 Now, there is another question that comes. So, basically, what we are saying is, this is, if you like, you can call this as the anti-derivative. A function capital F, A B to R is called anti-derivative of a function f. So, one introduces, if f dash of x is equal to f of x, for every x belonging to A B. So, that relation that we are saying here, so one gives a name that capital F is the anti-derivative of small f. So, the question arises. So, fundamental theorem of calculus is applicable whenever there is an anti-derivative for a function. So, the question is, what are the functions which have anti-derivative? What is the class of functions or is there some, we can say that, if a function has this property, then it will have anti-derivative. So, we want to answer that question. So, question is, which functions have anti-derivatives? Keep in mind, anti-derivative is not unique because we are just saying f dash should be equal to f. So, if you take f plus a constant, that derivative also will be same. So, it is a class of functions. Two anti-derivatives differ by a constant. So, to answer that question, let us first observe a small result. So, to answer this, what shall I call? Proposition. So, let us say f is a continuous function. Suppose it is a continuous function, then we know it is integrable. Then the claim is, there exists a point c belonging to a, b such that f of c multiplied by b minus a is equal to integral a to b f x dx. So, let us try to understand what is this theorem saying. Because f is continuous, the right hand side integral exists, every continuous function. But what is the left hand side? This is f of c into b minus a. So, this is the area of your rectangle with base as a, b and height as f of c. So, if you interpret it geometrically, it is very nice in the sense that if this is a and this is b, this is your function and you are looking at this area below the graph of the function, that is your integral. So, this is the right hand side. So, this is the right hand side. What is the left hand side? It says there is a point c in between. There is a point somewhere here, c. So, if you look at this height, that height is f of c. So, this is a rectangle. So, look at the rectangle. So, area of that rectangle is same as the area that is the area below the graph of the function. So, there is a point c in between in the interval a, b. Say that if you look at that height and the area of the rectangle with that height, that is equal to the integral of the function. So, this is the reason why it is called, this is called mean value theorem for integrals. So, this is called mean value theorem for integrals. Why mean value? You can also interpret this. So, this the claim star, if you interpret it as f of c is equal to 1 over b minus a integral a to b f x dx. Then what does geometrically right hand side represent? Do you think the right hand side looks like an average of the function? f of x is a value at a point in a, b. Sum up the values. So, summation is integral a to b divided by the total number of values that is the length of the interval. So, right hand side can be called as the average value of the function on the interval a, b. Right hand side can be called as the average value. So, the theorem says if f is continuous, the average value is attained at some point in between. The average value is attained at a point c in between a to b. So, that is another way of saying. So, this is we can call it as the average value. So, the proof is quite straight forward because f continuous implies integral a to b f x dx exist. If I look at this number average value, this is less than or equal to capital M times b minus a. Because what is that? That is, you take the partition with n points a, b only. m is the value. So, supremum capital M on the interval a, b. So, m times the length. There is the upper sum with respect to the trivial partition and this is b minus a. We have taken it down. So, let us do not write it there and less than or equal to small m. Is that okay? m times b minus a is less than the integral upper sum, lower sum is less than the integral less than the upper sum. The interesting thing is, if I look at this value, this is a number which is caught between the smallest value and the largest value of the function. So, what does intermediate value property say? This must be attained at some point. That is all. So, by intermediate value property, there exist a c belonging to a, b such that f of c is equal to 1 over b minus a a to b f x dx. So, that implies whatever we want to say that f of c times b minus a is equal to integral a to b f x dx. So, there is a point inside the interval in the open interval a, b. While writing the theorem, I should have said in the open interval a to b. You can specify. That is what intermediate value property says. If there is a value alpha, there is a value beta and something in between, then that must be attained at a point in between. So, that is intermediate value, mean value theorem for integrals. We will see an application of this in fundamental theorem of calculus part 2, which says the following that let f be continuous. Let us say f is a function which is continuous. See, actually, what we are trying to do is find out an anti-derivative for the function. What could be an anti-derivative for a continuous function? Fundamental theorem of calculus itself gives you an answer. It says the relation between f and f. If I know a small f, I can define what is capital F from this. Let us write that straight away. That equation, let f of x be defined as integral a to x f t d t. x belonging to a, b. For every point x in a, b, let us, because f is given to be continuous, so it is integrable. This function capital F is defined, then f is defined and f dash of x is equal to f of x. So, this is the function. So, this theorem says every continuous function will have an anti-derivative. As soon as you are able to recognize the anti-derivative, you can get back the integral. So, our class of all continuous functions have anti-derivatives and for such functions, computation of integral is straight forward by fundamental theorem of calculus. Let us prove that. This is the case. Proof. I have to find a derivative. Let us take a point x belonging to a, b. How do I compute the derivative of capital F? What is the definition? f of x plus h minus f of x divided by h, limit h going to 0. So, let us look at f of x plus h minus f of x. So, what is that quantity equal to? That will be equal to integral a to x plus h minus integral from a to x. I have not stated those properties of integral, namely that the integral is additive over the intervals u. So, integral from a to c plus integral from c to b is same as integral from a to b. So, I should have stated that first, but anyway, let us use that for the time being. Namely, this is integral from x to x plus h ft dt. This is integral a to x plus h minus the integral from x plus h and this is a. So, integral from a to this minus the integral from a to x. So, what is left is integral from x to x plus h. So, that is what we are saying. So, f of x plus h is integral a to x plus h minus f of x that is integral from a to x. So, what is left is integral from x to x plus h. So, let us continue with that. So, f of x plus h minus f of x is equal to that quantity. Now, look at this quantity. f is a continuous function. So, it is continuous in the interval x to x plus h also and just now we proved the mean value theorem. So, there must be a point in between x and x plus h. Say that this integral is equal to f at that point into the length. Length is h. So, let us write before this. Let us just now. By mean value theorem, there is a point. It depends on h belonging to x to x plus h. Here I am taking h as positive. Just for illustration, h could be positive or negative. x plus h could be on the left side or on the right side. It does not matter actually. So, c h such that integral x to x plus h f t dt is equal to h times f at c h. So, put this value in this. So, we call that as 1. Call this as 2. 1 and 2 imply f of x plus h minus f of x is equal to h times f of c h. So, that implies that the ratio f of x divided by h is equal to f of c h. To compute the derivative, we should take the limit as h goes to 0. So, take the limit. So, implies limit h going to 0 of f of x plus h minus f of x divided by h is equal to limit h going to 0 f of c h. And what is that equal to? c h is a point between the interval x to x plus h. h goes to 0. f is a continuous function. So, c of h will go to x. So, f of c plus h must go to f of x by continuity of the function f. So, that is equal to f of x because of the continuity of the function f. So, that proves. So, implies f dash of x exists and f dash of x is equal to f of x. I just said this h could be positive or negative. If x is the interior point, if x is the end point, then what will happen? You only have on the right hand side. So, you only have the right hand derivative. So, taking care of that, those are minor things. So, you can take x in the open interval, then plus minus does not matter. And at the end points, you can just have one sided limits. So, you can prove for everything. So, derivative at the left end point means it is the right end derivative. And derivative at the right end point means it is only the left end derivative possible. So, every continuous function has an anti-derivative and for every function which has anti-derivative integral can be computed. So, that is what is the importance of fundamental theorem of calculus. So, let me give you one application of this fundamental theorem of calculus, which is quite useful or of importance rather. So, let me guess application. Of course, it is of great importance. This theorem allows you to compute integrals without going into the limit operations. And it has implications in historically in Fourier series and so on. But let us look at a simple application, which we will all without knowing Fourier series, you will understand. I think we start using a function called the log function from our school times. I think probably send it is 8, 9 or somewhere, a log function. What is the definition of log function? We do not know. How is log defined? For us, log is defined by the log tables. Or another example is what is a trigonometric function, sin theta. We define sin theta as the height over base. But if you want to define sin function as height over base and you want to prove that function is continuous, how do you prove it is continuous? How do you prove that function is differentiable? Because we start using these facts that sin x cos x, all are defined functions, well defined, periodic and so on. Sin is continuous. In fact, differentiable derivative of sin is cos. So, if you take the definition of those geometric definitions, you do not get those properties easily. You have to assume those properties. Also, the log function is a function which has some nice properties. What are the properties of the log function? Normally, we have log of, what is log of a, b? Log of a plus log of b and then we have log of say a raise to power n is equal to n times log a log of a by b. Whenever we define, we say it is log of a minus log of b and so on. Is there any such function with these properties? Does such a function exist at all or not? Not only that, when we come to slightly higher classes, we say log function is differentiable and log derivative is 1 over x. We start using that also without any proof. The reason is that these functions are not easy to define. For example, you will look at the polynomial functions. Constant function is a constant polynomial. Function fx equal to x, x square x cube. They are sums, linear combinations. All are polynomial functions and you sort of feel comfortable with them. You can define them rigorously. You can define a rational function p by q, where p is a polynomial, q is a polynomial. But there are functions which cannot be obtained from polynomial functions in any way by any algebra. You have to go to the concept of limits. That is the reason these cannot be defined. For example, you must say that I can define sin x where the series kind of a thing. But that is an infinite operation. It does not stop somewhere. There is a limit involved in it. We will come to series later on. Let me just try to illustrate how a fundamental theorem of calculus can be used to define what is called the log function. We will not go into all the details of this, but we will at least initiate so that you can try to prove it yourself. It is reasonably easy. The clue lies in this thing. The derivative of log is 1 over x. If I integrate 1 over x, I should be able to get back the log function by fundamental theorem of calculus. One defines ln of x to be equal to integral. What is the value of a log of 1? That is taken as 0. Let us start as from 0 and go to x, 1 over t dt for x bigger than 0. If it is less than 0, what shall I do? I can write x to 0. So, x to 0, but with the negative sign, log of 1 is 0. What is log of a negative quantity? 1 is less than 1. What do you think of log of half? It is minus log of 2. That is why this negative sign is coming, x to 0, 1 over t dt. This automatically says log of 1 is 0. I will not go into all the details, but let me just say what is obvious from this? You are right. Bigger than or equal to 1, less than or equal to 1. Of my oversight, log of 1 is 0. For negative quantity, log is not defined. It is only defined when x is bigger than. So, this is a function defined. ln is a function defined and 0 to infinity taking values in R. It is a function defined only for positive numbers. Log is defined only for positive. greater than, less than 1. x is less than 1. What correction she has said? I should have applied it here also. log of 1, you want 0. So, 1 to x integral, that is, when x is bigger than 0, when it is less than 0, you can write 1 to x, but normally you write lower limit to upper limit. So, with a negative sign. So, that is what. So, this is a function defined for all positive real numbers. What is obvious for this? 1 over t is a continuous function. 1 over t is a continuous function. So, ln of x is defined. Continuous function is integrable. So, ln x is defined. One consequence. Immediately it is defined. And what is the derivative? What is the derivative of ln x? Fundamental theorem of calculus. Again, derivative of ln x is 1 over x. That gives you the derivative also. It is differentiable and derivative is 1 over x. Is that okay? 1 over x, 1 to x, derivative is, that is fundamental. Anti-derivative. Just now we said that f of x was the integral a to x. So, it is 1 to x. So, derivative is 1 over x. If the derivative of, so it is a differentiable function, derivative is 1 over x, x is bigger than 0. The derivative is everywhere positive. 1 over x, derivative x bigger than 0. So, ln x is a function with positive derivative. Go back. I know something about the derivative. What does it say about the function? If the derivative is strictly bigger than 0, the function should be monotonically increasing. So, ln of x defined this way is a strictly monotonically increasing function in 0 to infinity. And derivative is 1 over x. So, what is the second derivative? If you like, 1 over t, derivative 1 over minus, what is the derivative? Minus 1 over x square. X is bigger than 0. Anyway, x square derivative is everywhere negative. So, it is a concave down function. See, all of your calculus, all those results that we proved, derivative helps you. So, it is a continuous function. It is a differentiable function. It is concave down. At 1, it crosses the x axis. And what happens as you go on increasing? What happens to ln of x? As x goes to infinity, one can use this itself, 1 over t to show that as x increases, the area keeps on increasing. It goes to infinity. So, it keeps on increasing. And one can also prove those other properties that ln of 1 over x. They are very nice actually. One should try to read the proof that ln of, even all those properties, ln of a, b is equal to all these properties, can very easily be proved for this definition and give beautiful applications of calculus basically. So, one observes that the graph of this function ln of 1 is 0. So, the graph should look like this. It never meets the horizontal and never there is no asymptotes of, this is asymptote, but there is no horizontal asymptote. It keeps on increasing. So, on this side, as you go this side, the function keeps on going up. So, as a consequence of this, ln x is a bijective function from 0 to infinity to r, which is differentiable. So, it should have a inverse function. And what is that inverse function? It is the familiar exponential function e raise to power x. So, that is a precise definition of what is the exponential function. It comes from the definition of ln of x. So, exponential is the inverse of the bijective function ln of x. And one can use to show that, because this function is, ln x is a differentiable function, inverse function also is differentiable and derivative is 1 over the derivative of the original function. So, you get exponential is a function which is differentiable with derivative itself. Derivative of e raise to the power x is e raise to the power x. All these come because of this fundamental theorem of calculus. So, we call this as ln of natural base e. Then, you can define ln of other bases and so on. So, that is definition of what is called transcendental functions and one of the transcendental functions is log and the exponential function. Same way you can try to define what is trigonometric function sin. We can try to apply the same kind of trig. What is the derivative of sin? Cos, but I do not know cos. Derivative of cos is sin. So, I am stuck. But what you do is to define sin, it is good enough to define sin inverse. I can define sin inverse in an appropriate interval. So, minus pi by 2 to pi by 2. There sin is sin of minus. It goes the value is a one-one function. So, if I can define sin inverse, I can define what is sin as inverse of like here we are doing. Exponential is the inverse of log. So, sin inverse, what is the derivative of sin inverse? 1 over 1 minus x square. So, that probably we can integrate. So, one tries to integrate that to get sin inverse and show it as all the nice properties and the inverse of that is taken as the sin derivative of sin is taken as cos and all these properties are proved. So, that is the definition. The way one uses rigorously fundamental theorem of calculus to define functions which cannot be defined algebraically. So, I am just giving a glimpse. If you feel interested, you should try to read some book. If you want to know a reference, come and ask me. I will tell you a reference where you can read these things. So, basically, we have tried to define integral. So, let me just go back and look at, we looked at integration via upper sums and lower sums. There is another way of defining this integral. So, basically, what we did was, intuitively, we tried to capture the area below the graph of the function in lower and upper sums and we had to have the condition that if you want to define upper sum, you want to define lower sum, then your function must be a bounded function. Otherwise, infimum and supremum does not make sense at all.