GMT20220521 145652 Recording 1920x928

Good morning, let's wait a couple of minutes now, please. No one can write. No one can. I want to. I have good news. Of course. Well, we can't wait any longer. How is this Mateo? Mateo Solórzano Vega. I think there is only one Mateo. No more here. Yes, only one. Mateo I don't know how you are Mateo It's cold where Mateo lives If it's not an indiscretion the area just don't tell me at the bottom by Carlos izaguirre at the bottom like by the sea there Carlos yzaguirre izaguirre is the one that starts here near the Panamericana But you go to the bottom something like that and that area is near the sea and since it's tide it's cold here you live at block 42 of Perú Avenue more or less it's nothing else well what I wanted to say was to consult I already did it with the other section, look, for the midterm exam I am supposed to move forward and do extreme values, that is, maximums and minimums.

But what I want to do is the following since it does not involve the issue of maximums and minimums, we only do up to tangent plane and derivatives of gold Superior up to that point to be the partial that says what happens that extreme values ​​is a bit of a long topic as well as additional derivative we still haven't finished deriving it is inhale I already threw it But the most difficult for the student is when it is in implicit form Then I evaluate them Only up to the tangent plane and derivatives of Higher order and and extreme values ​​we leave it for practice 3 what they say I think so because I am noticing that the topics are piling up and and we I am aware that we are returning In person, you have the problem.

For example, writing, that is, suddenly you have the idea but sometimes you can't express something like that and you doubt yourself, as an anecdote, well, yesterday we talked at another University and they told me, Hey, pepelucho. The students now are no longer as they were before, they had not even plagiarized, well in person but not now, it is not that some of them do not have a notebook, well, and everyone is used to the fact that everything is the cell phone and today, well, in person, the cell phone is not used. well then there are the anecdotes and that's what I was thinking of telling them through just a little message But since we are here we also have to be respectful of people Let me just make a tangent plane and we do a couple of examples one where the one that almost I is Cartesian I do not ask and the other where it is implicit What do you think and there it is just Yes for an hour at least yes And next week we do derivatives of Higher order and we have to do some examples that are also missing additional to derive that and week 8 already midterm, right, I also don't know if I'm saying harm by giving them the whiteboards.

Why? Because you don't write, so it makes me want to review notebooks, even if it's worth 5 points. It's not mandatory either, but yes, no, but I'm going to continue sending you the whiteboards. but when you do the calculations you have to do the calculations because that is the problem I asked your teachers someone left before the time they didn't tell me the question one I changed it was first but I said there they are going to get trapped so I put it last and a question is something that you have asked in class, it is not to Find the kl and the least to do it, that question about the rate of change is also to ask you classics, those questions, well, let's not talk about yesterday, now, now let's Let's see how we do this now, let's justify why I didn't do it or not.

He doesn't usually make a tangent plane only with partial derivatives, you'll realize, look, how old is Alexis? 19 20 Alexis Quiroz, that's another matter, no, no, I asked him how old he is. Let's see if I'm guessing not in quotes 19 a little more a little less listen to me you can't listen to me years well Let's see let's see here let's start with the following let's see let's see let's promote the matter here like this when we had Here is a function, a curve that you could make and equal to f of a It didn't necessarily have to be a function, it could be a curve no You wanted there a tangent vector at this point here and you found that using this first of all the derivative and you already created the equation of that tangent line and the other thing was it immediately came out what would happen if you wanted to find a line perpendicular to that curve then the curve here a vector is perpendicular just in case to a curve yes it is perpendicular to its tangent vector it is already at each point So that problem was not complicated, possibly you even solved it Cartesianly and the issue is the following which is the topic for us today and I hope that the thing is clear, the problem is the following What happens if you have a surface here a surface we have a surface here like this and we want to find this surface we do not know what its equation is like let's not get used to putting Z equal to f okay It can be an equation that depends on three variables so that's the thing because when you see the books Z equals f of x and then you choose a point or they tell us at a point Not at this point p Sub Zero I want to find a tangent plane So what makes the directional vector with the straight line with the partial derivative with respect to y for example the vector a with respect to y is 0 1 f f and the other directional vector with respect to X according to this is called the imbé and we had with respect to x 1 0 and F su Of each of you I asked you yesterday I think that a derivative I think it is Suller or if there is your The problem has to be more general, so I solve it more or less in this way.

I solve it in this way. Look, first of all, we want to know and be clear about the following, up to this curve that was above. We want to solve the following. We want at this point. here from Sub Zero How to find a vector perpendicular to that surface How to find that vector perpendicular to that surface for a given surface in general x and z equal to zero How to make that vector perpendicular to that surface We intuitively know by definition that that vector a It has to be perpendicular to this curve that passes through here.

Let's see, this curve that passes through has to be perpendicular to all those curves, so it will contain all the tangent vectors and we are going to call that the plane, the tangent plane, that is the plane. today's problem It is ready and we call this its tangent plane but at the point and that point satisfies the equation of the surface before I did the two cases that is case 2 and from there I went to case 3 but the situation makes that we are profiling and we have to do it in another way the resources we cannot assume that you are Happy Well no that everything is fine now that you solve that problem the next result Look here we are going to write it We agree that we have the idea of ​​what we want to solve I forgot to ask you that we don't want to send or have the equation of a plane tangent to a surface that surface can be in Cartesian form as it is there or it can be in implicit form like the surface of the exam that I was taking yesterday on purpose There was a question where I had part there in one I wanted the partial derivative with respect to that when you evaluate it gave integers or it gave zero or something like that that fits well because if not, imagine giving a student to give you decimals, they have to operate that was a question for you, not to see the problem, this is the central problem, you imagine that when in the pandemic until 2000 211 I took the midterm exam in three parts, look at that job that That that that job that I took there was one where the moon was setting it was I think five points and I'm not mistaken I took it into account the in the virtual uni regarding the virtual one our platform we are supposed to be able to do why don't we make a plankton type because for emergencies this can be taken to the luma and five points being even at home being in a university and him answer or that it is in a team of two Well, not that it is worth two points now they will tell me but only one is going to work some do not want to work in a team etcetera that passes into the background the issue is that they get used to working that way So I tell him I prepared here at the unitedtual but on virtual Monday I had a window, disadvantage to everyone that no no no no it did not allow me to design a question it is good that they know that I design questions in another university they give you the problem you provide a solution and You are asked part by part in such a way that you have to develop it in one way or another and here you are answered there.

You write your answer that doesn't matter it can't be sometimes there is in degrees sometimes in radians And That doesn't matter what the issue is, it's true that it was difficult for me during the pandemic. They gave me the challenge. I took it up, but in general use, you can't. There were only two types of questions and the third part was when they gave this virtual one where they sent your file what you have given last cycle you did not send your file brother Yes I am fine and yes what you gave the first but I divided it like this into three parts this here was difficult for me because I had to make a bank of questions that is Each student had a different question so we had to design it.

I remember here the teachers had it two three in the morning that's what I wanted No, they told me that it has to be asynchronous And synchronous so we have to comply But the veterans sometimes don't They just wanted to take all the files well we continue here we delete this Sorry here we delete here we want the next little problem the next theoremite I'm going to do one just you will intuitively realize that case 2 is a particular case So the theoremite is like this suppose you have a function F of three variables Yesterday it is already it can be like that no that function has contour lines or contour surfaces that function talk about surfaces level surfaces contour surfaces time contour line has the one that has two independent variables actually has three not two independent variables and a dependent one that already exists the theorem is thus derivable and it already exists then the gradient of that f at the point p Sub Zero is orthogonal to the level surface to the level surface of F But and here I put it in green No What happens through the point that is a zero?

It's that simple. Look, it's that simple. And that's what justifies the results we want. Because look, what happens if we had a surface, so to say, like this, no more and square minus 1 equals 0 and that's it you have a surface there to explain what is here I am going to make an example even in a Cartesian way but the objective is it is implicit just in case Now then I am going to do that In the end, so that it stays there, here there will be a difference that if they ask you that there is the equation of the tangent line with the greatest slope to that surface, that is one item and the other item is what they ask you there in the equation of the plane tangent to that surface there is a very substantive difference between the two problems and you yourself will immediately realize if you are reading, eh, keep in mind that for the partial things are already accumulating, there is a directional derivative that there is a question of at least five points not that is related to partial derivatives differentialities etc. there is a tangent line of greater slope all of that to see to see for the surface that is there and p Sub Zero that is the point for example 111 that point satisfies the equation of the surface Yes instead yes yes yes yes yes satisfies not zero So why do I do it that way then you caress this function here you are going to call it F of x and z This is going to be x y z cubed minus xz plus y squared minus 8 Ya is then I ask you That is a level surface of that capital F at level 0 if I have a function here How are its level surfaces at the level here is not F of x and z equal to a k practice on that is good That is that's it So what is the answer of course now some of you could tell me why I don't need one I can pass it to the other side and nothing happens right now let's see the detail because yes it can be but you have to be careful there it is, we are there The answer is yes So you say, calculate the gradient here, we calculate here the gradient of f at point 111, calculate that gradient for me F sub x F sub y F sub Z, calculate that gradient for me, it's done, I get 0 3 y 2 professionals 0 3 and 2 3 and 2 let's see zero when it is derived with respect to Z I don't want zero eh No with respect to Z it gives you 3x times 10 times Z squared minus x that is, it would be 3 - 2 no to three minus one no And that is 2 with respect to that point you and you found the vector perpendicular to that surface which is the gradient of fd11 that's it the speculations about this look I'm telling you this because once a cycle they asked me a question yes they put me here nodathe surface here El Paso until they made an effort to put it I think that with exponential even I did not ask them there in the equation of the tangent plane and I put a point here The student is already there when he calculated the partial derivatives you know that -1 has nothing to do with nothing is clear Yes when you calculate the partial derivatives then in luxury calculate the partial derivatives and there what I asked Well the vector Teresa It is not true the point has to satisfy the equation of the surface of agreement because the result is like that in that point is orthogonal to things because how are you going to say that it is orthogonal to the surface in a point that is not on the surface for example.

In other words, you have to be careful, the point has to satisfy the equation of the surface so that they are not Saying teacher, it is indifferent to put minus one eight or 40. No, if the matter has to be clear, it has to be at a point on the surface, that's it. So what is the big result now if you wanted to find, for example, a straight line with a greater slope here and We are in the directional derivative part so you have to assume that Z depends on find a vector perpendicular to this surface here this function that is here this function here the and 0 3 and 2 2 now yes the application of this no and we do a couple of examples of this another thing from the blackboards the result is not a tangent plane and a line people do not notice it this result absorbs those who only use partial derivatives already and This is faster, a tangent plane and a normal straight line, that's it.

So let's do the following, as we say, here, let it be a level surface, level surface DF, according to a variable ft of x and z, no more, that is, we are here. Don't go thinking. that axis they give you like this directly well no right now we are going to see why then here we have the the the surface is already this the surface This is the surface is and we are going to give a point p Sub Zero that belongs to that it can be this little point here What does the theory say what does the theory say?

The theory says the following: that the gradient at that point is perpendicular to the surface, that is, it means that it is perpendicular to all the tangent lines, and then it will cause here that there is a plane, that plane is the tangent plane is the tangent plane which possibly when a vector a when is perpendicular to a surface will be perpendicular to the perpendicular for example you make a is perpendicular to c sub-2 is perpendicular to all those curves but the one that is perpendicular So one is perpendicular This tangent is perpendicular to this tangent.

In reality, that vector is perpendicular to all the tangents. That is why this gradient vector with capital letter F is the normal of that tangent plane. Now the equation that you present is going to be presented in the form in the way you want What is always reading the form that is requested What happens is that a plane presentable in vector form sometimes cannot be measured much because the student can write it in various ways the vector upwards the vector downwards, what do I know? I don't take another point but in this case, well, and here we write then the equation of that plane, tangent plane, we were here at the point this Z and this is x0 and it is a 0 and z sub Z.

So the tangent plane would be the sign p times the gradient vector of f in weight that would be the equation of the tangent plane What is the other issue that they ask of us is the equation of the normal line the equation of the normal line so here we put the normal line it is assumed in that point p Sub Zero no then the equation of the normal line the equation of the normal line at that point we use the gradient as directional vectors so here it will be p equal to p sub 0 plus r times the gradient of f in p Sub Zero now there are the two situations if you realize all this is the problem It is only in the normal it does not come to the surface then I am going to do 12 12 examples suddenly I do not do all the calculations But the idea is that one is left with cartesian and or vector and the other that the surface is given implicitly Now I ask you this result that you have means that you can find a vector normal to a surface in this way it is a separate question let's see here she finished the blue You have to finish them, the question is the following: you have a surface, that one is already there and here you have Z equal to f of a capital F F of x and z So the gradient of that capital F that who is is f subx f sub ye and F sub Z therefore you would have like this bfx F sub y -1 that would be the normal the normal vector to this surface that South is Well, that is, it means that the functions of two variables, the normal to that surface, are very easy to find, that's why I didn't spend time there, even those who use partial derivatives arrive at that vector FC with this result there it Abarca even this result already involves the the surface given in this implicit way let's see a couple of examples Hopefully Let me Search here the two of the two types we are finding that maximum extreme values ​​no longer Abarca because as this is being recorded this We already have fewer topics, we write not but I can tell you and like when we coordinate this well in other institutions We coordinate with the teachers this Raise your hand they speak no Or we speak it has not shocked us as much as you are there because we are in exam something like that look let's make two examples about that and one that I'm hardly a saint of devotion of that type because it's more than anything it's calculations and knowing what doesn't have there are two types of problems one the types of problems are more or less at the same time cinema Ah, let it be clear that when you are going to talk about a tangent plane you need a function of three variables so that you can find the gradient of that function of three variables that is clear, however, in a tangent line from greater to the slope No, since Z depends of x and y because you need that vector on a tangent line with the greatest slope you need here the Z its x between the module of the gradient of Z you need a y between the module of the gradient of Z and the third component is the King's module That is to find the tangent line with the greatest slope, on the other hand, for us you need a function of three of three variables, there is the difference and there may be problems where you lack the point of tangency.

It can happen where the function is given explicitly. It can happen where the function is given. implicitly when you give implicitly you have to be careful with the point that you apply and with the point that gives us data we are going to see an example we propose it to fall asterisk there Determine I am not going to find the equation of the plane tangent or well now the equation well of the tangent plane to the surface s which is there is y says Y that contains the line and that contains the line l of is in explicit form agree is in explicit form So we have more or less the little problem like this we want to find the equation of a plane tangent to that surface is x and z is already here question this I can at some point say this this one is here now bothered equation 1 and up to that is the surface and you want to find the equation of the tangent plane to this surface here it is, we want to find the equation of that tangent plane the issue here is Z, we don't have the point Well, there is the zero we want to find the equation of the tangent plane to that surface then we can begin to say the function of three variables it could be this one here F of x and z it could be this x squared over four plus y squared over 3 plus Z squared minus 6 that's it so here we are going to have we the gradient of that capital gradient function at the social point p is already there and we want to find the equation of the tangent plane at a point x and z But what is the additional data that it gives that says that this plane contains a line there it contains that line this l That's it.

So if it contains L, it contains all its points in particular, it contains this point that I'm going to call, which is 10 2 root of 2 and here it has its direction vector, it contains those points of the line, let's call it a directional vector, so for we, the directional vector is this 8 - 1 0 and the equation of the tangent plane is already there, we have written it like this, not the equation of the tangent plane. It should be p sub zero p times the gradient of f in weights 0 ready, what would be missing to find that equation There Now they say that the surface equation is missing, it is explicit.

I need to calculate the gradient and the gradient and we can do the gradient, but the gradient. At what point are you going to evaluate it at point 0, the point is the one that is missing. there agree that of the point you are replacing it with the equation of the straight line that is, full If you have been told the tangent plane to the surface SS of equation 1 at point 11 such Zetas easy you calculate the gradient and evaluate it at that point then here the little problem is to find the zero weight and we are going to put it like this better we are going to put like this the zero weight of If we are going to understand each other here, first I can say that the p Sub Zero belongs to the surface so here we would have x observed squared over 4 plus 10 sub 0 squared over 3 plus Z Sub Zero squared that is equal to b we will need three data from agree we would need three data What is the other data we need the gradient we are not calculating we need the normal we are going to express it the normal of f at the point p Sub Zero would be its partial derivatives it would not be 4x over 4 no So we are going to write x sub 0 squared over 2 the other would be 2 10 sub 0 over 3 and the other would be 2 Z sub Z ready then the second condition could be so the vector a that we have made with red if we multiply it by the gradient What should not give us too Because that line is contained in the tangent plane, then its directional vector of the line has to be perpendicular to its normal, by definition of a plane, not a plane, it is also already there, so here we would have another equation, we would not have it, this one is 8 - 1 0 And that you multiply it by a half from It would be 2 x sub 0 minus two thirds of ye Sub Zero and there we have an equation So we already have two equations here the equation here that I'm going to call 2 the teacher there In the green that your zero over two not over four x0 in The gradient on the side of the beach in blue is fine, I'm green, no, yes, blue is fine, no.

It should be about two, so it will vary here. It is x sub 0 over 2 because it is two quarters, not over two, and here it would be 4, it would be. That's it, we would need an equation from which we could obtain the tangent plane. You can write it like this, otherwise x minus x sub 0 and minus 10 sub 0 and z minus Z Sub Zero is already for the gradient. Not here it would be about 3 and 2 Z Sub Zero equal to 0 the point a here the point a belongs to the tangent plane so here we are going to have 10 we have put no now 10 minus x sub 0 here It should be with two stars no 10 here it is 2 minus 10 sub 0 And this here is the root of 2 to rosettes And we multiply that by x0 over 2 2 10 sub 0 over 3 and 2 Z And that must be equal to 0 done So why does the matter practically end there because if we knew the zero weight We already have the normal vector okay.

So we already multiply by this here I'm going to put here four no's of 2, 3 and 4, you get the point p Sub Zero would already be solved Well I wouldn't put a lot of operation on it too long there you have to solve when in the in the third one that is with green if you give yourself account when you multiply you will have I'm going to calm down with the following. Let's see here. I think he came once. I'm going to ask him to do a calculation, but do it right. He says here. This is the little problem.

Here he says, let's see. Here he says here. do it from the front here you have the surface the surface s which is equal here Z equal to f of where f1 of X Sub Zero is equal to 1 F sub 2 of x0 is equal to minus 2 and F sub 3 of one one zero is coming then the questions look to begin with the point of tangency they are giving us there is in the general equation of theplanetangent to that at point 110 This is the here this here is the second partial derivative But of this function F sub 2 We agree and this is its third partial derivative this point x Sub Zero here Don't confuse it with the point that is on the surface x0 and is sub 0 and z Sub Zero is not Here I'm going to give you a simple p Sub Zero okay That's those components here are from this function here the big question is the following I want to calculate the vector perpendicular to this surface what is it I have to do I need a function in such a way that this is its level surface what did we do when a one appeared here we passed it to the other side what are we going to do the same What happens when we had 7 equal to f of x and what happened When we had set equal to f of at this moment it is not going to be the next one we are going to consider so be F of x y xz x squared y minus Z square minus Z We are going to call that an F for ready ready So ​​what do you have to calculate the gradient of F This goes to be F sub x F sub and F sub Z that is what we have to calculate that is going to be the vector that is the normal to the tangent plane that is perpendicular to the surface that they are giving us here ready now how do we calculate it we have already done already in the previous parts no so we do like this well then F of x and z if you like we do it like this F of A B and C minus Z where a is x times and B is x times Z and c let it be x squared and minus Z squared okay ready now you calculate that gradient of capital F okay and evaluate it taking into account that this F sub c is equal to 2 that F su a is equal to 1 just in case you already understand the suggestion then what are you going to do Hello, yes, I want to calculate.

I'm interested in letting you know so that she doesn't fall into God. I'm going to tell you the following. You know what the students did to this thing here: they calculated the gradient and said that this was the normal one. No, well, the equation is everything everything and you have to get a function in such a way that the surface is a level surface which is why you call everything by that name if you don't want Efe calls the example but the gradient of this one here is the norm a see there is in that that professor in the derivatives we equal it at the point about 110 no more Excuse me rather tell me That is the vector someone came up with that vector Hey someone came up with that vector when they derive with respect to a certain be careful not because you derive the F let's see very quickly here when we differentiate F with respect to of x and the other one which is minus Z which is 0 note that here the variables are independent just in case, for example here we have Efe su a for a with respect to X which is and plus F sub V B with respect to X which is zero too more F subcc with respect to be with respect to this is the derivative with respect to x of capital letter but with respect to Z it is F its a with respect to Z plus F goes up with respect to Z plus F sub Z Z Sorry, here that happens with the derivative of c with respect to Z minus 1 was minus 1 because there is a minus 7 there, everything is being derived well then we agree with the vector 5 3 - 3 someone no longer once we have that there now well we already have the equation of the tangent plane not the tangent plane and we have a normal which is in this case it is 5 3 and minus 3 and the point by the way We already have it there, which is 1110 and maybe we write Oh, it doesn't ask to generate So it would be 5x more yes I think the fs is also minus 7 I don't know if I was wrong we already do it So we do it here we do it we already do it here we do it here with with brown we also do it F sub Z is the same here is F su a the derivative of a with respect to Z plus F sub B the derivative of B with respect to Z Plus C the derivative of c with respect to Z no less 1 that's it F its a we leave it here a with respect to Z that that is zero no more F subpp with respect to Z which is x plus F sub c and C with respect to of Z would be would not be minus two Z minus 1 So now we replace no F sub Z is equal to f its a which is 1 Well this 0 f2 would be minus two times one no That is to say x is 1 and it is one and z is 0 so here it would be here plus this professor Well up there the gradient put one one one no the Z is 0 in Guatemala Yes yes yes Excuse me there another question if you make the change then also the fsx gradients also change if right now we correct that right now school First this one here yes then there is less 3 left yes we are here the F sue x no something like that the axis its x here would be here F subx would be F su a with respect to X plus F sub p with respect to x and this would be F is a c and C is already with respect to xy is not good and now we replace it we replace here F sub x It should be f1 which is one by one no more F sub 2 which is minus 2 by 0 no more this F sub C Which is 2 3 2 okay thanks Very good for observations That's it and there you get the but the the top here is the next thing that I sometimes ask my colleagues to do other types of problems what they do sometimes sometimes you have to do problems change it no for example here put 12 F of review some things and there they are ready for their partial exam and without extreme values.

When they review partial exams I am going to look for at least a couple of partial exams But that have to do with the in-person type and not with the blackboard anymore Because yes You are going to see 2020 one 2020 two this one you are going to be surprised that the exam is not complete because I gave it up to three parts I think that would be all for now I was thinking of giving it as a little message but I better did something tangent plane now additionally some are still missing tell me Abraham a question in the exam there was a question yesterday that in question 2 exactly asked for the rate of change of the height and volume when the radius of 6 and I interpreted it as asking you for the reason for change and volume But perhaps it could also have been that he asks you for the reason for the change in height and volume, but since that was not there, I interpreted it as volume and I don't know if which one you considered.

It seems to me that the teacher when He presented it to me, he wanted to actually have two problems there. I mean, first you find out the reason for the change in height, that's what I think with him with the lateral area that is constant, not because you had The radius is fine and you interpreted that for the volume. You don't see that in the volume you need both Of course H no, but it shouldn't have been the question and calculating the rate of change of height and volume doesn't appear so right now I don't have the question, I'm going to look at it and I'll It is going to happen to the teacher that he manages two scenarios and as best as possible.

In other words, he makes his rubric considering several scenarios according to the readings and that seems good to me and that he grades according to that, right, yes because when there is sometimes there is a reading, no. Well, sometimes not, but I understood that when he told me this, I even had the idea that he should give him the lateral area, the area, well, the formula and what happens if he doesn't remember that it is my power because of the generator, he is not going to be the problem there Well then you have to give it then there more or less you understood problems they want two parts It is that there were two two problems in one there one and with lateral area and the other with volume that was the problem Just in case okay Let's take them thank you for observing you but he should take advantage Excuse this question too I think they confused the fact that the lateral area was 60 I think in the end they corrected it with 60 pi Ah he's no longer studying no I don't know yet no they haven't asked the teacher yet Yes yes They corrected it Yes because if it was with 60 the height came out negative it would be with 60 pirgas all output would be 8 I think however Yes because sometimes there is no time to correct everything sometimes they have given me some problems which is why when you give them a job I will not change the function to change that is, I ask you to change the data but you solve it and interpret it, not as I say, sometimes I had some problems that it was about designing a Colosseum I don't know when until in the end a Colosseum came out that was in ellipse shape but I think it was 200 meters but the height was 20 kilometers But well now I'm going to see that there because I'm sure he's going to tell me that there has been a correction but I asked him the bad thing is that they said it on average hour after the exam In my case in my case I had started with question 2 and that was it already you say your Christmas gift not the same I don't know if I do believe that but in the midterm I hope there aren't many errors confirm more than so many errors There have been, but don't say so many errors too.

They forgot about centimeters per second, I think, but that wasn't so much another correction minutes later as about 10 minutes after the same problem, that is, after one correction, another correction arose. Well, I'm coming. to ask her to be careful and fair I took it for granted because that problem has come several times that problem came several times but with other information up to her everything is called to the other do not tell us that about self-esteem no no no no no no What has done it is that they should reach me and everything has been resolved.

I am very respectful of the student, that is, for me there should not be those types of errors, including that last question, the teacher who reached me, that they calculate the derivative at the zero point. zero I tell him because you are offending an album Now I'm going to talk to the teacher about the question that no no there shouldn't be that project I asked yesterday I called and I told him there has been a problem no that's why there hasn't been any problem the partial yes I'm going to be there because you can't say too much to your teacher either because sometimes you know without wanting to return it to the teachers and for me it doesn't.

In other words, we have to respect it and if they tell me that there have been corrections, it's a good time to take it. You You have seen that I get upset when you correct me, tell me well, that is, when you call me teacher, it says, What do I do? I'll solve it. It's not right, so it's just that it should be like that, well, I mean, no, no, no, you've made me think. I'm going to see how I make it as Solomonic as possible, for example, a point to everyone that was going away due to time, I'm going to see.

Stop that, let me see it like this, but I'm not going to say that you have noticed, just in case. We agree, no, we agree, normal, nothing more. What's happening is that I was calling that my mother-in-law had surgery yesterday, well it seems that things have gotten complicated. Acevedo Acevedo present teacher Stephanie carhuaca president Gustavo because present teacher Luis de la Cruz grows as a teacher Diego de la Cruz present teacher David devidosa Gabriel Figueroa then Roberto Carlos Florescente teacher Miguel García Miguel García Sandra Gil Vicente teacher Linda Huamán Jesús Juárez present teacher dyla Mejía Uriarte Edward I'm not out teacher Pablo César Are you present Luis Abel Padilla Alejandra Peralta president Romero president teacher Gonzalo Sánchez Alejandro Sanz William Sierra present teacher Gisele Silva Mateo Solórzano Edwin Soto is ready What does Solórzano say then we will not see this the next class Now thank you rather have a good shot you Thank you