Good afternoon to one and all. So today in this session, we are going to conclude our graphs. It has been a chapter which we have been dealing with the last three weeks now. So fourth week is running on. And today I want to conclude this chapter. And under this chapter, we need to finish off two types of functions one being exponential function. In fact, we can take exponential and logarithmic functions together. And there's a reason why we are going to take it together. Exponential and logarithmic functions. So this would be our first part of the session early before six o'clock. I should try to wind it up. And in the second part of this session, the much awaited bridge course chapter introduction to calculus. That is what is going to be done in the second part. Okay, I don't think I'll be able to do much introduction to calculus. Probably I'll be introducing you to the concept of limits. And let's see if time permits will be able to touch upon derivatives. Okay, so this is going to be our agenda for today's class. So let's start. Let's start. We don't have even a single minute to waste. Okay, exponential and logarithmic function. I'm sure exponentials are not a new concept to you. Okay, you have been introduced to exponential functions since class seventh exponential laws you have done. But I just want to know how many of you have been introduced to logarithmic functions. Let me just take a quick poll. Just say yes, if you have been introduced to logarithmic functions before. And know if you haven't been. Just to give me just to get an idea about how comfortable you are with the logarithmic session because there are some schools with autonomous syllabus who have introduced logarithms in their junior classes like NPS Rajaji Nagar. Okay, so nine of you say, sorry, a seven of you say haven't been introduced to logarithms yet. Okay, no problem. No problem. Okay. Okay, never mind. Never mind. I'll start from scratch. Don't mind. Never mind. Yeah. Now, now first let us talk about let's talk about exponential function. Now exponential functions are basically functions of the nature, right, a constant race to some power. Okay, a constant a race to some power. So let's say if f of x is an exponential function, it is a constant a race to a power of another function of x. Okay, now here we normally have this base a to be a number which is greater than zero and should not be preferably one. Right. If it is one, it becomes a constant function because it will become just a straight line f of x equal to one. Okay, you can write it as why you also never mind. Okay. So an exponential function basically has a constant race to the power of a variable. The simplest of all exponential function that we can come across is something like a to the power of x. Right. So here g of x is basically your x. This is the simplest exponential function you can actually see. Okay. Now, any questions here. I've not started anything new. I've just the first slide. Okay. This is the first slide of the session, the previous slides. I didn't say anything important. Okay, but try not to join late. Okay, because if you miss out the initial part, you lose the track. Okay. So now the first question that arises in everybody's mind is why we have put this restriction of a to be positive. Okay, of course, not equal to one is something which I already told you because if you keep it equal to one, it will become one to the power anything is one. So that becomes a constant function. So we can't call constant functions to be, you know, you can call it as an exponential function, but it's a limiting case of an exponential function. Now, why do we keep a as greater than zero? Now, see, if you don't keep a as greater than zero, let's say if I keep a as a negative quantity, let's say I have negative one to the power of no P by Q. Okay. Now I've taken X to be some kind of a rational number right now. You'd realize here that you can find answers to these expressions only when your Q is odd. Okay, for example, you may find answers to this when I say answers, my dear friends, please let me know. I'm talking about real values as the answer. We are dealing here with real valued function that is something which I would like to highlight in the beginning of this course itself, that all the functions that we are going to talk about they would be real valued in nature. Okay, and I've been telling this since the first class real valued are those functions, which will take real inputs and give you real outputs. So we are not going to talk about any such functions which are going to deal with the non real inputs or going to give out non real outputs. Okay. So when we talk about exponential function, or for that matter later on, you will talk about logarithmic functions also in the similar sense. We are going to only talk about those, you know, functions which take in real inputs and give us real outputs. So if your Q here is not an odd number, that means if it's an even number, you realize that the output of it will come out to be imaginary. So in this case, only we get a real output. Okay. Right. But in case like, in case like minus one raised to the power half, where your base here is a, we call this as the even through it. We have already seen in the past while we were doing dealing with irrational functions that when there is an even through it, the answer for this may come out to be an imaginary quantity. Right. And we cannot plot them on the real graph. We cannot plot them on the Cartesian plane. Right. And yes, yes, correct Anusha. So that's primarily the reason why we ensure that our domain and range of these functions remain real. Then only we can plot them as there is no point, you know, talking about, you know, plotting them on the Cartesian plane. Okay. So that's why your base of the exponential function that is your A has to be greater than zero and preferably I'm again repeating this word preferably not one. Okay. Now, when you have a greater than zero and not equal to one, it divides a into two zones. Okay, so let me write it over here. If your A is greater than zero and not equal to one, it divides your A into two zones. One zone is where your A would be between zero and one, and the other zone will be when your A will be greater than one. Okay. It's obvious, right? Okay, so what I'm going to now show you how does the graph of exponential function a to the power x look like for these two different scenarios. Okay, so let's talk about the graph of the graph of y is equal to a to the power x under these two different scenarios. Okay, so let's first, you know, try to plot them by choosing some x and y values because initially when we don't know the skeleton graph is good to, you know, learn the graph by plotting them. So for the first case, let us say an example a being half. So what I've done, I've taken a simple example of a base between zero and one half is an example. You may take anything you want one, one by three, one by four, one by five, one button, whatever you feel like you can take. But just for the purpose of simplicity, I have taken half as an example of a number lying between zero and one. Okay, no problem with that. I remember the skeleton graph is going to be more or less the same, even if you take any other quantity between zero to one. So basically we are learning the structure of the graph right now. Okay, however, there are few, you know, changes in their, you know, bending and all will take that into account after we have plotted the skeleton graph. Right. Now the first thing that I would like to ask you is suggest me a few points on this graph. So if I take x is zero, what should be my y? If I take x is zero in this, what should be my y? One. Okay, so the graph passes through zero comma one. So let me just choose a point on the y axis, zero comma one. Okay, let's say, let's say I take the value of x is one, what do I get? Half. Okay, so at one I get a half. So basically the point will be here. At two, I will get a one-fourth. At two, I will get a one-fourth. So the point will be here. Right. At minus one, I'll get a two. At minus one, I'll get a two. Right. The point will be here. At minus two, I'll get a four. So at minus two, this will, oh my God, it'll go up. Okay. Now, we have sufficient enough points to know what is the, what is the going to be the graph of this type of function. So let me connect it by a dotted line. Okay, just let me connect this by a dotted line. There you go. There you go. So this is how the graph of an exponential function whose base is between zero to one, more or less, more or less my dear students will look like this. Okay, of course, if you take it as one-third to the power x, the structure will be the same, but does the bending of this will slightly vary? I'm going to ask you that next. But before I do that, couple of things that I would like to pinpoint over here. Number one, this graph will never touch the x-axis. This graph is never going to touch the x-axis. That means your x-axis is going to be an asymptote to this graph. Okay. Remember, any exponential function for that matter, whether it's, you know, between zero and one or whether it's greater than one, that will always be positive. Under no situation, an exponential function will ever, ever touch the x-axis. That means its value is never going to be zero. Okay. Now, I'm talking about this case. If I tweak with this poorly, it may come down. Okay, so don't take my word like, you know, if I change this, it cannot come down. For example, if I do a to the power x minus two, you know, function, of course, it may come down. It may come down to the negative side. Right. But this is not a pure exponential function. So I'm just talking about exponential functions of this nature. Okay. And you'll also realize that this graph is basically rising on the negative side and falling on the positive side of x. Okay. So this is basically trying to say that if you increase the value of x, that means if you go to very, very high value, why will start touching the x-axis? That means it will tend to zero. Now, today I'm going to spend some time on tending to aspect that is the aspect of limits in calculus. Okay. And if your x starts going towards minus infinity, why will also start going towards plus infinity? So why will start going up? Is that fine? Any question with respect to the graph of an exponential function where the base is between zero to one? Please ask. Any query? No questions. Very good. Okay. Now, let us take an example of the base being greater than one. Let's take two to the power x for the sake of convenience. Okay, two to the power x. Why I've taken two to the power x because it is matching with half. Okay. In such case also, if you start plotting the graph, let's say if I put x as zero, you know, why is going to be one? So again, it is passing through zero comma one, right? Again, it is passing through zero comma one. If I take x as one, what will happen to y? Y will become a two. Correct? If I take x as two, y will become a four. Correct? If I take x as minus one, y will become a half. If I take x as minus two, y will become one fourth and so on. Now, if you connect them, you'll realize that this is how the graph will look like. Please do not forget to put arrows at these ends because it is still moving. It is still moving. It hasn't stopped anywhere. Okay. Now, again, a quick analysis here. You'll realize that this graph again doesn't touch the x-axis, right? It comes very close to it, very close to it, very close to it, but doesn't touch it. That means the x-axis is again an asymptote for this graph. Okay? Secondly, as x becomes very large, y also becomes very large, right? And as x becomes very large in negative sense, y almost dies out at zero. Almost dies out at zero, but does not become zero. Right? Okay? And between these two curves, the way you see here, this is the way it is, you know, bent in this part and this is the way it is bent in this part. Between these two limiting cases, there is one special case when it is exactly flat when your A becomes one. Okay? So, as you can see, the changing point is A equal to one over here. So, before one and after one, the graphs are different. So, I would like you to see this on the GeoGebra tool. So, there is a GeoGebra tool. I'll just show you. When you take A to the power x, how does the graph actually change? Okay? So, right now, as you can see, it has already taken A as one by default. If I increase the value, okay, do you see that rise? Okay? And if you decrease the value lesser than A, you can see the other end will lift up. There you go. The other end is lifting up. Okay? Of course, when you go negative side, see the graph will disappear. Now, disappearing means it has gone, some of the values have gone to the non-real domain. So, we cannot plot them on the Cartesian plane. So, only from exactly at zero, it will be zero. Okay? And that too, you can see it is only in the positive side of x. And if you are increasing the value, see this is the way it is going to fluctuate. See the way it is dancing. It's as if like there's a fulcrum, you know, at one. Okay? Do you see that? Yes, any question, anybody? Yes, we can plot it in the argon plane. But let me tell you, argon plane deals with a different type of scenario altogether. Of course, we can use argon plane to plot the imaginary things. But argon plane is kept for a different purpose altogether. It is basically, argon plane is a way to make complex numbers take a real shape. Right? It's an effort to study complex numbers by imposing a real shape to it. Getting my point. But something which is non-real is like you cannot see it. You cannot feel it. Okay? Alright, so with this, let me tell you my dear students, all the laws that you have studied for transformation, all the rules that you have studied for transformation, shifting right, shifting left, shifting up, shifting down, will be equally applicable to exponential functions as well. Right? Okay? Would you like to all do a quick question and move on? Okay? And then we'll take up logarithm because I'm going to spend a considerable amount of time on logarithm today. So I would just like you to sketch this simple graph for me. The graph is, let me give this graph, y, let's say mod y plus 3 is equal to e to the power, let's say mod x minus 1 plus 2. Will that be fine or do you want me to complicate it a little bit more? Okay, that's fine. Okay, fine. Okay, Ansh has a question. If a is minus 1, then most of the values are real. No, no, no, no, no, no. No. When I say most of the values, what do you mean by most of the values? See, between any two, any two value of x, there will be infinitely so many numbers. Ansh. Right? Let's say your x is between 1 to 2 itself. Between 1 to 2, you'll have infinitely real numbers coming. Correct? Infinitely, fractions coming which has got even number basis. What will you do with those answers? Those answers will go on to the non-real domain. So the consortium of mathematician decided that let's not have that complexity because we don't want to plot the graph only for few selected values. It will become very, very difficult to deal with them. There will be a lot of complexity in plotting the graph. I understand where you are coming from. You're saying that for most of the values we may get. I should not say most of the values. For few selected values, I will get a real answer. Why don't we plot them? If you do that, it will become very, very, very critical. Between, let's say, 0 to 1 itself, there can be so many fractions which has got odd number basis. There may be so many of them which have even number basis. It's humanly impossible to sit and plot them. Got your answer and why we don't mess up with or why don't we entertain negative values of it. So here is the question. Let's quickly do it guys. We have learned so many rules of transformations of graphs. Let's supply them one by one. So which is the skeleton graph that you are going to deal with? How many of you do not know anything about E? E, the APS constant or the Euler's constant. Okay, Muskan doesn't know about E. Okay. Now, let me tell you E is a special type of an irrational number. It's a special type of irrational number whose value is approximately 2.718. Now, how does this E come? Just like pi, what is the significance of E for us? Okay, that's something which I'm going to explain you in the next 5 minutes. So after that, only we'll take up this graph. But just remember, it's an exponential function whose base is greater than 1. You can treat it for practical purpose like that. Okay, now what is E? Now, consider a term here, 1 plus 1 by x to the power of x. Okay, consider this expression E, not the small e, capital E. Let's say this is the expression capital E, which is 1 plus x to the power x. Okay, now people who have calculators with them or phones with them, I would like you to give out the following answers to me. Tell me when x is 1, what is the value of E my dear? That you don't need a calculator for that. When x is 1, yeah, Shrita is correct too. Okay, now tell me when x is 10, what is the E value? Basically, I'm asking you 1.1 raised to the power of 10. 1.1 raised to the power of 10. You may use your phone, calculator, whatever you have. I'm sure you must not be having a calculator because you have never used it before. But if you have a phone, you can use it. How much? Approximately, give me 4 significant figures. Give me the answer to 4 significant figures. Okay, so Parvati has given me the answer to 4 decimal places. Okay, never mind. I'm fine with that Parvati. Now, do this. Put it to, let's say, 1000. Sorry, I'm jumping a lot. From 110, I've straight away gone to 1000. Tell me what happens for 1000. 1 plus 1 by 1000 raised to the power of 1000. No, no, no, no. I'll not give you 1. Okay, so Ansh has given me 2.7169. So Ansh, I'm assuming that your answer calculation is correct. Okay, now put 1 lakh. Okay, put 1 lakh. If at all you can. 1 lakh, 1 plus 1 by 1 lakh to the power of 1 lakh. Now I can see you have started realizing this figure, 2.78.  Okay, your E value is going to tend towards a value which we call as E. Okay, now this type of notation you might have experienced while, you know, studying compound interest in maths, right? But as a science student, right, compound interest may be an application for a commerce guide. As a science student, let me tell you, E has a very vital role to play. It is so useful for us that we are going to use it very, very widely in calculus. Okay, now where do you see E in nature? We see E in nature in the radioactive substances. Let me give you a brief idea about where do we see E. Normally the radioactive substance that is there in nature, whether you talk about thorium or uranium, et cetera. Of course, yes. Shritish, they're half-lives. Okay, now these radioactive substance, they all decay. Okay, and the rate at which they decay is given by, or the expression for the mass of the radioactive substance at a given time t is given by M0, M0 is basically the mass at t equal to 0, e to the power minus lambda t, where 1 by lambda is the average life of the radioactive substance. Okay, normally, let's say for example, there is a radioactive substance, X, whose average life is, let's say, 5,000 years. Okay, let's say this X substance has an average life of 5,000 years. Okay, now what it says that in one 5,000 year, let's say if you started with one gram of X, in one 5,000 years, this substance will become 1 by e times its original weight. That means if you start with one gram, it is going to become 1 by e gram. Okay, another 5,000 years, it's going to become, another 5,000 years, it's going to become 1 by e square gram. Okay, right? So this is how in nature we can see where does my e exist, where does this irrational number e exist. Okay, and this has been figured out experimentally. Okay, through accelerated mode. Now you must be thinking, oh, people waited for 5,000 years to see that. No, radioactive substances are decayed in an accelerated way. Okay, in reactors. And from there they came to know that this is the rate at which they are decaying. Okay, anyways, you are going to study more about it in a chapter called chemical kinetics. In a chapter called chemical kinetics, especially when you are studying first-order chemical kinetics. First-order chemical kinetics. I'm sure you're going to study that in class 11 itself. Gaurav Sir is going to take your class on that. Okay, there your e is basically used. Okay, anyways, coming to the question, my dear friends. Treat as of now e being a number greater than 1. So 2.718 is the value of e. Okay, so let's plot this graph. So how will you first plot this graph? Let me again copy this. Mod y, what was the graph? What are the functions? Sorry, I forgot. Mod y plus 3 plus 3 is equal to e to the power mod x minus 1. x minus 1 plus 2. Plus 2. Okay. What is the skeleton graph you're going to start with? Anybody? y is equal to e to the power x, obviously. Correct. So the graph is going to look like this. So it's going to be, okay. Sorry for drawing it in yellow. I should have drawn it in white because my function is written in white. Actually, one question I would like to ask you. Okay, let me take you back to GeoGebra. Okay, sorry, I forgot to ask you that. Let's say, let's say I plot y is equal to 2 to the power x. Okay, listen to my question very carefully. What will happen if I plot y is equal to 3 to the power x? Will it bend more or will it bend less? That is my question. Okay, will it be like, you know, more bent like this? Okay, sorry, I'm not drawing it here because it's going to merge with that. Or is it going to be bent less like this? Okay, what will happen? Option A or option B. Let's have a quick poll. Let's have a quick poll. Okay, just choose A or B. Do you think 3 to the power x is going to be shaped like A or is it going to be shaped like B? 30 seconds for that. 30 seconds, not more than that. Wonderful, wonderful. I'm getting phenomenal response from everybody. Not able to see the chat? Don't worry, you can type it out. Okay, we'll discuss last three seconds. One, two, three, everybody please go on. Go on to vote. Okay, so 92% of you say A. Okay, the answer is yes, it is going to be A. Okay, so let me show the graph to you. Y is equal to 3 to the power x. Y is equal to 3 to the power x. Okay, as you can see, it is more bent as compared to 2 to the power x. Common sense here, common sense. X is equal to 1, the previous one will give you 2, but this blue one will give you 3. So for the same value of x as 1, now you are going to have a higher value, which is going to be 3. That's why it has to be more bent. So, you know, it's basically obvious that if you increase the value, it will become more bent. If you decrease it, it becomes lesser bent, lesser bent, lesser bent, lesser bent, and it becomes a straight line like this when your A is 1. And if you decrease it further, it starts lifting it from this end. It starts lifting it from this end. Okay, that is the transition. So, e to the power x will be somewhere in between. Now, this is just for your understanding. You don't have to exactly, you know, sketch it. Sketching is just a rough estimation. So, e to the power x is slightly in between this red and this blue graph. Okay, anyways, any questions here? Parvati, never mind. It's okay. Even if you don't see the poll, you can just type your answer on the chat box privately to me. Okay, no worries. Now, what are the next steps that you're going to do, my dear? Anybody, any suggestions from you, your side? Auro? Which is the next step that you're going to take? Shruthij. Shruthij is saying he is going to replace x with x minus 1. Then, Shruthij, tell me how are you going to put a mod on x minus 1, dear? Good that you realize your mistake. And everybody who's saying x minus 1, I want my dear students, you all to realize your mistake. If you put x as x minus 1, how are you going to put mod around it? Because mod can only be introduced with an x. Auro, y. Okay. So, Mriganka, Oshik, you are absolutely correct, dear. You're not going to put your x as x plus 2. Okay, that's the next step. Of course, Auro, we can do y as y plus 3 also. But let's take care of x part first. So, if I'm doing that, my graph is going to just shift 2 units to the left. It's going to shift 2 units to the left. That means the same value I'm going to now realize at minus 2. So, this value will come to this point. Okay, so your graph would look like this. Now, remember at 0, it will have attained a value of e square. So, it is 0 comma e square now. Okay, anyways. Next, what do you want to do? Next, what do you want to do? Now, of course, you will say let's mod x first. Okay, so let's do that. Let me take a green sketch. So, third step is let's do a mod of the x. Okay, modding x means whatever you have drawn in the positive side, sorry, drawn in the negative side, erase that off. So, basically this part is going to be erased off. And whatever you have drawn on the positive side, that is going to get reflected about the y-axis. So, more or less, this is how this structure is going to look like. Okay, okay. Next is the fourth step. What is the fourth step? Anybody changing? Changing? x with x minus 1, right? So, I'm not going to make x with x minus 1. So, when I do that, everything goes one unit to the right. Okay, so your graph is going to appear like this. Okay, basically it's a kinky parabola. Kinky parabola means having a kink over here. Okay, it's not a smooth, it's just a kink. Okay, next, what are you going to do? Next, if I have a u, I would change my y to y plus 3. Okay, so fifth step. When I change my y to y plus 3, I have to bring everything down by 3 units. Remember, e square, right now it is at 1 comma e square, right? e square is square of 2.7, right? Which I think conveniently will be like 7-ish, correct? So, if you bring it down by 3, it's not going to go down to the negative side. So, it's going to stop over here somewhat. Okay, sorry for making it so cluttered, but I am sure you can see my orange graph. Can you see my orange graph? Okay, now finally, modding y. Modding y means whatever is below the x-axis, remove it. But there's nothing below the x-axis, okay? And whatever is above the x-axis, you need to reflect it down. So, your graph will look like this, my dear students. Let me draw it in the, you know, same color. It's going to appear like this. Okay, so it will have a kinky parabola on top, kinky parabola below the x-axis. Can we just check this out? Can we just check this out?  Mod y, mod y, oh, sorry, where is it writing? Mod y plus 3, plus 3 is equal to e raised to the power of, e raised to the power of absolute value of x minus 1, okay? What else was required? Sorry, I forgot, x minus 1 plus 2, right? Okay, plus 2, plus 2, plus 2, plus 2. There you go, okay? Let me just zoom in a bit, okay? Please ignore the distortion in the figure, okay? Because, you know, Jiu Jitsu is a very, very heavy software, so there is some distortion at times, so please ignore it. So, basically it's like two kinky parabola, okay? Like this and like this over here, okay? When you start seeing that your Jiu Jitsu is giving you trouble, switch to desmos.com, desmos.com, okay? That's also a very, very good online platform for sketching the graphs, okay? Okay, so Auro is asking me to give some questions on GIF, okay Auro? Let's have some exponential questions on GIF, okay? One more we'll have it, sketch the graph off, sketch the graph off. Yeah, let's sketch the graph off, y is equal to, y is equal to 3 to the power x minus 1, x minus 1. Okay, and let's make a curly bracket along x, okay? Let's make a curly bracket along x, okay? And let's make a GIF around the whole thing, okay? And let me mention also, this represents the fractional part, fractional part, and this represents the greatest integer function. Now please note, never treat any bracket to be special bracket until unlisted. Until unlisted, don't start treating any square brackets or curly brackets to be special brackets. Of course, not in your assignments, in your assignments I have probably not mentioned this, but in the assignment all the square brackets are GIF, all the curly brackets are fractional part, okay? Let's try to attack this question, how will I do this? In fact, I will also, you know, struggle along with you. Let's solve it together. The first step that you're going to do is you're going to first plot 3 to the power x, okay? 3 to the power x, we all know it's a graph like this, okay? No problem with this graph. Any questions? Now, next that we are going to do is, let's say this was your first step. Second step is you're going to change your x with x minus 1. What happened when you do that? What happened when you do that? The entire graph shifts one unit to the right. So basically your graph will be like this, okay? Is that fine? Okay. Next, what do you do? Next, what do you do? Next, what do you do? You are not taking a fractional part of x. What do you do when you're doing that? What are you doing when you're doing that? Basically, you are taking the part of this graph which is sandwiched between 0 and 1 and you are replicating it. Getting my point. So this part will be replicated everywhere, okay? It's better to see it on a gg graph. Then it will make more sense to you, okay? So y is equal to 3 to the power of x minus 1. Okay. So this was your function, correct? Now let's replace your x with x minus, just give me a second. I'll just replace it with x minus floor x, okay? This is what happened. This is what I was trying to talk about, okay? So whatever is the part of the function which is trapped between 0 and 1 of the x, that is going to get repeated. Just recall it was just done last class with you, okay? Now next, what do we do? We try to mod the, sorry, gif the whole function, okay? When we gif the whole function, remember what was the approach? We used to make horizontal lines, right? We used to make horizontal lines at unit distance apart. So let me just zoom this a bit, yeah, okay? So basically you are going to make lines like y is equal to 1, okay? Ah, but here if you see, here if you see, the first line itself has gone above. And whatever is below, whatever is sandwiched in these zones, they're all going to fall on the y-axis. So everything is going to fall down on the floor below it. So it's going to fall down on y equal to 0. It's going to all fall down on this line. I'm showing that with orange. So whatever is between y equal to 1 and y equal to 0 is going to fall on the floor below. As in it is all going to collapse and become a straight line. So that's what I think Shirtij was trying to say. Isn't it Shirtij? This is what he wanted to say. Okay. Okay. So let me just do that also so that everybody is convinced that we are going the right way. So y is equal to floor of your p function. As you can see, the floor of p function is going to be right on the x-axis. That means it is going to match with the x-axis. So everything has gone. So it is going to be just a line like this. Just a line like this. Okay. Is that fine? Is this fine everybody? Any questions? Any questions anybody? Okay. Yeah, yeah, of course. There are going to be holes when it attains integer values. Okay. So 3 to the power, 3 to the power this minus 1. Right. The moment it achieves the value of 1, those values are going to stay there. Those values are going to stay there. Okay. Right. That means when this guy, just a second, for this to be 1, this guy has to be 0. Correct. And for that to happen, this is going to be 1, which is not possible. So no, no, there will not be any holes. Very good question asked. There will not be any holes. There will not be any hole, Shirtij. Okay. Is that fine? Any questions? Okay. So with this, we are going to now launch ourselves to logarithmic functions. Guys, very, very important for the next one hour. I want everybody to pay attention if possible.  Okay. Don't worry about the class notes. It will be shared with you on the group. You can, you know, our arms is sit and copy at home. Okay. Just listen to this. Okay. Logarithm is basically a function which was devised by John Napier. Okay. John Napier was the guy who gave us this type of functions. Okay. And this function is basically the inverse of what exponential function does. Okay. So we have just now learned y is equal to a to the power x. Correct. Now, when I say exponential function, sorry, when I say logarithmic function, it is log of some quantity to the base of A. Correct. This is the expression for a simple looking logarithmic function. That means the meaning of this expression is x is a to the power of y. X is a to the power of y. The same thing is represented in a different form, which is y is equal to log x to the base of A. Okay. So both mean the same things. Both of these then mean the same thing. So you may say that logarithm is doing what the inverse of what exponential function is doing. Okay. I'll give an example. Let's say, if I say 5 to the power, let's say x and I put x as 2. What value does y take up? Y takes up 25. Correct. 5 to the power 2 is 25. If you feed 25 to a log as an argument of log, this is called the argument of log with a base of 5, we are going to get back this input. That is how a inverse function behaves. Let me give you an idea about inverse of a function briefly, even though we are not going to talk in detail about it because it's a class 12 subject matter. See, I'll give you a simple example. Let's say there is a machine. Okay. This machine takes some input. Okay. And it gives some output. Let's say I call this machine as the exponential or the function f. Okay. Later on I'll talk about exponential and all. And there's another machine. Okay. What it does, it takes the input that machine number f gives. Okay. So let's say this is a machine which where you feed in, let's say, okay. And it gives you a chapati. Okay. So what does f machine do? It takes art as an input side and gives you chapati.  Then we say that these machines are playing inverses of each other. These machines are playing as inverses of each other. A typical example is sine x and sine inverse x. Right. So if let's say this is sine x machine, then it's inverse is your sine inverse x machine. Now before I move on, there is a small clarification which I want to give everybody. The symbol minus one does not mean one by f my dear. I know it's a human psychology to treat every term raised to the power of minus one as one by that term. Please note. Now you will be introduced to something called linear algebra where you will be learning how to find inverse of two elements. Now inverse function is a aspect of linear algebra. It just means this function performs the inverse of what this function does. For example, sine x. Okay. In sine x, you know that if you feed 30 degree, what is the answer that you get? What is sine 30 degree everybody? One by two. Correct. So inverse of this function that is sine inverse, if you feed half, it will give back 30 degree to you. Just think it like this. So it takes the input of f to generate the input of f. Getting my point. The same way log also behaves. So log is the inverse of the exponential function. In fact, vice versa, both are inverses of each other. For example, what e to the power x will do, the inverse of that log x to the base e will do. Let's take an example. If I put e to the power x value as let's say five, this will give me e to the power five. Correct. If you feed this in place of x over here, it is going to give you five back. It is going to give you five back. Okay. Now before I go on to such exercises, many of you may not be very clear with this definition. So let me give you some illustrations. See, when you say log of x to the base a, it means what power will you raise on a to get x? Think like that. So if I say log 100 to the base 10, it means what number will you raise on 10 to get 100? That means which value of this question mark will satisfy this equation? Of course, answer is this question mark carries two. Correct. So if I say log 64 to the base four, it means which power will you raise on four to get us 64? What is the answer? Which power will you raise on four to get us 64? Everybody tell me three. Absolutely. Okay. So log 64 to the base four is three. Understood how log behaves. Correct. Now you can know that it is actually behaving as the inverse of an exponential function. So this fellow is basically log x to the base four. Okay. And it is behaving as inverse of four to the power x. Let me give you more examples. I want you to be very, very comfortable with this. Tell me what is log one by 27 to the base of three. Just one person is answering. Very good. Very good. So answer is minus three. Very good. Okay. Now you've got a hold of it. Okay. Now let me ask you this. What is log? Log. Let me ask you like this. Log 25 to the base of one by five minus two. Very good. Very good. Okay. What is log one to the base of thousand? Excellent. Zero. Okay. Tell me log minus two to the base of five. I understand why you don't have clue because we cannot find it. There is no such real numbers. This doesn't belong to any real number. We cannot figure out any real number which gives this answer. Okay. Try this out. Log minus two to the base of one. Again, no clue. There is no such number which you can raise on one to get a minus two. See, I'm asking you these questions for a certain reason. I'll tell you the reason in some time. Okay. Okay. Tell me what value of this argument should be there such that x is real and it satisfies this criteria. Log of x to the base minus one is half. Tell me a real number which satisfies this criteria. Nothing exactly. There is no real number like this. Right. Because if you raise minus one to the power half, you end up getting a imaginary number. Okay. Now the reason why I told you all these things is because boys and girls, when you define the log function, there are certain restrictions. There are certain restrictions which we need to obey. Okay. What are those restrictions? Let me tell you first. Whenever you are defining a function log of f of x to some base g of x. Okay. Just for a purpose of, you know, taking a general example, I've taken f of x and g of x. Okay. There are two restrictions which log must obey. Number one, the argument of log must always be positive, not even zero. So whatever you are feeding to log, this is called the argument of the log. This is called the argument of the log. This is called the base of the log. The argument of this log must always be positive. We cannot feed negative. We cannot feed zero to log. Because if we do that, the same clueless situation will come in your mind for various values for various values. I'm not saying for all values. Right. It's the same way, you know, that we had, you know, experience in case of exponential functions. Okay. So log of a negative number is not going to give you a real answer. Is not going to give you a real answer except for a few cases. Okay. So what the consortium decided was not to entertain any argument given to log, which is less than zero or less than equal to zero also. So everything that you feed to log should be greater than zero. Okay. Second restriction that we normally put is on the base. Base is just like the number A in the exponential function. It is to be greater than zero and it cannot be one. And precisely, you have seen that in the previous example that we took, you were clueless about when I asked you what is log minus two to the base one. Right. You were clueless about that X value when your base was no negative number. So we never ever entertain any inputs to the log function which is less than or equal to zero. We never keep the base of log function negative or equal to one. In fact, negative or equal to zero or equal to one. Okay. So these are the two restrictions that we need to abide by if we are defining a log function. This is very, very important. Do not forget it. Unch has a question. Y is equal to A to the power X and written as X is equal to A to the power Y. Okay. Can we consider just switching Y and X? Unch, you actually, you know, took away what I wanted to say in after a few few minutes of my class actually exactly wanted to convey the same thing. I was just waiting for people to understand log first, then the graph of it and then come to this property. Okay. Yeah. Shitish has a question. Sir, if the base is negative argument can be negative argument can be negative. Right. Again, the same problem will come. It will selectively work. Shitish, it will selectively work. It will work for some values. It will not work for many, many values. That's why in order to avoid that complexity, we removed it all together. I understand your concern. For example, you may want to write log minus one to the base of minus one as let's say three, right? You may have, you want to write, right? But again, there will be several other values which we will not be able to address. Okay. So if this number here doesn't have, let's say or have, you know, even number in the base, it will become very difficult for us to answer such cases. Okay. I understand your dilemma. I understand your concern regarding this, but unfortunately we don't entertain such situations. Yes. When your base is one, the only value that you're going to entertain is one. Okay. Because one to the power of one is going to be one. So only for that one particular value it will work should be. Okay. So we are not going to, you know, change the rules for these bits and pieces values that are going to be real in nature. So we just stick to this strong, strong, you can say restriction on the log function. Okay. If you want to see your log as real valued, if you want to see your log as a real valued function, we must abide to these two important constraints on the function. Okay. Now talking something about the base. Okay. Okay. Talking something about the base, normally we deal with two types of bases. We deal with the base E, which we have already discussed about 2.718, etc. Log with the bases E are called natural logs. Okay. We also call it as the neperian, neperian logarithm after the name of John Napier. Okay. Mostly you'll see that in calculus we use log to the base E. Now all of you please pay attention here. In calculus if you realize, let's say tomorrow you are studying some calculus concept and you just see a log X, just see a log X without any base. Don't get surprised if there is no base. That means the base is E. Okay. In calculus, if they just say log X, it is to be read as it is to be read as log X to the base E. Okay. In the pronunciation sense, we call it as ln X. Right. Written also as ln X, ln X. Okay. The other type of base that we normally deal with is the base of 10. Okay. And log to the base 10 are called common logarithms. Common logarithms. Okay. Many a times it is called the Brixian, the Brixian logarithm after the name of, after the name of Henry Briggs. He was an English mathematician. It's the name after Henry Briggs. Okay. So Brixian logarithm basically you will find in your log tables. Okay. A Clark log table is available in the market. You can also find an online soft version of that. So when we are doing certain kind of calculations in maths or for that matter in chemistry, we'll be using log tables and that log table is actually written in the base of 10. So we follow Brixian logarithm there. So this guy gave a different nature to the log function that originally John Napier had discovered. Now, when we talk about these two bases, there is a formula which converts log to the base e to log to the base 10. The formula is log to the base e, let's say log of x to the base e is 2.303 times log x to the base 10. This is a very important formula that will help you to convert log to the base 10 to base e and vice versa. Okay. Now, there are a few values that I would like you to remember also log 2 to the base 10. This value is 0.3010. Okay. Now, guys and girls, please listen to this very carefully. For any competitive exams, you are not allowed to use calculators. Okay. So if at all, I mean, I know you have not been using calculator for your junior classes also, but if at all you are a person who has a habit to grab a calculator or grab a mobile phone for doing such calculations, please stop it. Please stop it right now. If you have a calculator in your house, whether it is scientific or non-scientific, ask your parents to keep it away from you. Ask your siblings to keep it away from you. You are not allowed to use calculators except for one part two or part B of kvpy exam and that two calculator which is present on the computer. That is what you can use. But for all preparations, whether it's your CBC board exams or whether it's for your, you know, any competitive exam in India, not no calculator is allowed. Another value that I would like you to remember is log 3 to the base 10, which is 0.4771, roughly. Okay. So just a question for you all. If log 2 is this, what is ln2? If log 2 to the base 10 is 0.3010, what is ln2? First, give the answer to three significant figures without using calculator. Two significant figures. Just a simple calculation. What is ln10 also? That's correct, Shatish. 0.693. Very good. Very good. What is ln10? ln10 you should tell me in one second only here. 2.303. Isn't it? If you put 10 over here, ln10 will become 2.303 into log 10 to the base 10, which is 1 only. Correct, Shatish. Why 2.71? Okay. So with this, with this, we are going to now see the graph of log functions. Okay. Now remember, I talked about the base, right? So we were talking about log x to the base, let's say a. So we know that the base has this restriction just like the exponential function had the restriction. So this actually divides the graph into or we can say it divides the concept into two different cases.  So let's talk about these two types of bases. Remember, the graph nature will be different in both the cases. Let's take for example, a base which is half just for, you know, plotting purpose just for knowing the skeleton graph of such cases. Remember, you can take any value between 0 and 1. The structure or the skeleton would more or less remain the same. Okay. Now when we don't know the value of, when we don't know the skeleton graph of something we mostly rely on, on plotting it. Okay, so let's plot it. So let me just ask you a few questions. Yeah, I think more than sufficient we have. Tell me what is the value of y when x is 1? First, I will not give you more than 5 seconds for every question I ask. Value of y when x is 1. Value of y when x is 1. Zero, very good. Okay. Tell me the value of y when x is half. Half. Half. One, absolutely. Value of y when x is 1 fourth. Two, very good. Value of y when x is 2. Very good. Value of y when x is 4. Minus 2. Very good. Okay. So we have enough number of data points to plot the graph of log x to the base of something which is between 0 and 1. Okay. So the first thing is the graph is passing through 1, 0. Now this is something very important, my dear students. Please remember this. We had a very similar trend seen in the exponential. In exponential, if you recall, the graph was passing through 0, 1. Here it is passing through 1, 0. Now slowly start comparing those two graphs. Then I will ask you the final verdict from your side. Next, when x is half, y is 1. So basically you are at this stage. When x is 1 fourth, y is 2. So you are at this stage. When x is 2, y is minus 1. So you are at this stage. When x is 4, y is minus 2. You are at this stage. So we have enough number of points. Let's see how the graph look like. Okay. Just, okay. Yeah. Sorry for this freehand drawing, but this is going to be just smooth. And this is how your graph actually looks like. Okay. Remember it is going indefinitely in both the directions. So put arrows. Now, other comparison that you would see is that this graph will have y axis as the asymptote. Now dear friends, let me tell you y axis is the asymptote only for log x to the base of half. Okay. Or log x to the base of any number between 0 and 1. If you change your x, asymptote is going to shift. Right. Many of us have a very wrong notion about asymptote. I think that asymptotes are immovable structure. No. No. Asymptotes can move. Okay. So if I say log x minus 2 to the base half, then the graph will move 2 units to the right. In that case, x equal to 2 will be your vertical asymptote. Okay. Right. Okay. Now again, there is a very, very stark resemblance. You know, between the exponential curve and the logarithmic curve in this case. Okay. Let me complete the entire process and then we'll make a final, you know, comparison of the graph and see how can the graph of exponential function and logarithmic function be related to each other. Okay. And another thing that I would like to highlight over here is that the graph will never ever touch the y axis. But as your x tends to infinity, y will go to minus infinity. Okay. So please remember this nature of the graph. Now coming to, coming to a case where your base is greater than 1. Again, for example, we can take a simple case log x to the base of 2. Okay. Again, I've taken 2 for simplicity. Okay. Okay. I'll ask a question after this little later on. Let's plot this. Let's plot this. So for that, I need some sample values. Let's have some sample values. Okay. Let's have x, y. Again, what is y when x is 1? Quick. Quick. Let me take it here. Absolutely. Absolutely. What is y when x is 2? 1. What is y when x is half? Minus 1. Very good. What is y when x is 1 fourth? Minus 2. Very good. Okay. In fact, 1, 8, it will be minus 3. Okay. Now, let's try to plot this graph as well. And you can see just like the previous graph, this graph will also pass through this 1, 0. Okay. And as I increase the value, let's say 2, it will become 1. Okay. 4, it would have gone at 2.  Sorry, minus 3. And when you plot it, this is what you see. This is what you see. Okay. Again, y axis is behaving as the asymptote. Okay. And again, y axis is going, the asymptote is going to change if you change the input function, if you change your argument of the function. And you can see here is that when x is going to infinity, y will also go towards infinity. Okay. Now, dear friends, I would like you to see this on the, on the GeoGebra as well. Let's appreciate this on GeoGebra as well. So in GeoGebra, you can actually choose your base. See, there is a function log if you type, it will prompt you. Okay. The first of this function you can see, here you can choose the b value and you can also choose your argument. So let's say if you choose a half as b and let's say x as the argument, this is how the graph will look like. This is how the graph will look like. Okay. Okay. So I have a question for all of you. How would this graph change if I change my base to let's say one fourth? Will it bend more or will it bend less? That means will the graph look like this? Or will the graph look like this? Okay. Let's say I give you the option. Option A. Option B. Let me switch on the poll button. Okay. In case you're not able to see the poll, please type it out. Will it be more bent or will it be less bent? Oh my God. 50-50 response. Okay. Last 10 seconds. Please conclude your voting in the next 5 seconds. 5, 4, 3, 2, 1. Everybody please vote. 4 of you are still not voted. Okay. Do you see that response? Can you see my reaction? Exactly 50-50 voting has come. Okay. We did it. Okay guys. So see common sense. If you apply your common sense, you can resolve this issue very easily. Okay. You're talking about these two functions. Log X to the base half and log X to the base one fourth. Okay. If I were you, I would choose the value of X. Log X to the base four. Okay. Now you know that log four to the base one fourth will give you a minus one. Whereas log four to the base half will give you a minus two. Correct? Yes or no? Okay. So log four to the base of one fourth will give you a higher value as compared to log four to the base half. Now this is already log four to the base half. So option A must be the right option. The graph will bend more. The graph is going to bend more. Okay. Let's see that. Let's see that. Y is equal to log one fourth of X. See, it has been more. Okay. So more lesser you go, that means you're going towards zero more bending it will show. Okay. And more you go towards one, let's say if you take three fourth, it will less bend less. Correct? And you realize that exactly at one, it will become a vertical shape. Okay. So see, Y is equal to log. If you take, let's say, seven by eight X. See, it'll bend less. Okay. Let's say if you make it one by, by any chance, let's say log to the base one, which actually we don't entertain. Okay. That's why it is not giving you any results for it. You make it very, very close. Let me make it as point nine, nine, nine, nine, nine, nine kind of a thing. Okay. It'll almost become a vertical line. Okay. So again, just like the exponential function from more bend, it'll become less bend less bend, it'll become vertical and then it'll start bending this way. Now I'll show you what happens when you do log X to the base of two. So it has not bent the other way. So appreciate the transition. See, the transition is like this idea from this bend, the bend will start becoming lesser and at one stage it will become vertical. And after this, the bend will go this way. Getting a point. Just like in the exponential function earlier, the bend was like this, then became flat and then started bending the other way. Okay. That is what you need to appreciate. Okay. Now another thing that is the most important part for us is if you compare, if you compare the exponential and the logarithmic functions, if you compare the exponential and the logarithmic graphs, see what will happen. Let me draw, let me draw these two graphs on the same XY axis. Okay. Let me draw the same graph. Two to the power X graph, as you all know, it's a graph like this. Okay. And log X to the base two would be a graph like this. Okay. If you see very closely, you would realize, I think a little while ago, somebody was pointing this out, these two graphs are like mirror images about this red line, which is basically your Y equal to X line. Okay. It is mirror image about Y equal to X line. Right. Do you realize that? Should I show you correct order? Let me show you on the Desmos, sorry, on the GeoGibra tool as well. Let me remove this. It is too muddled up. I'll pick up a new instance. Y is equal to two to the power X and Y is equal to log X to the base two and Y is equal to X. See, appreciate. Look at the graph. How symmetrically they are placed. If you want, I can show the mirror image about Y equal to X reflect object about this line. Okay. You can see it is exactly overlapping with this. So even if I switch off the log X to the base two graph, it doesn't change. Even if I switch it on, both are overlapping on each other. You can see them. I'm switching it on and off. Okay. This is something which is a very, very important revelation for us. It tells you that when you want to have the positions of X and Y swapped. See, what is happening? Y is equal to two to the power X and Y is equal to log X to the base two, which is indirectly saying X is two to the power Y. Basically, if you see these two, you can notice that the X and the Y values are swapped. X and Y positions are swapped. Okay. So when you are swapping the position of X and Y in any function, you now know what to do with them. You just have to reflect the one about Y equal to X to get the other one. Okay. So this is another rule that I would like you to add to your rule list that if let's say Y is equal to f of X graph is given. Graph is given. Okay. And somebody wants to plot the graph of X equal to f of Y. Then what do you do? Right? Then what do you do? This graph is obtained by, let's say I call this as one is obtained by reflecting reflecting the graph of one, the graph of one about Y equal to X line. Right? It's a very, very important tool because many a times what happens when we see a question, you know, we sometimes feel that oh, had X and Y positions been in where, you know, interchanged, I would have easily plotted the graph. When such kind of feelings start coming in you, it basically, you know, this particular rule is going to help you in that regard. Yes, that's what I wanted to say the graph of a function and its inverse will be mirror images of each other about Y equal to X line. This is something which you're going to study in next year in the function chapter. But if you know it now itself, nothing like that. Okay. So on this rule, I'm going to ask you a simple question. Sketch the graph of X equal to Y squared minus three Y plus two. Okay. Yes. Unch. So what I said was in any function, if you are swapping the position of X and Y. Okay. And you want to know the graph of this new function obtained. For example, let's say this was the original function and you swap the position of X and Y to get this and you want to obtain the graph of this, all you need to do this, you take this graph and reflect it about Y equal to X line. Your job is done. Understood the rule. Unch. Oh, inverse one. So see what see this fellow is the inverse of this fellow. Okay. Y is equal to f of X. Right. And X equal to f of Y. These two functions are inverses of each other. Okay. So what she was saying that if you want to get the graph of inverse of any function, you just have to reflect that function about Y equal to X line. Same thing. Okay. But let me tell you the inverse concept is not as simple as that, you know, at our time chapati example that I gave you there are a lot of restrictions that we need to exercise. Not every function in this world is invertible. Okay. A function in order to be invertible must be a bijective function. What is a bijective function, et cetera, et cetera. As of now you will not understand it. We'll talk about it when the right time comes. As of now my immediate question to all of you is sketch this graph. Sketch the graph of X is equal to Y square minus 3 Y plus 2. Is it done? Okay. So now whenever you see whenever you see such kind of a question, the first thing that comes in your mind. Oh, how I wish X and Y positions were interchanged, right? Okay. So don't worry, interchange them. Okay. Get this simple looking graph. We all can plot it. It's basically an upward looking parabola. Okay. And it basically cuts the X axis at 1 and 2, right? We all know it's factorizable, right? We all know it's factorizable as X minus 1, X minus 2. Okay. Now how do we change the position of X and Y? How do we plot X equal to Y square minus 3 Y plus 2? Okay. So first you need to understand that we have to reflect such graph about this line. Okay. So we need to reflect it about this line. Okay. I'm making it dotted because this is not a part of our graph. It is just a mirror for us as of now. Okay. Now how would I reflect this? Okay guys, let me tell you many of us are blessed with this capability to imagine the reflection. Many of us can sketch it without much problem. Okay. I as a student, I was not blessed with this capability. I found it extremely hard to reflect any graph about Y equal to X line. Okay. And I assume most of you may be also feeling that same pain. Okay. We are not that you can say artistic kind of people that looking at it, we can reflect it. No. Even I was a person and I felt miserable when somebody asked me to reflect graph about Y equal to X line. But I never gave up. I figured out a mechanism to reflect any function about Y equal to X line. And I would like now to share that method or mechanism with you so that if at all you are a person who is not able to imagine reflections about oblique lines, you can, when I say oblique line, I mean Y equal to X line, you can use that methodology to achieve your task. So what I followed was this method. Okay. Listen to this method very carefully. So let's say I have a question here that I want to reflect this white parabola about this yellow line about Y equal to X line. Okay. So first thing what I used to do was I used to first reflect it about X axis, reflect the graph about X axis. Okay.  I used to draw the reflection of this graph. I'll show you how I used to do it. So let's say this is a small piece of paper. Okay. Let's say this is a paper which I have torn from the back of my book and I used to just draw the reflection of this graph, very small graph I'll make like this. Okay. Okay. This is my step number one. See reflecting about X axis is always easy. Right. There's no problem with that because you can see, okay, this part will go up. These two arms will come down like this. So reflecting about a horizontal or a vertical line is always easy. Okay. Reflecting about oblique line is challenging. That's what I'm trying to address over here. Now the second step what I used to do is, okay. I used to rotate this paper, this piece of paper. I used to rotate it 90 degrees anticlockwise. Okay. I used to rotate it 90 degrees anticlockwise. And when I used to do that, whatever figure I used to get, see this paper, I just, you know, rotate it like this says that my X axis becomes Y axis. Okay. And whatever graph I used to get, for example, in this case, you'll end up getting a graph. You'll end up getting a graph like this. Okay. This used to be your final answer. This graph is going to be your final final answer. Okay. So people who are not able to imagine reflection about Y equal to X line can use my method. Now, later on, I came to know the reason for this, you know, why this method actually worked. Okay. Now, you're saying, wow, right, you will explain me why this method is working. You have learned all the rules, right. And trust me, when I learned this, I never knew these rules, which I'm going to teach, which I'm teaching you right now. Yes. When you rotate it 90 degree Unch, you are making X axis as Y axis and you're making Y axis as negative X axis. Now, anybody, not only Siddish, sorry for taking your name, Siddish, but it is an open question to everybody. Tell me when I am going from Y is equal to f of X to X equal to f of Y. How does these two step help me to achieve that? You can unmute yourself and talk. Anybody, Gurman, Arabi, Arya Dilip, Auro, Reganga, Muskaan, Oshek, Pradyun, Parvati, Rubav, Shubhi, Priyam, Pradyun. Arya Dilip, my question to you is why these two steps convert this graph to this graph? That is my question. Okay. See, what did I do in step number one? I reflected the graph about X axis. Correct?  Y sign you're changing? Correct? So, when you did the first step, my dear, you actually converted this guy to this guy. Correct? Yes or no? Okay. Second step is what? Second step is you're rotating the graph 90 degree anti-clockwise. 90 degree anti-clockwise. Just now I told Ansh that you are making your X axis as the Y axis and Y axis as negative X axis. Right?  So, if you make that change over here, your Y will be replaced with an X. Sorry, your Y will be replaced with a minus X and your X will be replaced with a Y. Correct? So, if you simplify this, don't you end up getting X equal to FY? That is what I was wanting. That is what I was looking for, isn't it? That's why this rule worked. And trust me, I did not know this for at least a month. Okay. When somebody, you know, taught me about these rules, then I correlated, okay, what I did really worked. Okay. And then it also shocked me that if I reflected it about Y axis and rotated it clockwise, then also I can get the answer, isn't it? Okay. So, if you reflect, there's another method, method number two. If you reflect it about Y axis, reflect this graph about Y axis, then actually what are you doing? You are changing the sign of X, correct? And second step is when you rotate the graph 90 degree clockwise, what are you doing?   And there you go. We end up getting the same graph once again. Okay. So, even this rule works. So, whatever suits you, you can do it and trust me, it is a universal rule. You can apply it to any function where you want to reflect that function about Y equal to X line. It works. It really works. You really feel the need of this in a next year topic called inverse signometric functions. Inverse signometric functions, we will recall such rules. Okay. That's why we are dealing with this now only in the bridge course. Okay. That time I will not teach you. I will directly ask you the rules. Okay. Next thing that we are going to talk about my dear students are the properties of logarithms. Again, let me tell you it's very unfortunate that CBSC has designed mass curriculum in a very, very weird way. You have to deal with log functions at every stages. You'll be dealing with log function in calculus. You'll be dealing with log function in coordinate geometry. You'll be dealing with log function in, you know, so many places. But there has not been a single dedicated chapter on log. Okay. I don't think so. Any one of you would have studied log in class 10. Have you? Has anybody studied log in class 10? Of course, in the other book you have studied. But NCRT syllabus, have you studied log? Not in school syllabus. And they expect you, the moment you enter your school premises in class 11, they'll expect you to know log also, correct? So it is very important that we need to learn these concepts in the bridge course because we'll not get any other opportunity to learn them. Okay. So having introduced the graph to you, having introduced the restrictions of log to you. Now, the third thing which is obviously left is your properties of log. Let's talk about them. The first property that many of us, you know, do not know about log. Even the class 12 students make a mistake in this. If you have log of a quantity M to the base A, whole race to the power of A, the answer to this is, any guesses? Very good. Okay. This particular identity is called the fundamental logarithmic identity. This is called the fundamental logarithmic identity. And you'll be surprised that many of your seniors will also don't, they forget this particular identity. Why does called fundamental logarithmic identity because it comes directly from the definition of log itself? Right? Let's prove it. You'll come to know why it is coming from the definition. So let's prove it quickly. Let's say I call log M to the base A as X. Okay. So let log M to the base ABX, right? It means M is A to the power X. It comes from the very basic definition of log. Correct? And what is the X itself? X itself is this term so I can say M is equal to A to the power log M to the base A. Okay. So this property is very important. You'll find that, you'll find this property used in several stages. I've seen many questions on integration also. In order to make it complicated, they use these log properties. Second property of log is, log actually converts products into sum. This property I'm sure many of you would be knowing. And this is a property which is widely used in calculations. Okay. See, why it is used in calculations is because, go back couple of, not couple of, at least 10 years down your line, you will recollect, even you can ask your parents that the first operation that you would have learned in mass is addition. Okay. Any child after counting the first operation he learns is to add things. He never learns multiplication. He never learned division. Okay. So multiplication comes very late in our course, right? So we as humans are more comfortable doing addition, right? Our human mind is more, you can say, designed to do addition faster as compared to multiplication. So this is what log facilitates us. It converts multiplication as you can see, it converts multiplication to addition. And that's why it finds its uses in log, in calculations. The moment we start having physical classes, I'll teach you how to see log tables also, how to read characteristic and mantisa. And this is an extensible property. You can extend it to let's say M and P. So it will become log M to the base A, log N to the base A, log P to the base A. Provided these logs are all defined. Okay. So you can extend this to any number of product. Now how do we prove it? Prove is very simple. In the interest of time, I'll be proving it. See, if you want to prove it, you can start from the RHS side. You can take log M to the base A as X. So let's call this as X. Okay. And let's call this as Y log N to the base A as Y. Okay. This means M is A to the power X. This means N is A to the power Y. Okay. Let's multiply it. Let's multiply these two. When you multiply these two, means you are multiplying A to the power X and A to the power Y. Now, I'm sure we all know exponential laws. Class 7th onwards, we have been studying exponential laws, correct? So when the bases are same, we know powers get added up. So this is as good as saying log of MN to the base A is X plus Y and your X and Y are basically those log M to the base A and log N to the base A. Okay. And hence these properties are two. Any question with respect to this? Very, very simple property. In a similar way, in a similar way, which property number was this three, no? Okay. In a similar way, if you're dividing two quantities, even log will help you there. It converts them to subtraction. Again, we know that we have done subtraction in our life earlier than we had learned division. Okay. So this is again useful in calculations. Useful in calculations. Okay. Again, the proof is very similar. So I'm not going to waste your and my time doing the proof. Next is log M to the power N to the base A is N log M to the base A. Okay. Can you prove this property? Anybody? I'm giving you a minute. Please write out on your notebook, prove this property where N can be any real number. Okay. It need not be a natural number. It can be any real number. Your N can be any real number. Of course, subject to the fact that all these functions are well defined in real terms. Try to prove it. Okay. Very good. Anybody else who could prove it? Okay. In the interest of time, I'm going to help you with this. Let's say, let's say log M to the base A is X. That means M is A to the power X. Yes or no? Okay. Let's raise both the sides to the power of N. Now, we all know it is as good as saying A to the power NX. Correct. Now, not that you have got M to the power N as A to the power NX. Okay. It is as good as saying by definition of log that this to the base A is NX and X itself is log M to the base A. Remember, X is this term. Correct. And there you go. We are done. Okay. Now, many people tell me, sir, why to do this? Couldn't we write log M to the power N as M into M into M into M N times? Okay. Now, please note, this is, I can understand your intentions are right over here, but it will not work if N is not a natural number. It is fine when N is a natural number. Okay. Then you can say it is as good as writing log M to the base A N times. So, you are writing log M to the base A M times by the previous, I think property number three, sorry, property number two, we did that and it is as good as N times log M to the base A. But remember, this type of approach will only work when your N was a natural number. This is a generic approach. It can work for any N value. Okay. Next property is, which was this property number four. Next property is log M. Now, this time the base is raised to the power N. This time, the base is raised to the power N. It is given as one by N log M to the base A. Okay. Again, in the interest of time, I'll be proving this quickly. It doesn't, it'll not take much time. Okay. So, let's say I take log M to the base A as X. Okay. Which means M is A to the power X by the very definition of log. Okay. I can also write this as A to the power N raised to the power of X by N. Right. So, I did nothing. I just put N power on A and divided by M. So, more or less, N and N will get cancelled and it is as good as writing this term. Okay. I can do that. Now, from the very definition of log, we can say log of M to the base of A to the power N will be X by N, which is nothing but one by N times X. X itself is log of M to the base A. And there you go. Done. And it's proved. Okay. So, please note that you may get both the property number four and five together also sometimes. Okay. So, be aware of that. Property number six is what we call as the change of base property. Change of base property. So, listen to this property. This property is very, very useful property. Meanwhile, any questions? Oh, yeah. Sure. Sure. Just I'm going back to the previous screen once. I think Ansh wants to note down something. Let me know once you're done. Okay. Dear students, many a times I forget to look at the chat. Okay. So, in case you find that there is not responding to your question, please speak out that time. Okay. So, please interrupt me. Feel free to interrupt me. Yeah. What is change of base property? It says log M to the base N could be written as log M to the base A by log N to the base A, where A is basically any base suitable for you to evaluate it. Okay. For example, let's say if I want to evaluate what is log, let's say 81 to the base of 27. Okay. Normally, to answer such questions, if you try to go to your power tables, let's say 27 to the power 0, 27 to the power 1, 27 to the power 2, you will realize that 81 will not appear in those power tables. So, how do we tackle these kinds of questions? So, what we do is we choose the number on whose power both 81 and 27 appear. So, I think that number is three. Correct. Other numbers also that depends upon how convenient you are. Some people may take nine also. Okay. So, just take a number on whose power you think 81 and 27 both can appear. So, you can write this as 81 to the base of 3 log 27 to the base of 3. Now, these numbers you can easily evaluate. This is 4. This is 3. So, the answer is going to be 4 by 3. Not only in this cases but also in case of calculus, you will realize that in calculus you will be only told the derivative of ln x or ln x. And the question setter may ask you the derivative of the common log x. In those cases, you have to change the base and solve the question. How we'll discuss when we learn derivatives. Don't worry about it. But this property is very, very important. And a derived format of this property is log m to the base n when multiplied to log n to the base m. This will always give you a 1. Right? How? It's very simple. If you see the left hand side, you can write this as log m to the base b by log n to the base b. Similarly, this term you can write it as log n to the base b by log m to the base b. You can cancel off these terms giving you a 1. Right? Which actually means that if you flip the position of the base and the argument, it gives you reciprocal of that log. This is a very important property. Very, very important property. If you flip the position of the base and the argument, it will give you reciprocal of that. For example, let's say log 4 to the base 2 is 2, then log 2 to the base 4 is going to be half. Right? I don't have to tell you why. Any questions with respect to change your base property? Okay. The next property is m to the power of log n to the base a is same as n to the power of log m to the base a. That means if you swap the position of m and n, nothing happens. Nothing happens. The result is still the same. I'll show you an example. Log 4 to the power of 8 to the base 2 is same as 8 to the power of log 4 to the base 2. You can check how. This is 4 to the power of 3. Right? Log 8 to the base 2 is 3 only. Correct? Which is 64. And this is 8 to the power 2. Log 4 to the base 2 is 2. Right? So this is also 64. So that gives you the same result. But this is about verifying it. Can you prove it? Yes or no? That's correct. Can you prove it? Okay. I'll just quickly prove for you because I have to give you two problems also and we need to start calculus before the break. See, it's very simple. I'll give you a proof for this. Let's say log m to the base a is x and log n to the base a is y. This means a to the power x is m. This means a to the power y is n. Okay? Now, anybody can guess that this is going to be a correct expression. Correct? Because ultimately the power of a will be xy eventually. Correct? Right? So we all know that this is true. Okay? Now what is a to the power x? a to the power x is m. Why is what? Why is log n to the base a? Right? So this is equal to a to the power y which is actually your n and x is what? This is your x. Okay? Which is log m to the base. And therefore this property n is true. Okay? So please remember whenever you have to solve few questions where you feel like flipping the position of this m and n, you can very well do it. So now if you have understood these properties, I would like all of you to solve a question for me. Okay? By the way, log assignment has been uploaded on the bridge course. Okay?  But let's have a question. I would like you to evaluate this log of under root of 3 minus root 5 plus under root of 3 plus root 5 to the base 10. Okay?  And your options are half, one fourth, 3 by 2, 3 by 4. I will switch on the poll. Okay? After you're done, please press on the poll button. Two minutes for this time starts now. It's almost going to be a minute. One more minute. Nobody has voted so far. Of course, you're not allowed to use log tables. You're not allowed to use calculators. So how do you solve this? I can see one of you has responded. Last 15 seconds, my dear. Two minutes are done. Two of you have voted so far. Now I'm giving you another 10 second grace. I want everybody to press on the poll button. Don't worry. I don't come to know who has pressed on what poll button. Okay? Just press on the poll button so that we can discuss it. Okay? At the count of five, I will end the poll. Five, four, three, two, one. All right. So now maximum of Janta has gone for a C option. Okay? This is the result. Okay? Then 14, 14% each for A and B and no vote for D. Okay? Let's see. Let's see what is the answer. See, first of all, if you see this expression. Okay? I'll write it down again for you. Let's multiply and divide with a root 2. Let's multiply and divide with a root 2. How many of you will ask, sir, how do you come to know about these steps? See, when I see under root or something, I want to make it a perfect square. Okay? I want to make the inside term as a perfect square because I want to get rid of this under root symbol. Now, since I see a root 5, correct? That means one of the expressions was root 5 and the other expression has to be 1 and this root 2 will help me realize this. For example, if I multiply with root 2, it will make it root 6 minus 2 root 5 and here also 6 plus 2 root 5. Okay? Now you can see that this expression over here is under root of root 5 square, 1 square minus 2 into 1 into root 5. That's how you end up getting 6 minus 2 root 5. In a similar way, this expression is under root of root 5 square, 1 square, 2 into 1 into root 5. Okay? Now what? Now, when I do that, I can easily break up this under root and write it as root 5 minus 1 and this term will be again root 5 plus 1 by root 2. Okay? So is it clear the transition from this step to this step? Any questions? So the quantity within the under root symbol was actually a perfect square. So square and perfect square, right? They gave me a mod of that quantity but root 5 minus 1 is a positive so mod was removed. Now remember guys, this is a very important step which many people mistake. When you are taking out some square from an under root sign it will come out as mod x. So ideally I should have written it like this. Right? Please keep this in mind. This is something which is very, very essential. It may make a difference between you getting the answer right or wrong. Right? Let me tell you. It may make a difference between you getting the answer right or wrong. Okay? Why is it root 2? See root 2 actually helped me to complete a square inside those under root terms. Getting my point. Root 2 was an agent that helped me to complete a square because when I introduce a root 2 that 2ab thing comes into picture. Correct? And thankfully 6 is basically square root of, square of root 5 plus square of 1. That's why this root 2 is introduced. Don't worry. You'll come to know about all these minor, minor tricks as you solve more and more questions. I don't think so. In this case Anusha, there would be another way to figure out how to do that. It's because I had, I mean I'm talking from a teacher's point of view. From a student point of view if I have to give an opinion it's only when you see many expressions of these types. Then you'll be able to see that oh this expression was actually a perfect square. So you have to get exposed to more and more questions. Okay? Yes. How do we further solve this? That's what I'm doing Anusha. So when you have this term, remember mod becomes immaterial over here because both these quantities are positive. So 1 and 1 goes off so it have 2 root 5 by root 2 which is nothing but root of 10. Okay? So ultimately what you see here is that you have been asked to evaluate laws of root 10 to the base 10. Correct? And we already know one of the properties that this power can be brought in front so it becomes log 10 to the base 10 and that's half. That means option number A is correct. Option number A is correct. Okay? So well done to those who answered with A. Any questions here? Other than the fact that how do we come to know? See math is very very unconventional. Math is like you know are trying to reach the result from right to left not from left to right. Right? It becomes very abstract at times. There is no fixed methodology that will help you get the result. You have to be a lateral thinker. You have to be an unconventional thinker.  When you start doing you know advanced level stuff towards the you know next to next year may not next to next year. Next year may you realize that there can be problems which will blow your mind. You will be like forced to think how can a problem set a thing like that. Okay? Welcome this is Maths for you. One more question I would like to take up before we go for a break. Okay? Evaluate. Evaluate. Okay? This goes all the way till infinity. The base here is 9 by 4. Options are minus 2, minus 1, minus half. Minus 1 full.  Okay? Don't be scared. It is not that difficult as it appears to be. It's an infinite series. 1 by 2 root 3, under root 6 minus 1 by 2 root 3. Again under root 6 minus 1 by root 3 again. So basically you are you know continuing the process on and on for infinite term. Let's have 2 minutes exactly at 6 p.m. I will discuss it and then we will take a break. And on the other side of the break calculus is waiting for you. Once you are done also please press on the poll button so that I know what is your response. It is already 6 but I am going to give you 30 seconds vote. Only 3 of you voted so far. So I am going to give you another 15 seconds please. Everybody please vote irrespective of whether you have solved it or not. Let's vote and then we will discuss it. So in another 5 seconds please finish off the voting. 5, 4, 3, 2, 1, go. Go. Everybody. Okay. Let's end the poll here. This is the result. Maximum Janta thinks B. Maximum Janta thinks B. Okay. Let's see whether B was correct or not. Okay. I think all of you got intimidated by this expression. See this is not a new thing to you. Remember in your class 10th you used to get this question. Evaluate root of 6 minus root of 6 minus root of 6 minus root of 6. Remember you used to get such expression. What do you used to do in those cases? Go back just a couple of months. You used to all this expression as X and from the part where the expression used to get repeated used to replace it with X again. So you should treat it as X is on the root of 6 minus X. Something like this used to do. Yes or no? In the same way I am going to deal with this expression, this ugly expression. I am going to deal with this in the same way. Okay. Many of us we get scared by looking at infinite things. There is nothing to be scared of. So what I am going to do is I first need to figure out let's say I call this as X. Let's say I call this as X. The whole thing as X. From where onwards you think that this X is repeating again. I can see it is repeating again from this stage. You see that? From that term onwards it is again X. Isn't it? So can I say X is 1 by 2 root 3 under root of 6 minus X itself? Are you happy with this or not? I will not proceed till you say you are convinced with that. Just write CLR if it is clear. Because if you don't understand this there is no point going forward. Everybody? Right? Muskaan, Shrita, Shruba, Parvati, very good dear. Okay. Next is simple. I am going to make a quadratic out of it. Let me take this base 2 root 3 on the other side. Okay. Let's square it. Let's square it. When you square it you end up getting this quadratic equation. Okay. Now is it factorizable? I hope so it is factorizable. 72. 72 can be broken as 9 into 8. Yes. Yes, it is factorizable. So let's do that. Here I will take 3X plus 3 minus 2. I will take 4X plus 3. Okay. So I end up getting these two factors which eventually gives me 3X minus 2 as 0. That means X could be 2 by 3 or it gives me 4X plus 3 is 0. That means X could be negative 3 by 4. Okay. Now if you check minus 3 by 4 cannot be your answer. The reason being it is under root of something. Under root of something multiplied to a positive number. 1 by 2 root 3 multiplied to under root of something is going to be a positive answer. So this answer is definitely rejected. Now first question that arises in everybody's mind. If it was not supposed to be an answer how did it end up coming in my answer? Now this is something what we call as the extraneous roots. Okay. I will talk about it in detail in the chapter theory of equation. See I will explain you why these extraneous roots come into the system. See many a times when we are dealing with irrational term. So this is an irrational term. This is an irrational function. Correct. We tend to square both the sides. Don't we? Right. Sometimes we cube also. Sometimes we raise it to a power of 4. Whatever you know cubing, squaring, etc you are doing. When you do that remember you are introducing extra roots into the system. Right. You are most welcome to do it. Nobody is stopping you from squaring both the sides. In fact sometimes you have to square it. Without squaring it you can't solve the question. Right. But when you are squaring it remember boys and girls. You are introducing extra roots into the system. And when you are putting extra roots into the system unknowingly towards the end exercise or towards the end you need to remove those extra roots. You have to select them and remove them out. See I will give you a simple example. If I say x-1 equal to 0 what is the root of this? I am sure most of you will say 1. Correct. That's absolutely right also. Okay. 1 is the root of this. But let's say there is a student Akhil who does this. Okay. He squared both the sides. Okay. And then again he brings this 1 to the left side. Okay. And then he factorizes it. And then Akhil realizes that oh I have now produced 2 roots. I have got 1 and minus 1 both. Right. What went wrong here. In fact there is no problem with squaring. Okay. He did nothing wrong when he squared both the sides. But when he did that when Akhil did that he actually introduced an extra root into the system. Okay. So it is his duty to check whether this root satisfies the original equation or not. Since it will not satisfy he has to remove he has to discard that root from the system. Is that fine. Okay. Now once you have got your x value as 2 by 3. Basically what are you trying to solve my dear students. You are trying to solve this problem reduces it reduces itself to evaluating log of 2 by 3 to the base of 9 by 4. Right. Anybody can answer this. What is this. What is this. What is this. So your answer number C is going to be correct. Option number C is going to be correct. Okay. So with this I will allow you to take a break. Let's break. Okay. And what time you want me to resume at. What time you want me to resume at. 15 minutes is good enough. Okay. 15 minutes is good enough. 30 minutes. No, no, no. I can't give so much break. Let's have 625. Okay. Let's meet at 625. Any questions. Anybody. Meanwhile. Can I also take a break and come back. Telescopic means what is telescopic. It is dead. It saves something for calculus here. A lot of things are going to come your way. Okay.  So calculus. Let's start with calculus. Now again, just a quick poll. How many of you have been introduced to calculus already? Okay. I'm just going to launch a poll. Just say yes. If you have been introduced to calculus before and know if you know nothing about calculus. Not in any form. You have never been exposed to calculus in any form, whether through foundation course or something which you have learned on your own. Little bit derivatives. Okay. Fine. Never mind. 60% of you say you have not been exposed to calculus in any form, either through any kind of foundation course or any self-learning also you have not done. Okay. No worries. Just to know your status as of now. It has nothing to do with my lecture of calculus. So calculus. Let me begin with a basic introduction. Okay. First of all, just a brief idea of what are we learning through calculus? Why did we need this field of mathematics? Right. As you would see the evolution of mathematics, we started with geometry where we started talking about figures and shapes, triangles, circles, et cetera. Slowly it evolved to algebra where we talked about equations, where we talked about variables. Then slowly, you know, transition to, you know, what we call as analytical geometry, which is coordinate geometry. Okay. So we mixed algebra along with geometry and what we got is analytical geometry. So the coordinate geometry part that you are learning in schools is basically what we categorize as analytical geometry. And then came the, you know, a very, very modern field of mathematics called calculus. Okay. The word itself has been derived from the translation of the word small pebbles. Okay. Probably meant for computation or meant for counting. What is this field of maths? Why is this field of maths actually required? Wasn't algebra or wasn't analytical geometry or geometry sufficient enough for all our needs? The answer to that was no. And hence this modern day field of mathematics was required. Okay. Now, give you a simple example of why we need calculus. Let's say we have a function. Okay. We know this is your input, right? And you know, why is your output? Okay. Now, if you want to study the changes in the output, when there is a very, very small change made in the input. Okay. What we call as infinitesimal change. If you make a very, very, very small change in the input. Okay. How does my output get affected by this small change is what actually is a subject matter studied under calculus. Okay. Now, there are a lot of application of calculus, but this is how primarily it was originated through the works of two famous mathematicians, one being Newton and other being Gottfried Leveny's. So these two gentlemen were instrumental towards the invention of this, you know, field of mathematics, which we call as calculus now. Okay. Newton was more instrumental in working towards a segment of calculus, which is now called as differential calculus. Okay. I'll talk about what is differential calculus. It's basically a calculus with deals with very, very small changes in input or output side of any functionality. Okay. Differential, the word itself means a change which is as good as no change at all. Now, how did he end up working on differential calculus? I'm sure you would have heard of a guy's name called Edmund Haley. Have you heard of this name? Edmund Haley. Yeah. Yeah. Haley's comment. Absolutely correct. Okay. This guy was super rich. So one day he came to Newton and he said, see boss, I'm going to give you a lot of money. Okay. I want you to work on, you know, the concept of the gravitational force between celestial bodies. So he basically came for, you know, some kind of theories, which he wanted Newton to develop on, you know, how the, the gravitation force between these celestial bodies change, you know, with respect to their distances. Okay. Later on, he was the pioneer in the field of gravitation also where he worked on a lot of gravity related concepts. Okay. So the other day I showed you a slide that during the, the morning played plague in 1665, he worked on his theories of gravity. Okay. So that time he was working on how does the gravitational force between two planetary bodies change with respect to their distance? Now, of course, we know the formula now that is GMM by R square. I'm sure when the research takes up the gravitation chapter with you, we'll come to know about that formula as well. So what happened is that between two planetary bodies, okay, the gravitation pull, okay, dependent on the distance between them. So if this distance changed by a very small amount, that is what I call as infinitesimal change, infinitesimal means infinitely small, infinitely small. So this word is an amalgamation of these two, you know, small, two words infinite, small infinitesimal. Okay. So when this gap change by a very, very small amount, how does the force change in that respect? So how does the output change when the input, let's say force is a function of the distance between these two planetary bodies. How did this force change when that distance between the two planetary bodies change infinitely? That is what Newton was actually working on, on the request of Edmund Haley and there he, you know, stumbled upon a lot of concepts of differential calculus, right? On the other hand, Godfrey Leibniz was working on the concept of accumulative calculus. Accumulative calculus means when you add these small things, right, how do you figure out what is the bigger thing? So he was working on something which we call as the integral calculus. The word integral calculus came from the word integration. Integration means putting together very, very small quantities. For example, you can say this earth is an integration of dust particles, correct? Our body is an integration of all these cells that we have. So his field of calculus was something different from that of Newton. But nevertheless, the works of both these mathematicians were linked by something called fundamental theorem of calculus. It was linked by what we know as fundamental theorem of calculus, okay? I'll introduce this concept to you in class 12th, okay? Guys, there's something very unfortunate again, okay? Through this video, I would like to, you know, put a request to the government secretaries to all the officers sitting in the education department. Please, please do not treat physics and maths separately in class 11th and 12th. Yes, it is a very ironical thing that you are studying vectors in physics now, but you will be studying vectors in maths in class 12th. You will be studying calculus in physics now, but officially the 70% part of the calculus is studied in class 12th, okay? So there is a very, very bad correlation, or you can say a very, very bad mismatch of concepts of physics and maths in 11th and 12th. Now, I do not understand the rationale for the CVSE board to do that, but I speculate that they probably have done this to make your, you know, a curriculum lighter. Okay, so they have taken a deviation from the logical flow only to make your 12th more lighter because it's a board year for you. Okay, so they want to keep it as lighter. Okay, never mind. See, these are things which are beyond our control. So unfortunately, or I mean, fortunately, we have to abide by the rule set in by the government. So I'll just give you an idea. What is there in calculus for you for this entire two years? As I already told you, calculus is, you know, divided into two parts. One is the concept of differential calculus. Under differential calculus, you people will be studying, under differential calculus, you people will be studying primarily these three concepts. Number one, the concept of limits, which probably I'm going to start today. Then you're going to study the concept of derivatives. And then finally, you are going to study the concept of application of derivatives. Okay, out of these three, the first chapter is what you're going to study in 11th, but that to not completely. Okay, you will study only one aspect of limits. Okay, the major aspect you will study in class 12th, along with a chapter called limit continuity and differentiability, what we call as LCD. Okay, but remember, without limits, you cannot understand derivatives. So we say derivative is an application of limits. The next concept is the concept of derivatives, which half you're going to study in 11th. And again, a major part you're going to study in 12th. For example, in 11th, you will be studying just the methods of differentiation. Like what are the basic rules which are useful for differentiating a function. But you will learn how to differentiate complicated functions only in class 12th. And application of derivatives is a chapter which is completely in 12th. So as you can see, we touch upon only half of these two chapters in class 11th. And major part of the differential calculus lies in class 12th. Okay, the second aspect of calculus is your integral calculus. Integral calculus comprises of four major chapters. And they are indefinite integrals, indefinite integrals, which is completely studied in class 12th. Okay, unfortunately, you'll start using integration in physics. I think very soon in kinematics, in let's say rigid-mod emotion, in question related to SHM, in chapter related to moment of inertia, etc., you'll start applying integrals. But again, in maths, you're going to study in class 12th, sorry. Your seniors are actually studying it right now. Next is definite integrals. Again, you're going to study this chapter in 12th. Now, many people ask, are these two related to each other? Yes, you can say there is a bit of overlap. But remember, indefinite integral basically is a raw form of integration. Definite integral basically works on properties. There are a lot of properties which we'll be talking about, which will help us to solve definite integral problems. As a prime of SE, the way to distinguish it is definite integral has limits, has upper and the lower limits, so if you integrate something from A to B, you are performing a definite integral on it. But if you just integrate without any limits of integration, we are performing indefinite integrals on it. Okay, we are going to touch upon few of these concepts in the bridge course. Then comes application of integration, or application of integrals. This chapter basically helps us to find area under some peculiar functions. See, this is another application of calculus. When I ask you, what is the area under this square? I'm sure everybody will take not even a second to say A square, correct? Or let's say area of a circle, right? Pi r square. But if I say what is the area under this function like this? Let's say from 0 to A, then probably you will not be able to answer it without the knowledge of calculus. For that matter, you will not be able to find out the length of curves like this, correct? There is one typical flyover in Indra Navya. I don't know whether you have seen that. It has got this typical kind of bending in that. With our analytical geometry, we are not able to find the length of such curvilinear structures. So we need to use calculus for that. Of course, one way is you take a string and go along the path. But that itself will have some errors. Because when you are taking it through strings, you may be having some kind of linear term replacing an actual curve. So here are to answer such kind of questions. That is what we call as accumulative calculus, which is covered under application of integrals. But in your class, sorry, this is also a class 12 topic. And in a class 12th, we will be only talking about one such application which is the area under curves. This is what we are going to talk about. AUC. Finding the length of the curves is something which probably would be required in your APs. People who are writing advanced basements, you would require the idea of such length of arc, what we call it. No, no, we don't do length of arc. We even don't do the volume of rotation of a surface. We only do area under curves in your CBC class 12th curriculum. And finally, we have a mix of both integration and differentiation which we call as differential equations. Where you will be learning how to solve equations which are made up of derivatives. So that is the last part of the calculus part. And this is also in 12th. So let me tell you 95% of calculus comes in 12th. And almost your 70% of the topics in class 12th would be made up of calculus. So you must be wondering why are we doing this chapter in the bridge course? Because yes, you will be needing it in every small concepts in physics and to a certain extent in chemistry as well. So with this brief understanding, let me start with the concept of limits and derivatives. Now, before I start into this concept, I would like all of you to pay attention to a scenario. So I have a question for all of you. Let's say you are travelling from Bangalore to Mysore. So let's say on the y-axis we have displacement, displacement signified by S and you are travelling from Bangalore to Mysore. Let's say this distance here is 130 km. And you take 5 hours to reach. This is your time axis. So you start from t equal to 0 from Bangalore and you reach at t equal to 5 and unit is hours. And this is how your displacement versus time graph looks like. Let's say this is how you... So this is exactly this. Now answer these two small questions that I am going to ask you. Number one question. What's your average speed? You must be wondering if that is asking such a simple question. Absolutely. So basically you all gave me the answer as 26 km per hour. Now this was not a challenging question for you because all you did was what is the total change in the displacement from B to M divided by what is the total change in the time as you went from B to M. So this displacement was 130 km and you took 5 hours for that and there you go you get 26 km per hour. So no doubt about this. Nothing very very difficult about this. But now my second question is slightly tricky. My question is what was the speed of your car at t equal to 3 hours. Let's say this is your t equal to 3 hours. Add this instant. Add this very instant when your clock exactly... Let's say you started your journey at 12 noon and you reached Mysore at 5 pm. I want to know how much was the speed of your car from this graph when the clock struck 3. Yes Ansh. This is just an indication for you because you know it. I want this question to be answered by all of you. Not only Ansh. How did you figure that out? You gave me a value. How did you figure that out? You can unmute yourself and talk. No, geometrically. See this curve is given to you. From this curve do anything you want. You have been given a scale. You have been given every instrument that you want to. From this graph you measure out the velocity of the car that you were travelling on exactly when the clock struck 3 pm. Now this is something which many of us may not be able to answer right now. Why we are not able to answer this question right now is because we have always been used to finding the speed by taking the change in the displacement or change in the distance by change in the time. But here I am not allowing you for that change because I am asking you the velocity or the speed exactly at a pin point. Exactly when your clock was showing 3 pm what was the speedometer of your car reading is something which baffles most of us. It surprises most of us. It takes most of us to our back foot. Because to know the speed we have been always been taught that there has to be a change in the displacement and there has to be a change in the time. So when I say 3 pm how much change will you make from 3 pm? That is something which is confusing right now. Will you see how much your car has travelled between 3 pm to 3.01 pm? Will you see that? So many of us would be thinking let's say between 3 pm to 3.01 pm how much is the displacement of the car that we will figure out and we will divide this by 1 minute. We will divide it by 1 minute and we will get the speed. But do you think this is going to give you the right answer? Because somebody else can argue why did you take a minute? You could have taken a second. You could have seen how much did my car move from 3 pm to 3.001 pm. That means in that 1 second it may be a very small move, very tiny move. How much did my car move in that 1 second? That distance divided by 1 second will give me the speedometer reading or the speed of the car. But somebody else can argue why even 1 second? Time is a continuous quantity. Take 1 microsecond. In 1 microsecond how much did your car move divided by 1 microsecond? That should be the speed of your car. And trust me guys there is no limit to how small an interval you can take for that time gap. Somebody can say take 1 nanosecond somebody can take 1 picosecond, 1 femtosecond. Right? Now understand the problem here. The problem situation is how much should I allow my car or how much time should I allow my car to move? And that time gap that you can allow your car to move can be a very very infinitismal time gap. That means if you find how much did your car move? Okay? In a very very small time gap where this time gap is as good as tending to 0 this probably will give you the speed of the car at 3pm. Okay? And later on you will come to know that it is basically obtained by creating a tangent over here and the slope of this tangent will give you the speed of the car. Okay? So I'll address this question I'll justify this solution of mine in the later part of this chapter. But try to understand what are we studying? We are trying to study something which we call as instantaneous change. So now it's a world where we live in instantaneous changes. We are trying to study something which we call as instantaneous change. Your junior classes you were studying average change. Now grow up. It's not a world anymore of studying average changes. It's a world of studying changes which are happening at every instance. You are growing in height as I speak to you even though you cannot see it happening but all of us we grow or let's say I'm becoming older or I'm becoming thinner as the moment I speak to you. Right? So most of the things happening around us they are instantaneously changing. Any small government decision taken will instantaneously you know fluctuate the sensex market. People who are interested in stocks and all. Okay? They fluctuate on real-time basis. That's why we need a graph for it. Okay? We cannot afford to now study average changes. Right? It's not like okay now let me sleep today I'll purchase a stock tomorrow. Who knows within one you know Modi makes one decision and stock markets crash out. Right? So as a person who is working whether in the field of science or in the field of economics you need to be aware of instantaneous changes happening around you. I'll give you an example. Let's say you are an IS officer. Okay? I'm sure some of you would be having you know IS officers in your family. This is the you know biggest government post anybody can occupy as an executive other than of course the MLAMBs and all. Now these guys they need to take decisions. They need to take decisions. Right? Now you know that the food market is heavily dependent on the price of fuel. A farmer in order to cultivate his crops would need diesel would need petrol would need kerosene to run his motors. Right? Of course it depends upon so many other factors like price of seed, price of fertilizers, the price of water, the price of machine. So let's say one of the factors I take is where your food is a function of function of diesel. Okay? Now let's say as an IS officer you know that what is the relation which governs food and the price of the diesel? Correct? Now if you want to see how much does the price of food let's say any food item it may be a dal item or it may be a rice item or it may be a wheat item. Let's say if you make a small change in the price of the diesel. Okay? Let's say you make a small change in the price of the diesel. Okay? How much does the price of the food change because of that? If you want to study this change you must be knowing calculus in that sense. Right? So even a person who is sitting on a post which is non-scientific even in the administrative post administrative post you should be aware of how is your output fluctuating with respect to the input. So if you increase the price of diesel by one rupee today how will a poor farmer get affected because of that? How would the price of cultivation change because of that? And how would the price of food that eventually people like you and me will be having we have to purchase the food from the grocery stores how will we be affected by that? So he will basically do a sensitivity check. He will see how much is the change in the price of the food with respect to the change in the price of the diesel happening when this change is very very small. So first he will note this data. Okay? Let's say this data is a huge value let's say it is 100 which means that if he makes one paisa change in the price of diesel it will lead to one rupee increase in the price of food. Right? And trust me many of us in our country will go sleep will sleep without having food because of this one paisa increase in the price of food. So as a person who is at a powerful position like this you need to know how sensitive is a commodity with respect to another commodity and for that you need calculus my dear friends. Okay? Such is this important concept for us. It is not just about studying speed and time relation. Right? It is about studying how is an output? Right? Changing with respect to input. So if you change your input to delta X how is your output getting changed? Sorry. How is your output getting changed? So it's a field of mathematics which tells you how is this quantity you know, affecting your decisions. Is this fine? Okay. So with this example basically I wanted you to bring your notice to the field of instantaneous changes. Okay? And given that I have written something like tending to zero basically this concept itself comes from the concept of limits in calculus. So let us talk about it for the next half an hour or 36 minutes of our class. The concept of limits. Now I would request all of you to put your pen down. Put your pens down. Okay? Okay. Nothing should be there in your hand. Just listen to me. Whatever I'm going to write down wherever it is required I'll ask you to copy it but just listen to me because in the first go this concept may be slightly confusing. Okay? It's by experience from the last 12 years I have been teaching this and I realized that in the first go many people don't get this right. I'm not saying you will also not get it. God forbid that you get everything. Correct? You should not be struggling with anything.  When I say limit of a function as x tends to a what do I mean from that? I'm not explaining you the meaning of this. This is a term which will come across a lot. The moment you open the first chapter of calculus you will see this expression. So let me explain you what does this mean and before I explain you what does this mean? What does this mean? I would like you to understand what does this mean? This is also a concept which many people don't get it correct. Okay? Correct? When I say x tends to a first of all get the pronunciation right? x tends to a many people call x approaches a what does it mean? It means that x is coming close and close and close and close to a fixed value a but never achieving it. Let me take an example. When I say x is tending to let's say 2. Okay? So x is a dynamic variable. It's changing every time and as it is changing it is trying to come as close as possible to 2 but never achieving 2. For example, it can take values like 1.9 it can take values like 1.99 it can take values like 1.999 it can take values like 1.9999 and so on but it will never achieve a 2 so it is still moving towards 2 but not achieving 2 it can also take values like 2.1 2.01 2.001 2.0001 etc and it will slowly move towards 2 it will slowly move towards 2 but will never achieve 2. Please remember you have x will never achieve 2. Please get this right. Any questions? So what I am trying to say is that on a real number line let's say 2 is here then you are trying to approach 2 from left side this is what we call as approaching 2 from the left side of 2 left side means from values which are lesser than 2 towards 2 or you can approach 2 from the right side of 2 let me show it by the blue color sorry okay when you approach 2 from the left side of 2 that means you can take these kind of values we say that you are approaching 2 minus or what we call as you are approaching 2 from the left hand side of 2 get these terms correct okay I have not yet talked about limit I am just explaining you extending to a means what okay when you are approaching 2 from the blue arrow side that means you are approaching 2 from the right hand side of 2 and we write it as extending to 2 plus please note it doesn't mean 2 minus doesn't mean minus 2 don't get me wrong 2 minus means you are approaching 2 from a lesser side of 2 that means 1.99, 1.99, 1.99, 1.99, 1.99, 1.99, 1.99 like that okay when you are approaching 2 from the positive side or you can say from R H L right hand side means you are approaching 2 from a higher side of 2 let's say 2.1, 2.01, 2.001, 2.001, 2.001 like that okay but in both the cases you are not reaching 2 it is still a motion that X is performing X is still in motion you can say it is still dynamic okay this is what we call as the right hand side approaching 2 from the right hand side any question regarding right hand side left hand side by the way they can only be 2 sides because a real number line is you know a single direction yes Ansh what I meant from the entire explanation was I am very close to 2 but I am not equal to 2 okay it's like however great a batsman I am I am very close to Sachin but I cannot become Sachin okay okay now try to understand so extending to A is clear to all of you right no doubt about it now what is this full expression what is this full expression if you say limit of a function as X sends to A is a value L it means that all of you get this it means that as X approaches A whether from left side of A or right side of A doesn't matter your f of X will either approach L or will be equal to L this is the meaning of the limit of a function as X sends to A is equal to L any question I will take a pause over here L is the limit L is the limit see I have written it as L no so what I am trying to explain here I am repeating my words once again my dear students see X is the input f of X is the output if X is approaching A f of X is also trying to approach something or it may also achieve something I will tell you when does it achieve something when does it approach something so as X is moving f of X is also moving so as X approaches A f of X will approach a quantity which is what we call as the limit of the function so L is a fixed value just like A is a fixed value correct so limit is a fixed finite value since the function is trying to go or will achieve as X is going towards A are you getting my point like so it's like if I am trying to go towards a dried well correct my feet are let's say X and my destiny is let's say Y so as my feet approach the well my destiny approaches death right sorry to take a very sad example like that but this is how you can understand as your input is approaching A output is approaching some fixed value and that value is what we call as the limit so Anusha has a question unless X is at the point how can Y be equal to L okay that's the whole point X is not at a point even f of X is not at a fixed point both are moving correct in many case even when X is moving Y may not move for example a constant function okay let me ask you this let me give you a simple scenario Anusha let's assume that you have a constant function like this okay f of X equal to 2 if I ask you what is your function trying to what is your function f of X approaching towards as you approach 3 then say sir even if you approach 3 from the left or right your function always remains 2 so 2 is your limit 2 is your limit in that case but many a times there are functions which are moving for example instead of a straight line I had a let's say a linear line like this let's say the line was like this okay let me take an example of a line X plus 1 okay now if I say as X tends to let's say 4 okay as X tends to 4 what does this Y tend to or what does this f of X tend to that means I am evaluating this limit you will say as you come very close to 4 okay whether from the left or the right your value of Y will be very close to 6 sorry very close to 5 right so this 5 becomes your limit try to understand this okay now you must be wondering why did we need limit for this you could put 4 over here and got 5 why sir is selling all these kind of coming from left and coming to right see this is not as easy because you cannot put 4 into this because you cannot achieve 4 you are either slightly lesser than 4 or slightly bigger than 4 but you are not 4 exactly X is not exactly 4 but the function is still tending towards the fixed value and that is 5 I will give you more examples let's have patience right now I know at this point of time lot of questions come in our mind what is he saying tending to and he is putting the value just relax it's a phase which everybody undergoes I have also undergone the same phase your seniors also underwent the same phase probably any senior of yours who has learned this will also go the same phase so have patience right now slowly and slowly unwind the entire story as of now whatever I am saying is that clear or not no f of x may tend to that value may achieve that value also sometimes to answer your question see in this case it will never achieve 5 but in this case it has achieved 2 so it may tend to also it may achieve also that depends upon scenario to scenario does that answer your question Ansh ok things will be more clear to you as I proceed with this concept ok as if not just listen to what I have to say ok so basically what I want to highlight here is that as x is moving your f of x is also moving so as x is moving towards a fixed quantity a f of x is also moving towards a fixed quantity L and that L is called the limit it may achieve it it may just tend towards it this is what I have spoken so far in this discussion of limits any question here please highlight yes should be tell me ok you had a question on top so if you put that arrow for me I don't see all your chats together ok so can I say that one is the limit of the fractional part function again we will come to it I don't want to dump you with too much of information in the first introduction ok see you don't know everything about a person when you meet it for the first time right ok you have to give that person some time in order to know more about it so let's teach this limit is like a stranger for me at least for people in this session let us learn more about it many things will be clear ok of course Oshik I will give you when does the function achieve a limit that also I will tell you Auro what is that question no needn't be x may be still moving Auro but f of x may achieve well I gave you know that example constant function ok wait wait many things will be clear when I tell you the next step next things ok now all of us let us try to understand first thing here limit is basically the value that the function achieves if you approach x from either the left side of A or the right side of A ok you must get the same value irrespective of whether you are tending towards x from the left side of A which what we write it as this or if you tend to the function or you tend to the value of A from the right side of A so these two values must be equal first of all and whatever they are equal to that is your limit value ok that is what we call as the limit of the function as extends to A now in order to show you this first of all I will give you some cases ok cases where your limit will exist and what is the value of that I will tell you and cases where your limit will not exist ok so somebody was asking does the limit of a function always exist the answer is no when does it not exist let us try to figure it out from these two examples let me first begin with a very simple example that most of you would be able to understand very easily let us say I take a function x square minus 1 by x minus 1 ok extending to 1 ok oh sorry I should not write here so let us say this is your function and let us say I want to evaluate I want to evaluate limit of this function as extends to 1 minus that is the number one question for us number two question is I want to evaluate limit of this function as extends to 1 plus and finally I want to evaluate what is the limit of this function as extends to 1 so for the purpose of better understanding what I did was I am separately evaluating what we call as left hand limit so this is what is called as the left hand limit why we call this as left hand limit for the simple reason that we are now approaching x from the left side of 1 ok so the answer that you get is what we call as the left hand limit ok this is what we call as the right hand limit and remember what did I tell you just now when both your left hand limit in short form I can call it as LHL when you both LHL and RHL are equal and whatever they are equal to that becomes your limit that becomes your limit so we will be evaluating from the result of the first and second what is your limit finally ok let's do this exercise yes or what is that question why what is the complete question why are we doing it you are asking that Anusha you will get all your answers ok let me put a thumb rule let's not ask any question till I am done with this exercise ok whatever I have on my screen exactly at 7.25 I will stop whatever I am going to tell and I am then going to entertain your questions if at all you will have any ok so I bet you will have no questions till I finish this exercise yes if LHL is not equal to RHL we say limit doesn't exist so let me write that down also if you want me to write so that is number 2 if LHL is not equal to RHL it implies L does not exist ok and I am going to give you instances of that as well in this example Anusha I can very well understand what is your question just wait for me to get on over with this exercise ok now before I start evaluating this I would like you all to you know look at this function very carefully very very carefully many of us you would have already noticed that your numerator is factorizable correct you can write it like this and many of us would have a feeling or would have this tendency inside us let's cancel them off my first question to all of you is can we do that can we cancel out x-1 from the numerator and denominator guessing no no ok the answer is you can you can do so if you know your x is not equal to 1 but you cannot do so if your x is equal to 1 that means if your x is 1 your function will actually become an undefined expression getting this point this is very important so when I redefine this function again recall the word redefine I have been using this word ok if you redefine this function f of x this function f of x is behaving as x plus 1 when x is not 1 and it is behaving as undefined that means I don't know what to do with this function exactly when x is equal to 1 is this convincing enough for everybody are you happy with this ok now why I cannot cancel it the answer to that is very simple you cannot cancel 0 from the numerator and denominator guys if you are doing it you are committing one of the biggest blunder in mathematics for example let's say 2 into 0 is 3 into 0 you cannot bring this 0 down and say let's cancel 0 and 0 and you will end up getting the biggest discovery of mankind that is 2 is equal to 3 right so when you are cancelling x minus 1 and x minus 1 please remember you could not have done that if your x were 1 but if your x is not 1 if your x is not 1 definitely cancel it off ok now somebody was asking me about a whole in the graph so this is an answer to your query if you actually plot this function graph you would realize that this function is behaving as y equal to x plus 1 line sorry for that I will draw much better line than that let me draw it in white your function is behaving as a y equal to x plus 1 line at all the points but just at one point which is x equal to 1 there is a hole I am putting a hole over here so there is a hole in this function at x equal to 1 now those who were asking can we even evaluate the limit when there is a hole at the function at that point the answer to your question is yes we are not going to do that we are not going to evaluate this limit because I have nothing to do what is happening at A in this case I have no business whatsoever at whatever is happening to the function at x equal to 1 whether it is defined whether it is undefined whether there is a corona virus sitting over there whether there is a bomb blast over there I don't care what is happening at A because x never reaches A is this tending to A so what is happening just before that point just before that point whatever is happening and just after that point whatever is happening I just care about those values and are they tending towards a single value and that value is my limit is what I am worried about I am not worried about what is happening at one in this case does that answer your question even if there is a hole what do we do now trust me Anusha 99.99999 you can say tending to 100% questions will be of this type to you where you don't know what is happening to the function at that given point and this clarifies and this clarifies the second out of students who believe that or who wrongly believe that limit is all about substituting values it's not that easy guys if it was substituting values why would we have studied it why we would make our own life so difficult so you can't evaluate answers by substituting x value so if I have to evaluate this limit I can't put 1 over here in this function because if I do I get 0 by 0 which is undefined so it clarifies 2 of your questions I am sure now finally how do we evaluate it for that I would like you to first evaluate the first part left hand limit let me complete that then I will answer any further questions so for left hand limit I am going to make a small table for you I am now approaching 1 from negative side of 1 and I am going to se