Transcription
So now we start getting into the heart of calculus and uh start looking at derivatives. So first, we will consider the one-dimensional case, and later on, in a future lecture, we will start looking at derivatives for multi-dimensional uh vectors. So first, we'll start with the definition. We'll try to define what a derivative is and then see some of the properties of uh derivatives.
So, definition. So let's consider an interval uh which is uh call it U. This is an interval in R. So this is an interval, and we will define a function on this interval. So the function is defined from U to R. Then we say the function is called differentiable. Is called differentiable. Now, differentiability is a local property. So we have to uh talk about differentiability at a particular point. Um, so differentiable at a. So we will later on generalize this and talk about differentiability uh over the entire function. Uh, but first, we define differentiability with respect to a point. So this is differentiability at a particular point A, and here this point A belongs to uh U. And this function is called differentiable if it satisfies the following. So I'm going to call this f- of A. This is the symbol that we'll be using. Uh, we have the limit where H goes to zero. We look at the fun, we look at the following function: we have F of A + H - F of A / H. We say that this limit actually exists, then this function is differentiable.
Okay, so just to recap here. So we have a function which goes from this interval U to R, and we call, we say that this function is differentiable at a particular point A within this interval U if the following limit exists. So this limit is given by F of A + H - F of A / H. So this is the quotient of F of A + H - A and H. So if this limit exists, then this function is called differentiable. And we denote uh this uh uh differentiable uh at A through another symbol. U, so we [Music] often we write this as f- is equal to DF by DX.
Now let's look at intuitively what this is saying. Um, so intuitively, um, this derivative is nothing but the slope of the function at a particular point A. So if you have a function like this, um, let's say we have a function like this. So we look at a particular point A, maybe for example, this is A here, and we look at A + H, which is somewhere here. A + H, that's going to be here. And uh, the definition of this derivative uh is we look at the difference in the in the function value between these two points. So that is this difference here, the the height of this rectang, of this triangle here, divided by the width of this triangle here. So the width is simply going to be A + H - A, which is H, and the height is F of A + H - F of A. Now we're going to, of course, we want to look at this in the limit that H goes to zero. So what does that mean? That means we have a uh sequence which we define A_n as A + H_n, and this H_n is a sequence, and this H_n converges to zero as n goes to infinity. Okay. Um, and we also want this H_n to be greater than zero. So this is the limit from this one direction. Limit from the left side. Of course, you can also define the limit uh from the right side, and we will, we will actually uh see this later on. Um, uh, but in in this definition here, I'm assuming H_n is greater than zero and approaching zero from uh the left side.
So let's actually see another picture to see why this is uh the slope of the function. So, um, so we have another picture, maybe right. Uh, let's draw a function which looks like this. Okay, and we are interested in looking at a particular point A, for example, here A, which is the point here on the function. So now let's say initially, maybe our H is not close to zero. H is some very large number. So it's going to be, um, our H is going to be somewhere. Okay. Uh, so maybe this is A + H is very far away, and I have a my function value over here. So I'm going to, uh, approximate this, uh, sorry, I'm going to compute the derivative according to the definition by considering this triangle here and taking the quotient of the height and the width. Okay, so that's what the uh derivative would be according to our definition. Now H is very far away, right? Uh, but what this derivative then says is that I'm going to look at this value or look at this uh uh height uh divided by the width as the value of H goes to zero, right? So that means that we have this H uh getting smaller and smaller. So I'm going to get closer and closer to zero, and that means my triangle is going to get smaller and smaller. So let's say I take one of these points, points here, then my triangle is going to uh be this very small triangle here, and I'm looking at the slope of this triangle here. And as this H gets closer and closer to zero, that means I get very close to A, then I'm looking at the slope of this line over here, right, at the point A. So in in plain English, basically the derivative, or informally, derivative is the slope of the function at the point A. Okay. And uh, or more, you can, you can also say that this is actually the slope of the linear approximation of a [Music] function at A.
So what does this mean? Well, if we, if we take the function F of X and we, uh, actually let's take the function A, or rather, actually we can, uh, so take the function F. I'm going to approximate this at the point A by a linear approximation. So that means F of A + X - A times some number B over here, which would be our, which would be our slope, and it's also our derivative. So, so this is the slope, and it is the derivative at A. So we're taking a function F, uh, we're taking the function and we are approximating this by a line at the point A, and we look at the slope of this line, and that line, that slope is precisely the derivative of the function at that point A. All right, so this is, uh, uh, this is the definition of a, of the derivative of a function.
Now let's at at a particular point A, right? So now let's say, uh, the function, the function, the whole function is called differentiable. So now we are looking at, uh, the whole function, not just at a particular point. A function is called differentiable if it is differentiable for all values of A in our interval U. Furthermore, it is continuously differentiable. Continuously differentiable. So this is the concept we're trying to define here. Continuously differentiable if it is differentiable, differentiable, and the derivative, the D, the derivative is actually continuous. So it is differentiable, and the function which is the derivative, the function F dash, which is our derivative, which is going from U to R, also when A maps to F dash of A, the derivative is is continuous. So this is the, the definition of for the, of a differentiable function. So a function is differentiable if it is differentiable at all points, and it is continuously differentiable if it's differentiable and the function that you get after differentiation is also continuous everywhere.
Now, of course, uh, this is just differenti, differentiable once. Uh, but you can also, uh, define higher derivatives. Call them higher derivatives by simply applying the same concept on the result of the first derivative. For example, ODS many derivatives. So we can basically, we can repeat this process. We can repeat the process of [Music] taking derivatives. So we have f- is equal to DF by DX. Uh, then we can do f double dash, which is taking the derivative again, the second time. But so now we are taking the derivative of f- with respect to X. In fact, we can keep doing this, f triple dash, f, you know, four dashes and so on. Uh, a simpler notation to Def for writing this, uh, higher order derivatives is to use this notation N, where you take the derivative N times. So the N, the number N denotes the nth derivative. So this denotes the n derivative, again, if it exists, right? If it does not exist, then you cannot keep taking as many derivatives as you want, but you can keep taking derivatives as long as they exist.
Okay, now that we have defined, uh, what a derivative is and differentiable function and so on, um, let's actually look at some important theorems which we will not be proving, but I will be stating them here. So these are just important theorems related to derivatives. Um, the first one here is that if a function is differentiable, it implies that the function is also continuous. Of course, the other way does not hold, but let me write this down. So theorem, uh, basically differentiable implies continuous. And as I said, the other way does not, this way does not hold. That's not true. If it's continuous, it doesn't mean it's differentiable, but if it's differentiable, then the function is continuous. So let's, uh, formally state this here. So let F, let F be differentiable at A, at a point A, then there exists a constant which we call C_A. The constant is dependent on the point at which you're taking the derivative, A, C_A, such that on a small ball around A, we have F of X - F of A is less than or equal to C_A * X - A. Okay, and then, so this basically means, in particular, F is continuous at A. So this is the statement of the theorem. Basically, it says that, uh, if a function is differentiable, then there is a constant C_A such that such that the, uh, difference of the function F(X) - F(A) is bounded by a constant C_A * X - A. And, uh, this statement is very similar to a local Lipschitz continuity, where the constant C_A is the Lipschitz constant. And we have seen this concept of Lipschitz continuity in a previous lecture. And this, although the statement is, uh, uh, defined for a particular point A, this statement also holds both pointwise as well as globally too. Which means that, if a function is differentiable everywhere, then it is also continuous everywhere. So this statement, in particular, is saying if it's differentiable at a point A, it is different, it is continuous at the point A. But in general, this is also the case for globally. So if the function is differentiable everywhere globally, then the function is also continuous everywhere globally. And this constant C_A is like, in some sense, the slope of the function. Uh, and as long as this function, the slope is a bounded finite number, then we, this this property, uh, holds. So this is one important, uh, theorem, uh, for differentiable functions.
We have another one, which is the the intermediate value theorem, but for derivatives. We have a theorem. This is the intermediate value theorem for derivatives. So let's see what this says. Basically, this says that if I have a function which is, uh, on differentiable, it's a continuous function which is once differentiable in an interval A. So this one over here, if you remember, is is the notation that we use for a continuous function that is once differentiable. So maybe let me write that down here. That is, functions that space of functions [Music] on A that are once continuously differentiable, differentiable, squeeze that there, then the, then there exists, then there exists a, uh, Zeta belonging to A, B, such that F of A - F of B, no, that's a typo, it should be F of B - F of A / B - A is F dash of Zeta. Okay, let's see a picture to understand what this is, uh, essentially saying. So, uh, let's say we have a function, um, some function like this, and, uh, we have this interval A and B. So this is A here, this is B. So we are looking only at this interval. Okay, and, uh, this is corresponding to F here, this is corresponding to F(B) here. All right, so then, uh, we look at the slope between this A and B. So if we look at the slope of this line over here, so this part, the vertical part is F(B) - F(A), the horizontal part is B - A. So we're looking at the slope, uh, of F(B) - F(A) / by B - A. Then the theorem basically says that there is another point somewhere in this interval AB. It has to be inside this interval AB, for example, let's say over here, and this is our point, uh, Zeta, where the slope, where the derivative of the function at this point Zeta, that means the slope of this point Zeta, let's say over here, this slope actually matches the slope F(B) - F(A) / B - A. That means if I have this interval AB, if I take the two extreme points and I look at the slope of the function between those two, then there exists another point Zeta in this interval AB such that the slope of the point Zeta at the point Zeta is equal to the slope of the function between the two extreme points AB. So, uh, that's that's what this intermediate value theorem, uh, says for derivatives. Now, of course, there's some, this only works, uh, for continuously differentiable functions because otherwise, F dash is not continuous, and the intermediate value theorem does not apply for non-continuous functions. If you remember from earlier, the intermediate value theorem applies for continuous functions. So that's why we need F dash to be continuously differentiable for the the intermediate value theorem to apply to the derivative as well. So this is the, this is one theorem.
We have another, uh, important theorem, the last one for, uh, derivatives here, or the last one that we will see for derivatives. So this is about exchangeability of the limit and derivative. So exchanging limits [Music] and derivatives. So let me first state the theorem and then we can, I can explain what it is. So let's say we have a function F_n, which is in the interval A to B, from A to B to R, and this F_n is also, uh, once continuously differentiable in the interval AB. Once continuously differentiable. If the limit of the sequence of functions, so this is the limit of the sequence of functions, is limit as n tends to infinity, F_n of X, if this limit exists for all X in this interval A, B, and the derivative F dash converges uniformly, then F is and F is continuously differentiable. F is continuously, continuously differentiable, and we have the following, and we, we have that, uh, F dash of X is equal to the limit F_n as n tends to infinity dash of X, and this is the limit of F dash_n of X. Okay, so, uh, let me, let me see, let me explain what this part over here, the first part over here is. So this one is saying I first take the limit, and then we take the derivative. Okay, so first take the limit of F_n, then, so after we take the limit, we obtain F, the function F, which is the limit, then compute derivative. While in the second case, what we will do here is first we compute, first compute the derivative of F_n, and then take the limit. So we compute the derivative of the sequence F_n, we then take the limit, yeah, then take the limit of the derivative. And this says that these two things are equal. That means I can exchange between the taking the, uh, limit and then derivative, and exchange that with taking the derivative and then taking the limit. So I can do this. Okay, so this is the, uh, property that I have, U, if the function converges, if F dash converges uniformly, right? Um, and F is continuously differentiable. So for this property to hold, which is that the exchanging between limits and derivatives, it's very, very important, critically important for uniform convergence or uniform continuity. Uh, without this, uh, we would, we would not be able to, uh, have this theorem.
Now, let us, let's see how to compute, uh, integrals of functions. So first, uh, intuitively, what is an integral? It's essentially the area under the function. So if I have a function, let's say like this, in an interval A to B, then the integral is nothing but the area under this curve. So it's the, this, this whole area under this, this particular curve. So that's the intuition for what an integral of a function is, trying to compute the area under the curve. Now, there are many different ways to compute integrals. Um, uh, one of the simplest, uh, way is the Riemann integral. It's not always, uh, useful to compute integrals, but it's a very simple method for computing integrals, uh, that we should all know about. So let's learn about Riemann integrals and, and in in subsequent lectures, we will talk about other types of integrals. So let's formally define, uh, what integration is and how Riemann integral actually works. So we consider a function, and this is going to be a function on a bounded domain, so F is a function on a bounded domain A to R, and we're going to assume that the function itself is bounded, that F is bounded. And what does, uh, what does bounded mean? Basically, it means that there exists a value L and U belonging to R, uh, such for all X in this interval AB, we have L less than or equal to F(X) less than or equal to U. So L is the lower bound of the function in the interval AB, and U is the upper bound of the function in the interval AB.
Now, the way Riemann integral works is by taking by dividing this function into small rectangles and then computing the area of the rectangles and adding them up. So we can actually see a, uh, illustration here. So let's say you, you're going to consider going to break this into many rectangles, and these rectangles may not all need to be of the same size. This, uh, this, uh, they can be of different sizes, but nonetheless, you divide this function into small rectangles like this. Maybe I can just draw the rectangles here, and we're going to consider, uh, we're going to compute the area of these rectangles. So area like this, maybe area this, and so on, and we're going to add up all of these, uh, areas of all the rectangles, and that's how we're going to arrive at the integral of the function. So this is the high-level idea. Of course, uh, there can be two different ways in which you can create this rectangle. So we're going to look at a lower bound of this rectangles and an upper bound of this rectangles and show that both of these converge, then we have a Riemann integral. So that's the basic idea, and that's where we are going at. Uh, but now let's actually define this, uh, formally. So first, we need to introduce this concept of these rectangles. So we need to partition our, uh, domain AB into small partitions. So, uh, we consider, we consider, uh, these points X0, X1, and so on, up to Xn, such that the first point is A0, is A, is X0, uh, and these points are ordered. That means X0 is less than X1, less than or equal to, uh, actually not equal to, they have to be strictly less than X2, and so on, up, less than Xn, and Xn is basically the other, uh, limit of the interval B. So this is going to be equal to B. So these points, these points that we just defined, they introduce a partitioning of the domain AB. So the points introduce a partition of AB into N intervals. We can write each interval as I_k, which is going to be, it's defined as the region X_{k-1} to X_k. So for example, you could have an interval over here, uh, so this interval, this could be an interval I_k, for example. So one of these partitions, one of these rectangles over here, is, uh, defines an interval. And now we're going to define the concept of taking, uh, sums of these integrals, of these rectangles, and we're going to introduce the concept of a sum that is, uh, from above and a sum from below. So let's actually see what that is. So first, I will draw this. So let's say we have, uh, some function which is like this, and we're going to consider a rectangle. We're going to consider two rectangles. So first of all, we're going to consider the two endpoints of the interval, so let's say X_{k-1} and X_k, and from here, we're going to consider two rectangles. So one rectangle would be a lower bound of this, of this, uh, function, or a rectangle that is completely contained within or completely under the function. So it would look like this. So this is one, uh, rectangle. The other rectangle would be the one that is out, that is the upper limit of this. So it's going to be a rectangle that's going to take the maximum value and then it's going to create that rectangle, L, like that. So the full rectangle there. So, so we basically have two rectangles, one is being defined by the lowest value of the function in this interval I_k, and the other is being defined by the largest value of the function in the same interval. So capital M_k. So now we can, uh, now that you get the idea, so we have two kinds of rectangles, one defined by the smallest value of the function, and the other defined by the largest value of the function. Now we can formally define these. So we're going to define in terms of supremum and infimum. So we're going to define, uh, small m_k as the infimum of the function, inimum of F in this interval I_k, and we're also going to define capital M_k as the supremum of the function in the same interval I_k. And, uh, we know that these two, uh, supremum and the infimum exist because we assume that the function F is bounded. So this exists since F is bounded. So now this is, this is, uh, defining the, uh, smaller and the bigger rectangle for a one interval. So now let's define how to compute the integral of this function. So we're going to define first the lower sum. Define the, what is called the lower sum, small s of, actually, we're going to define the lower sum, small s of the function F using the partition, using the points that we created, the partitions that we created, from all the points X0 to Xn. This is defined as the sum K = 1 to n, let's say there are n intervals. Now we know that the area of each rectangle is is simply the height times width of the rectangle. So the width of the rectangle is simply the, uh, difference between X_k and X_{k-1}. So it's the length of this interval I_k, which is X_k * X_k, X_k - X_{k-1}, times the height. And the height for the lower sum is going to be the smallest value of the function in that interval. So m_k, just to be clear. So this is going to be length of I_k, which is X_k - X_{k-1}. So this is how we define the, uh, lower sum. Similarly, we will now define the upper sum. So we define the upper sum using the supremum of the function value in the interval. So we define the upper sum, so upper sum, capital S, again, this is with the function F and the points X1 to Xn, which define the partition here. So this is K = 1 to N, the sum of all these, uh, rectangles. So again, the width of the rectangle is I_k, uh, so the length of I times the height of the rectangle is capital M_k, which is the largest value of the function in the interval I_k. So this is the lower sum and the upper sum.
So now we define a quantity. Now we define two new quantities, something called, uh, J star with a subscript, the star in the subscript. So this is going to be the supremum over all possible partitions. So we just, in the example that we drew, we just drew one possible partition. There could be many, many such partitions. So among all possible partitions, we're going to take the supremum of the lower sum, so s of F, comma, partition. And we're going to define another quantity, which is J star, which is going to be the infimum over all partitions, but now we're taking the infimum of the upper sum, capital S. Okay, so we define two new quantities. So, uh, the first quantity is the, uh, sum from below, and the second quantity is the sum from above. Uh, and we're looking at the largest sum from below and the smallest sum from above. So let's, uh, maybe take a look at a picture in order to understand, uh, this concept of partitions and supremum over partitions. So maybe you have a function which is like this, and now you can have a very coarse partitioning of this function. So you can have, uh, this is maybe one value A and B here, and maybe I just have one, two, and maybe just three partitions, that's it. Okay, so this is actually a very coarse partition, and I can compute the, uh, the area, the sum from below, right? So by considering these rectangles here, so I'm going to compute this area, this area plus this area. So this is the sum from below. So this is a coarse partition, and we're looking at this from below. And I can compute this. This is just one partition. And this for the same function, I could have another [Music] partition. Maybe it looks like this. Let's assume that's the same exact same function, but now I have a partition that is much more fine. So I'm going to divide this, uh, interval AB into many, many, many small rectangles, and I can still compute the, uh, sum from below. So I can still com, I can still consider small rectangles like this, the lower rectangles. So I, I take these regions here. So I can take a finer partition from below, and I can also look at, uh, similarly, I can look at partitions from above. So let's maybe consider one example here, just one, one drawing, and hope I can draw the same function. Let's say assume that's the same function again. I have the interval A, I have the interval B, and let's say I have some partition like this, but now I'm going to consider, uh, functions, uh, from above. So just going to look this, this, and like this. So I'm now considering this area here, and this would be a partition from above. So there can be, uh, many partitions, as I showed here, there's a coarse partitioning and a fine partition. Now you want to compute this lower sum and this upper sum for all possible partitions, and then look at the supremum of the lower sum and the infimum of the upper sum. So those are the two quantities, J subscript star and J superscript, uh, star.
Now we are ready to define the Riemann integral. So we call F Riemann integrable, inte, grible, or in, in, grable, if we have that the sum that we looked at, J star subscript, super, subscript J subscript star, and J superscript star, that means the lower sum and the upper sum are both equal to each other. Then we denote this J star through the symbol which we are all familiar with, as the symbol for integration, A to B, F of T DT. So when we compute the sum from the lower rectangles and we compute the sum from the upper rectangles, and if both of these sums converge to the same thing, when they're both equal, that's when this function is Riemann integrable, and the quantity that you get from the sum is the Riemann integral of the function.
Now, uh, this is the definition of Riemann integration. Now let's actually look at some, uh, look at a theorem which shows, which talks about some properties of this Riemann integration, and then we look at some shortcomings of Riemann integration. So first, uh, theorem. So if I have a function from A to B going to R, and this function is monotone, then it is integrable. And this is, this is not just Riemann integrable, but this is integrable in general through other methods too. So it's general integration. Uh, what does monotonicity mean? Basically, it means that if I have X1 less than X2, means that F of X1 is also less than F of X2. This is is monotonically increasing, but you can do the same thing for monotonically decreasing. So both are both are fine, but it has to be a monotonic function. The second property is that if, if I have a function F, and this function is continuous, then again, the function is integrable. In fact, uh, this is actually, uh, true in a weaker form too. So actually, even this is a true even if F is continuous everywhere except at a finite L many points. So what this says is that even if the function is not continuous everywhere, but it can, it can afford to be not continuous at a finite many points, then the function is still integrable. So these are both, uh, theorems about general integration, not not something specific to Riemann integration, but they also, of course, equally hold for Riemann integration. These are two nice properties. So a function is monotone, then it's integrable, and the function is continuous, uh, then it is also, uh, integrable.
So now let's look at some shortcomings of Riemann integration. Um, in fact, there are many shortcomings, and that's why we, uh, there are other, uh, definitions of other ways of computing integrals, and we will see that, uh, in a future lecture, but let's look at what the shortcomings are. So shortcomings. One, simply, uh, many functions are not even integrable, and in which case, Riemann integration does not work, right? So we can look at an example of such a function, but in general, many functions are not integrable. So one example is a Dirichlet function, which is defined as F(X) is equal to value of 1 or 0, depending on whether X is a rational number or not. So to, in order to see why this is not integrable, you can actually try to draw this function, and the set of rational numbers would be, let's say, a bunch of numbers like this. So these are all rational numbers, and at which the value of the function is one. So these are rational numbers, Q, and for all other values, uh, that are not rational, again, there are many of them here, the value would be zero. So these are functions that are not rational, so R - Q. So this is a scenario where the function is not integrable. In fact, the function is, uh, keeps jumping up and down between these rational numbers and, uh, between 1 and 0. U, and no matter how small of an interval that you take, because rational numbers are dense in R, no matter how small of an interval you take, you will always have points that are, uh, in where the function takes a value of 1, and the function takes a value of 0. That means they always find points where which will have rational numbers and also not rational numbers. So, uh, to simply state this, U for any interval I_k, which is between, let's say, X_k and X_{k-1} or X_{k+1}, the upper bound is 1, and the lower bound is going to be 0, right? And then we will have that, then the lower sum, J subscript, uh, star is going to be less than or equal to, less than the strictly less than the upper sum. So that these actually do not converge to each other. So the lower sum is simply B - A * 0, and this upper sum is going to be B - A * 1, and these are not equal to each other. So that's why this is an example of a function that is not even integrable, uh, and in which case, uh, of course, you cannot use Riemann integration.
Another drawback is that one cannot prove theorems about exchanging integration, uh, with limits with these, with Riemann integration. So one cannot prove theorems about exchanging integral with limit. So that means, for example, if I have the limit as n tends to infinity, integral of F_n DT, then is it the same as integral of the limits F_k DT? We cannot, we cannot exchange the, the position of the limit and the integration, uh, if you're looking at the Riemann integral. And another drawback is that this is hard to extend to other spaces. So this can mean many things. So other spaces could be, for example, so first of all, the Riemann integral is defined on R, on which there is a notion of ordering of numbers. Uh, so we have X1 less than X2 less than X3 and so on, and we use that ordering notion in order to define the integral. But on higher dimensional spaces, for example, you're in a two-dimensional space or a three-dimensional space or N-dimensional space in general, there is no notion of ordering. And how do I order the points? And so if you, without a notion of ordering, you cannot actually define this. Uh, so the other space could be, for example, spaces with no notion of ordering of points. Uh, another example is basically when you go to higher dimensional spaces. Okay, so simply higher dimensional spaces, even if you could somehow create a notion of ordering in this higher dimensional spaces, you will need an exponentially growing number of partitions in order to compute this integral, and that is not scalable. It's exponentially growing with the number of dimensions. So that's going to be hard in order to compute. So these are some drawbacks of, uh, the, uh, Riemann integral. And, uh, later on in the Le, in the course, we will be looking at another form of integration, which is called the Lebesgue, Lebec integration, which will overcome some of the shortcomings of for Riemann integration. So this is the one which is more modern and more, uh, useful form of, uh, integration that we will see later in the course.
So far, we have seen two concepts in calculus, taking derivatives and then computing integrals. So now we look at the most important theorem in calculus, which is called the Fundamental Theorem of Calculus. And this theorem tries to provide the relation between derivatives and integrals. Now, just to recap, we looked at derivatives, and we said derivatives are intuitively nothing but the slope of the function at a particular point. So let's say I have a point X over here, the derivative is nothing but the slope of the function at this point X. While integration is, if you take the same function here, integration is actually trying to compute the area under this, area under the curve. And it's not obvious how, what is the relation between these two, or why these two things are related to each other, or how you go from from one to the other. So the relation between these two is not very clear, and this is what the Fundamental Theorem of Calculus is going to be about. It's going to try to relate the gradient of the, or the derivative of the function with the integral of the function. So we're going to state the theorem in two parts and then we're going to try to prove both parts of the theorem. So let's get going.
So we're going to look at the first part of the theorem. So theorem one. So theorem one basically says that if I have a function on a bounded domain A to B going to R, and this function is Riemann integrable, Riemann integral, and let's say the function is also continuous at at some point Zeta, which is in our domain AB, and we let C be another point in the domain AB, then the following function, then the function which we're going to define as follows, F of X, capital F of X, which is defined as this integral of F of T DT, the limits of the integration being the point C that we chose up to the point X. So this is the, this is the new function that we are defining, F of X is going to be this integral from C to, uh, X. So this function is differentiable at the point Zeta, and not just differentiable, but the, the value of this derivative is the same as as the value of the function small F at the same point. So, and so if F is, um, continuous in this interval A, then cap F is once continuously differentiable in the same domain A, and F dash of X is F of X for all X belonging to the interval AB. So this is the first part of the theorem. So let me first maybe note this that this symbol here is continuous, and this symbol here is continuously differentiable, once continuously differentiable. So if you look at this theorem, what it is saying, first of all, it defines this function F of X, cap, cap F of X, which is the integral of the small F from, uh, a point C to the value X. And if I take the derivative of this cap X at the point Zeta, then it is equal to the value of the function, function small F, at computed at Zeta. So that's, and, uh, this is true not just at a single point Zeta, but it also true for all the points in the interval AB. So this is the first part of the theorem. Uh, we're not going to prove this yet. We will prove this, uh, in just a second, but let's actually look at the second part of the Fundamental U Theorem of Calculus.
So we have theorem 2. So this is saying now let's assume that I have a function F, cap F, from A to B, and I'm going to assume that this is already continuously differentiable. Then I have that the integral of from A to B of the derivative of cap F is going to be equal to the difference of the value of cap F at B minus A. So F of B minus F of A. So if I have the function cap F, it is continuously differentiable, then I take its derivative, and but then integrate it from A to B, it's going to be equal to the value of the function at B minus the value of the function at A. So, uh, this is, this is, this is the Fundamental Theorem of Calculus, it's in two parts over here, uh, defined in two parts. So what this Fundamental Theorem of Calculus really says is that the integration and taking a derivative are inverses of each other. And this is not very obvious, because, uh, well, uh, taking a derivative is related to the slope of the function at that point, and computing the integral is related, is the area under the curve, and it's not very obvious that taking the, the slope of the function at a point is sort of the inverse of the under the curve and vice versa. But that's what the theorem says. That's that's why this is a fundamental theorem. It relates these two, uh, concepts.
So now we will look at the proof of, uh, these two, uh, parts of the theorem. Um, so we'll start with the first part over here. So proof for the first part. So what, what do we need to do? We need to, we need to prove that F is differentiable at this point Zeta. We need to [Music] prove that the quantity cap F of X that we defined is differentiable at the point, uh, differentiable at Zeta. So how do we do this? Well, let's go ahead. So first, we will consider another function, A of H, going to define this as F of Zeta + H - F of Zeta / H. Basically, we're taking the definition of a derivative, and, uh, we're considering the function that looks very similar to the definition of a derivative. Uh, and now we can basically substitute the definition of cap F of X into this. So this is going to be equal to 1 over, it's going to be equal to 1 / H times the integral from C to Zeta + H of F of T DT, minus integral from C to Zeta, F of T DT. And this we can write it simply as 1 / H integral from Zeta to Zeta + H of F of T DT. And now what we want to do is want to prove that this F of Zeta, this converges to F of Zeta as H goes to zero. So we want to prove this quantity converges to small F of Zeta as H goes to zero.
All right, so now let's, uh, try to prove that. So again, want to prove, we're going to consider A. I want to prove that this goes to zero as H goes to zero. Basically, all right, so what are we going to do? We're going to, uh, add a new, uh, uh, term to this. So we're going to first, uh, in order to get F of Zeta, what we will do is, yeah, so this is nothing but we want to prove this goes to zero, uh, A of H - F of Zeta goes to zero as H goes to zero. So we're going to take, we just saw what A of H is. It's basically 1 / H integral Zeta, Zeta + H of F of T DT minus F of Zeta. Now, what I can do to F of Zeta is, I can actually write this as, so F of Zeta is a constant here. So I can write F of Zeta as an integral from Zeta to Zeta + H of a constant F of F of Zeta DT. So F of Zeta is a constant, does not depend on on T. So if I actually want to compute this, uh, this is simply going to be 1 / H * F of Zeta integral Zeta to Zeta + H of 1 * DT, which is 1 / H F of Zeta * Zeta + H - Theta, uh, that would, so that would be H, and that which will cancel out with the denominator over here, and this will simply give you F of Zeta. So we can write this constant in terms of an integral of a of this constant. So that's what we will do here. So we'll going to write this as, um, 1 / H of integral Zeta to Zeta + H of F of T DT, minus 1 / H of integral Zeta to Zeta + H of F of Zeta DT. So now I can write this as 1 / H integral Zeta to Zeta + H of F of T minus F of Zeta DT. And now the intuition from here is that because F is a continuous function, um, then this difference over here, F of T minus F of, uh, Zeta, as H tends to, as H tends to zero, this difference is going to be very, very small, uh, again due to continuity, uh, at the point Zeta. So F is a continuous function. So the intuition, this is small as H goes to zero, since F is continuous, F is continuous at Zeta. So now let's actually try to do this more formally. So basically, as H goes to zero, uh, this difference itself is going to get really, really small, which means this integral is going to go tend towards, uh, zero or converge to zero. Now we can actually formally, um, uh, show this using the definition of, uh, convergence. So, so formally, so given an Epsilon greater than zero, we can find H greater than zero such that F of T minus F of Zeta is less than Epsilon for all T belonging to Zeta to Zeta + H. So then we have that 1 / H integral Zeta to Zeta + H of F of T minus F of Zeta DT is going to be less than or equal to 1 / H integral Zeta to Zeta + H, uh, first, we're going to look at the absolute value of the difference rather than the just the difference, because this could be both positive or negative. Then this would be less than or equal to 1 / H * Zeta to Zeta + H, so this absolute difference is going to be less than or equal to Epsilon, right? Because that's what continuity says, that if, uh, given an Epsilon, I can find an H where the difference is less than Epsilon. So this is Epsilon, not not Zeta, but Epsilon DT, which is going to be 1 / H * now, Epsilon is a constant over the integration. So this will be 1 * DT, which is simply Epsilon * H, which is equal to Epsilon. Okay, so this shows that the difference between A of H, so which implies that A of H - F of Epsilon is less than or equal to, sorry, F of Zeta is less than or equal to Epsilon, and this will tend to zero as H goes to zero. So this proves the first part of, uh, the theorem.
So now we will look at the second part of the theorem and we will use the proof of this, uh, first part, or we will use the first part of the theorem in order to prove the second part here. So we look at the second part of the proof. So what is the second part of the proof? Well, here we know that, uh, F dash, or the derivative of cap F, is continuous. Um, so we know, or let's actually, uh, go back here. So for the second part of the theorem, we want to, uh, basically show that if I take F, F, the cap F, and if I take this, uh, derivative, which is F dash here, and integrate the derivative from A to B, then that value is equal to F of B minus F of A. So the value of the function itself at the two ends, B and A, and I take the difference between those two. So this is what we're trying to prove in the second part of the theorem. So for this one, we know already that F dash is continuous. Then by theorem 1, which we just proved, we're going to consider a new function, which we are going to define as G of X is a function which goes from A to X of F dash of T DT. We know that this function is differentiable. Again, by the first part of the theorem, and we can make some observations about this function. So the first observation is that if I compute the value of G at A, that means my limit X is A, that means the lower limit and the upper limit for my integration is the same value, then this is going to be zero. Okay, why? Because by definition of G, because that's how we are defining G, we are defining G as an integration from A to X, and if X is equal to A, then this integration is going to result in a zero. The second property that we have is that if I look at G dash of X, that is going to be equal to F dash of X on the interval A. Why is this true? Again, this is true by theorem 1, we just proved this in theorem 1. So these are the two, uh, uh, properties of this function G that we defined. And now we're going to define yet another function. So we're going to consider, we're going to consider yet another function. This function is going to be H of X, and this is going to be defined as F, cap F, of X minus G of X. Okay, so we're going to consider this new function, cap F of X minus G of X. Now, uh, let's analyze this function, uh, a little bit. So by the second property that we have, which is G dash is equal to F dash, we know that H dash of X is going to be F dash of X minus G dash of X, which is nothing but zero for all X, why? Because G dash of X is simply F dash of X, so F dash of X minus F dash of X will give us zero for all X. Now we know that the derivative is zero for H, uh, so H of X is equal to 0, which means that H of X is actually a constant function. Hence H is a constant function. So now that we know it's a constant function, that means it's equal to some constant, let's say C. So now let's actually, we can estimate what this constant value is. So we also know that H of A, we know that H of A is equal to F of A minus G of A, by definition. But then G of A is equal to zero. Why? Because you look at the first property earlier, so we go back here, we see G of A is equal to zero by the, by the definition of a G. So what does this mean? This means that H of A is simply equal to equal to the constant F of A. So, so we have the third observation here is that H of X, and we already know that H of X is equal to a constant, and we know the value of H at one, at one point A, which is equal to F of A. That means the function H of X is equal to F of A everywhere. And this symbol here with three equal, with three horizontal lines, basically means it's a constant. So now we will consider the other end of the domain. So we'll consider X = B. So we know that, so then we consider H of B, which we already know from the third property that H of B is nothing but F of A, because H of X is a constant everywhere and it is equal to F of A. And by definition, H of B is F of B minus G of B, which again by definition is going to be F of B minus the integral from A to B of F dash of T DT. And now we are basically done. If you read this from left to right, what we have is F of A is equal to F of B minus F of, uh, the integral of F dash of T from, uh, A to B. Okay, so if I put all of this together, then I have F of A is equal to F of B minus integral A B of F dash of T DT, which I can then move the terms around, and I'm going to have integral A to B F dash of T DT is equal to F of B minus F of A, which is exactly what the theorem 2 says. So that proves the whole Fundamental Theorem of Calculus, both the first part and the second part. And in the proof for the second part, we actually used the first part of the theorem.