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See why this surpising result is true. Created by Sal Khan.
Video transcript
Let's say that we've got some function f that is continuous on the interval a to b. So let's try to see if we can visualize that. So this is my y-axis. This right over here, I want to make it my t-axis. We'll use x little bit later. So I'll call this my t-axis. And then let's say that this right over here is the graph of y is equal to f of t. And we're saying it's continuous on the interval from a to b. So this is t is equal to a. This is t is equal to b. So we're saying that it is continuous over this whole interval. Now, for fun, let's define a function capital F of x. And I will do it in blue. Let's define capital F of x as equal to the definite integral from a is a lower bound to x of f of t-- let me do that in-- of f of t dt, where x is in this interval, where a is less than or equal to x is less than or equal to b. Or that's just another way of saying that x is in this interval right over here. Now, when you see this, you might say, oh, the definite integral, this has to do with differentiation an antiderivatives and all that. But we don't know that yet. All we know right now is that this is the area under the curve f between a and x, so between a, and let's say this right over here is x. So f of x is just this area right over here. That's all we know about it. We don't know it has anything to do with antiderivatives just yet. That's what we're going to try to prove in this video. So just for fun, let's take the derivative of f. And we're going to do it just using the definition of derivatives and see what we get when we take the derivative using the definition of derivatives. So the derivative, f prime of x-- well this definition of derivatives, it's the limit as delta x approaches 0 of capital F of x plus delta x minus f of x, all of that over delta x. This is just the definition of the derivative. Now, what is this equal to? Well, let me rewrite it using these integrals right up here. This is going to be equal to the limit as delta x approaches 0 of-- what's f of x plus delta x? Well, put x in right over here. You're going to get the definite integral from a to x plus delta x of f of t dt. And then from that, you are going to subtract this business, f of x, which we've already written as the definite integral from a to x of f of t dt, and then all of that is over delta x. Now, what does this represent? Remember, we don't know anything about definite integrals or somehow dealing with something with an antiderivative and all that. We just know this is another way of saying the area under the curve f between a and x plus delta x. So it's this entire area right over here. So that's this part. We already know what this blue stuff is. Let me do it in that same shade of blue. So this blue stuff right over here, this is equal to all of this business. We've already shaded this in. It's equal to all of this business right over here. So if you were to take all of this green area, which is from a to x plus delta x, and subtract out the blue area, which is exactly what we're doing in the numerator what are you're left with? Well, you're going to be left with-- what color have I not used yet? Maybe I will use this pink color. Well, no, I already used that. I'll use this purple color. You're going to be left with this area right over here. So what's another way of writing that? Well, another way of writing this area right over here is the definite integral between x and x plus delta x of f of t dt. So we can rewrite this entire expression, the derivative of capital F of x-- this is capital F prime of x-- we can rewrite it now as being equal to the limit as delta x approaches 0-- this I can write as 1 over delta x times the numerator. And we already figured out the numerator. The green area minus the blue area is just the purple area, and another way of denoting that area is this expression right over here. So 1 over delta x times the definite integral from x to x plus delta x of f of t dt. Now, this expression is interesting. This might look familiar from the mean value theorem of definite integrals. The mean value theorem of definite integrals tells us there exists a c in the interval see where-- I'll write it this way-- where a is less than or equal to c, which is less than-- or actually, let me make it clear. The interval that we care about is between x and x plus delta x-- where x is less than or equal to c, which is less than or equal to x plus delta x, such that the function evaluated at c-- so let me draw this c. So there's a c someplace over here-- so if I were take the function evaluated at this c-- so that's f of c right over here. So if I were to take the function evaluated at this c, which would essentially be the height of this line, and I multiply times the base, this interval, if I multiply it times the interval-- and this interval is just delta x. x plus delta x minus x is just delta x. So if we just multiply the height times the base, this is going to be equal to the area under the curve, which is the definite integral from x to x plus delta x of f of t dt. This is what the mean value theorem of integrals tells us. If f is a continuous function, there exists a c in this interval between our two end points where the function evaluated at the c is essentially, you could view it as the mean height. And if you take that mean value of the function and you multiply it times the base, you're going to get the area of the curve. Or another way of rewriting this, you could say that there exists a c in that interval where f of c is equal to 1 over delta x-- I'm just dividing both sides by delta x-- times the definite integral from x to x plus delta x of f of t dt. And this is often viewed as the mean value of the function over the interval. Why is that? Well, this part right over here gives you the area, and then you divide the area by the base, and you get the mean height. Or another way you could say it is, if you were to take the height right over here, multiply it times the base, you get a rectangle that has the exact same area as the area under the curve. Well, this is useful, because this is exactly what we got as the derivative of f prime of x. So there must exist a c such that f of c is equal to this stuff. Or we could say that the limit-- and let me rewrite all of this now in a new color. So there exists a c in the interval x to x plus delta x where f prime of x, which we know is equal to this, we can now say is now equal to the limit as delta x approaches 0. And instead of writing this, we know that there's some c that's equal to all of this business, of f of c. Now we're in the home stretch. We just have to figure out what the limit as delta x approaches 0 of f of c is. And the main realization is this part right over here. We know that c is always sandwiched in between x and x plus delta x. And intuitively, you can tell that, look, as delta x approaches 0, as this green line right over here moves more and more to the left, as it approaches this blue line, the c has to be in between, and so the c is going to approach x. So we know intuitively that c approaches x as delta x approaches 0. Or another way of saying it is that f of c is going to approach f of x as delta x approaches 0. And so intuitively, we could say that this is going to be equal to f of x. Now, you might say, OK, that's intuitively, but we're kind of working on a little bit of a proof here, Sal. Let me know for sure that x is going to approach c. Don't just do this little thing where you drew this diagram and it makes sense that c's going to have to get closer and closer to x. And if you want that, you could just resort to the squeeze theorem. And to resort to the squeeze theorem, you just have to view c as a function of delta x. And it really is. Depending on your delta x, c's going to be further to the left or to the right, possibly. And so I can just rewrite this expression as x is less than or equal to c as a function of delta x, which is less than or equal to x plus delta x. So now you see that c is always sandwiched between x and x plus delta x. But what's the limit of x as delta x approaches 0? Well, x isn't dependent on delta x in any way, so this is just going to be equal to x. What's the limit of x plus delta x as delta x approaches 0? Well, as delta x approaches 0, this is just going to be equal to x. So if this approaches as delta x approaches 0, and it's less than this function, and if this approaches x as delta x approaches 0, and it's always greater than this, then we know from the squeeze theorem or the sandwich theorem that the limit as delta x approaches 0 of c as a function of delta x is going to be equal to x as well. It has to approach the same thing that that and that is. It's sandwiched in between. And so that's a slight-- we resort to the sandwich theorem-- it's a little bit more rigorous-- to get to this exact result. As delta x approaches 0, c approaches x. If c is approaching x, then f of c is going to approach f of x. And then we essentially have our proof. F is a continuous function. We defined capital F in this way, and we were able to use just the definition of the derivative to figure out that the derivative of capital F of x is equal to f of x. And once again, why is this a big deal? Well, it tells you that if you have any continuous function f-- and that's what we assume. We assume that f is continuous over the interval-- there exists some function-- you can just define the function this way as the area under the curve between some endpoint, or the beginning of the interval, and sum x-- if you define a function in that way, the derivative of this function is going to be equal to your continuous function. Or another way of saying it is that you always have an antiderivative, that any continuous function has an antiderivative. And so it's a couple of cool things. Any continuous function has an antiderivative. It's going to be that capital F of x. And this is why it's called the fundamental theorem of calculus. It ties together these two ideas. And you have differential calculus. You have the idea of a derivative. And then in integral calculus, you have the idea of an integral. Before this proof, all we viewed an integral as is the area under the curve. It was just literally a notation to say the area under the curve. But now we've been able to make a connection that there's a connection between the integral and the derivative, or a connection between the integral and the antiderivative in particular. So it connects all of calculus together in a very, very, very powerful-- and we're so used to it now, and now we can say almost a somewhat obvious way, but it wasn't obvious. Remember, we always think of integrals as somehow doing an antiderivative, but it wasn't clear. If you just viewed an integral as only an area, you would have to go through this process to say, wow, no, it's connected to the process of taking a derivative.