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# Taylor polynomial remainder (part 1)

AP.CALC:
LIM‑8 (EU)
,
LIM‑8.C (LO)
,
LIM‑8.C.1 (EK)

## Video transcript

let's say that we have some function f of X right over here and let me graph an arbitrary f of X so that's my y-axis that is my x-axis and maybe f of X looks something like that and what I want to do is I want to approximate f of X with a Taylor polynomial centered around centered around X is equal to a so this is the x axis this is the y axis so I want a Taylor polynomial centered around there and we've seen how this works the Taylor polynomial comes out of the idea that for all of the derivatives up to and including the degree of the polynomial those derivatives of that polynomial evaluated a should be equal to the derivatives of our function evaluated a and that polynomial evaluated a should also be equal to that function of value today so our polynomial our Taylor polynomial approximation would look something like this so I'll call it P of X and sometimes you might see a subscript a big end there to say it's an nth degree approximation and sometimes you'll see something like this sometimes you'll see something like n comma a to say it's an nth degree approximation centered at a actually I'll write that right now maybe we might lose it if we have to keep writing it over and over but you should assume that it is an nth degree polynomial centered at a it's going to look like this it is going to be F of a plus F prime of a F prime of a times X minus a plus F prime prime of a times X minus a squared over either you could write 2 or 2 factorial they're the same value I'll write 2 factorial you could write a divided by 1 factorial over here if you like and then + go to the third derivative of F at a times X minus a to the third power I think you see where this is going over 3 factorial and you keep going I'll go to this line right here all the way to your nth degree term which is the nth derivative of F evaluated at a times X minus a to the N over N factorial and this polynomial right over here this and degree polynomial Center today it's definitely F of a is going to be the same or F or P of a is going to be same thing as F of a and you can verify that because all of these other terms have an X minus a here so if you put an A in the polynomial all of these other terms are going to be 0 you'll have P of a is equal to f of a let me write that down P of a is equal to P of a is equal to F of a and so it might look something like this and it's going to fit the curve better the more of these terms that we actually have so it might look something like this I'm try my best try my best to to show what it might look like and what I want to do in this video so this is all review I have this polynomial that's approximating this function the more terms I have the higher degree of this polynomial the better that it will fit this curve the further that I get away from a but what I want to do in this video is think about if we can bound how good it's fitting how good it's fitting this function as we move away from a so what I want to do is define a remainder function or sometimes I've seen some textbooks call it an error function and I'm going to call this let me call this let me call this I'll just call it an error so let me just just so you're consistent with all the different notations you might see in a book some people will call this a remainder function and sometimes they'll write a remainder function for an nth degree polynomial centered at a sometimes you'll see this as an error function some the error function is sometimes avoid it because it looks like expected value from probability but so you'll see this often this is e for error E for error R for remainder and sometimes they'll also have the subscripts over there like that and what we'll do is we'll just define this function to be the difference between the difference between f of X and our approximation of f of X for any given X so it's really just going to be in the same colors it's going to be f of X minus minus P of X minus P of X where this is an nth degree polynomial and degree polynomial centered at a so for example if someone were to ask you or if you wanted to visualize what what are they talking about if they're saying the error of this nth degree polynomial senator today when we are at X is equal to B what is this thing equal to or how should you think about this well if B is right over here if B is right over here so the error of B is going to be F of B minus the polynomial at B so f of B they're the polynomials right over there so it'll be this distance right over here so if you measure the error at a if you measure the air at a it would actually be 0 because the polynomial and the function are the same they're f of a is equal to P of a so the error a is equal to zero let me actually write that down because that's an interesting property and will help us bound it eventually so let me write that error the error function at a and for the rest of this video you can assume that I that I could write a subscript this is for the nth degree polynomial centered at a I'm just going to not write that every time just to save ourselves a little bit of time in writing to keep my my hand fresh so the air at a is equal to F of a minus P of a and once again I won't write the sub n sub a you can assume it this is an nth degree polynomial centered at a and these two things are equal to each other so this is going to be equal to 0 and we see that right over here the distance between the two functions is 0 there now let's think about something else let's think about let's think about what the derivative let's think about what the derivative of the error function evaluated a is well that's going to be the derivative of our function at a - the derivative of our the first derivative of our polynomial at a and if we assume that this has more than if this is higher than degree one we know that these derivatives are going to be the same at a you can try to take the first derivative here if you take the first derivative of this whole mess and this is actually why Taylor polynomials are so useful is that up to and including the degree of the polynomial when you evaluate the derivatives of your polynomial at a they're going to be the same as the derivatives of the function at a and that's what starts to make it a good approximation but if you took a derivative here this term right here will disappear it'll go to zero I'll cross it out for now this term right over here we'll just be f prime of a and then all of these other terms are going to be left with some type of an X my it's a in them and so when you evaluated a day all the terms with an X minus a disappear because you have an a minus a on them this one already disappeared and you're literally just left with P of a P prime of a will equal to F prime of a and we've seen that before so let me write that so because we know that P prime of a is equal to F prime of a when we evaluate when you evaluate the error function the derivative of the error function at a that also is going to be equal to zero and this general property right over here is true up to an including n so let me write this down so we already know that P of a is equal to f of a we already know that P prime of a is equal to F prime of a this really comes straight out of the definition of the Taylor polynomials and this is going to be true all the way until the nth derivative of our polynomial is going evaluated at a not everywhere just evaluated a is going to be equal to the nth derivative of our function of our function evaluated a so what that tells us is that we could keep doing this with the error function all the way to the nth derivative of the error function evaluated at a is going to be equal to well that's just going to be the nth derivative of F evaluated at a minus the nth derivative of our polynomial evaluated a and we already said that these are going to be equal to each other up to the nth derivative when we evaluate am a day so these are all going to be equal to zero so this is an interesting property it's also going to be useful when we start to try to bound this error function and that's the whole point of where I'm going with this video and probably the next video is we're going to try to bound it so we know how good of an estimate we are we have especially as we go further and further from where we are centered from where our approximation is centered now let's think about when we take a derivative beyond that so let's think about what happens when we take the n plus 1 derivative so let me what's a good place to write what can I have some screen real estate right over here so what happens if I take the what is the n plus 1 derivative of our of our error function and not even if I'm just evaluated a if I just say generally the error function e of X what's the n plus 1 derivative of it well it's going to be the end post want the derivative of our function is going to be the n plus one derivative of our function minus the n plus one derivative minus the n plus one derivative of our we're not just evaluated a he or either let me write it X there or our function - I'm just I'm literally just taking the the n plus one derivative of both sides of this equation right over here so it's literally the n plus one third derivative of our function minus the n plus one third derivative of our nth degree polynomial of our nth degree polynomial the n plus one derivative of our nth degree polynomial I could once again I could write an N here I could write an A here to show it's an nth degree centered at a now what is the n plus one derivative of an nth degree polynomial and if you if you if you want some hints take the second derivative of take the second derivative of Y is equal to X it's a first degree polynomial take the second derivative you're going to get zero take the take the third derivative of Y is equal to x squared the first derivative is 2x the second derivative is two the third derivative is zero in general if you take an N plus 1 derivative of an nth degree polynomial and you could prove it for yourself you could even prove it generally but I think it might make a little sense to you it's going to be equal to 0 it is going to be equal to 0 so this thing right here this is an N plus 1 derivative of an nth degree polynomial this is going this is going to be equal to 0 so the so the N plus 1 let me write this over here the n plus 1 derivative of our error function or a remainder function we could call it is equal to the N plus 1 is equal to the n plus 1 third derivative of our function and so what we could do now and we'll probably have to continue this in the next video is figure out at least can we bound this can we bound this and if we are able to bound this if we're able to figure out an upper bound on its magnitude so actually what we want to do is we want to bound its overall magnitude we want to bound it's absolute value if we can determine that it is less than or equal to some value M so if we can actually bound it maybe we can do a little bit of calculus we could keep integrating it and maybe we can go back to the original function and bound that in some way if we do know some type of bound like this over here so I'll take that up in the next video
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