Transcript for:
Calculus: Continuity & Differentiability

Calculus explosion with continuity end differentiable hello everybody i hope that you are all good and today I brought hmm class 12th maths continuity end One short video of Differentiable is over We will do this whole lesson in just one video Due to special request from me and the kids There will be lots of them in this video questions and so many questions By solving this you will definitely get this much You will definitely gain confidence that the text book You can attempt these answers yourself If you find me then I am Roshni from Lan Hub The Free learning platform where you can Physics, Chemistry, Maths, Biology, Everything You can read it absolutely for free at hub.com so everybody is ready this is going to happen It's a long way but it's a lot of fun so let's get Started now with the story of calculus for kids It had started in class 11th itself when We studied Limits and Derivatives Right there we learned that man limits what are derivatives how do we Let us find out the fact that we did it in class 11th I also learned some basic things like in a way that solves trigonometry functions find derivatives such as theta k how do theta tan3a etc come to the polynotes find the derivatives as x to the how to find the derivative of power n right so these kind of basic things we have there Now all those children who had covered it Limits and derivatives not read properly This is my request to all of you in tha 11th that in this One Shot of Limits and Derivatives You must watch the video very much It will not take your time, but your The concept will be crystal clear. so what are we going to read here look here we will read that continuity and differentiability are what is the thing between these two what is relation and most Important Here we will learn how we Derivatives of many other functions You can take it out, like in 11th we had Just trigonometry functions and polynotes we saw the function but now we will see the evers Trigonometry Functions Logarithmic Function Expo sial function and many more special Functions means the agenda is clear brother ok but today we start the story The first question we will ask is about continuity It means that study continuity Why do I need to read Continuity because dude What is our end goal? An end goal is accomplished In calculus, the whole calculus is not meant At least this is our end goal in this lesson Finding derivatives of differential functions is fine Now a particular function of any The derivative at the point p exists only if If the function is continuous at that point means if that function is continuous only then will its derivative exist This means that long before we find the derivative We should have this knowledge of continuity Now the kids must be getting irritated a lot that ma'am what is this derivative we have Limits and Derivatives Class 11th I had not seen the video yet, I sat down immediately I don't feel like watching it, I'll see it later Right, so for now, you tell me that what is a derivative thing right so let's go Let us understand what a derivative is You will understand it in a little while But it is more important that the derivatives Remember why it is so useful for us In physics we study about instantaneous Speed ​​Instantane Necessary Velocity Instantane In neous acceleration chemistry we study are the instantaneous rates of reaction and this all the things that we calculate there But we use the infact derivative I am pretty sure that many of you kids has already used derivatives In these topics of Physics and Chemistry okay but what is our agenda today we will learn how to solve any given function How do we calculate derivatives So if we take an example from lets say that we talk about velocity okay when we If we find the instantaneous velocity, we how do you get it out, think a little, think how we bring out what we say what is velocity change in displacement with time so if we talk about the graph To find the velocity, which one should we use? The graph uses displacement time The graph with y axis is displacement and is in the x axis Time is right now here we are at Velocity Let's talk about two velocities, one is Average velocity and other instantaneous Velocity is absolutely correct when I say average velocity so what i want to say yes bhaiya the time is from t1 to t2 I did it from s1 to lets go to s2 so much If displacement is done then its average What is the total velocity? The same displacement it did with Lets delta s divided by this over time which is delta t right now from t1 to t2 that this whole time it had delta s displacement kya so delta s / delta t is its average velocity but I said no, I am not average velocity I want to know you in an instant I want to know how much velocity it is, that is I want to know the instantaneous velocity How do we define instantaneous velocity? Let's think a little about this, if we Have you discussed the example then you Basics of Limits Derivatives are Clear It will be okay so how do we get it out we say dude first we'll calculate delta t I was talking about this much from t1 to t2 How much displacement did one cover now You are talking about instant, this means which delta t is mine that delta t has almost reduced to crying because now I I am talking about one point, now this point But what is delta t for any large time interval? So I am not talking about just one instant so what did we do Mathematically, we get the limit delta t from V. tends to 0 delta s / delta t and this This is what we learned about how we define In 11th we define it as edge so that's why we are very logical We will not think, how will we think about maths think like this means we will see graphically we will see things in such a way that we How to calculate, so study maths now So you will think like maths, right? And you will think like maths. To think like this you need a focused mind And a little is required for a focused mind A little walking, a little jogging, a little exercise and A little relaxed mind it graphically think that all these quantities are Whether there is instantaneous velocity Be it acceleration or instantaneous rate off reaction you must have seen it whenever we Their values ​​are calculated graphically so what do we do I said at this point I need to find out instantaneous velocity on p so what do we do at that point let's make a tenge and that tenge eight which Whatever slope happens, that is what happens to me instantaneous velocity right this is for everybody I know, because we've done all these things This is the basics of physics, friend. So all this things right so what are we doing basically what basic sacrifice we are doing is that making one on a particular point what is tange and slope of tange perpendicular by base so as soon as its we're taking out the slope of a tange at is t theta as soon as you find the slope If so, what am I getting from that? has instantaneous velocity value correct so from this we get one more thing this It turns out that the derivative is what actually happens if we think graphically so it is basically the value off the slope of the tenge now the second question which What might be coming into your mind is that ok ok I understood this but it is like this Why is it that any function has a The derivative at a particular point p is then will exist when that function is at that point will be continuous why is it that will be continuous only then its derivative If it exists then the best way to understand it is to First let us understand the concept of continuity Whenever we teach these concepts to kids If you see the terms of the graph then don't give it to us You will get to see some types of functions The functions will be like this when we represent them graphically You will see that we can kill them in one stroke You can draw functions from your pen without lifting i.e. assume that you mean like me I drew this function so what should I do I had to pick up my pen once in between No, I got drunk in one go, I guess This function didn't even make me lift the pen once I haven't read it yet, suppose I drew this function did I have to bear the brunt even once otherwise this There are such functions which we can call the same You can draw the pen in one stroke What kind of functions can it perform without lifting Continuous Function Continuous Meaning assume that something is going on continuously You are going on a road, that road is continuous It does not mean that he doesn't go in the middle The road is ending, no it's not like that You can go safely on that road it's continuous okay on the other side there's something There are functions that have no intersection between them There is a break because when we draw them If we go to do it then in between we have to put our pen you will have to lift it, suppose it is there is a function, you go till here then there is a break then hand had to remove it back from here below Right, so what are these functions called? these are what we call discontinuous functions okay now let's take a look here The thing is that if you use the discuss function focus on them and what will you notice There is obviously a break in between but if you Look at the starting part Like the part from here to here So it was continuous, meaning from here to here See, I was able to make it in one stroke Where could you not make it to this point? basically the discontinuity that is at a The point is there is nothing like that which is a function Every point of his is discontinuous, it is not like that nothing means it is like it is depending on whether that function is a particular Is the point continuous or discontinuous saying ok means that you are going ahead you will go and see that whenever we go for some function We always talk about continuity Talk about a point means this function is Whether it is continuous at the point or not, this is The function is discontinuous at this point or no right so continuity or Discontinuity is always with respect to a point for a function so kids let's see we define continuity mathematically How to do it? Suppose we have a function is f lets say f x f x what does this mean The input values ​​of the function are That is x means it is dependent on x you will keep changing the value of let's define it on fx8 so let's suppose that This function is a point in its domain is c okay domain means any number of its possible input values ​​are ok so let's than c is a point such that I want to tell that c But whether this function is continuous or not then this Function fx8 c will be continuous only when limit x Tendencies to c fx3 When this condition is satisfied only then we Let us say that this function f x c is at a point Now this condition is continuous what does it mean that the point c but the value of that function will be at point c It will have the same value in the neighbourhood of x tends to c means the value of x is something like this which is very close to c but not equal to c okay means its right next to c The value of this is the same as the value of c So that means the function is continuous, okay? Let's take an example, look here, you have two The graphs are shown in the first graph if Let me talk about the first graph. Let's talk. about a point x = 1 okay so x When the value is one, look at the value of f x What is -1 is visible to everyone right when the value of x is If the forest is fx-100ms is -1 i.e. on punt which There is value in one's neighborhood as well, the same value i.e. what kind of function is this Continuous function occurs at x = 1 then x = 1 At this point, this function is continuous okay now let's look at the other okay on the graph there's something else going to happen here Here also you focus on x = 1 When x = 1, see what your value is right so if you look at your value at x = 1 so what is that value that value is again the value of the right function is -1 but now If you look at its neighborhood, as soon as you If you look at the neighborhood on the left its value is -1 but as soon as you The value of the neighborhood to its right If you look at it, you will see what its value is. It has a very positive value What is the value of this + 3 is the value of the function value then this means the function at x = 1 the value it has and the neighborhood The value is not the same, that is, this what kind of function is this discontinuous The function is and the point of discontinuity is x = 1 means the point at which the function it is discontinuous we point it out discontinuity says okay now look The definition that we gave earlier was that Create continuous function in one stroke we can look at the first one once I can make it without lifting the pen But look at the second graph We can't make it in one go, first We will make the bottom line and then the top line If we make it right then we have to lift the pen will have to do so that function is a discontinuous function so let's see some examples So here's our first one for continuity. For example, suppose we have a function is f or x whose value is 3 if x The value less than or equal to 0 is zero and The value of this function is four if x The value is greater than 0 so let us Suppose this function is defined in this way OK, so now we have to use the function We need to discuss about continuity Obviously, how can we tell What we just saw is that any function A point is continuous when it its value at that point and at that point be equal in his neighborhood ok now look here it's so simple I have taken the function that it clearly has It can be seen that if we consider x = 0 what does it say when the value of x is row that the value of this function will be three, that is where is the function this f x is basically y so y We have taken it on axis, if it is okay then y On the axis, suppose this is one, this is two, this If it is three then it is saying that How long will fx3 be when the value of x is row or Then less than weep ok the graph will be something like this Meaning the value of x may be zero -1 -2 -3 whatever happens the value is always there There will be a graph like this on the other side saying that if the value of x is greater than 0, then The value of f one will be 4 i.e. where zero If the value of f is greater than then the value of f is one how much will it be f okay let's say this is point f now now look here is a special thing this is above This is the line, look I haven't seen it yet Why did you not touch the axis? Because at the time when x = 0, y If the value of is three only then it is exactly zero what will be its value after If it starts becoming four then that means this one which this is the point of this How to know if there is a discontinuity point If we look at it graphically, look at one Can we create this whole graph in stroke? will be able to not even draw at all After doing this you will have to lift the pen and then draw it back from here If you have to write it down then it means lifting the pen I am having trouble reading which means what kind of thing is this This function is a discontinuous function okay now look at the other side if we Read the mathematical definition and talk about it too So according to that also if we talk about x = 0 right x = 0 lets say f is of 0 then what is its value when x is 0 If it is equal then the value of f0 will be 3 The value is 3 but as soon as we say let us that goes into its neighborhood that limit x tends to let 0 + f x take its value What will happen means a little more than crying If the value is a little more, just one step If you come here then what is my value will become four because as soon as x If it is greater than 0 then my value will be four If it goes then are these two values ​​equal is not equal so this function is it discontinuous and at which point is discontinuous at x = 0 let's see Second example, let us suppose we have a The function is f x = 3 if x is not equal to 0 end f x = 4 if x is = 0 ok so something like this the way this function is defined ok let's go Let's make a graph of this as well it will be easy means whenever the value of x is not zero x is anything other than zero if the value is then what will be the value of y three so let's say this is 1 2 3 and four Meaning the value of y is always three except at the point when the value of x row when the value of x is 0 then the value of y is How much will be the four i.e. at the time x = 0 its value will be four so now you tell me looking at this graph whether this It is a continuous graph, just look at the graph Look at the pen I am going to destroy this in one stroke Am I able to make the whole graph here I am able to make it till now but here I am not going to do it cannot draw it straight right because when x = If 0, then the value of y is not three but four So that means I had to lift the pen I put the pen back in and draw this side. So I can tell you clearly what is happening what kind of function is this is a discontinuous function ok ok now discontinuous kiss point But it is obviously visible which one this is It is discontinuous at the point, let us look at it People don't denote it by a ring like that What the ring means is this continuation I'm not going to go here it just went to a But at this particular point it goes beyond this not happening that is what this ring is means okay so which point is this is discontinuous at x = 0 then x = 0 its There will be a point of discontinuity brother The kids said that maam there are too many questions So let's try some questions There are so many questions that you will practice absolutely continuity and differentiable which is No, your crystal must be clear Let's look at the questions with a pen and paper get ready let's see the first question which is Question of your NCERT exercise 5.1 Number one proves that the function f x = 5x - 3 is continuous at x = 0 and at x = 5 ok If yes then to do the continuity proof tax What is our simple condition that at Point C Any function at point C is continuous only when f c e equal to limit x tends to c f x means if that The value of the function at that point and the value of the function at that point is equal to the neighborhood of then we say that that function is continuous okay this is my If there is a condition, let's focus first At the first point, that is x = e 0, okay So now our focus is x = e 0 so what do we do First we will do the function at this point We will find out what is the value, that is, f If we take out 8 0 then what will be 5 * 0 - 3 that is equal to -3 now we'll find this on the neighborhood of the point, i.e. the limit x tends to 0 f x will find its value means the value of this function when x is It's in Roe's neighborhood, Roe has it, so this what will happen limit x tends to 0 5x - 3 means the limit x tends to 0 5x my limit x tends to 0 3 now three is a constant so that was It will remain three only and here as soon as x starts crying If it gets closer, what will be the whole term This whole term will become row, so that means it will be 0 - 3 means it will be minus so now look at this match both the value of f of 0 is also -3 came and the limit x tends to 0 f one If the value is -3 then what does it mean? this means f of one which is this is Continuous at x = 0 Now let's talk about the second case that is at point x = 5 So the steps will be followed here as well Firstly, we will find the value of this function When the value of x is 5 then it will become 5 * 5 , 3 Now we will find the neighborhood of this point That means if x tends to 5 f x then its value is limit x tends to 5 f x will be replaced by we can write 5x -3 so we can write this limit x tends to 5 5x - limit x tends to 5 3 so here if the value of x goes towards 5 then 5 * 5 will become 25 and Here, th is a constant so there is no value of x If there is no role then it will remain three only So this will come out to be 22 now both of them Compare the value of the function at 5 and The value of the function lies in both the neighbourhoods of 5 I am equal to there four we can say that at x = e 5 this is question number two which is Question number six is ​​from exercise 5.1 find all points of dis continuity of f where f x is given as this f x defined I have lost all my points Discontinuity needs to be removed, which ones? This function is discontinuous at the point means it is not continuous okay so most First look at the way the definition is given tell me this has happened x which is from when to is greater, then this is the value of x when if it is less than two or is on two then this one is of it The value of f means from this definition we get It is so clear that this is The function is defined at the points on the Real The line is fine, now suppose we are on the real line Let us assume a point, let us suppose C B A Point on real line so on this line we have a point c assumed okay now we are here This C, the value of C has three categories I can fall down maybe it's C's The value of c may be greater than two less than value and it is possible that the value of c is to Equal If so, these three situations can happen to us We have these three possibilities in front of us Let us assume any one number C. Now our What is the target here now we will see that these three cases This function A in this is continuous or if not, then which ones If it is not on point then we can easily understand it I will come okay so let's go one by one Let's first see c > 2 Now see c If there are more than 2 then fxxx.pro If logic remains then what will be f c here on c greater than 2 i.e. 2x - 3 If we take the value and put c in place of x then this will become 2c - 3 now the limit x tends to c You could have put the value of the function in the the limit x tends to c will become 2x - 3 so we can write this as limit x tends to c 2x - limit x tends to c 3 Now whatever is constant is constant and here if x approaches c what will happen if 2c - 3 does this Let us compare both the values, both are same yes it is absolutely yes since both are same so we can you say there are four f of se equal to limit x tends to c f x so that means that f is a function Continuous for all x gr 2 because here But we had considered c as a point, now this If there is any point like c which is too much for all of them fx2 now when c happens lesson 2 we still have f c Let's find out the value of x when lesson 2 is So what will be the value of this function 2x + 3 so what will this be 2c + 3 because of x We will put c in its place, now let's find the limit the value of x tends to c f one so here y will be the limit x tends to c f of one in place of 2x we can write 3 so this will be equal to limit x tends to see 2x plus limit x tends to C 3 then it will become 2c p 3 if both of them If you compare, the value of both is 2c + is 3, which means it is the same, which means f x which is this is Continuous for all x < 2 OK, so now we have understood this, let's see In the third situation we have Let the value of c be equal to two then In such case also we will first find out f c In the case when the value of x is equal to two, then also the value of f x is 2x + 3 then This will become 2c + 3. Now in this case also we will add 2 You will see its value in the neighborhood of Right So we can give the right to two in the neighbourhood as well and can also be given in left neighborhood can see means limit x tends to c - f x can also see limit x tends to c + of f x you can also check its value right so if this function is continuous then All these values ​​are equal to the value of f c It should be right, so look here now. what is said 2 exactly P2 Suppose this is it Now look at the question clearly He is saying that anyone greater than two will remain two If anybody stays on this side then at that time what will happen the value of the function will be 2x - 3 is less than 2 if any one remains, it means two If someone stays on this side then in that case The value of the function will be 2x + 3 then its I mean that this is point two of this the neighborhoods on either side of the neighborhood in which the value of the function is that is not the same right c minus means smaller than c side right means this side so this one What will be its value on the side 2c P 3 If we talk about the other side, then its What will be the value of 2c - 3 correct 2c mine will become 3 here now What we have ascertained is that the value of c if it is exactly two then like in the case of f c Also if we put the exact value of c If you give this then it will become 2 * 2 + 3 that is 4 + 3 that is 7 now see what happens in their case is coming out here if the value of c I put in sen the value of c if I to If I put it in, I will get 2 * 2 4 + 3 that is 7 what will happen in this case 2 * 2 4 - 3 that is 1 so is this value same or not Infact in class 11th when we studied limits I had read this limit where x tends to c minus we consider We call this the left hand limit we say right hand limit right so here But what we are seeing is that the left hand limit and right hand limit which is they are not Equal so that means x equals 2 which is this is the point of discontinuity so let's see the problem Number three which is NCERT exercise 5.1 ka question number nine find all points of discontinuity of f where f x is defined like this okay so f x key The definition is given that if x < 0 then Its value will be x / m x x greater than and If 0 is equal then its value will be -1 ok It is better to solve this question before this what does mud x mean mud x is Basically the absolute value of x means this mud If you enter a negative value inside So what result will you get? Is it positive? I will get it okay okay let's go now whatever function I have I have given you a pass, don't underestimate its value Let's simplify it further, friend fx -100ms everything comes out positive anyways right but if the value of x is less than 0 then means the x that is inside the mud x that is actually going to be - x correct so that means what will be the value of f x x / - x that is equal to -1 ok so when such a value will happen when the value of x is less than 0 okay and when the value of x is greater than or equal to If to be 0 then it is mentioned in the question then the value will anyway be -1 okay so now Notice one thing, before calculating anything Before that notice one simple thing that x When the value of is less than row, it is equal to row and fx is greater than row in every situation What is its value -1 means its through out The value is -1, which clearly indicates It is quite clear here that We, according to our mathematical definitions, we will prove it from here only okay so let us assume that s is any real number okay so What do we take out first? s c is any real number okay so f what will be the offer what will be the value of fs will be -1 only, right because if we get x Neither of them is visible anywhere the value of f c is -1 only, right? The value will also be -1 if I limit x trends Even if I ask you to find the value of two c f one The value of f is always -1 then both are equal which means f is one what is this is continuous so question ne I told you that friend its points of remove the discontinuity friend but when If the function itself is continuous then its point is off What will be the discontinuity, no point of it discontinuity let's see problem number Four which is from NCERT exercise 5.1 question no. 11 find all points of disk t of f where f x = e x k − 3 if x is < = 2 if x is greater than 2 then f x e x S + 1 ok look now I understand the approach Whenever I am getting any function continuously whether it is there or not it needs to be proved or not Points of discontinuity need to be removed First we will see whether this function All points are defined, see here but what do you see when the value of x is two is equal to that which is defined as greater than 2 it is defined for him less than it is for him it is defined for everyone so Which means, it can be known from its definition that f x is defined for all points on the real line ok that is, its definition for all points is So brother where will it be and what will be its value we know okay so now here's what we will let us assume which is a point on the real line okay now look at c what all situations can happen with c which is can be smaller than two c which is smaller than two can be equal to and c which is this to It can be greater also so these three are possible there are cases now we will see here that in these What is my function in all three cases? is it continuous if not then point off what is discontinuity okay so the most First let's see c < 2 ok now let's see Look at the definition when the value of x is less than 2 then the value of f x is x q - 3 then what should i take out here one is f s I will take out the value and I will take out another one The limit x tends to c f x value is ok f what will happen to c - 3 this would be a limit x tends to c f one what will happen this will happen limit x trends to c x k -3 so this becomes limit x tends to c x cube minus limit x tends to c 3 now this So it is constant so as it is -3 only and here c will come in place of x, so c a - 3 Now look at the value and limit of f of c the value of x tends to c f x is equal to which means that function fx2 now let's talk about x is 2 okay so In such a case, what will be the value of f c - 3 According to the definition, when the value of x 2 even then the value of f x is equal to -3 Now in place of c you put the value of u Two so 2 to the power 3 -8 means -3 means 8 - 3 f i eq 5 okay now here also we will take out limit x trends to c f off one okay because here we have a pick point If we talk about edge point two then will take it out in the neighborhood So Its value is slightly higher than so2 and in the neighborhood behind two We will also find out, meaning both right hand limit And we find both the left hand limits we will take it okay so see when c does it when x does it The value should be slightly more than C, meaning from two If it is a little more, then what will be the value of f one will go to x s + 1 okay so that means here what are you will go here it will become c s + 1 okay and what is the value of c 2 then 2 s that is 2 * 2 4 + 1 that is 5 when the value of x is c it is a little less than means when it is a little less than two So f x is what x k - 3 so write it down can c k - 3 then many people will say c s was +1 here c is -3 so remove it here That is, it is not continuous but it is like this can't say because c is exact here if it is two then put the value of two in place of c that's why you will calculate it here also if you put the value of two then 2 to the power 3 is 8 8 - 3 is 5 so look at the end of the day Joe The value I got for the right hand limit All these values ​​of the left hand limit are the same what does it mean there's four f x which is it is continuous for x e 2 okay so let's do the second case as well It's sorted out, now let's look at the third case the third case says that c is this two is greater than ok so in such case f is what will happen greater than 2 in that case f The value of one is x s + 1 so we call it we can write c s + 1 and here if we find the value of limit x tends to c f x So what will this be c s + 1 because Here also it will be x s + 1 and in place of x we ​​will If you put c then see both the values ​​are equal is exactly equal to so this means that fx2 then look brother fx2 is also continuous for x = 2 is also continuous and for x > 2 is continuous, that is, the given function This is completely continuous and there Four it has no point of discontinuity let's see question number five which is Exercise 5.1 question number 17 is See I solved a lot of questions at random I am going to get you to answer the questions that are left You must solve them yourself is find the relationship between a and b so that the function defined by f x below is continuous at x = 3 so look here It so happens that f x is a function whose If x is less than , then the value of x will be is greater and the value of x is Now I have been told that friend I should say I am here so the question is saying that brother fx8 x = 3 so if this question is saying that f x is continuous at x = 3 means What the question is saying is that this function The value of this function will be whatever is at three will be the value when we multiply its value by three will take it out in the neighborhood Right so according to the definition that's what he should say want the value of this function to be at 3 and it and the value of this function in the neighborhood of 3 these two are equal okay so Let us find the values ​​of both of these First we figure out f3 okay so when x If the value of f x is 3 then in that case the value of f x is how much will be the value ax1 so in place of a x we ​​will put 3 + 1 so this will become 3a + 1 Now we take this out, okay, so limit x Tends to 3 f of x ok infact this also more detail If I see, brother 3 ps means that right hand side neighbor and left hand side of r In both these cases the neighbour on the side The value of the function will be the same, which means we can call it we can say that the limit x tends to 3 p and limit x tends to 3 - f one all this Equal will be Okay so let's x tends to 3 ps if talk means more than 3 then at that time What will be the value of the function bx3 so b * If we put 3 in place of x then it will be + 3 will go to 3b + 3 okay in the same way if we do this If we take out x tends to 3 minus then we get you will get 3a + 1 back if you want can try that out now look here we have what did you say that since this function is x e e 3 is continuous, which means these values will be equal see this is i equal to sign So that means this is 3a + 1 and this is 3b + 3 Both values ​​will be equal that means 3a + 1 will be equal to 3b + 3 so we can write it as We can say 3a - 3b = 2 or else we can say a - b = 2/3 so let's look at the next question which is from NCERT exercise 5.1 Question number 19 is saying this question that shows that the function defined by g of x = x - inside the third bracket x is Discontinuous at all integral points okay so this function g is saying that is every integer point on the whole integer But it will be discontinuous, this is proof to us I have to do it, okay, how will you prove it, friend? what is our rule, what is that rule At any point c, if the value of f of c his neighborhood i.e. limit x tends to c f of x is equal to f of x if both these values If equal, then the function will be continuous If it is not equal then it is discontinuous If it happens then we will prove it with the same logic Now the point to note here is that what is this x inside the bracket denotes greatest integer less and understand this equal to x very carefully This is the tough part of this question, what is this This value is the greatest integer less than and Equal to x means suppose the value of x is 3 just suppose the value of x is three then The value of x inside the bracket is What would be the greatest integer which either is less than or equal to three so what is the next integer smaller than three? and what is the integer equal to three? three Who is the greatest among the two? That is, it is the value of x inside the bracket will happen in this case too 3 OK, if you understand what I mean, then this is what The x inside the bracket is its value will be an integer which is either equal to x or smaller than x but both of these The greatest among them is its value It must be okay, I understood it, okay, amazing So let's start solving the question now so here's what we do Since we have to prove the integral on points so let's assume that n is what this is an integer okay so let me give you the proof What we have to do is that this function is n is discontinuous at the point then for that We will first find the value of n then After that we will find the value in the neighborhood of n that is, the limit x tends to n p g of one We will find the value then we will limit x trends to find the value of n - g x if then this If all the values ​​are equal then this function The integral will be continuous at the points if If these values ​​are not equal then this it will be discontinuous okay now let's go to them Let's calculate g of n, that is, of x put n in place of x inside the bracket too put n in place of x so if this n is the sum of x If value is n then it is an integer smaller than n what would happen suppose such a just for For example, let's take the number line n here then what will be the next integer smaller than n - 1 Similarly, what is the next integer greater than n? it will be n + 1, right, so here this n is in brackets what will be its value An integer that is either equal to n or Then n is smaller than n and whichever is larger of the two then what is equal to n n is smaller than n What is n - 1, which is the larger of the two? is n then that means its value will be n and n - n What will happen? Did you understand? Amazing, now look at him talking about the neighbourhood that when the value of x is n plus means now The value of x is not equal to n from n It is a little bit more, just think of it like this x x Whoever is there, he brings it here somewhere okay so this is not equal to n but ok so what will happen to me in such a case I can write this function as limit x tends to n + g instead of x I will write x then the minus limit x tends to n + inside brackets x ok so now see what are their values Now the limit will be x tends to n + x so its what will be the value of n okay because x What is it that is slightly smaller than n in the neighborhood of n so we can write it as n okay now its Tell me what the value will be. Many children will say its value will also be n but its the value will not be n because look what will happen is this value here is n It is plus, meaning it is slightly greater than n, so What will be its greatest integer? It will be n right now if you say maam then She is confusing me on purpose I am doing this so that you can understand that My x is here, I brought it here somewhere where I'm making a cross okay now where am I making a cross with this Which of my greatest integers is coming first if n is my greatest integer then Here its value will be n i.e. The answer will be 0, so far it seems that this function is continuous but stops Go now the story of x tends to n minus If there is anything left then its story will be something like this that the limit x tends to n - x minus the limit x tends to n inside my bracket x so here this limit x tends to n minus so That is, where is the value of x right now here Somewhere means a little before n a little more than n this value is small okay so when we will limit x tends to n - x so this how much it will be it will be n only but now when limit x tends to n inside my bracket think what will happen if you do x now see this Focus on the yellow cross, this yellow one Who was the greatest integrator before cross? is it sa or n or no so we reached till n it is not right because this is the greatest It is an integer, it must be such that it is less than and if x is equal to x then my current he is sitting on the yellow cross is sitting then this x What is the greatest integer at least right here it's n - 1 so that's why here it's The value of the limit will be n - 1 so what is the total? n - n + 1 So this is equal to 1, so that means these three the values ​​are not equal 0 0 and this one it has been corrected so that is what it means here My left hand is limited and the one on my right hand There is a limit, it is not equal, right, this is mine My right hand limit was this, my left hand There was a limit and these two are not equal which means that this function g x this is not Continuous at x = n what is n integer That is, whatever integral points there are, But this function is discontinuous I understood the concept of continuity We want to ensure continuity of any function Let us define a particular point ok so now with the help of the same concept we What would one call Continuous Functions? A function that takes all its possible inputs Continuous be right for values ​​means The way we are talking about a point c Similarly, whatever domain he has, The number of possible values ​​in the domain for all if that function is continuous This means that overall the function is continuous Simply right, the meaning of saying it is something like this that brother the highway is just a highway It is not smooth for VIP vehicles The highway is smooth for all vehicles Any number of vehicles entering that highway There will be a path for everyone You will get continuous directly you will find a smooth way okay so like this If the situation arises then what is my function Continuous function is Hey questions so If you keep asking questions, you will solve so many of them that the concepts will become crystal clear let's look at question number one which is Question number of NCERT exercise 5.1 Part C of 3: Examine the Function for continuity at x i e x is not equal to-5 where f x = x s - 25 / x + 5 look These are the small mistakes that happen many times In the exam we get the answer x is not Equal to -5 is given but I suddenly When I read it, I read x = -5 So let's see what we will do is there any number c okay and let's say that for any number c that is not equal to -5 what will happen limit x tends to c f x we ​​will find its value after that We will find the value of f c if both are equal so this means the function which is of this c what is it for it is continuous okay let's go If you remove it then it will become limit x trends to c fx2 - 25 / x + 5 So we can write this as limit x The trend to c x s - 25 can be written as x s - 5 s that is a s - b s so we call it we can write a + b * a - b and divide it by So x + 5 and x + 5 cancel out. if it happens then what will happen now to x If the value is going towards c then it is The value of limit will become c - 5. Okay now. what will be the value of f c c f of c Meaning if you simply put c in place of x then this will be c s - 25 did by c + 5 see now I can also write cs - 25 as cs - 5s hmm that is a s - b s that I can write a + b * a- b c + 5 c + 5 will cancel out so this will come out will c-5 come so see what happened to these two The values ​​came out exactly the same, this means limit x tends to c f of x = i f of c which means that the function is Continuous for all x that is not equal to -5 So kids let's see the problem now Number two which is exercise 5.1 D part exam of question number three The Function for Continuity f x = mud x - 5 Ok Wherever the mode is given, do not Write the value of the function in a slightly different way take it because whatever is inside the mode There is a value, it is positive and negative There are different rights for the values ​​like that here at but if the value of x is less than f If it happens then what will be the negative value inside if it would be right then in that case this fx5 - x means the negative of this, okay so This is how my function is defined So now look at me with this definition what is known is that the function which fx5 pay five pay more than five pay less It's defined at every value, so f x is Defined for all points On Real If the line is fine then what will you do as usual we will assume one point let's see A Point On Real Now this line, the value of c has three There can be possibilities C which is from F can be as low as c5 and c can be greater than f, so this Three Possibilities and as usual what i have to do is In all three possibilities it has to be shown that is the function continuous or not c If we talk about < 5 then in this case first so we find f c what will happen when f c See when the value of c is Lesson 5 then f c What will happen to 5 - c limit x in such case tends to c f x what will be its value when If x is less than 5 then its value is 5 - x and x tends to c then Its value will also be 5 - c for both the value is the same there four function which is that is continuous for x < 5 so for all x < 5 This is continuation now of the second case Let's talk about when c is equal to 5, like this In this case, we first find the value of f c we will find out when the value of c i.e. when the value of x if the value is equal to 5 then my function what happens x - 5 so it's going to be c over here - 5 and we're saying that the value of c If it is exactly 5 then 5 - 5 means 0 now Here we take it out in its neighborhood i.e. that the limit x tends to c + f x is also find the value and the limit x tends to c - f x both find its value also If we find out then it means the value of the function when The value of x is slightly greater than c, right there is a little bit more c what is it here Five means the value of x when is less than five If c is greater, then what is the value of the function will be x - 5 so in this case this will be the limit x tends to c + x - 5 but in the lower case When the value of x is slightly less than c, i.e. if it is a little less than five then in that case my What will be the value of the function The value will be 5 - x okay now here We can put c in place of x so it will be will become c - 5 here it will become 5 - c c If the value of is 5 then 5 - 5 that is 0 here also 5 - 5 that is 0 so basically what are we looking at The value of all this is being appreciated, everyone is crying the value is coming same which means that is the function f that is continuous for all x = 5 Now it is the turn of c gr 5 so in this case f The value of c will be found first when the value of c If the value is greater than 5 then the function by definition it should be x - 5 that is c - 5 okay now let's find the limit x what is the value of trends to c f x f x when the value of c x is greater than 5 According to the function it should be x - 5 so this also it will become x - 5 if you put c in place of x then c - 5 Both these values ​​are the same, exactly the same there are four functions which it is Continuous for all x greater than 5 then total that we are seeing that this function x < 5 is also continuous for x equals 5 is also continuous for x and x is greater than 5 is also continuous for this, that means overall this function is Continuous so let's look at the next question which is This is a question from NCERT exercise 5.1 Number 24 Determine if f is defined below is a Continuous function so here's a function it's defined let me tell you that this whether it is continuous or not is ok then most First see where it is defined x = 0 when the value of x is 0 and when the value of x is If the value is not 0 then it means in a way fx9 line zero as well as the one which is not zero pe bhi so it is defined on all points On the Real Number The line is fine, after this our next step As always what will happen let's be real The number is ok now what are the possibilities for c c What all can obviously happen here Look at the conditions given here The value of c will either be zero or non-zero If not, there will be only two options here I have either c is not equal to 0 a Option 2 or c is equal to 0 So these are the only two options I have. First let us see when c is not equal to will be 0 in that case what will happen in that case I will find the value of f of c and what I will find the limit of x trends to c f x I will find out the value, if both are equal then this It is continuous, okay so what is f of c when x is not equal to 0 then fx1 sa 1/1 so I'll write this as c sa 1 ba c I can okay okay now when limit x tends to c then what will happen if limit x tends to c instead of f x i can write x s 1/1 okay I can write it as limit x tends to c x s in limit x tends to c ok this is absolutely clear That's cleared up, let's look at the second one now What happens when the value of option s is zero? It will happen here too we will do the same thing, first we will f We will issue the offer and after that we will issue it limit x tends to c f one and the same The same old story will continue so what will happen to f as soon as you calculate x Now instead of x you are assuming x to be zero if we are right then what will be the value of x Zero is saying straight away that this is Ro So what will happen to f even if it is zero then this will be It came out very easily, now let's see another one in other case what would we say will say limit x tends to c now many The kids will say that ma'am we are here too fx2 c here we are applying limit we c Trying to find value in the neighborhood of If you are doing it right then immediately mark it as zero you can't put it, that's why limit is x tends to see means x is equal to zero The neighborhood is moving around zero Now x is okay so what do we write here is around zero means x is not equal to If the situation is 0 then that sum is right So what do we put inside here? ssa it is zero right means right now we are saying that x is somewhere very close to zero ok so how can i write this Now see how I can write it There is a way of thinking but now look at the maths In this way we complicate things which They might have some tricky questions This is how the approach changes here now sa theta's dizziness has come sa theta's There are a few things we know about Theta has a minimum value, which means its The values ​​are -1 and +1 Right keeps rotating between the values We already know all these things, so Meaning, whatever is here between -1 and +1 My sin is 18 and its value is also the same What will be the right 90 between the two it happens one happens right it's about 180 If you do it, it will be correct, that is, from -1 to +1 it keeps rotating in the middle ok ok now what do we do this is this what is this you are watching the whole thing this is one in equality one The equation is right, what did I do about it what if i multiply x by s will become - x s < i x s s 1/1 < equ + x s Now you will think that I have done this why did you do it so that friend this x ssa 1/1 I can bring the term now why am I doing all this I am doing it, check it out right away and you will know it's ok Now what should I do, if you put a limit on everything If I say limit x tends to 0 - x is I lesson equal to the limit x tends to 0 x s Saw ba ek i lesson equal to limit x tends to 0 x I can write it exactly like this Now see how I can write it further can hmm further limit x tends to 0 o - x s what will happen this cry will be right because the value of x you will see straight away If you put it then 0 will be less than equal to limit x tends to 0 x s it will be right 1/0 is not defined correct So that's why we didn't put it there, now look at it How can we express this value in another way? So now look at us What did I get that my limit x tends to 0 x is sin18 this is saying that its value is greater than or equal to row if smaller or equal then on both sides If you look at the pallu, what does it want to say? What I am trying to say is that this The middle one is my term, isn't it's value? How much is its value if you look at it then know this it is going on and it is telling us that its The value is either greater than row or is it equal to cry from here it is saying that or So is it smaller than Ro or equal to Ro is correct then it means limit x tends to 0 x s sin18 what is its value row so now See, the value of f c at c = 0 was also zero and its value is also zero so there's four f x what is that is Continuous at x = 0 so what do we find out Is this a continuous function at all? It is a continuous function, now we will see Problem number four which is from exercise 5.1 It is question number 30 so it means 5.1 I did more than half the questions Well now at least those who are left are on their own Do it, let's read the question find the values ​​of a and b that are true function is a continuous function ok It says that a function is defined It has been done and it has been said that brother This function is a continuous function We are not asking you to do this continuously function is you tell me if this is a continuous function so here this is a and b is what will be their values ​​okay So what will we do? We will just do what we want It's the same old rule, the same [ __ ] this is the definition which we paste everywhere if we live it's okay then we'll say okay if what you say is true and if fxxx.pro to that is the point where this function There was a possibility of it being discontinuous no in the same way look here less than 10 Here it means equal to 10 or more than 10 10 is again a point where There was a possibility of discontinuity Although this function is continuous, it's ok So if this function is continuous, then this the point with oo is the limit x tends to 2 + x tends to 2 - fx2 means the value of the function at p2, and Its value is all in the neighborhood of P2 should be equal okay okay let's go now Let's put in the values ​​if x is equal to 2 It is in the neighborhood but two to 2 plus Right, let's say if this is my number line hey let's say here there is two 2 p meaning this point is somewhere here a little bit more to the right than t so now If x is greater than to, then its What will be the value of the definition of this function According to this its value will be a b x then we will put t then a * x p b if x tendons totu minus means if it is totu It's a little earlier, which means it's a little less than two. then its value in that case is How much will it be if it is exactly two Look in that case also, less than equal to In case also its value is five then Basically from here I got a relation that 2a + b = e 5 we call this an equation Now this is exactly how we will talk about this 10 for 10 rupees so this value of 10 rupees is also in the neighbourhood of this point and this The value of this function should be the same at the point should because fxxx.pro tends to 10+ fx1- the fx10 is ok it should be because it It is continuous so let's see its values Let's put x to 10 + means if we assume This is 10, 10+ means a little more than 10 this point is greater than 10 what will be the value of the function 21 this is equal to x tends to 10 my means it's a little bit less than 10 so x If it is less than 10 then what is its value will put ax1 + b else f1 if equal to 10 If it happens then in that case also its value will remain 21 If it is right then here it is 21 on both sides so what relation do I get from this 10a + b = e 21 So, now we can find the value of a and b you can take it out, you can take it out boldly What is this? This is a pair of linear equations These two variables which we have discussed in class 10th itself I learned it right, so let's solve both of them How will we solve this quietly Subtract equation one from equation two What will we get from the equation 2a + b = 5 one was 10a + b = 21 this is equation two okay now change these, the b ones will get cancelled this will become - 8a and this will become -1 so minus minus cancels out to a so a's The value of to is what is the value of b You can take it out from any expression will become 5 - 2 * 2 that is 5 - 4 that is 1 So, that means the value of a is 2 and the value of b is Forest So now let's learn the algebra of Continuous functions algebra means like It's the same old maths stuff, like addition Subtraction Multiplication Division These let's see related things, suppose we have There are two continuous functions f and g and these Both functions are at some point c It is continuous, okay so what will happen f + g whatever it is will also be continuous f - g that too continuous will be f * g that too continuous will be f divided by g that too continuous would be at point s okay just f divided by By g, this condition will definitely be applicable that g is not equal to 0 because brother is the denominator If it becomes row then overall it is not defined if it is going to happen then it means that if Anytime you find two functions that both If the functions are continuous then both of them have addition to both of them subtitles to both of them Division of both products, what is all that It will also be continuous at that pert point But okay, so this is our simple rule. it's a very simple rule right but when we If you apply these in questions then How will you apply, oh those are questions You will find out by doing it, let's just put pen and paper on it Be ready, there are only questions Let's look at the first question, which is the exercise 5.1 is the first part of question number 21 Discuss the continuity of the function f x = sinx-cosx Now look what this function shows Us We see that there are two different functions which are added with the edition sinx-cosx We will say okay brother let's go sinx-cosx will be the function on g gx9 ok so let us assume c is real Number means this statement which is there in every we are using on question so that is kind of becoming the format theek hai maan Let us assume that c is a real number. Now what should be the value of c here? maybe right meaning it might be coming to your mind What could be the value of c? How do we relate c to x, meaning x is for any general x any input value we can catch c from that we say if x is equal to c then let x = c + h okay means the real number we took If you add h to the same C then you'll get x from late okay so its This means that if x tends to c then Meaning if the value of x gets closer to c is going to be almost equal to c What does this mean for the value of h? if the value of x equals c will be equal when the value of h becomes row Meaning as the value of x increases, the value of c increases. The closer she comes to me the more she is mine The poor guy will keep getting closer to zero okay I understand now why I did that okay I will understand because I Proving continuity is right continuity I need a point to prove it Continuity will be proved only at one point so that's why I took that point as c okay Now what do I have to prove, as always the same old rule of function The value at C is the same as the value of the function It should also be in C's neighbourhood so what does it mean the value on sea and on neighbourhood If the value of both is same then it means The function is continuous okay so on c What will be the value of x if you put c two then this will become sa s x tends to c ki When we talk it will be something like this limit x tends to c sinx-cosx So, we can write a + b as a sa b sorry sa a k b p k a sa B Now here's a little thing that I also I did not pay attention and you people also paid attention you must not have given it see when I said x trends to c g x x as it approaches c then Towards which side will my h move, towards which side That means I can also write this on the limit here can that h tends to 0 here also i I can write that h tends to 0 here as well i can write h tends to 0 write sorry I can't write here, poor thing if there is no entry of that then here I'm just gonna write that x tends to c okay Meaning, where h is entered there I am not going to talk about x a bit there If I write it in terms of h then h tends to I'll write two zero okay now look at this break it, what can we write about this Limit h tends to 0 sa s h plus limit h tends to 0 k c sa h okay now h is going towards zero h if zero if it goes towards then what happens to k 0 and here But if h goes towards ro then what will be the 0 it happens 0 okay so overall its what will be its value will go Like c is right because the second one is the complete term Anything with zero will become zero If you multiply, you will get zero and the first term in me c will be saved only because h then it will become one so what do I care about here value gota sa c so see whether these two the value is equal to exactly equal to it means whatever is of x that is Continuously, at least we have proved one thing Curry will now focus on exactly this method on the other thing that is h of a function on h what is one k x okay so now here too What should I take out once I get the h of s find the value and once I reach the limit x tends to c h let me find the value of one okay Now let c be a real number x i.e c + h all these something is still true for this means here till now this is from here till here this is common for this age well okay so h of c what will happen if you put c in place of x here two so this will become k c okay now here what will happen limit x tends to c this is will go to x ok now what will we write in place of x h If we write it in terms of, we will write it as k In place of x, we will write c+ h now how can we break this a + b we can write this as a minus sa a sa b okay now this is in two terms I should take a break because it is over now. There is only one right so we can write it as limit h tends to 0 as k h minus limit h tends to 0 sinc-sind means h of x which is the function is this function is Continuous and brother, this is what I had to prove Proved that g is continuous h is continuous then g + h is also continuous If it happens, then from there itself we told that this The f x is also continuous, let's see The second question is from Exercise 5.1 question no. 34 find all the points of Discontinuity of f defined by f x = mud x - mud x + 1 so look here too are functions mud x is a function of itself let's assume this is g x mod x + 1 You have a function let's say it is h x ok so we will do the same thing here too g it is continuous we will prove it h continuous yes, we will prove it and if it is proved So g - h will also be continuous, okay? Let's first focus on g x. look what is g x mud x right that means its The value px is when the value of x is positive is zero or greater than zero and its Value – one when the value of x is negative i.e. it is less than zero then do something like this will be defined by off one okay but this It is clear from the definition that of one which is Defined for all real The numbers are okay so let's assume let sea be real The number is fine, what is possible with C These are the values, look at this definition and you can tell It is possible that c could be greater than zero can be equal to zero can be less than zero okay so the value of c is greater than zero The value of C may be as low as zero may be and the value of c is greater than zero can Is ok so these are three possibilities We have the correct one so let's try it c When we talk about < 0, first of all we We will find g c then we will find the limit x tends to the value of c g x both values If it is equal then it is continuous it will go okay then g c what will happen less than is 0 then this will be - c is okay because In lane 0 it has a value of -6 Now in case of limit x tends to c what will be the value of off one in this - x then what will be its value - S So look, if both these values ​​are same then it means that at least in this range then this is continuous Now let's talk about the function c = 0 here If we talk about ofs then I equals two In the case of this is p x, that is, its The value will be there, let's talk now limit x tends to c now when it is equal to zero If we are talking about right hand and left hand then Both hands take the limit, which means zero Let's talk about the neighbor routes on both sides of the then x tends to c p g of even one and x tends to c minus g of one we will take it out as well we will take it out both of them so okay X tends to c plus means lighter than zero more so in that case it will be + x that means ps will come and the value of c here We have taken zero in fact this also The value will be zero because the value of c is exact zero now x tends to c my we talk about it means it is a little smaller than ro So its value will be - c c here is zero, then its value will also be zero Look, the values ​​of all these limits are coming out to be the same. that means i am in this range too The function is continuous now that c is greater than 0 Let's talk about g c in this case if I Let me find the value so that is c greater than 0 the value x comes in it and if its take it out in the neighborhood i.e. limit x If I find the value of tends to c g x then it will the limit come to x trends to c g x in place of this we can write x so this will also come out to be c then both these values ​​also is equal to, which means it is here as well If it is continuous then what did I learn from it that this function g x is continuous So okay the game is over for one person, okay Now what's left for me to do now The next part is that I need to find the h x I have to do continuous proof friend it's been so late I have got so many questions done to prove it that brother where all are continuous right so i think this you can do so let's do this I give you your homework sometimes yes i am homework so let's take it as homework But you can see for yourself that hx8 but continuous came out so in that case this The f x function will also be continuous and it has no point of discontinuity If it happens then children will learn Algebra of Continuous We will discuss in connection with functions Another Theorem Well suppose we have There are two functions f and g and now we will talk about we will do some function which we will call It is a composite function, what do we do in it We fit these two functions together like this Let's do that one function over another function it depends, okay just assume like These f and g are two different functions but we have We combined the two and made a function f now it's not f fog f means f of g x is fine, that is, as the value of x there will be change gx-1 what does our rule say right now The rule of continuity states that if gx-1 f is continuous when be gx3 because f is on top of what is dependent gx8 then if f is continuous when what is it okay it means that this which overall is my function that is f which is the function that is also continuous at x = c I understood the point, let's take an example, okay First of all, a composite function Let's take an example like sinx-cosx is g okay so two functions but we have these two functions in this way is bounded such that the value of g as The value of x will change as the value of s changes the value of sa of x s will change accordingly it will keep happening okay so this is a composite The function is good, our rule says that If x is continuous for some point c, then and if g means for x when x is the sum of the squares The value c is s i.e. the sa of c is if It is continuous, which means that it is overall The function that is continuous at x = c OK, I understood it and understood it well I will come when we ask some questions let's look at problem number one which is The question is from NCERT exercise 5.1 Number 31 shows that the function defined by f x e c x s is a continuous function so look Till now we have solved so many questions I hope you understood the approach How to approach Continuous Butt here attention to function proof The thing to note is that this f is one There are actually two functions hidden in it Such that the function of a is a function from let us that this is my g function okay off is one okay and x is a function of itself Let us tell him that this is h off there's a function okay so I have these two The functions are hidden okay right so now If we look at it in one way, then this g This is a function isn't it? Whose function is this? Actually g of h is what means overall this function in which format is it in g format means g off h one is in that format okay okay let's go I also understood this, that in a way We can write f x = k x s , that is, that this this is g of x s that means this is g of h of x means this is g If you understood the point then first of all this What format is a convinient function in? okay okay now you understand the format so that means If I prove it here, it means Simply put, what do I need to do we have to prove that the g x that is continuous which is h x that is also Continuous If I prove it then Obviously the composite function of these two the g will also be continuous right This is my target so let's go first will focus on that if I write this as c x that means here cosx-sinx number okay and we say that What is c? Let's define it as c + h = x which means that if x tends to c means if x is getting closer to c then its means h will tend to 0 if x and c are equal Both of them have to happen so obviously h what would have to happen for h to be equal to rho So now by that logic the limit is x tends to c g of x if we take out then how much will it cost limit x tends to c g x in place of cough we will take out x and write it We can write the limit h tends to 0 as x what can we write in place of c + h so it will be will go to c + h k a + b how can we write this k a k b - sa a sa b h tends to 0 k 0 how much is it and here also h tends to 0 sa 0 How much becomes 0 i.e. this whole term What will happen, this whole term will become zero so what will I be left with c okay so I took this out limit x tends to c g of x now what do i have to remove We have to find out what is the value of g of s If both are equal then it means This is a continuous function so g of s See what will come straight away so what we got to see was that g of s which is the value and this limit x tends to c g The value of x is equal to both which means what is this function this is a continuous function so g x we ​​have Proved that a function is continuous It has been proven that now it's whose turn is next now it's h of x's turn which is x's now Prove that x is a continuous function function okay so let's say here we have Let us assume that let k be a real number fixed Let's take k instead of the real number c If I have taken K then we have to prove it that h of k and in the neighbourhood of k, that is limit x tends to k h of one of these two whether the value is equal or not if equal then This is continuous okay so h of k what it will be done simply put k in place of x limit x tends to k h of one means x s what will this be this will also happen so look at both the values ​​are equal its exactly equal what does that mean h of x which is that is also continuous now h of a continuous is g one is also continuous there four f one jo hai that is also Continuous Okay so now we need to use Continuity Now that you have fully understood the concept, Whose turn is it for differentiable then? What is differentiable, it is a simple thing that if there is any function whose How to calculate derivatives we extract the function of any these are differentiable if I take this from another In other words, if there is a function like this whose derivative exists derivative exists, which means we call it differentiable means that it is differentiable and this is called Differentiable means if a function has The derivative exists at some point This means that the function is differentiable. Now the question arises that mathematically How do we find the derivative of any function? Let me find out, I am talking mathematically now. okay so let's say we have a The function is fx8 c let c be the coefficient of that function is a single point of the domain okay so if I need to find the derivative of this function f x ho at point c then how do we find out Let's define it like this, we say The derivative of f at c is equal to the limit h tends to 0 f of c+ h - f c / h panic no, I am explaining by looking at the expression This expression is saying that you are do that find the value of this function at c + h then find the value of this function et c okay is the difference of the two divided by h limit A Tendency to Cry First of all think that limit h tends to 0 why do i want to say this that the point which is c and the point which is c+ h yes, these two are very close to each other, that means h tends to h zero means between c and c + h the gap that is there is very less that is what it is to try to learn mathematics If this limit of ours exists, then we will We can find the derivative of a function if you can find the derivative of that function that function is differentiable Now you have understood the matter, so look at everything We have connected the limit differentiable Continuity everything is connected now ok okay now everyone will be angry at me I will say that ma'am in the beginning of the video So you said that any function what is the derivative slope of the tangent Ate this is what I said and now you have given this a heavy Oh friend, I have written a heavy formula now Graphically it remains to be seen when we see it If we look graphically, we will see that this is nothing but the slope of the tense exactly to Let's look at its graph now we will see it Let's look at it graphically: Okay, what did I do? Who took fx8 let us assume y = fx8 is y on the axis or fx is on the x axis I have x, what did I say, point c which is what is this it is a point in the domain that is, there is a particular value of x c ok now what did my formula say that you find its value at c then c + h But I kept telling you to take it out, but I what did you say that derivative means slope of the tenge so I stick to my point I will stay here, what do I have to take out Draw the derivative at point c then at point s I made a tenge. Okay, now this tenge is I need to find the slope when I need to find the slope then what will i do then i will marry these two Identification of points Let's see how much its value will be look at the y axis f of c + h - f of c then That gap is the difference that is basically Your if dy8 then what is the difference between the two so what is it, it has come, what has come I said it right so basically this is my The mathematical definition of the derivative was that is basically nothing but the value of slope of the tense at so now it is to be seen that we how do you denote derivatives ok So suppose we have the given function that is fxxx.pro how will we denote it, we can write it are d / dx2 then draw a stick and write it below We will give you c, this will mean that what we have The derivative of this function is found, which one is it is drawn at point c at point or we can write this as f' of c okay so Overall, children, what do we understand so far? I came across derivatives that if I have any function given fxxx.pro means if I have fxxx.pro We are bringing the limit with the same concept This limit is coming, okay, everyone has reached here It is clear because now we will continue to do the same It is going to be used for all different types of functions okay as well as this definition We also came to know from this that any A function that is differentiable is a function There is also a continuous at that point so in A way every differentiable function is Continuous So let's quickly see that What are the situations when someone The function is not differentiable in C At the point, firstly when that function is do not be continuous at the point now you can see this graph Just look at this graph straight away friend isn't it this is discontinuous right and the Is it discontinuous at that point? You will be able to draw a tange at the point because tange at any one point is a there is a unique tension at right but here this graph is broken there so you will not be able to have a tenge which is tenge If it is not there then where will the slope of the angle come from take it out right so that's why in this case we If you cannot find the derivative then the function would not be differentiable if we had no A graph that has exactly one vertex It should be steep like this vertical line What will happen in case of vertical line When you go to find its slope, Your tan3 is actually theta 90° and tan0° is not defined so slope If your not defined then brother derivative Where will it come from, these are some situations where your function is a particular is not differentiable at a point, then Children in previous class i.e. class 11th In this article we have discussed some basic functions Learned to find derivatives such as Paulino function if x to the power n then what will be the derivative of this n x to the n – 1 if sinx-cosx When we were taking it out we told you about some rules I have also discussed the rules which I will tell you further. It is also going to be useful so let's do a quick recap Let's see which rules we learned So what we learned was that suppose we have There are two functions u and v ok so u + v which is the derivative means that d / dx2 order derivative this is equal to u d pvd ok means d body one of y pv = This body is a plate dvx200 rule product rule why because Look at the two functions here. The product is happening and d/ on that product dx8 is left okay means to write this one way of and it could also be that d / dx-data-grid plus u* dvd-r doing it first u did v to edge it is it left on the second term or ko as it I left out the is and put the v differentiable does that is if u of bawi What do we say if we want to remove the dash then we say yd v minus yv d divided by v s and where the obvious We never make the denominator zero. Then my whole not defined will be done And we call this rule the Kosh rule and many more pay close attention because these there are rules too, product rules, rules etc There's a lot of work to come ahead okay So did you understand the question rule u / v dash when If you take it out then in the first term You will also remember to keep the denominator as it is I have to keep it okay because what is the minus in it it is a sign so v is kept as it is u d In the second term, put u as it is and put v diya ok and divided by kiya v s ok so these are the rules, practice them do it, remember it but they need to be in your Finger tips so dear kids all ready to learn how we can behave in different ways If we find the derivatives of the functions of Our type one we'll start with Composite functions Composite functions which have A while ago I also talked about was that these are the functions where More than one function evolves Meaning clubbing two or more functions By doing this we get functions like this like suppose that sinx-cosx we are looking at the function f This bit of mine is dependent on t so that means that t is another function whose change If my f function changes then how will this happen function this is a composite function if Let us take the example of another function Let's take 2x + 3 and what do we see here? that 2x + 3 is a multiple of itself is a function that changes as x changes the value of will change to right and this hole It's a cube, it's a function in itself. So for example here also if 2x + 3 is Let us ask you this: fx3 means the value of the function as t increases the value of the function f will change And how can we find the value of this t function? will change as the value of x changes hogi right so we can call these types of functions It is said that composite functions are infact sometimes then more than two functions are involved it is like this look at the example sinx-cosx this is so much fun right in this What is happening is a function in itself which is depending on the function If the value of x changes on cos-x space then What will happen we derive cosx-sinx from this and answer the question The answer is chain rule. I love chain rule very much. Let me explain it in clear and simple words, okay, accept it Any composite function you ever have within which more than one function is evolved So what will you do? You can do very simple things. Suppose you will do it, let us understand by taking an example From let's we have a function We will assume another function sinx-cosx lets say x is t this we have t So now what can we write for f? I have founded sinfam.com with their help than i am df1 dx1 dx1 dx5 / I have already removed it, so that it is correct Gaya, this is the way we and our children will be taken out there is no need to panic ever Two-three-four functions may be involved but we We will apply this simple and beautiful logic of ours Now let's try a few questions There are also some simple ones If we start with something complicated then let's go brother type If you get some practice of one problem this is our problem number one which is Question number one of exercise 5.2 sinx-cosx so from here we have ok so i can easily calculate df2 hmm so that will be equal to c okay so I could have easily taken these two things out and I took it out but what is the question from me is asking the question so I need this f function To It is differentiable, meaning it is asking a question so you find out df1 x so the question is this he is asking right because f is the product of x if f is a function then we can define f with respect to x only I have to find the differential now see What did I take out df1 dt2 / dx5 / dx5 / dx5 Children will say this is my answer but this answer can't be because t is what it is to me It was not there anywhere in the question so I I assumed that's why we returned the value of t If I have to put it then what is the value of t what did we assume x s + 5 so If you put xs + 5 here then it means df1 What is the value of one 2x k x s + 5 and this is ours Let's see the answer, the problem number is Exercise 5.2 question number two now this is also a composite function since k has its own you have a function sinx-cosx sinx-cosx will become x now we have to put back the value of t so t is like x then this will become minus sa sa x in k x Okay so this will be my answer let's go Now let's look at problem number three, which is Exercise 5.2 Question Number Sis K x K * sa s x to the power 5 so look at things now A little more right what kind of function is this obviously this is also a is a composite function look at your cube which isn't it true that x is itself a function the function s is a thing in itself now x x to the power of 5 is a function in itself So here a lot of things have evolved okay so how do we solve this First, let's see how to do this Think like this then look at it cosx-sinx * cosx to the power 5 this second part is ok so i mean this is that form in that let suppose this is u and this is the whole v okay so if let's assume that this is f So if I have d f then dx8 will come in d / how can we help u * in term sin2 x ^ 5 as it will be d / dx3 okay now look at this also The differentials are pretty tricky, right? Because / dx2 x ^ 5 means it is composite in itself function okay so let's say this one which differentiable this one which Differentiable Let's take the right out so that there is no complication If it doesn't grow then let's focus first The First Ones is a differentiable function that is inside the brackets yes, I accept this, so what did I do Let t = x to the power 5 then from here I So what happens in that case is p = sin2 t If it is done then let's go first dpd.co.uk right because sa theta sa t its you have a function in the whole square which is that is a function in itself Come on let us talk about this as well it will have to be right, whenever such small are composite functions so they can be used again and again Better than adding variables one by one we'll take a shortcut to this what is the shortcut whenever it happens that the hole of the t If it is square then first see that sin2 is the mistake Get that in If a polynomial function of anything is if x of it was 2x of it then what would it be The derivative of 2x is right-hand side of x^n when What will happen if you take out the derivative? will be n x ^ n - 1 so in the same way here the sum of n Place that the derivative of whatever was inside what was it x it was not there what was inside t then sa is the derivative of t by itself and what will be its derivative c t okay so this is our shortcut when We will get small composer functions Then we can use such a shortcut and we will take out the derivative, okay that's complete No need to assume variables okay okay so should I take it out now I had to take it out dpd.co.uk so then dpd.co.uk x to the power 5 okay so this is what dpx3 cube this was to be taken out ok so man let's take that let's assume that this whole thing is a It is a cube, right? Let's assume it to be q. to keep things a little easy now let's do the trick inside there's a composite function so we've got x cubed what did I assume to be t i.e. t Let x be the cube, then dt2 okay and this whole q here That means that d k d k ba dx9 / dt-2425 So let's enter the values ​​and it will become like this - sati in 3x st is equal to my 3x instead of ssati we will write x is that ok Whose value is this? This value is Now if we find the values ​​of one and two of number two If you put in this expression then I will get the answer I will get it okay so now my third step what will be the put values ​​of 1 and 2 ok so what do I get will get df1 x e e what was here cosx–sinx ^ 5 cosx–sinx plus sa sc x to the power 5 it already was look this term was already into us After calculating in the second one, we got that that is this that means -3x s a So if we look at this value then This is going to be 10x to the power of 4 or x to the power of 4. power 5 k x k cosx–sinx K sinx-cosx like we did here I made this in the beginning and then separated the one It was separated from the forest and the forest was calculated separately. I broke the two apart and then used the two calculated separately and then the last I added them all together okay so Break them like this, step by step If you have to solve it then kids, now we will talk We are going to do an explicit and implicit sit what are these about functions We will understand the explicit meaning of this word from the word means directly expressed meaning Any relationship which we directly If you are able to express such functions where we have the relation between y and x can express it directly We call them explicit functions For example, suppose there is a function y = 2x now Look, this function clearly shows that what y is depends on x keep changing the value of y It will keep happening ok let's say there is a function y = sinx-cosx that y is dependent on x if x changes If you do y it will keep changing right so it means y writing in terms of x is very easy in Case of explicit functions right second On the other hand there are some functions where We know that the relationship between y and x there is a relationship but y is expressed in terms of x It is very difficult to write, as a Example: x + sinx-cosx I understand this when I look at that there is a relation between x and y, change x if you do this y will change but here I have to do y to write in terms of x if one say so that is quite challenging for me to We call these kinds of functions implicit sit functions infact not even in real life We use both these words explicit and implicit I use it a lot in what situation For example, let's say there is a guy named Rohan, everything is fine for him Ram has a friend Rohan who needs a lot of money if needed okay then there can be a situation that Rohan goes straight to Ram and He goes and says brother I need money So in this case Rohan has directly expressed what he needed or else what of his feelingless It was a police function, which means it was an explicit The second scenario is this It could be that Rohan needs money But Rohan went to Ram and said no I accepted this, I assumed that my friend Ram He is such a good friend of mine that he must know I might need the money right then In this case he directly expressed If you didn't do it right then what kind of situation has happened If this imply situation happens then now we what are we going to do now Derivatives of differentiable set functions We know how to take out peacock and lace but the They have implicit functions how do we find the derivatives because They look a little tricky so let's go Children, this is our type two Functions that imply sit and Derivatives of Explicit Functions Now we will take out our approach here What will be the approach? It is very simple and straightforward. our approach is that whoever is our we will have a given expression to that whole to the expression differentiable okay means like let's say that we have a given relationship between x and y that x - y = pa is ok I need to take it out will go to dydekop.org and after that what will happen here yes it is constant differentiable you will realize that as soon as You can write the whole expression in Differentiable I wanted dy1 dx1 which is from NCERT exercise 5.3 Question number one is find dydekop.org 2x ps d body ek off 3y i equal to d body a off sa x so this will become 2 what will become here will become 3 is constant and what happened to yhwa will go d or ba d the difference of one i equals one x is equal to one x okay so look what I found here one time d got a t of one and i what did I have to take out i had to take out dydekop.org dx8 x + by2 = c y what we will do it on both sides x y is mixed so its complete to the equation Differentiable if yes right then it will be d/ dx1 dx2 = e d / Kaswa so this will be a here this is b If you take the constant out then this will happen d ba d aqua s Now look sideways I will show you what will happen d ba d ekwa s What is it is y scale now y scale in general Differentiable what would have happened if x was s then it would have been just 2x But since this is y and we are talking about x, In Respect Differentiable to -sawa now look here also ky If we take the differentiables together then it is will go 2 bwa d wa body a plus cyva d body a e eqv mine a so here Parva body can take one common then inside there will be 2 biva ps saiwa wa body a i ikt mine a der four d or body one = - a / 2by+ sin3 which is exercise 5.3 question Number Six Find dydekop.org differentiable ok i was doing it with respect to x to x after that see what happens here x y we have to focus a little here There are two functions, x is a function and y is a function There's a different function, okay, so what do we do here? we will put it here we will put the product rule How to apply product rule, look at a term will leave the first as it is the second term Differentiable are differentiable on the third term, then if If we look at the third term, here also there are two is a function one is x one is y then here But we will also impose product rule first Others will leave the term as it is to the term Differentiable OK, now let's move on to the fourth term. So the fourth term in this case is d / dx3 y what will happen to 3y a will become 3y But since we can define y as Differentiable Let's see further and the calculation will be like this 3x s + x s d body a point in d body one off ace square what is its value if it goes 2x then this will happen 2x plus look here d body aqua s what it will happen it will happen 2y inva body a pwa scale d one body one ev p 3y s d wa ba d one i equal to 0 now d that body one ones we will put all the terms together and in which d is The body is not the same, we will send them to the other side such as this it and so these are the terms and one this is so these We will send all the three terms to the right hand side if we can then what will be left here x sp 2 Aqua P3y S Inva Body What will be on one side and what will be on the other side Will remain -3 x - 2y x miva s so There four we can say that body one e ect mine 3x spa 2 xva pva sk divided by x sp 2 aq p 3y s okay so what is this it is this Went dydekop.org + d / dxcpl.exe body of a file squa is okay so san squa is basically This is what the whole square of Sawa means So let's assume that Sawa is t so means t e ect sava so from here we dt2 ba d wa don't want so ejumis from that This is complete, we should accept it as complete f Let's assume it's okay then should I remove it? Is df2* sinx-cosx Now in place of t we'll put 2 Saiva Kaswa d wa Badi Well, my first term is over now In the same way, we will now remove the second one what is the term second term d body of cos xva okay so this xva here Let us assume that o t e e xy2 dx2 = 2 dx1 ba dx1 ba dx1 okay so I got this good now this what is the situation f what is the matter here in this case I f is k ekwa right so f is c so from here we can find df1 f will be equal to my sa t okay so then d fdf2 in x d ba d One what can we write in place of pwa so t aqua is mine sa aqua in aqua body One pwa okay now what we're going to do is these values into this expression so let let's put here the first term what do we do you can write 2 Sawa Kaswa d dx-data-grid d is not a d right If you send it to the side then there is only one term like this in which this y is one of these then we call it d wa ba d a iva sa aqua divided by 2 sawa kava 2 sa theta co theta i sat theta we can write minus x sa Aqua So This Is Ours So kids, now we will discuss our answer Type three function i.e. evers How Trigonometry Functions Correct the Differentiable Trigonometry Function It is very interesting and worth watching The approach is so simple, suppose I have it A function is given, sin-1 x, then here what approach should be used to solve sin-1x If it is differentiable then we can do a simple We will approach and say it's okay If I assume sin-1 x to be y then now sin-1 x = y We can also write it like this that x x = e sin.com trigonometry function so let If we try this then our children will According to our approach, we will take it as y Okay, so the average of x is given by the value of y. So it means x = two what happened sins.com with respect to x ok like this if you do this then what will come on the left hand side dx1 d one of sawa sawa will become kava But here we have y and we are talking about x Differentiable be equal to this will become one so this will be will go / kasva okay now what is y so we have ajyu whom did you ask sin-1 x So let's put in its value so 1 / k Saevers x has it arrived has it arrived dydekop.org this value is defined will happen only when this denotes is not equal to zero Because if Dino Minute becomes zero then The whole not defined will become right So this means that we can say that this dydekop.org What is the value of Ro which value of row is equal to is equal to cos-1 will also be equal to 0 right then What does this mean I know that K off + - pa / 2 which is equal to 0 but I have to Off, this value of sin-1 x is not 0. should that means i never want i.e. here the value of sin-1 x This is minus 5/2 or else pa ba 2 because As soon as sa inverse x - pa ba 2 or + pa ba 2, so this is the inverse of x, which is zero. If it happens then this is mine The value of dy1 x is neither pa nor pa If Ba2 is neither there nor Pa Ba2 then it is ok In a way what I am trying to say is that the one who x what should not be the value of x Meaning if sa inverse x is not equal to - pa / 2 then x should not be equal to sa of - pa ba 2 now sa 90 how much is one right so that means the value of x is this should not be equal to 1 and -1 is ok means minus bit off - Pa / 2 How much will it be -1 sa of pa / 2 how much will it be will go to 1, which means the value of x is These two values ​​should not be -1 and +1 It should not have these two values It can have any value in between okay so that means x which is it will Belongs to -1 to 1 open interval means this In the interval, -1 and one, these are the last two points are they are not included so x in Any value in between the two The value can be okay, it's clear till here okay let's see how we do after this will take out So now what can we say in such a situation You can say that my expression was on which I need to work on this expression if it is correct then I have to write it before this expression I just found out that the value of x is It ranges between -1 to +1 exactly now see how we can find a simple solution It is a logical thing that sin-1 x is will be one will be right sin-1 x right side off sign evers rat so this will be one so now Suppose if I talk about cos square, what do i write for squa so cos squa can I 1 minus sine square cos square theta p sine square theta ive we all know so I can write this as 1 minus sinva ka hole square right what can i write this am 1 Minuses sign off wa what is wa we starting I had already assumed that Sign Everest is an Ex The Sign of Sign Evers one and the hole of this whole Square what did I just say that sign off sine inverse ace which is its value How much will its value be x correct ok here i mistakenly typed one I had written the sign of Cy Evers x then sa and sin-1 cancel out each other's effects if it does then it will become x so here it will become x s so from here we can say that cos1 = √1 - xs Now we go back to In this relation, we can say that can that I found the value of y from dydekop.org Lee So 1 / √ 1 - x S was a little TDS Why because in this we are in terms of x You were trying to find out the answer, so that's why this It was necessary to find out who brother Ivers was What could be the value of k What could be the value of what it can't happen and based on that we came to this conclusion now look here pay attention The thing is that this trigonometry aren't they like fixed functions The derivative of sin-1 x is always 1 / √ o 1 - x It will happen but we solved it just to show you where these values ​​come from okay yes, in the same way we will take another example And we will take the tan-1 of x that if I have If we want to find the derivative of tan-1 x then we what will you do the approach will remain same consider it y what will happen if you take it x will be equal to t y then on both sides Differentiable of t yd / dx2 x but here t is y so this will be to s y and together with a dydekop.org ok now look at it tan1 tan1 x then instead of this we write tan1 x and the whole of this is square okay so 1/ 1 + tan1 x tan1 cancel out the effect of each other If you make it disappear then what will remain is x and the square of x It's okay so ours just came out The answer is d / dx1 whenever x is found It was sold, how much was its value 1/1 + x S to Kids Evers Trigonometry Approach to finding derivatives of functions You would have understood it then, but now what happens is that when we go further with more functions we will find the derivative we will see that friend sin-1 of x or tan-1 of x or the inverse of If I want to find the derivative of x, then it The value will always come out to be the same, right? That's why for our easier calculations we use What will we do at least these six basic ones everts trigonometry function that is sin-1 x cos-1 x tan-1 x cot-1 x sec-1 x and these six derivatives are cc-1 x We will remember what happens next Will it help in faster calculations and what ok but it was important to understand the approach because Many times you may be asked this question, brother Find the derivative of sin-1 x and show that You must know the approach, now the formulas are ready If it happens then these are our formulas now It is very easy to remember as we can see sin-1 and k inverse are exactly the same The formula is that there is a plus sign in the cos-1 is the same as tan1 and cot-1 if Let us see that the exact same formula is tan1 In case of is there a plus sign In case of cot-1 is there a plus sign The minus sign is similar to that of the inverse and cosec2x evers from the exact same formula plus in ivers and minus in coke ivers so Basically you just remember three formulas to do that is sa inverse tan1 and from of Evers because the other three were automatically remembered If it happens, we are ready for the questions for this with pen and paper of course because today So brother there is going to be a barrage of questions So let's solve the question on evers trigonometry functions so come on kids let's look at problem number one which is Question number nine of exercise 5.3 is y = sin-1 2x / 1 + x s Well, we can solve this what can we write can we write this sin2x / 1 + x s okay on both sides we have Differentiable Another term will come dydekop.org this is equal to now look here this kiss is in the form is in the u / v form and What do we know whenever we do d / dx1 v if yes then how do you do it v*d/ dx-data-grid will skip 2x as it is d / dx1 + x s and divide the whole thing 1+ x Scale Screw okay so what is this it will come it will come Kava dydekop.org 2x so here * 2x / 1 + x sss it is ok if you solve it properly So do this 2 + 2x s - 4x s / 1 + x s So this is equal to 2 - 2x s / 1 + x s ok so now what should we write for this we can write that dydekop.org y was also lying there If you bring it then it will become 2 * 1 - x s / ba 1 + x s s * 1 / c y so now the value of this y What value can we write? Look here. But if we focus on this first step then at this step I what is sin3 and what is theta Perpendicular by hypotenuse me we have to figure out what is theta of theta base by hypotenuse okay this is what happens ok so let's apply the same logic now Sawa here I know what it means is perpendicular and this is the hypotenuse then Perpendicular and hypotenuse if any Let us know the dimensions of a right angle triangle This is perpendicular this is hypotenuse And this is base, so we can remove the base easily can this will be equal to √ o h s - p s so this is going to be root over hypotenuse Square that is 1 + x x x - p x i.e. that of 2x If the square is correct then what is the value of the base will go √ a s + b s + 2ab - 4x s then it will come to √ o x2 power 4 - 2x s + 1 okay now look a s + b s - If it is in the form of 2ab then this will become 1 - x skw hole square now hole square and the square root will cancel out so this it will be 1 - x s okay what is this This turned out to be the base This turned out to be just the base right now Now that the base is out, I can take out what? hmm what will happen base bye hypotenuse that means 1 - x divided by by hypotenuse is 1+ x now This is the value which we call this value What will I get if I put it here? if I go I will get 2 * 1 - x s / 1 + x ss * 1 / c y ie 1 + x s / 1 - x So this and this will get cancelled, this and If one gets cancelled from here also then the answer will come will become 2 / 1 + x s now we will see the question Number two which is the question of exercise 5 3 The number 10 is y = t e 3x - x div 1 - 3x So look, this question is a little bit It's a different question, how is it different now? You will know what our first step is In Evers' case we would look at it like this We will write that t y e 3x - x divided by 1 - 3x s now the first thing that comes to mind In such a case, the numeration is the denominator if it is in u/v form then let we apply the dictionary rule but if we You can apply the dictionary rule here if you want. So try it and see, it is very long will go because the numeration above is and Whatever the denominator, both mean a lot. If the length is there then our calculation is enough it will become lengthy so that's why always this Wondering if there is such a trick? Is there any shortcut way by which this can be done a little If it gets solved easily, then if it is solved very easily, then If you look carefully you can see it here here the cube is visible so is this This expression of Nava, is it from anyone it is giving result from formula, think a little exactly doing the formula of tan3 theta remember tan3 theta i equals two 3an my t why a divided by 1 - 3an scale a this used to be And look at this, if you compare these two do what is my given expression and what The formula for tan3a is if we find both When we compare, what we see is that One way we can find out is that x is not is behaving like no a correct so in a way it should be as it is it means instead of x we ​​have written ne look if you write a in place of x it will be is it like exactly the same thing or not right but when we look at the left side so what we notice is that instead of y which yes we have written 3a means instead of y we just haven't written a right so Somewhere we have to forget about this whole format I'll have to set it a little bit okay so that Both of these should come in exact format And what can we do to make that happen we can do that friend left hand The one on the side here is t y so here also It would be better if there were no a then what would we do we will do it let's do it if it doesn't so wherever there is a there will be I'll have to do a ba 3 because here I have 3a divided by 3 okay so so In that case it becomes 3an a ba 3 minus now if we use this expression Compare this earlier expression with So if we look at its right hand side so what we see is that x is tan3 so x is in the format tan3 and what is this a / 3 where is this a / 3 This is what came from tan3 is what we mean here in place of a what is a instead of y here correct then x will be equal to t y / 3 So this is the relationship I got from here i dy1 I can take it out on both sides d / dx2 = 2 dx1 what happens to dx3 tan3 from s theta so this will be from s/ 3 is ok but y/3 is a single thing in itself is a function then its differentiable that's why it is see dydekop.org dy365 1 p n squa ba 3 can be written like this now look no squa ba 3 what is this here We wondered whether we did not wonder By matching both the expressions we did it It was written that nava ba 3 = x so that means this what can we write 3 div ba 1 p x s and this will be the answer so kids now We are going to talk about Expo Shcial and Where do I go about logarithmic functions? But we use such functions when We talk about population growth How fast is the population increasing brother Or when we talk about a How much in particular investment If you earn interest then it gets complicated like this where there are growths or There are complicated calculations there But let's come into the picture ExpoSial function or logarithmic function so the beginning do an expo function then one What the expo sial function looks like Mathematically it is y = b to the power of x where But b is the base of this exponential function whose value is greater than one means b is always greater than vo okay so this y = b to the power x this defines a the expo function is ok we call it we look at it graphically because from the graph You will understand it better if you If you look at it graphically, what you'll notice is that Its domain means whatever is possible The input values ​​are the values ​​of x what could she be x could be any real Number is positive negative any real number can be x but y i.e. the output i.e. What is the range of this function? since my function is y = b to the power x b since it is greater than one means b is always positive so brother you want power Put anything, even if it's power, put it positive put negative but overall the value of y we get It will come out positive only, right, that's why The range of expo sial functions is It's always a positive real number, okay? OK now what do we notice when we look at the graph? Let's do an expo function The graph is ever increasing, it is always like this Look, it keeps increasing as the value of x increases The graph is increasing, it is rising like this going right the other interesting thing that What we notice is that when x is The value went towards a very large negative it goes means look towards x As the value of x increases, the value of y increases what is happening with the value y is basically the function right then the value of y Look it's coming very close to zero right because these things are in your text It is written in the book but some how you understand it I do not understand what they are trying to say when will it come when you relate it to the graph if you do it then look at the graph when x is Absolutely - going towards x is going towards the negative value of x then what is the value of y going to be zero The right one is getting closer, let's talk now Here are the types of expo functions: But we can see two types of expo functions We discuss a lot, one thing is common Expo cial function is second, natural Expo sial Function Common Expo sial A function is one in which the value of b whose base is 10, that is Common Expo sial Function is something like this y = 10 to the power x natural expo A real function is one whose base is e that is y = e to the power x in fact expo As soon as we hear about angel function, we it feels like hey there's e to the power something will remain right although expo function name was read because expo Ant means power and see what is in this function is b to the power x so this is to the power of x It is installed, that is why it is named Expo cial function okay so e to the power x formats that are expo sial functions what do we call them natural Expo cial function is now the one that comes to mind The obvious question that will arise is whether this which thing is e we call it yours constant or Napier constant so this is a constant value obviously now e has to be There are many ways to define it, look at all of them There are ways to define e but these take any method, any formula take it but take out the value of e comes approximately 2.71828 It is now seen that mathematics and This e in engineering, this one It is very valuable, it was very useful It was used very repetitively so that's why This approximate value was taken so that were ever we need this value we can directly put e there ok so this way since then we started using e for end thatchi something like this is obviously a log of something it will look like this right then you will notice that Whenever we write something like this in which log to the base b a means log of a to base b we read this as the value If ever a is to the power n then this is called How to read a to the power n the same way if b is under the log it means log to base b whose log of a log of a okay so keep this in mind We have to keep in mind that while reading this we are in a state of stupor off a or else log off a to base b now if ever also l of a = x is given okay so its what does it mean in away is that b to the power x = e a means if ever such what happens if b to the power x = a This b is the base of the log so log base b a e e x okay remember b for base that's why b whatever it is it's going to the base okay so If this is a relation then an example let's see let's suppose if we talk are 2 to the power of 4 e e 16 okay What is the power in this, four is equal two will become four okay what is the base here whose power is 4 to the power of 2, so that means The base of log will be 2 so log base 2 16 = I 4 Okay, so do the second one Let's take an example: 10 to the power of 3 = 1000 what can we write this look base what is my base is 10 so log to base 10 Whose log are we taking here? yes here we will take a log of 1000 1000 is equal to whatever power that is 3 okay in this way any expression which given in expotal or in the form of power Whatever has happened, we will write it in the form of a log okay this is a very basic thing If not this about the logarithmic function If you understand then proceed to solve the question I will face a lot of trouble okay now we You will see some properties of the log: It is important to know the properties because when we Find derivatives of log functions This is going to be very useful there okay So first of all we will see the change of base rule what does it mean if we have a I am given a log function and if I have to calculate its value then Suppose you want to change the base, I have it it has happened l p to the base a but i love this base if i have to change it then how can i do it I can write it down I can write it down Divide and divide, how can I remember what I am was in the base so that's why we put a below a So put a in the denominator so l p / l a can write true that numeration and The logs of both the denominators have the same base Like here what is the base of both logs b So look, in one way we have changed the base diya earlier it was a and now base has become b If it happened then that is why we call it change off The base rule is okay, there are two more There are important small rules like The second rule says that if log p q our If we have it then how can we write it l p + l q provided everyone's base is same, see every what is the place base b is ok so l off pqsljvl5ara So basically anytime if you have any such be a function that is log p to the power n then we can call it what can we write directly n l p simple was ok now the second rule is that If the divisor is in the form l p / q What can we write about this when the product He was in form so we added him Now if you are in division form then what should you do? I will subtract it exactly so l p / q = l p - l q and here also the base of everything should remain the same then expo function and We have taken Larimer, now let's see How to find their derivatives Two formulas for children, only two formulas You have to remember related to these firstly that d / dx2 power x is always what e to the power x this was so simple that is e The derivative of to the power x is always e to the power the power is x only secondly that d / x of l x is 1 / x then l is the derivative of x what is 1 / x so these two are simple Remember these small and cute formulas And now after this we will do questions now In the questions you have asked such easy questions don't keep your hopes because what normally happens is that now when you see that the functions that you They are given, aren’t they Use of combos and exponential functions Sometimes it is Evers Trigonometry If there is any work of function and log then all these will it be given in combos ok but Do not get confused and do not panic if you are interested then let's get started Let's ask questions so I think we're ready To solve some questions is the first question Exercise 5.4 question number one e two power x / If it is sinx-cosx then in which format is it it is in we can see clearly u / v If it is in the format u / v then whenever we of d / put dx8 is / This is what our Ksh rule told dx2, so here but we will apply the Ksh rule only, okay so let's go Let's guess what will happen The denominator is v which is my sinx-cosx okay and sinx-cosx We learn from different people that we Differentiable where you will find Trigonometry The function also appears multiple times invars Trigonometry is also seen in Expo The function also looks like logarithmic function If it is visible then it means everything is mixed together you are asked okay let's see Problem number two which is Exercise 5.4 Question number 2 of now look here also to the power sa ive x so here is the expo There is also the Hashem function and the Evers function is also a trigonometric function, so this is also a There are combos. This is also a composite function. Inside the right e key function we have done this we have inserted the evers function so here we have but what will we do, we will decide that it will be late As suppose we will take sin-1x as t, okay so what did i assume i assumed t I've done sin-1 x. Okay, so from here I'm going to what is the derivative of dt1 x in a little bit A while back we also worked out 1 ba √ 1 - x Okay, let's assume that this is the whole thing is given this is y so according to this question to me now dt1 dx8 is done so what is left dydekop.org that is e to the power t and dtdc.com it has become a log function also There is also a function of and there is also an expotal function So combine all of them to form a composite function We have given it so what will we do obviously chain We will have to set the rules right, so first of all we So what do we do, take things step by step Let's step it up, let's say this whole bracket I take it for granted that the thing inside is okay So what do I mean by that? to the power x that means t which is this in itself I'm a composition of two functions, okay So from here, if I So we will multiply it by e^x, okay? so this is my dt1 dx2 / dx-data-grid dx8 I've already done it, okay? What's left? what can we put in place of t the same as guessed that is 1 to the power x and it will be raised to the power of e x there is no minus here it is e to the power x is - e to the power x sa e to the power x okay so look - e to the power x sa theta what is theta of ba, tan3 so this is also we can write tan2 to the power x so this is My full and final answer will be given my dear kids now we are going to discuss our Type Five Functions or Type Five Kinds functions of where we can see them Differentiable Damiak Differentiable Here we will look at log functions. really here we're going to use logs Focus on differentiable functions Whenever we see a function like this which we can express in the form u to the power v where both u and v are are functions u of x is a function v of x is a function so if our given function f is if that can be written in the form of We apply logarithmic to uxx1 Differentiable dynamic Differential disposition says that both friends log on to the side meaning fxxx.pro [music] Right, so this was our logarithmic Differentiable meant that if ever We have a function that raises u to the power v if it is in the form of then what do we do We take logs on both sides, right and left On the hand side it will be log off y and What will happen on the right hand side? Log off u to the power v which we can write as v l u right and what is this v and u of x It's a function, so we can write it like this now what do we do after that Differentiable y and what will happen on the right hand side d , v l u okay if this is what happens then what should we do about it we can write l of y Differentiable will apply because look there are two things one v and the other log is u so what here For once we'll let v be as it is and we will say d / sounding y and then once log y as it is will let it be and say d ba dx1 dydekop.org So with this approach we are to the questions Differentiable number one which is Exercise 5.5 question number one is c x cos2x cos3x ok now look carefully at this function What type of function category does the function fall into? it is coming now you will say that maam this is for you Not exactly in the power v form it is absolutely this is exactly u to the power v he is not in form but think about this how do we implement the function Differentiable If you apply it, it can get very complicated why because there isn't two in it right If there are three terms then club two at once Then he will have to do it inside her again product rule will have to be applied right then meaning Product rule will not be a very easy option So an easy option here is logarithmic di There will be a difference in which we look at both sides would have taken it look how easy this thing is to log in will make it will be on the left hand side get a log off on the right hand side * b * c will be right so we have a*b*c can I write log off aps log off B Plug of C We read this rule right that if log if a is there then how do we write it l a + l b So here also if we take the whole log then log off will be a*b*c then attack it now we can write a log b + l c what shall we do now Differentiable of y doing it in terms of differentiable that's why dydekop.org because we cosx–sinx But if cos2x is there then the product of this 2x is also will have a derivative so the derivative of 2x what will happen *2 then do this 2 here also two okay look now let's do it in short cut If you are learning right then you are using chain rule There is no need to complete the butt now look at the third term of cos3x Its Differentiable B - sin3x Now this is the coefficient of 3x as well It will be a derivative of itself which will be 3 So * 3 Okay, so that's it, now look from here me a dydekop.org 3 so there farwa body one what will happen wa inu what is this whole thing wa it is cos cos 2 a cos 3 ex this multiplied by if the minus sign If you take it out then don't get a pin 2 exp 3an 3x end It is a problem, how will we take it out, see whenever I told you, look at this ghazal Differentiable format or when u * v * w * x Look at the previous one in such type of format We also saw in the question sinx-cosx to the power 4 * 1+ x to the power 8 right So here is the format of all the products But we'll again take logs on both sides So on the left hand side the law will be f What will be on the right hand side: Log off in The product of all four we can write as log off a log off b p log off c so now we wrote it like this so now See if I can make it Differentiable of 1 + x If it is differentiable then how much will it be Its differentiable value is 0 of one and so it means It will be into 1 only, ok, no problem After this look at the lag of 1 ba a square will become 1 by 1 p x but the same thing is that this is 1 p x s There can also be a difference and its How much will the derivative be 2x because one it will be zero right brother d ba d one ofv e 0 p d ba d a 2 a sch is 2x then log off p 1 to the power 4 what is this It will go to Differentiable gonna be 1 times 1+ x to the power 8 * x to The Power of 8 differentiable so it is like 1 by 1+ x + 2x Ba 1+ x s 4x Ba 1+ x to the power 4 + 8x to the power 7 / 1+ x to the power 8 okay so now df1 x means what will become of f ' x f * this the whole thing f what was it f this whole The thing is, let's write down the value of f as 1 + x * 1 + x s * 1+ x to the power 4 * 1 + x to the power power of 8 means this one which is in the denominator f he came here on the right hand side came in numeration okay and in brackets what was already inside 1 / 1+ x + 2x / 1 + x s + 4x k / 1+ x to the power 4 + 8x to the power 7 / 1 + x to the power 8 so this is f ' of x What value do I need to extract now? f-1 means wherever x appears in it put a forest there when you do this and If you calculate, you will get 16 * 15 / 2 so this will become 2 * 8 16 that is equal to 120 so 120 will be your answer so kids Now we'll look at our type six functions So we call these kind of functions Parametric form functions are functions that what is it in parametric form what are parametric forms The word parametric comes from parameter Many times we will get functions which have Their relationships inside are a little different would be of the type like suppose if we have There are two variables x and y Let x be the function f of t I can write y in terms of I am g of t I can write it in terms of right means x which is another function, means one The function f is y which is dependent on a on another function g but f which is that is a function of t g also that is a function of t so basically t is a parameter which This is common in both functions right so this What do we call these kinds of functions? Now let's look at functions in parametric form. is that when it is in this format our functions so how do we find their derivatives Let's take it out, let's see, look at all of them There are things we do in a roundabout way parametric form i said look at one function x which is f t another function y which is g t t but look t which is Both are dependent on whom? It is dependent, so what do we call this t there are parameters this is one such parameter in which both which are mine are x and y It is connected ok so in such case assume If I have to take it out dydekop.org in let's talk on dt1 so what is it If I want to find dx2 then I will use this logic I can, in fact, if you pay a little attention so look at this is what we're doing no that is basically like we are in chain rule We also do chain rule, we start a small function inside a function at t value take it then Both are dependent on what they are is dependent i.e. t is my parameter ok here so in such a case what should I do first let me write what I want from an equation I can that dx8 is a constant so t becomes 2t That means that 4at ok this is dx1 compared to the other one what can i take out i take out Am dydekop.org 4at k ok that was easy But the question is what do I need to figure out? he is saying that he wants to remove the question Is dydekop.org so t is the answer you will see Problem Two which is Exercise 5.6 Question number two is x = eb k theta y = b look at theta of both x and y The expression shows that both of them is dependent on theta so that means theta here is the parameter so what should i do What can I write with the first expression dx3 ee a * - sa theta because a theta of derivative - theta is the other one what can i write with expression dydekop.org - a bit theta so sa theta sa theta will cancel out minus minus cancels so the answer is b / a so finally children for different functions The process of extracting derivatives starts here It's over but this lesson is not over yet We are going to discuss the second order derivatives whatever we were doing till now what was that first order derivative I mean I had a function given to me Wright was finding its derivatives, such as that we used to calculate instantaneous velocity so what we used to do was just take a derivative we used to find the edge in we used to find only dx2 So we would have got instant nice velocity but when we find out the acceleration if we were there then how did we take it out? You would say ah we Even when we took the derivative, the velocity means we if there is formation then acceleration and What is the relation between displacements think acceleration is So that's my first order derivative if him fard differentiable d s f / dx2 so this is my second order If it is a derivative then things will remain the same here How to find the derivatives, that is the story it will stay the same just that if you need seconds If you want to find the order derivative, you must first Once the derivative is found, it is again Differentiable such as second order derivatives we can denote by f du x or d can be denoted by f / dx2 or d can also be denoted by s so these are three There are different ways in which we can do second order denote derivatives so let's have some Let me try the questions, let's see Problem number one of exercise 5.7 Question number one x s + 3x + 2 so value Let's take this value, this is the value of y so here I have the second order derivative We have to find out that is d square ba d is a square I need to find its value, okay? what will you do first, first of all brother first What will we do to find the order derivative? do d body one off this whole thing x span 3x p 2 so what can we write this as d body one off x span d body one off 3x p d body one of 2 ok constant what is a derivative what is it will become 2x and what will be its derivative 3 Now we will find the second order derivative that is d s y / dx2 which means now I am d / Let me find the further derivative of dx1 which I got. If I take out d / dx2 x + 3 then this what will happen d / dx2 x + d / dx3 What is the derivative of the constant rho So what will be its derivative 2 so two This is the answer. Now let's look at problem number two. which is the question number of exercise 5.7 Three is x * c x then suppose this is y y the value of is x * c x and what do I need to know To find out the age of second order We are currently learning to calculate derivatives So we have to find the value of d s y / dx2 What will you take out first d y / dxcpl.exe off x ok so what is this will go to x * d / dx-data-grid will do so d/ dx-data-grid will become d / dx6 after that Once sinx-cosx then - sa x because dx1 x is 1 and after that - x ok so it will become - x c x - 2s x and this is our full and final Answer: Now let us see the problem number. Three, which is the question in Exercise 5.7 The number six is ​​e to the power x * sin5x okay this is my y right here is a has an expo sial function and a there is also trigonometry function here i What do I need to do to get the second order derivative? i.e. d sw / dx1 first what will we do first order derivative Okay, we will take out the first order of the whole We will find the derivative, suppose this is y and this is v then apply product rule If you apply product rule then once e to the power x as it is d ba dx5 a then sin5x as it is d ba e to the power x is exactly the derivative of sin5x what will happen to sa theta here what is theta is 5x then what is the derivative of theta that theta but this 5x is going to come back The derivative can be which is phi to 5 multiply it by ok plus sin5x d / dx2 what will be e to the power x The Power x Okay so this is my dydekop.org Now I am ready to find d s y / dx2 ok for that then what will happen d/ dx5 e to the power x cos5x + e to the power x sin5x okay so let's start with the first term what will we take out is the five constant So this will become d / dx2 power x cos5x plus d body x of e to the power x Sign 5x okay so in these two separately we We will apply the product rule, okay, so the first term what will we talk about here let's do this y and this is v so what will this be a bar we will keep y as it is and do v we will give differentiable times v as it is will keep that is cos5x as it is and e do Differentiable and/or dx2 to the power of x okay so let's figure it out So this is going to be 5e to the power of x of theta what happens is - sa theta theta over here 5x then the factor of 5x differentiable will be cos5x but of 5x Fard Differentiable p e to the power x 5e to the power x cos5x + e to the power x sin5x Now look where sin5x is e to the to the power x sin5x e to the power x sin5x this If we put both of them together then it is -25 + 1 that is -24 e to the power x sa 5x now look here e to the power x cos5x e to power x cos5x so 5 + 5 will become 10 so 10e to the power x cos5x So this will be our answer, so come on kids Let's look at problem number four, which is 5.7 The question number of the exercise is at tan-1 x Now if I want to extract it in seconds derivative you will say that ma'am its The first derivative will fall in a line because we have kept it in mind and it's datchi because it's that much frequency Cont rule is ok so what do we do v as it is will put d ba d one off u will do minus u As it is I will keep you d / dx1 is the square of skew okay now constant what is the derivative of rho so this is the first My entire term will become zero, okay? what will happen in the second term -2x and what is the denominator of 1+ x Hole Square End that's it so now let's finish this lesson Before doing this we have two very important things to consider Very simple, very simple two theorems We will discuss whose first theorem is Roles While reading the theorem from the book it feels like this That friend, so much is written, there are so many diagrams I did not understand anything but actually roles Theorem is a very simple theorem this is a very She says a very simple thing, okay let's understand The Rolle's theorem says that if we have have a function fxxx.pro a which is a closed interval above it this is continuous okay what else fx5 force on a now look what this both in a There was a difference when I talked about continuous I said a closed interval third bracket The third bracket was applied, meaning a and b, this Remember both points are included in this If you studied with me in 11th, then set and relation function etc. in that I have this When I explained things to him, it was closed again There is a bracket, it means this it is like a window which is blocked she keeps things stuck okay so a and b are also its It is included inside, that is, this function It is continuous at point A and also at point B. Pe is also continuous and all in between It is continuous in points also, okay The second one is saying that if this function yes this is differentiable in a look at this We put the first bracket first Bracket means open interval first If we say brackets then it means both a and b values ​​are not included in this means that the function which is at point a and point b They are not differentiable but there is a line between them for all the points to be made it's different why did i do this when This is continuous on a and b also so this Think about why it is not differentiable there think think think what did i say The function is differentiable where But it is possible to find its derivative where But now we can find the slope of that t tent The end of any graph of any plot there are points, right, we never go there For him Differentiable you get it and that's it here we have I used the first bracket and understood it correctly and if f a = f b means this is the function The value of the function at both end points But it is equal to if this is the case for this function is ok then Rolles' theorem says that In case the graph of this function is any It should be of any shape but not somewhere between a and b There is at least one such point somewhere There will definitely be a place where f'c will be equal to 0 means f ' c means the derivative of the function f At point c means there is no line between a and b There must be a point where The derivative of the function will be zero. The derivative of the function The derivative is zero means the slope of the tangent Ro is the slope of the tange Ro is the meaning think how to find the slope of a tangent from tan3 means tan3 is 0 tan3 is 0 means theta The value of rho is theta The value of rho is mat means the tense at is that it is parallel to the one I understood the point, we are always on the y axis If we take the function, then it is very simple What Rolles theorem tells us is that if our is any function such that A is a function of this interval Pay is continuous at this interval is also differentiable and if their end points the values ​​of are same i.e. f a = f b then for these There will definitely be a point somewhere in the middle When my tension will be at x axis it will be parallel to that means think logically that Suppose I have a plot of land created like this okay so the value of f of a is the same value of f of b, then if Whatever be the shape of my plot starting key value and ending key value If it is the same it means that this plot has Somewhere or the other this graph took this turn It must be so, that is why it has come at the same value And where he took this turn what could have happened there what is my slope It must have been zero right there. If the slope is zero then correct it in this way So Rolle's theorem tells us that now At the same time, it is important to know here that Here it is not necessary that these are a and b there will be at least one point in between There may be more than one point For example, suppose if between this A and B If my pot is like this toy If you are a toy person then look here how many There are points where my slope is zero Look for yourself, there are four such points right Hence Rolle's theorem says that if All three conditions must be satisfied for any function for it to be continuous at that interval is differentiable and has end points at that If the value of the function is same then these two ends At least one line between the points There will definitely be a point where the function First derivative will be zero now rolls With the help of the theorem we can find the probability of any graph Can you tell me the minimum and maximum values? Let's see how to answer some questions So we can actually apply Rolle's theorem if we do then we will see a question which Question number one of exercise 5.8 is Verify Role's Theorem for the Function fxxx.pro The interval would also be given that in this interval Is Rolle's theorem true or not? So what do we need to do first for this We need to find the value of this function what are the end points at its end points are -4 and +2 i.e. the function key We will find the value of the function at -4 We will extract + on 2 and function key on both The value should be same, hence Rolle's theorem right will be applicable if we put -4 there are so-4 sqaure 2*-4-8 I mean basically this is fx-2175 what does rolle's theorem state Now tell me that Rolle's theorem states that If the value at the endpoints is the same then Means someone in between these two end points no one brings the same point which lines between -4 and u means -4 and t in The truth lies somewhere in between the two that add s e e 0 okay so that means now I have add c has to be found out now to find f' first First of all this is fxxx.pro let us say there is a point let there be a Point s is ok so instead of f I put c So this becomes 2c + 2 which is equal to 0 So from this we get 2c = -2 and c ei -1 can you check now whether this is c The value has come, is it between -42 look at the number line remember this is 0 and there is 2 here there is -1 there is -2 there is -3 there is -4 so This is saying that the value of c is -2 and minus should lie somewhere between -4 and +2 and the value of my c is -1 which is exactly I am lying somewhere in between these is so that means this c e -1 watch belongs to -42 That means Rolle's theorem is Verified so finally we have arrived On our second theorem, which we call Mean Value Theorem or in short you You can also call it MVT, maybe right so mean Value theorem which is a way of playing roles How is it an extension of the theorem Because this pretty much says the same thing. means the situation actually remains the same Rolle's theorem was then the mean value theorem says that if you have a function f x which is continuous at an interval which is differentiable on an interval, then the sum of a and b If there is any point in between let's say any also point c then first point c is at What will be the derivative of that function? You can find it using this formula right This gave a simple formula that f ' c e e f b - f a / b - a okay now you Will you tell me where did this formula come from? why should i agree look at basically no things It's very simple, what are we taking out? f' c Basically we can find the derivative at a point Finding out what is a derivative The value of slope of the tangent at right is and We are doing the same thing friend, this is the formula na this is nothing but the slope of the tenge now You can see it yourself in this graph if you If you make a tange then it will turn out like this now you are tange at Find the slope of or find the slope of a skewer What difference does it make? Many children are wondering you will learn what happens if you If you don't know then let me tell you that tange is such a A line is a line that intersects a curve at a point learn one like this that goes away by touching a line which divides a curve at a minimum two cuts at the point okay so here Look, if this is my tenge eight, then this is my there is seken and seken and tenge at because these If both are parallel then the slope of both is It is the same correct no matter what I say Should I find the slope or should I find the slope of the needle? it's just the same now learn the slope of it Try taking out tan3 and see what will happen Perpendicular by base now Perpendicular key what value is coming focus on y axis Do it basically f b - f a this is what is coming What is the value of perpendicular base b is coming - a right so what has come f b - f a / b - a and that was my formula so That means the Main Value Theorem has given very He said a simple thing which we know to some extent I thought that there is a function which works on an interval is continuous and differentiable then a and b At any point between if its I have to find the derivative so that is given by this is okay we will see the questions let's see So let's see a question which is asked by NCERT Question number four of exercise 5.8 is Verify Mean Value Theorem if f x = x s - 4x - 3 in the interval a where a = 1 and b = 4 ok so what is the first step To prove the mean value theorem, first So I need these functions at the end points I need to find the value that means I am getting a function I want to find the value of at one end function key I want to find the value of eight four, how is it ok Let's find out the given function in which Enter the values ​​like in this case enter one So 1 s - 4 * 1 - 3 so that is equal to -6 Similarly it will become 4 s - 4 * 4 - 3 which will become -3 okay so now I mean The value of this function is found out at the end of on points okay so now in between Its slope at any point c is How do we take it out fb login 4-1 then this value will be Okay, now let's look at the mean value theorem. Mean value theorem states that does that it has both end points there is a point see between them that Belongs to two and somewhere in between five and four Somewhere it brings the truth that add c which has the value that is equal to this value t e i 1 this is what it says na me The value theorem that if I have a function is defined between the points a and b then a A point c lies between b and a whose add s value we can find using this formula let's calculate okay so here I have calculated I went backwards and calculated the value using this formula I calculated it and now I said that brother there must be a c whose value If this comes out okay then how do you remove the f'c now? f is a given x s - 4x - 3 then here from this we can find out add x which will be 2x - 4 okay so what will become add c 2 s - 4 okay now what we're saying is that this 2s - 4 its value must be equal to one it should be ok so according to the According to Theorem 2c - 4 which the value that should be one means f ' c of The value must be one Mean Value Theorem According to that, c is the what will be the value 4/2 that is equal to sorry 5/2 okay so 5/2 is like 2.5 so will 2.5 fall between 1 and 4 Look where did you get 2.5 between one and 4 will it happen here then will it be between one and four It's coming right in the middle between one and four It is lying in the middle i.e. mean value theorem is Verified I understood so Rolles theorem and mean Value theorem seems very complicated but it is so complicated no it is ok This was our last video of this long video The question remains, we will come with this have arrived at the point of continuity and differentiability Till the end of this one shot video and I am pretty sure that video was too long Because there were a lot of concepts in it And we also solved many questions why but i am pretty sure that it was Definitely worth your time because we Entire concept covered, lot of questions Solved and if your concepts are clear If it is crystal clear then definitely write in the comments Let me tell you that the concept has become crystal clear Now what you have to do is to open your book Solve the questions given in the question paper by yourself You have to do it and if you are able to solve it by yourself If yes then please write to me in the comments Let me know and I'll see you soon then with a new topic and a new video till then stay home stay safe take care by Bye