Fundamental theorem of calculus and accumulation functions
The fundamental theorem of calculus and accumulation functions
Let's say I have some function f that is continuous on an interval between a and b. And I have these brackets here, so it also includes a and b in the interval. So let me graph this just so we get a sense of what I'm talking about. So that's my vertical axis. This is my horizontal axis. I'm going to label my horizontal axis t so we can save x for later. I can still make this y right over there. And let me graph. This right over here is the graph of y is equal to f of t. Now our lower endpoint is a, so that's a right over there. Our upper boundary is b. Let me make that clear. And actually just to show that we're including that endpoint, let me make them bold lines, filled in lines. So lower boundary, a, upper boundary, b. We're just saying and I've drawn it this way that f is continuous on that. Now let's define some new function. Let's define some new function that's the area under the curve between a and some point that's in our interval. Let me pick this right over here, x. So let's define some new function to capture the area under the curve between a and x. Well, how do we denote the area under the curve between two endpoints? Well, we just use our definite integral. That's our Riemann integral. It's really that right now before we come up with the conclusion of this video, it really just represents the area under the curve between two endpoints. So this right over here, we can say is the definite integral from a to x of f of t dt. Now this right over here is going to be a function of x-- and let me make it clear-- where x is in the interval between a and b. This thing right over here is going to be another function of x. This value is going to depend on what x we actually choose. So let's define this as a function of x. So I'm going to say that this is equal to uppercase F of x. So all fair and good. Uppercase F of x is a function. If you give me an x value that's between a and b, it'll tell you the area under lowercase f of t between a and x. Now the cool part, the fundamental theorem of calculus. The fundamental theorem of calculus tells us-- let me write this down because this is a big deal. Fundamental theorem-- that's not an abbreviation-- theorem of calculus tells us that if we were to take the derivative of our capital F, so the derivative-- let me make sure I have enough space here. So if I were to take the derivative of capital F with respect to x, which is the same thing as taking the derivative of this with respect to x, which is equal to the derivative of all of this business-- let me copy this. So copy and then paste, which is the same thing. I've defined capital F as this stuff. So if I'm taking the derivative of the left hand side, it's the same thing as taking the derivative of the right hand side. The fundamental theorem of calculus tells us that this is going to be equal to lowercase f of x. Now why is this a big deal? Why does it get such an important title as the fundamental theorem of calculus? Well, it tells us that for any continuous function f, if I define a function, that is, the area under the curve between a and x right over here, that the derivative of that function is going to be f. So let me make it clear. Every continuous function, every continuous f, has an antiderivative capital F of x. That by itself is a cool thing. But the other really cool thing-- or I guess these are somewhat related. Remember, coming into this, all we did, we just viewed the definite integral as symbolizing as the area under the curve between two points. That's where that Riemann definition of integration comes from. But now we see a connection between that and derivatives. When you're taking the definite integral, one way of thinking, especially if you're taking a definite integral between a lower boundary and an x, one way to think about it is you're essentially taking an antiderivative. So we now see a connection-- and this is why it is the fundamental theorem of calculus. It connects differential calculus and integral calculus-- connection between derivatives, or maybe I should say antiderivatives, derivatives and integration. Which before this video, we just viewed integration as area under curve. Now we see it has a connection to derivatives. Well, how would you actually use the fundamental theorem of calculus? Well, maybe in the context of a calculus class. And we'll do the intuition for why this happens or why this is true and maybe a proof in later videos. But how would you actually apply this right over here? Well, let's say someone told you that they want to find the derivative. Let me do this in a new color just to show this is an example. Let's say someone wanted to find the derivative with respect to x of the integral from-- I don't know. I'll pick some random number here. So pi to x -- I'll put something crazy here -- cosine squared of t over the natural log of t minus the square root of t dt. So they want you take the derivative with respect to x of this crazy thing. Remember, this thing in the parentheses is a function of x. Its value, it's going to have a value that is dependent on x. If you give it a different x, it's going to have a different value. So what's the derivative of this with respect to x? Well, the fundamental theorem of calculus tells us it can be very simple. We essentially-- and you can even pattern match up here. And we'll get more intuition of why this is true in future videos. But essentially, everywhere where you see this right over here is an f of t. Everywhere you see a t, replace it with an x and it becomes an f of x. So this is going to be equal to cosine squared of x over the natural log of x minus the square root of x. You take the derivative of the indefinite integral where the upper boundary is x right over here. It just becomes whatever you were taking the integral of, that as a function instead of t, that is now a function x. So it can really simplify sometimes taking a derivative. And sometimes you'll see on exams these trick problems where you had this really hairy thing that you need to take a definite integral of and then take the derivative, and you just have to remember the fundamental theorem of calculus, the thing that ties it all together, connects derivatives and integration, that you can just simplify it by realizing that this is just going to be instead of a function lowercase f of t, it's going to be lowercase f of x. Let me make it clear. In this example right over here, this right over here was lowercase f of t. And now it became lowercase f of x. This right over here was our a. And notice, it doesn't matter what the lower boundary of a actually is. You don't have anything on the right hand side that is in some way dependent on a. Anyway, hope you enjoyed that. And in the next few videos, we'll think about the intuition and do more examples making use of the fundamental theorem of calculus.