b for which the derivative f' evaluated at that point must equal 0. And notice, this also holds for the degenerate case, when, in fact, everything is just horizontal. If there's no movement at all, then pick any point you want, because, if I just have a horizontal line, any point has slope 0. So, in fact, that's fine, too. So if it's a horizontal, boring curve, you win, any point will do. If it's a more exciting curve, like this, then there will be at least one point that will do. And that is Rolle's theorem.
Now, in fact, Rolle's theorem is really great, because it actually inspires my favorite theorem. This is my favorite theorem, and it basically just involves taking Rolle's theorem and putting it on its side. There's my favorite theorem, and this is known as the mean value theorem.
Now, it sounds sort of threatening, the mean value theorem, but it's really actually a fantastic theorem that allows you to get under and over any kind of little pickly mathematical jar you get into. So, in fact, the mean value theorem is great, and it's just a generalization of Rolle's theorem. Rolle's theorem says if you pick two points and look at the line that connects them, if that line has slope 0, meaning that the values are equal, then, in fact, there's a point somewhere in between, for which the derivative is 0. And the mean value theorem just generalizes that.
So same hypotheses; what we have here is we have a function that is very smooth, so it's differentiable between two points a and b, very smooth, y = f(x). And I pick these two points, a and b, so let's pick a point right here. I'll call this a, and then maybe I'll pick a point right here and call it b. So this here would be f(a) and this point here will be f(b). I don't want to draw that line in, because it will make things a little bit confusing, but that's just f(b) right there, that height. This is (b, f(b)) and this is (a, f(a)).
Then what does the mean value theorem say? It says that, since you're so smooth, then, in fact, the Rolle's theorem must hold even if these points aren't equal. What would be Rolle's theorem in this context? Let's connect these two points with a straight line. If we connect these two points with a straight line, the mean value theorem says there must be at least one point somewhere in between, in this case it looks like it would be right here, c, so that the derivative at that point, the tangent line, will have the same slope as the slope of this line. So, in fact, these lines will be parallel. So the point is the mean value theorem says there will exist a point where the tangent line will be parallel to the line connecting that.
How would you write that out in math stuff? What you would say is - well, what's the slope of this line? It's rise over run, so it would be . That represents the slope of this line. And if that's going to equal the slope of a particular tangent line, that must mean that this would equal f'(c), and that's for some value c that lives between a and b. And so pictorially what it means is whenever you connect the two end points, whatever that line is, this line will have the same slope. You can find a point for which the tangent line will have the same slope as this. There may be many others, but there always will exist at least one, as long as the function is differentiable, and that's the content of the mean value theorem, which, by the way, allows us to prove a lot of really important and cool calculus theorems.
Okay, congratulations on three big theorems. Show them to your friends, enjoy them. I'll see you at the next lecture. `Bye.