 So, what we will do today is we will look at, I show you a demo, right, we will use hat functions and what we have seen so far is the intervals are important, the number of intervals that we have or the size of the interval is important for orthogonality and the fact that the intervals do not overlap is important for orthogonality, right, the size of the intervals so far did not matter, right, we took them as equal sizes, right, that is what we basically had. The number of intervals increase the accuracy with which we were able to represent the straight line for box function, if you think back to the example of the box functions, how well we were able to represent the straight line, it got better, the error reduces as the number of intervals that we use increases. So we will try to see if there is a relationship between the function that we are able to, that we want to represent, right, and the number of intervals, is that fine and I made use hat functions to do the demo, so this is a, so there will be a combination of things that we are going to get out of this demonstration. I am going to use a scripting language called Python, a particular version of it called IPython, just so that what comes on the screen does not sound strange to you. I am going to use two packages, I would suggest that you find for yourself some package with which you are able to plot, right, plot graphs and check them out on the screen, there are lots of packages out there. I am going to use a package called grace plot, which I will, this is as I said so that you understand as I pull up those packages what it means, what I am doing. And to represent arrays, I am going to use a package called numpy, numpy I guess for numeric Python, but anyway numpy, which is all that we basically need, right. So the objective is to see that if I take a function say for example sin x and I use 10 intervals on which to represent sin x, how well do I do it? And then we will change the function sin x, sin 2x, sin 3x, sin 4x to see whether it gets better, gets worse, you expect that maybe it gets better, gets worse, anything before we start? If I use 10 intervals and I try to use, I try to represent sin x as opposed or sin 2x or sin 3x, sin nx, you expect the representation to get better, representation to get worse, you expect the representation to get worse, fine, you expect the representation to get worse. And the reason why I pick trig function, actually numerous reasons why I pick sin x, one of them of course is that it is possible for me to scale it, I can change, there is an inherent length scale associated with sin x and by changing going to sin 2x, sin 3x, sin 4x, I am changing those length scales, okay. So let me start it off, the first time around I will set it up, maybe subsequent classes I would not go through as much of a setup, I am going through the full setup this time so that you see what I am doing, okay. So I am going to run I Python and I am going to set up my grace plot, initially it is going to be very intense, I will change the background color, I create the plot, it is going to be an intense white color, so I will quickly resize it and I have set up something that will give it a more not so bright, not so bright color, not so bright color, okay. Now what I will do is I will now get my numpy in, from numpy import star, you can go learn Python if you want to use this programming language, most of the stuff that I am doing I am not going to do anything unusual, I will create an array, so numpy has in it various things including definitions of pi, sin and so on, right. So I will create a row vector, peculiar way by which you do it going from a row vector whose values go from 2, 0 to 2 pi and I want 11 points that is 10 intervals, so if you say what have I managed to do, that is what I have managed to do, so I have x going from 0.628 so on to 26.28 which is 2 pi, right close to 2 pi. Now it is very simple, I create y which is sin of x and since my plotting utility is going to connect these points by piecewise straight lines which means that inherently underneath I am doing hat functions, I will just use it to plot, okay. Now if I were to use qubits, quadratics I have to be a little more careful, I cannot just use these straight lines, I have to be a little more careful, I have to actually plot the interpolating polynomial, right but in this case by coincidence it happens that it is going to join them by straight lines so I am using hat functions, okay. So I say g dot plot x, y and I get that, what do you say, are you happy with the function? You have to be a bit critical, is there any problem with this function? First of all it does not seem to reach the maximum, right, it does not reach 1, there is a small gap, there is a small gap, there is a small gap here, why does it not reach 1? Because I am not sampling the function at that point so I am using the word sampling here, I am not sampling the function at that point, I am not sampling the function at the peak, okay which is the reason why anything else? It looks reasonably like sin x, it has a reasonable semblance to sin x, okay, it has a reasonable semblance to sin x, maybe if we try, excuse me, higher number of grid points or whatever it is, maybe, right. So what about sin 2x, shall we try sin 2x? Sin of 2 star x, I do not have to keep typing this every time but anyway, what now? So is it any better, it is worse, it is worse, it really does not look like, I do not know, maybe the peaks also do not seem that great, the peaks also do not seem that great but actually if you look at it, it will turn out that the point where it crosses the 0s, the 0 crossings so to speak, these are the 0 crossings will turn out to be quite good, okay. On this graph of course, it is not that easy to find out but you can actually evaluate it and check it out and you will see that the 0 crossings are actually fine, okay, the point at which the function crosses the 0 is fine and the number of 0 crossings is fine, the number of peaks and valleys is fine, the number of peaks and troughs is fine, right. The function values seem okay, they are not great and with 10 intervals, this is not looking that good, so we will see what happens if I go to 3x, okay. So tell me what we have, I see a lot of smiles, tell me what we have, right. So here obviously that minima is gone, the minima and the maxima here are gone, okay and has it caught at least the frequency, does it seem as though it has caught the frequency is the number of 0 crossings that I am going to have, right divided by 2 in this case, so it is like this, so here to here, I can count, I can count, it looks like the frequency information has been picked up properly, the 0 crossings seem to have been picked up properly, amplitudes are totally off, okay. So now I am doing sin 3x with 10 intervals, okay, so there is clearly some relationship between the number of intervals that you are going to pick and say the underlying frequency or length scale that you are trying to represent, okay, so that is one of the things that we want out of this. Let us go ahead, I mean we will just try 4x and see what it does, sin 4x, now this is like nothing, I do not know, you tell me, I mean what do we have, do we have anything at all, do we have any, is the frequency information correct, is the number of 0 crossings correct, how many times does it cross 0, there is 1 here, 2 here, 3 here, 4 here, 5 here, 6 here, 7 here, 8 here, 9, 4x, that looks about right, okay. So in some sense the frequency content is there, right and the function representation, that is if I were to show you this, you would be hard pressed to say that oh it is actually sin of anything, sin of 4x or something of that sort, it is difficult to say that it is sin 4x, okay. Let us try the next one, what do you think will happen here, it has been getting bad, how bad can it get, how bad can it get, this is I should actually, this is, you have to look at the scale before you look at this, right, that is 10 power – 15, okay, 10 power – 15, just for the fun of it, just so that you can see the kind of stuff that you may have to do when you are running or exploring these programs, I will reset the scale here to – 1, plus 1 and that is what you get. So what happened, we sampled the functions exactly at the 0 crossings, now this is a bizarre situation, we picked up the 0 crossings exactly but we have nothing, right, we do not, the functions just totally disappear, we have not picked up the peaks anywhere, so we really have nothing, is that fine. So let us go on, we go on and see what we get, after all we have 10 intervals, 11 grid points, 11 points, 10 intervals, so what do you expect for 6, going to be the same as 4 is the prediction that I get, indeed it is the same as 4 or similar to 4, is it the same as 4, this is a sign change, this is a sign change, so you would expect and I think if all of you are familiar with this, I do not have to do all of, I do not have to go on, so if I go to 7, 7x, okay and we see that the pattern is repeated, the pattern is going to repeat, the pattern is going to repeat, this is what I expect, I go quickly to 8x and I plot that, right, 8x is getting better but it is not 8x, right, 8x is getting better but it is not 8x, 9x, what do we expect for 9, yes, a bit strange, it is clearly the sign is flipped and 10, 10 will be, 10 should be a straight line and of course I mean, I do not, I am not going to rescale it again but 10 should be a straight line but we will see if we get the same, actually we get the same value, it should be a straight line but remember we are dealing with numerics and yeah it is not quite the same but yes the scale is still 10 power – 15, so if I make it – 1 to plus 1, it will just be a straight line, the value is essentially 0, it is within machine epsilon, value is essentially 0, okay. So and if we go on what do you expect to happen, if I were to repeat this process what do you expect to happen, it is going to keep repeating, it is going to keep repeating, so the first lesson that we get out of here is if I have 10 intervals, if I have 11 grid points, there is a largest frequency that I will be able to represent with those grid points, right. If I have 11 grid points and I am using hat function, there is a largest frequency that I am going to be able to represent using those grid points, is that fine, does that make sense, okay, fine. Now we can try a different set of grid points just to see if we can infer anything else, there is a largest frequency, we had 10 intervals and the largest frequency was what, it was 4, so it looked like maybe right now you would say n by 2 – 1 or something of that sort. So let us pick some other number, let us say we take x to have going to 2 pi, how many shall we take, 21, you want to take 21, take 21, 21 grid points is about 20 intervals, it is 20 intervals, right. So we go through this business again y equals sign of x, check this out, 21 grid points is definitely better than, right, 21 grid points is definitely an improvement on, 11 grid points, 20 intervals is definitely an improvement on, right, 11, it looks like though it is pi, pi by 2, you know you are not going to actually sample it exactly at pi by 2 but we are quite close, we cannot make out that, so it looks like it is pick the peak well, the function representation is quite clean, okay. So this is one thing that we should bear in mind now, a lesson that we take from here that if you are representing a function, good idea sometimes either an engineering way to look at it would be to double the number of grid points to see whether there is an improvement, any improvement, this is something to bear in mind, right, it is something to bear in mind. I will, we will come back and visit this, that suggestion at the end of the semester, okay. So yeah, the function definitely looks a lot better, what about, what do you expect will have to happen to 2 star, 2 times 2x, this I would expect, this should be the same as sinx on the 10, right, so now it is very clear, this should be the same as sinx on the 10, right. So if I now go ahead, if I now go ahead, we will do a few of these and then I will sort of jump ahead after that, so if I do 3x, 3x of course has no equivalent there, right, so it seems to have picked the peak in some places, it has picked the peak in some places, it has lost the peak in some places, the 0 crossings are not there or I mean are good, otherwise yeah, you could say it is sin, I mean we can live with it, but I want you to bear this in mind every time, so if you are going to, if you are actually trying to represent a trigonometric function, right, it looks like, if you are going to represent sinx, it looks like if you want to capture the function well, right, it looks like you need at least 20 grid points, 21 grid points, 20 intervals, right, employing hat function, you need at least 20 intervals for a given wavelength, that is one lesson that you get out of this, you need at least 20 intervals per wavelength. So if I am talking and you say I want to model this in some fashion, right and you want to model it so that you pick up that signal, you actually pick up the pressure variations, then you need at least 20, you could live with 10, but 10 does not look that good, right, 10 does not look that good, okay. So let us try, 4x should again look familiar, 4x is again familiar, okay, but okay, let us skip ahead, which one shall we do, let us try 7x. So you can get, you can get, it is very clear that beyond a certain point the functions start looking very strange and they do not really look like sin x, sin 7x that you want, okay. That is the only reason why I am sort of bothering you with this, it also looks like, if you look carefully now, it looks like on top of this, if I look at the envelope over this, there is a sign kind of something like a sign that is riding on top, okay. So I am sorry about that, there is something like a sign riding on top of that, okay. Let us try 8x, 8x should be something that we have picked up earlier, 8x is something that we picked earlier, 8x corresponds to 4x that we had, however, of course the wave number is different, the frequency is different, normally when you do it in space, so when I am doing these, when I am, when we are talking about some of these kinds of problems you will see me use wave number and frequency in a sort of interchangeable fashion, okay, but normally when it is in space we say wave number when it is in time we say frequency, okay. So the wave number is of course different, but it looks very similar to 4x if you look at it on the 0 to pi interval and what happens to 9, why do I want to try 9, 9 should be the bad one, 9, yeah and it has that envelope again has that variation, so there is something riding on it, a sign x riding on it literally, okay. And what do you expect will happen with 10, 10 should be 0, each one has its own, it is 0 but each one because of the nature of the representation of the number itself on the computer, right, it should be 0 but each one is a little different and the magnitudes you can see that the maximum and minimum magnitudes do not necessarily see it to be the same and we get the impression that as the value as you go further and further down it is getting noisier and noisier that is it is closer, it is smaller value here, closer to the 0 value here and much worse at higher angles, okay, these are all observations that we can make just purely from looking at what we have got so far, right. So what do you expect will happen to 11, 11 will repeat, so it looks over, so this is for this reason 10 would be called a folding frequency, it looks as though the figures fold over, right, fold over in the sense that 9 is the same as 11 except for a sign, right, 8 is the same as 12 except for the sign, so it is as though we have folded, folded it over, right, we have folded it over that frequency and anything else that we can say, so if I give you n intervals or n plus 1 grid points, right, if I give you n plus 1 grid points then n by 2 seems to be the largest, n by 2 in fact seems to be the troublesome frequency, it is 0 there, so let me try a different set of grid points again, let us try something, we have for both the ones that we tried were odd, 21 grid points, 10 intervals, the number of intervals is even, why do not we see what happens when we change, okay, so the first question is do you expect any change, do you expect change, x equals you expect change, okay, how many shall I do, I do 20 or let us keep it at 10, that way we do not have to, of course the trouble with 10 is it would not look that good but it does not matter, that is not bad, the one improvement of 9 intervals over 10 intervals is that we pick the peak better, okay, it is not draw major conclusions from that, right, I mean it is possibly just happenstance that, it may just be chance that that is happen, the function does not look bad, right, the function does not look bad, what about 2x, again like last time we pick the peaks better, this peak is off, that peak is off, 3x, this is a nice sawtooth, we have picked the amplitude, right, we picked the frequency, right, the 0 crossings are right, it just is not sign, that is only problem, right, it got all that information, of course if you talk to your electrical engineering friends, they will tell you to convolve it with some sync function or something of that sort extract out the signal but it does not matter, right, for us as far as we are concerned we are not going to do complicated things like that, I just want the sample points and I just want to use linear interpolants to represent the function and it bothers me that this is what I am getting, okay, it bothers me that this is what I am getting, okay, 4x, give me a guess, 4x, what do you expect, do you expect it to be 0, okay, so it does make a difference whether it is even or odd, so the n by 2 where n is the number of intervals, the n by 2 seems to be significant, right, so at n by 2 if you have, if n is even, n by 2 is going to give me, pick up that 0 or put the other way, if you know you start off by saying that this is my frequency or wave number or whatever, you may make your choice as to what is the number of sample points that you are going to take, right, so you may take at least 2n sample points, 2n plus 1 sample points, right, you may sample the function at 2n plus 1 points at least, okay, is that fine, okay, so what do we have, what is the, so we have, what are the other conclusions that we can draw from this, pardon me, so frequency information is it lost here, frequency information is still there, the peaks are gone, right, so up till that point, so up till the point, so let me, so one of the big conclusions we can draw is that there is a largest frequency that can be represented, okay, there is a largest frequency that can be represented, so if you give me, I will call it a mesh or a grid, right, if you take an interval a, b you break it up into equal number, right, for now because that is all we have been doing, we cannot draw any other conclusion, equal number of intervals, right, n equal number of intervals, then n by 2 is sort of the highest frequency that can be represented, okay, I turn it the other way around for a given mesh, for a given mesh, whether something is high frequency or low frequency depends on the mesh, am I making sense, so if I were to pick a very large number, say 101 intervals, if I were to pick a very large number, right, so it comes out, sine of x comes out extremely smooth but like, I mean it is pretty obvious that I can actually represent say 250, I am sorry, say 25, 25x, sine of 25x, am I making sense, it is not a great representation, right, it is not a great representation but on 10 intervals I cannot pick it up at all, right, so it is obvious, right now what we have seen it is obvious on 10 intervals like but that is not, I want you to pick, I want you to, so when we talk, we talk about high frequencies and low frequencies, as we go along you will see that we will keep talking about the high wave numbers, low wave numbers, high frequencies, low frequencies, so the point that I want you to get out of this is, when you say a frequency is high, right, in our world when you are performing some computation and you are trying to determine a function, when you say a frequency is high, the frequency is high with reference to the underlying grid or the underlying mesh that you have used, you understand, so what is high frequency that is wave number 4, right, on a grid of intervals 10 is a relatively low frequency on a grid of 100 intervals, in fact we pick it extremely well, you understand, so the mesh will pick up frequencies, wave functions which are low with reference to the mesh, it will pick it up well and those that are high with reference to the mesh, it is not going to pick up as well, okay, okay, is that fine? So when I say, when we say high wave number, low wave number, when you are talking with respect to the, when you are talking with respect to a physical problem, you may have actually an understanding as to what is a high wave number, what is a low wave number, but when we are talking with respect to computation, when we say high wave number, low wave number, you have to ask for clarification saying what is a grid size, otherwise there is no sense unless I give you that other piece of information, what is high and low does not make sense, right, high and low or comparative terms, high with respect to what, low with respect to what, right, that information has to be given, is that fine, okay, are there any questions, okay, so I think I would suggest that you possibly can go on to trying out different functions, my suggestion is try out different functions with, right, with this, with hat functions as such, try out representing different functions with hat functions, I would also suggest that maybe you can try the quadratics and qubits, okay, and you may be surprised, so I will let you try quadratics and qubits yourself, then I will come back and maybe we will do a little demo again with that, is that fine, okay, are there any questions, okay, so there what I will do is I will stop my demo and I will go back to my regular, I will go back to my regular session of what we were doing earlier, what we had was we in the last class, right, is that fine, in the last class, is that fine, in the last class we saw that if we had or okay, why do not I continue first with, I will just say something a little something about this hat functions and so on and then maybe we will connect up to the last class, I just want to make a point here, so if I have a hat function, that is xi-1, that is xi, that is xi-1, that is x coordinate, I take one hat function, is that fine, I take one hat function and the question that I have is, we saw that we are able to represent the function reasonably well when we have sufficient number of intervals, in the last class we also saw that we may want to use differential, we may want to use Taylor series or something of that sort to represent derivatives, so then the question crops of how well can I represent derivatives here using hat function, okay, so we have looked at three classes of functions so far, we have looked at box functions, right, we have looked at box functions, we have looked at hat functions, okay, so representing derivatives with box functions, if you represent the function itself with the box function, then the box function is a constant, right, it is a constant on that interval, the derivative is always 0 in that interval, that does not work, okay, what about hat functions, hat functions, there is some hope, right, there is some hope but the only point is that the slope is a constant, the slope is a constant in the interval xi, xi-1, xi, okay, so the derivative you can represent the derivative and in fact if I were to take the derivative of the hat function that is centered at x, then what do I have, if I were to take the derivative of this hat function, let us restrict our interest to only xi-1 to xi, right, elsewhere, only in the support, elsewhere the function value is 0, so the derivative n prime, so if I have, let me, let me, let us say if you have f of x is fi ni summation over i, that is the representation, f sub i is the function f at xi at the grid points or the nodes, grid points or nodes, these are terms that you will see used for these points, grid points or nodes, so the derivative, if I represent the derivative as f prime x is in fact summation over i fi ni of x derivative, okay and as a consequence I am asking the question what is the derivative of, what is the derivative of n prime of some hat function n prime of i, okay, remember the function ni of x is 0 for x less than xi-1, ni of x is alpha of x for x, x in xi-1 to xi, 2 xi and 1-alpha of x for x in xi to xi-1 and 0 for x greater than or equal to xi-1, is that okay, so the derivative, the derivative of course as I said I am not really interested in these because these are 0 anyway but it is important they are 0, so n prime of i is 0 for x less than xi-1 equals alpha prime of x, you remember what alpha was, maybe I should remind you what alpha is, alpha of x is x-xi, x-1, xi-1, okay and I really need a subscript here, so if I make this i, either i and i-1 or whatever, I really need a subscript there, okay, so what is this going to turn out to be, 1 over xi-xi-1 and we call that h, right, h is like xi-xi-1 and otherwise it is minus 1 over h, sorry, h is like xi-xi-1, this is for x in xi-1, xi and this is for x in xi-xi-1 and it is 0 otherwise, so what does this turn out to be, what does the graph of this function turn out to be, the graph of that function, what is the graph of that function, is 1 over h on i-1 and i and minus 1 over h on i, i plus 1, okay, so given the relationship between this and the hat function, it looks like a box function but it is not quite the box function, right, it looks like a box function, it is not quite the box function, so we use the box function for convenience of this class for what we are doing and so on, right, we use the box function for that reason, so if you look here, we could take this interval, we could take another interval, they are orthogonal because the intervals were not overlapping, that was all very nice, right, for that purpose it was nice but it is very likely that often we will use the hat function, right, to do a representation because we are going to be solving differential equations and if you are going to be solving differential equations, it is nice to note that the derivative follows some function that we are using to represent other functions, right, so this may be preferable to the box function, it belongs to, it is named after h, it is called the h function, okay, it is called the h function, so it is clear that the derivative is a constant but it can be represented, what do we do at the node i, what is the derivative at the node i, there is a jump, is there a way I can estimate it, is there a way I can estimate the derivative at the node i, any suggestions, why do not I take these, so if I have a function that is varying in this fashion, it has a constant slope in this interval, constant slope in that interval, I know the slope on either side of this value, why not just take the average as a value here, maybe, maybe it works, let us see what we get if we do that, just for the fun of it, right, let us see what we get if we do that, so if this is fi-1, fi, fi-1, right, what is the slope of the first line segment here, what is the slope of that line segment there, fi-1, fi-fi-1 divided by h and what is the slope of the second one, fi-1-fi by h, I add the 2, I take the average and this seems to tell me that an estimate of the derivative f' at i is fi-1, fi-1 by 2h, sounds good, okay, so from this function it is possible almost everywhere that we will be able to get it, whether this is an acceptable value or not is something that we will have to figure out, okay, fine, right, so before we leave these h functions, let us get back here, before we leave these h functions, I just want to point out something, I want you to try out something, before I get back to Taylor series and so on, I want you to try out something, on the interval 0, 1 you consider a function that is a constant, just drawn this vertical line, it should be a light vertical line, consider a function that is constant and then consider, so I will call this h0, right and I will call something h1 of x, so up to 0.5 it has value 1, from 0.5 to 1 it has value of –1, so you try to find h0, h0, h1, h1, h0, h1, is that fine, okay, so far just to keep it familiar I have been calling these dot products, we should really call them inner products, okay, they are called inner products, you can continue to use the word dot product if you want, inner product, because we use terms inner product, scalar product, dot product, right, dot product, so you can and I call this h1 and I call that h0 because if you think about it, if you look at what is happening on the interval 0 to 0.5, it looks like h0 that has just been squeezed down to 0 to 0.5, so I can again define a small function there which is positive negative, right, I mean you can, so you can see that there is a hierarchy of functions that we can build on, if I define an h2, I would actually normally put subscripts also but it does not matter, so I can have something that is 1 up to 0.25, right and –1 up to 0.25 and actually I can define 2 of these functions, I can define a h2 subscript 1 that is here and I can actually translate that function to the right, right, this function just like, so it looks like we are getting a low frequency, high frequency, higher frequency, right, just like sin x sin 2 x sin 3 x, it looks like we are doing something like that, something very similar to that, is that fine, is that okay, right, so I just wanted you to be familiar, I just wanted you to get that idea, right now what we want is we will just tie up with the derivatives representation of function, representation of derivatives that is basically what we want because we want to solve differential equation, right. So in the last class, we saw that if I have, if I give you f at x that we could use Taylor series, prime indicates differentiation times delta x plus f double prime that we could use Taylor series to figure out what was the nature of the error that we are making, if we were to approximate f at x plus delta x as f of x, right, so in the last class we basically said well if you have a function value at some point, easiest thing for us to do is assume the same function value at a neighboring point, right, at an adjacent point or whatever. So that would be an approximation f of x plus delta x is approximately like f of x and what would be the error that we would make in that approximation, well we have an infinite series here and this infinite series we have basically truncated the infinite series in making this approximation, we have created what is called a truncation error, this is a repeat of what I did last time and the truncation error is typically indicated by the leading term. So the order of the truncation error is f prime of x times delta x and because the delta x has an exponent 1, right, this error is said to be first order. So basically as delta x goes to 0, as delta x goes to 0, this error goes to 0 in a linear fashion, okay, is that fine? What about a higher order representation, how can I get a higher order representation? I say f of x plus delta x equals f of x plus delta x times f prime of x plus higher order terms, delta x squared by 2 f double prime of x, I keep that plus higher order terms. It is clear that if I have the derivative which we saw when we did, when we used cubic, so if you have the derivative information it is possible for us to get higher order representation, right. So there is a connection between what we are doing there and what we are doing here but the trouble is what if we do not have the derivative, then we are in trouble, right. So if you have the derivative information, it is actually possible for you to look at what is, so if you say that this is my function value right now, this was my function value yesterday, right. So my bank account has so much money, yesterday my bank account has so much money, today then I can maybe predict what my bank account is going to get if I know the rate at which that value is changing, right. That is basically what it says, is that fine, okay. This also gives me a different clue. If I look at this, if I look at this equation, this equation tells me, right. So right now we have been looking at saying that what happens at x plus delta x, this equation tells me, so if I have f prime and if I have f of x, this equation basically tells me that I can predict f of x plus delta x, right in a little more, say earlier I was saying approximate but now I am saying, I have changed it around, I can actually predict. So if from a differential equation you are able to get the derivative value, you could actually predict what is going to happen at x plus delta x given the value at x and the derivative at x, fine, okay. That is one way to look at it. This is the argument we have been using so far. The other thing that you can get from this equation and once you have written this equation is you can write it as a way by which you can approximate f prime of x. What is f prime of x? f of x plus delta x minus f of x divided by delta x. That is the other possibility that we can actually evaluate the derivative based on the values at x and x plus delta x given that now you know f of x plus delta x, is that fine, okay. So we will see in the next class how to go about doing this, right and we will see what is the error. So we will now go about representing derivatives, how do we represent derivatives, how do we estimate derivatives, how do we represent derivatives, approximate derivatives and how do you find the error in, right, that we are making in that representation, fine, okay. I will see you in the next class.