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Let's say that we have some function f of x, and let me graph an arbitrary f of x... that's my y axis, and that's my x axis...

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and maybe f of x looks something like that...

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and what I want to do is approximate f of x with a Taylor Polynomial centered around "x" is equal to "a"

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so this is the x axis, this is the y axis, so I want a Taylor Polynomial centered around there

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You've seen how this works; the Taylor Polynomial comes out of the idea

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that for all of the derivatives up to and including

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the degree of the polynomial, those derivatives of that polynomial

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evaluated at "a" should be equal to the derivatives of our function

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evaluated at "a". And that polynomial evaluated at "a" should also

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be equal to that function evaluated at "a".

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So our polynomial, our Taylor Polynomial approximation, would look something like this;

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So I'll call it p of x, and sometimes you might see a subscript

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of big N there to say it's an nth degree approximation and sometimes

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you'll see something like this, something like N comma a to say it's an nth degree approximation centered a

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at a. Actually I'll write that right now... maybe we'll lose it

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if we have to keep writing it over and over, but you should assume

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that it's an nth degree polynomial centered at "a",

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and it's going to look like this; it is going to be f of "a" plus

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f prime of a, f prime of a, times x minus a, plus f prime prime of "a"

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times x minus a squared over (either you could write two or two factorial, there's the same value)

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I'll write two factorial, you could write divided by one factorial over here if you like.

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And then plus go to the third derivative of f at a times x minus a to the third power,

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(I think you see where this is going) over three factorial,

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and you keep going, I'll go to this line right here, all the way

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to your nth degree term, which is the nth derivative of f evaluated at a

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times x minus a to the n over n factorial.

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And this polynomial right over here, this nth degree polynimal centered at "a",

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it's definitely f of a is going to be the same, or p of a is going to be the same thing

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as f of a, and you can verify that, because all of these other terms have

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an x minus a here, so if you put an a in the polynomial, all of these other

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terms are going to be zero, and you'll have p of a is equal to f of a, let me write that down

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: p of a is equal to f of a. And so it might look something like this.

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It's going to fit the curve better the more of these terms

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that we actually have. So it might look something like this.

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I'll try my best to show what it might look like.

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And what I want to do in this video, since this is all review,

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I have this polynomial that's approximating this function,

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the more terms I have the higher degree of this polynomial,

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the better that it will fit this curve the further that I get away from "a".

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But what I want to do in this video is think about, if we can bound

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how good it's fitting this function as we move away

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from "a". So what I want to do is define a remainder function,

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or sometimes I've seen textbooks call it an error function.

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And I'm going to call this, hmm, just so you're consistent with

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all the different notations you might see in a book... some people will call this

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a remainder function for an nth degree polynomial centered at "a",

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sometimes you'll see this as an "error" function,

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but the "error" function is sometimes avoided because

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it looks like "expected value" from probability,

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but you'll see this often, this is e for error, r for remainder

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and sometimes they will also have the subscripts over there like that,

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and what we'll do is define this function to be the difference between

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f of x and our approximation of f of x for any given x.

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So it's really just going to be (doing the same colors), it's going to be

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f of x minus p of x. Where this is an nth degree polynomial

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centered at "a". So for example, if someone were to ask:

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or if you wanted to visualize, "what are they talking about":

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if they're saying the error of this nth degree polynomial centered at "a"

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when we are at x is equal to b. What is this thing equal to,

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or how should you think about this. Well, if b is right over here,

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so the error of b is going to be f of b minus the polynomial at b.

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So, f of be there, the polynomial is right over there, so it will be

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this distance right over here. So if you measure the error at a,

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it would actually be zero, because the polynomial and the function

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are the same there. F of a is equal to p of a, so there error at "a" is equal to zero.

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Let me actually write that down, because it's an interesting property.

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It will help us bound it eventually, so let me write that. The error function at "a"

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, and for the rest of this video you can assume that I could write a subscript for the nth

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degree polynomial centered at "a". I'm just going to not write that every

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time just to save ourselves some writing. So the error at "a" is equal to

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f of a minus p of a, and once again I won't write the sub n and sub a, you can just assume it

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, this is an nth degree polynomial centered at "a",

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and these two things are equal to each other. So this is going to be equal to zero

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, and we see that right over here. The distance between

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the two functions is zero there. Now let's think about something else.

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Let's think about what the derivative of the error function evaluated at "a" is.

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That's going to be the derivative of our function at "a" minus the first deriviative of our polynomial at "a".

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If we assume that this is higher than degree one, we know that

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these derivatives are going to be the same at "a". You can try to take the first

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derivative here. If you take the first derivative of this whole

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mess, and this is actually why Taylor Polynomials are so useful,

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is that up to and including the degree of the polynomial,

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when you evaluate the derivatives of your polynomial at

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"a" they're going to be the same as the derivatives of the function at "a".

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That's what makes it start to be a good approximation.

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But if you took a derivative here, this term right here will disappear,

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it will go to zero, I'll cross it out for now, this term right over here

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will be just f prime of "a", and then all of these other terms are going

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to be left with some type of an x minus a in them. And so when you

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evaluate it at "a" all the terms with an x minus a disappear because

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you have an a minus a on them... this one already disappeared,

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and you're literally just left with p prime of a will equal to f prime of a.

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And we've seen that before. So let me write that.

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So because we know that p prime of a is equal to f prime of a

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when we evaluate the error function, the derivative of the error function at "a"

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that also is going to be equal to zero. And this general property

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right over here, is true up to and including n. So let me write this down.

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So, we already know that p of a is equal to f of a, we already know that

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p prime of a is equal to f prime of a, this really comes straight

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out of the definition of polynomials, and this is going to be true

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all the way until the nth derivative of our polynomial is evaluated at "a",

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not everywhere, just evaluated at "a", is going to be equal to the nth

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derivative of our function evaluated at "a".

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So what that tells us is that we could keep doing this with the error function

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all the way to the nth derivative of the error function evaluated at "a"

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is going to be equal to the nth derivative of f evaluated at "a" minus

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the nth derivative of our polynomial evaluated at "a".

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And we already said that these are going to be equal to each other

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up to the nth derivative when we evaluate them at "a".

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So these are all going to be equal to zero. So this is an interesting property.

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but it's also going to be useful when we start to try to bound this error function.

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And that's the whole point of where I'm trying to go with this video, and

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probably the next video

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We're going to bound it so we know how good of an estimate we have

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especially as we go further and further from where we are centered...

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from where our approximation is centered.

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Now let's think about when we take a derivative beyond that.

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Let's think about what happens when we take the (n+1)th derivative.

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What is the (n+1)th derivative of our error function. And not even

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if I'm just evaluating at "a". If I just say generally, the error function

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e of x... what's the n+1th derivative of it. Well, it's going to be the

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n+1th derivative of our function minus the n+1th derivative of...

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we're not just evaluating at "a" here either, let me write an x there...

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of our function... I'm literally just taking the n+1th derivative of

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both sides of this equation right over here.

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So it's literally the n+1th derivative of our function minus

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the n+1th derivative of our nth degree polynomial.

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The n+1th derivative of our nth degree polynomial.

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Once again, I could write an n here, I could write an a here to show

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it's an nth degree centered at "a".

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Now, what is the n+1th derivative of an nth degree polynomial?

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If you want some hints, take the second derivative of y equal to x.

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It's a first degree polynomial... take the second derivative, you're going to get

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a zero. Take the 3rd derivative of y equal x squared.

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The first derivative is 2x, the second derivative is 2, the third derivative is zero.

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In general, if you take an n+1th derivative, of an nth degree polynomial,

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and you can prove it for yourself, you can even prove it generally,

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but I think it might make a little sense to you, it's going to be equal to zero.

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So this thing right here, this is an n+1th derivative of an nth degree polynomial.

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This is going to be equal to zero. So the n+1th derivative of our error function,

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or our remainder function you could call it, is equal to

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the n+1th derivative of our function. What we can continue in the next video,

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is figure out, at least can we bound this, and if we're able to bound this,

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if we're able to figure out an upper bound on its magnitude,

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actually what we want to do is bound its overall magnitude, to bound

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its absolute value.

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If we can determine that it is less than or equal to some value m...

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if we can actually bound it, maybe we can do a bit of calculus,

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we can keep integrating it, and maybe we can go back to

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the original function, and maybe we can bound that in some way.

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If we do know some type of bound like this over here, so I'll take that up in the next video.