Maclaurin approximations

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In summary: I'm sorry, I didn't quite understand your idea. Can you please summarize it for me?In summary, the conversation discusses whether or not it is possible to use Maclaurin polynomial approximations for a function f(x) that is known to be differentiable twice. The use of Lagrange and Cauchy reminder theorem is mentioned, as well as the requirement for a function to be infinitely differentiable for a full Taylor series. The difficulty in determining a useful expression for the remainder is also discussed, and the idea of using integration by parts to determine the remainder is suggested. The conversation ends with a question about the fundamental problem being addressed.
  • #1
estro
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Suppose f(x) is differentiable 2 times, can I still use Maclaurin polynomial approximations and write:

[tex] f(x)=f(0)+f'(0)x+\frac {f''(0)x^2} {2!}+R_2(x) [/tex]

If yes why? (Lagrange and Cauchy reminder theorem are using in this case the 3'rd order derivative...)
 
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  • #3
Are you sure about that?
I remember reading proof for a theoretical problem using this concept. (Can't remember where it was).

I hope I'll find it tomorrow and post it here.
Thanks for the quick reply!
 
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  • #4
I think if you do it on a polynomial, you will just get back the polynomial.
 
  • #5
rock.freak667 said:
I think if you do it on a polynomial, you will just get back the polynomial.
Yes, I understand and know it.

I asked this question from theoretical point of view.
f(x) is unknown function we only have the numerical information about f(0), f'(0) and f''(0).
 
  • #6
estro said:
Yes, I understand and know it.

I asked this question from theoretical point of view.
f(x) is unknown function we only have the numerical information about f(0), f'(0) and f''(0).

Yes I don't see why you can't use the series if you have those values.
 
  • #7
rock.freak667 said:
Yes I don't see why you can't use the series if you have those values.

But [tex]R_2(x)[/tex] is defined using [tex] f^{(3)} (x)[/tex] And we don't know if this function exists. [Lagrange Reminder Theorem]
 
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  • #8
estro said:
Suppose f(x) is differentiable 2 times, can I still use Maclaurin polynomial approximations and write:

[tex] f(x)=f(0)+f'(0)x+\frac {f''(0)x^2} {2!}+R_2(x) [/tex]

If yes why? (Lagrange and Cauchy reminder theorem are using in this case the 3'rd order derivative...)

You can always do that.
The difficulty comes the determination of a useful expression for [tex] R_2(x)[/tex]. Probably the easiest way to determine such an expression is the integral form of the remainder which comes from integrating by parts repeatedly.The result is an expression in whch the remainder involves the integral of the third derivative. So you need at least three times differentiability to get a McClaurin polynomial of degree two with the usual integral form of the remainder.

[tex]f(x)-f(0)=\int_{0}^{x}f'(t)dt[/tex]

= [tex] x f(0) - \int_{0}^{x}f''(t)(x-t) dt[/tex]
.
.
.
= [tex] f(0) + f'(0)x + 1/2f''(0)x^2 + 1/2 \int_{0}^{x} f'''(t)(t-x)^2dt[/tex]

Please excuse the sloppy Tex. I seem to be getting no correlation between the code, the preview and the post. The first expressions should be the remainder, not f and the expressions for the integrals are pretty much a mess. But I hope you can see through this disaster and recreate the proof.
 
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  • #9
DrRocket said:
You can always do that.
The difficulty comes the determination of a useful expression for [tex] R_2(x)[/tex]. Probably the easiest way to determine such an expression is the integral form of the remainder which comes from integrating by parts repeatedly.The result is an expression in whch the remainder involves the integral of the third derivative. So you need at least three times differentiability to get a McClaurin polynomial of degree two with the usual integral form of the remainder.

[tex]f(x)-f(0)=\int_{0}^{x}f'(t)dt[/tex]

= [tex] x f(0) - \int_{0}^{x}f''(t)(x-t) dt[/tex]
.
.
.
= [tex] f(0) + f'(0)x + 1/2f''(0)x^2 + 1/2 \int_{0}^{x} f'''(t)(t-x)^2dt[/tex]

Please excuse the sloppy Tex. I seem to be getting no correlation between the code, the preview and the post. The first expressions should be the remainder, not f and the expressions for the integrals are pretty much a mess. But I hope you can see through this disaster and recreate the proof.

I'm not sure I understood your idea.
Basically if I knew that f(x) is differentiable 3 times, I would use Lagrange Reminder Theorem to determine [tex]R_2(x)[/tex]. The problem arise what to do when I don't know such thing.
 
  • #10
estro said:
I'm not sure I understood your idea.
Basically if I knew that f(x) is differentiable 3 times, I would use Lagrange Reminder Theorem to determine [tex]R_2(x)[/tex]. The problem arise what to do when I don't know such thing.

The idea is that first, you can write any function as a quadratic polynomial plus a remainder. That is trivial, since the remainder is just the difference between the function and the polynomial. A Maclaurin expansion is useful only because there is a useful expression for the remainder -- which is "small" and therefore makes the approximation by the polynomial "close".

The problem is really one of finding a useful expression for the remainder. There are various forms for that remainder. What I showed (as best I could with problem that I was having with the Tex software) is a simple way to get the remainder by integrating by parts a couple of times. The process works iteratively and will produce a Maclaurin expansion of any degree that you like, so long as you can keep integrating by parts. That requires some level of differentiability -- it requires n+1 derivatives for an nth degree polynomial.

If you know nothiing whatever about the differentiability of the function then you are basically out of luck. That is because you have no good way to estimate the remainder.

Now, if you know something else about the function then depending on what that is one might be able to do something else -- I am not sure what at the moment.

Someone noted that a full blown Taylor series requires infinite differentiability. That is also true. In fact it requires more, It requires the function be not only infinitely differentiable but actually analytic, a much more restrictive condition. The point being that to approximate a function by a polynomial requires that you know something about the function.

What is the fundamental problem that you are trying to solve ?
 
  • #11
DrRocket said:
The idea is that first, you can write any function as a quadratic polynomial plus a remainder. That is trivial, since the remainder is just the difference between the function and the polynomial. A Maclaurin expansion is useful only because there is a useful expression for the remainder -- which is "small" and therefore makes the approximation by the polynomial "close".

The problem is really one of finding a useful expression for the remainder. There are various forms for that remainder. What I showed (as best I could with problem that I was having with the Tex software) is a simple way to get the remainder by integrating by parts a couple of times. The process works iteratively and will produce a Maclaurin expansion of any degree that you like, so long as you can keep integrating by parts. That requires some level of differentiability -- it requires n+1 derivatives for an nth degree polynomial.

If you know nothiing whatever about the differentiability of the function then you are basically out of luck. That is because you have no good way to estimate the remainder.

Now, if you know something else about the function then depending on what that is one might be able to do something else -- I am not sure what at the moment.

Someone noted that a full blown Taylor series requires infinite differentiability. That is also true. In fact it requires more, It requires the function be not only infinitely differentiable but actually analytic, a much more restrictive condition. The point being that to approximate a function by a polynomial requires that you know something about the function.

What is the fundamental problem that you are trying to solve ?

Oh, I think I get it now, and you answered the next question I was about to ask.

I don't have any fundamental problem I'm trying to solve, just want to get idea and intuition with Taylor approximations and the Reminder Theorem of Lagrange.

I think because of this "analytic" thing I have so much trouble feeling this idea with intuition. (I don't have problems solving exercises in this topic in my book, But I don't "feel" this topic to the extend I want)

Thank you very much for your detailed answer.
I will find myself over the moon if you can refer me to a good source about this topic.
(I'm self learner student with Calc 1-2 knowlage)
 
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  • #12
estro said:
Oh, I think I get it now, and you answered the next question I was about to ask.

I don't have any fundamental problem I'm trying to solve, just want to get idea and intuition with Taylor approximations and the Reminder Theorem of Lagrange.

I think because of this "analytic" thing I have so much trouble feeling this idea with intuition. (I don't have problems solving exercises in this topic in my book, But I don't "feel" this topic to the extend I wanted to)

Thank you very much for your detailed answer.
I will find myself over the moon if you can send me to a good source about this topic.
(I'm self learner student with Calc 1-2 knowlage)

The first source is your calculus book, assuming that I understand what Calc 1-2 are.

I am not a big fan of most calculus books, but Spivak's book is pretty good.

The next level would be an introductory text on real analysis. There a couple that I like. Those are Elements of Real Analysis by Bartle and Principles of Mathematical Analysis by Rudin. These are generally considered junior-senior level books for mathematics majors.

Analytic functions are usually encountered in a class on complex analysis. It turns out that if a complex valued function is differentiable even one time, then it is differentiable infinitely often and in fact is analytic -- locally representable by a power series. So to see this you would need to read a book on complex analysis. There are several good texts on this subject, but a classic is Complex Analysis by Ahlfors. But to read it you may need a bit more than just Calc 1-2. It is considered at advanced undergraduate or beginnin graduate level book.

As far as the proof that I tried to sketch, I am sure that I have seen it in a book somewhere, but most texts take a slightly different tack and I am not sure of a ready reference off the top of my head. But in any case it is not difficult to work out for yourself -- just fill in the small blanks in what I posted (It is just integration by parts).


Basically what you get is a function minus some polynomial as the remainder, and at that point you have to know something to limit the size of the remainder. Knowing something about its differentiabillity gives you some knowledge of that sort. Without any knowledge at all, anything can happen. So the important part of such approximation theorems revolves around showing how some piece of information regarding the function limits the size of the remainder.
 
  • #13
DrRocket said:
The first source is your calculus book, assuming that I understand what Calc 1-2 are.
I'm from Israel, Calculus here is some sort of mix between "Real Analysis and "Calculus 2" tending heavily towards "Calculus 2 [In US]"

I am not a big fan of most calculus books, but Spivak's book is pretty good.
I'm too, they tend to concentrate more on solving very similar problems rather understanding theory.

The next level would be an introductory text on real analysis. There a couple that I like. Those are Elements of Real Analysis by Bartle and Principles of Mathematical Analysis by Rudin. These are generally considered junior-senior level books for mathematics majors.
I'll try to get my hands on one of these.

Analytic functions are usually encountered in a class on complex analysis. It turns out that if a complex valued function is differentiable even one time, then it is differentiable infinitely often and in fact is analytic -- locally representable by a power series. So to see this you would need to read a book on complex analysis. There are several good texts on this subject, but a classic is Complex Analysis by Ahlfors. But to read it you may need a bit more than just Calc 1-2. It is considered at advanced undergraduate or beginnin graduate level book.

I have very long way to Complex Analysis.

As far as the proof that I tried to sketch, I am sure that I have seen it in a book somewhere, but most texts take a slightly different tack and I am not sure of a ready reference off the top of my head. But in any case it is not difficult to work out for yourself -- just fill in the small blanks in what I posted (It is just integration by parts).

I was never exposed to this approach, but I think now I get the idea and this is quite trivial.

Basically what you get is a function minus some polynomial as the remainder, and at that point you have to know something to limit the size of the remainder. Knowing something about its differentiabillity gives you some knowledge of that sort. Without any knowledge at all, anything can happen. So the important part of such approximation theorems revolves around showing how some piece of information regarding the function limits the size of the remainder.

About this I have to think more.

Thank you!
 
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What are Maclaurin approximations?

Maclaurin approximations are a mathematical tool used to approximate a function with a polynomial of a certain degree. They are named after Scottish mathematician Colin Maclaurin and are a special case of Taylor series expansions, where the center of expansion is at x=0.

How are Maclaurin approximations calculated?

Maclaurin approximations are calculated using the formula f(x)=f(0)+f'(0)x+f''(0)x^2/2!+f'''(0)x^3/3!+...+f(n)(0)x^n/n!, where f'(0), f''(0), etc. represent the derivatives of the function f at x=0. The more terms included in the approximation, the more accurate it will be.

What is the purpose of using Maclaurin approximations?

Maclaurin approximations are useful in many areas of science and engineering where it is necessary to approximate a complex function with a simpler one. They can help simplify calculations and provide insight into the behavior of a function near a specific point.

How accurate are Maclaurin approximations?

The accuracy of a Maclaurin approximation depends on the number of terms used and the behavior of the function near x=0. In general, the more terms included, the more accurate the approximation will be. However, for some functions, the Maclaurin series may not converge to the original function.

Can Maclaurin approximations be used for any function?

Maclaurin approximations can be used for any function that is infinitely differentiable at x=0. However, for some functions, the Maclaurin series may only converge within a certain interval and may not be valid for all values of x. It is important to check the convergence of the series before using it for a particular function.

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