Error or Remainder of a Taylor Polynomial Approximation Understanding the properties of the remainder or error function for an Nth degree Taylor approximation of a function
Error or Remainder of a Taylor Polynomial Approximation
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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...
- and maybe f of x looks something like that...
- and what I want to do is approximate f of x with a Taylor Polynomial 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
- You'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 at "a" should be equal to the derivatives of our function
- evaluated at "a". And that polynomial evaluated at "a" should also
- be equal to that function evaluated at "a".
- 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
- of big N there to say it's an nth degree approximation and sometimes
- you'll see something like this, something like N comma a to say it's an nth degree approximation centered a
- at a. Actually I'll write that right now... maybe we'll lose it
- if we have to keep writing it over and over, but you should assume
- that it's an nth degree polynomial centered at "a",
- and 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 two or two factorial, there's the same value)
- I'll write two factorial, you could write divided by one factorial over here if you like.
- And then plus 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 three 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 nth degree polynimal centered at "a",
- it's definitely f of a is going to be the same, or p of a is going to be the 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 zero, and you'll have p of a is equal to f of a, let me write that down
- : p of a is equal to f of a. And so it might look something like this.
- 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'll try my best to show what it might look like.
- And what I want to do in this video, since 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 this function as we move away
- from "a". So what I want to do is define a remainder function,
- or sometimes I've seen textbooks call it an error function.
- And I'm going to call this, hmm, just so you're consistent with
- all the different notations you might see in a book... some people will call this
- a remainder function for an nth degree polynomial centered at "a",
- sometimes you'll see this as an "error" function,
- but the "error" function is sometimes avoided because
- it looks like "expected value" from probability,
- but you'll see this often, this is e for error, r for remainder
- and sometimes they will also have the subscripts over there like that,
- and what we'll do is define this function to be 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 (doing the same colors), it's going to be
- f of x minus p of x. Where this is an nth degree polynomial
- centered at "a". So for example, if someone were to ask:
- or if you wanted to visualize, "what are they talking about":
- if they're saying the error of this nth degree polynomial centered at "a"
- 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,
- so the error of b is going to be f of b minus the polynomial at b.
- So, f of be there, the polynomial is right over there, so it will be
- this distance right over here. So if you measure the error at a,
- it would actually be zero, because the polynomial and the function
- are the same there. F of a is equal to p of a, so there error at "a" is equal to zero.
- Let me actually write that down, because it's an interesting property.
- It will help us bound it eventually, so let me write that. The error function at "a"
- , and for the rest of this video you can assume that I could write a subscript for the nth
- degree polynomial centered at "a". I'm just going to not write that every
- time just to save ourselves some writing. So the error at "a" is equal to
- 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
- , 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 zero
- , and we see that right over here. The distance between
- the two functions is zero there. Now let's think about something else.
- Let's think about what the derivative of the error function evaluated at "a" is.
- That's going to be the derivative of our function at "a" minus the first deriviative of our polynomial at "a".
- If we assume that 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".
- That's what makes it start to be a good approximation.
- But if you took a derivative here, this term right here will disappear,
- it will go to zero, I'll cross it out for now, this term right over here
- will be just f prime of "a", and then all of these other terms are going
- to be left with some type of an x minus a in them. And so when you
- evaluate it at "a" 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 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 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 and 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 polynomials, and this is going to be true
- all the way until the nth derivative of our polynomial is evaluated at "a",
- not everywhere, just evaluated at "a", is going to be equal to the nth
- derivative of our function evaluated at "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 the nth derivative of f evaluated at "a" minus
- the nth derivative of our polynomial evaluated at "a".
- And we already said that these are going to be equal to each other
- up to the nth derivative when we evaluate them at "a".
- So these are all going to be equal to zero. So this is an interesting property.
- but 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 trying to go with this video, and
- probably the next video
- We're going to bound it so we know how good of an estimate 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.
- Let's think about what happens when we take the (n+1)th derivative.
- What is the (n+1)th derivative of our error function. And not even
- if I'm just evaluating at "a". If I just say generally, the error function
- e of x... what's the n+1th derivative of it. Well, it's going to be the
- n+1th derivative of our function minus the n+1th derivative of...
- we're not just evaluating at "a" here either, let me write an x there...
- of our function... I'm literally just taking the n+1th derivative of
- both sides of this equation right over here.
- So it's literally the n+1th derivative of our function minus
- the n+1th derivative of our nth degree polynomial.
- The n+1th derivative of our nth degree polynomial.
- 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+1th derivative of an nth degree polynomial?
- If you want some hints, take the second derivative of y equal to x.
- It's a first degree polynomial... take the second derivative, you're going to get
- a zero. Take the 3rd derivative of y equal x squared.
- The first derivative is 2x, the second derivative is 2, the third derivative is zero.
- In general, if you take an n+1th derivative, of an nth degree polynomial,
- and you can prove it for yourself, you can even prove it generally,
- but I think it might make a little sense to you, it's going to be equal to zero.
- So this thing right here, this is an n+1th derivative of an nth degree polynomial.
- This is going to be equal to zero. So the n+1th derivative of our error function,
- or our remainder function you could call it, is equal to
- the n+1th derivative of our function. What we can continue in the next video,
- is figure out, at least can we bound this, and if we're able to bound this,
- if we're able to figure out an upper bound on its magnitude,
- actually what we want to do is bound its overall magnitude, to bound
- its absolute value.
- If we can determine that it is less than or equal to some value m...
- if we can actually bound it, maybe we can do a bit of calculus,
- we can keep integrating it, and maybe we can go back to
- the original function, and maybe we can bound that in some way.
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