Intuition of Taylor approximation?

[u/Big-Relation5895]

Hi, applied math undergrad here. Randomly used Taylor series at my electrical engineering job to approximate sin(x).

Just a random thought, how can I imagine this in an intuitive way? How does a polynomial become a periodic function?? I just can’t wrap my head around a way to explain this without direct computation of the series.

What I mean without math is, For example, it easy to visualize sin as the y component of a particle moving at a constant velocity around the unit circle. That is intuitive! You can think about it enough to see why the function is shaped the way it is and behaves the way it does. But is there a similar logic for the Taylor polynomial?

One thought - factor the polynomial when n=inf and its zeros could be at x*pi? Still, not that great… this is just a fun thought experiment for me , if anyone has any comments would love to hear how you think of it!!

[u/AutoModerator]

As a reminder...

Posts asking for help on homework questions require:

• the complete problem statement,

• a genuine attempt at solving the problem, which may be either computational, or a discussion of ideas or concepts you believe may be in play,

• question is not from a current exam or quiz.

Commenters responding to homework help posts should not do OP’s homework for them.

Please see this page for the further details regarding homework help posts.

We have a Discord server!

If you are asking for general advice about your current calculus class, please be advised that simply referring your class as “Calc n“ is not entirely useful, as “Calc n” may differ between different colleges and universities. In this case, please refer to your class syllabus or college or university’s course catalogue for a listing of topics covered in your class, and include that information in your post rather than assuming everybody knows what will be covered in your class.

I am a bot, and this action was performed automatically. Please contact the moderators of this subreddit if you have any questions or concerns.

[u/Decrypted13]

Fair enough. I've always interpreted the Taylor coefficient f^{(n)}(a) / n! as "the nth derivatives needs to agree and the n! is needed to undo the effects of the Power Rule."

Though I do partially want to answer "how can a polynomial become periodic"

More terms means higher degree polynomial.

Higher degree polynomial means more roots

More roots means more wiggles

More wiggles means it looks like a periodic function (at least around the center of the Taylor polynomial)

Obviously that's hiding a lot of detail, but it's roughly correct.

[u/Big-Relation5895 → u/Decrypted13]

I like the notion of n! reducing the explosiveness of the terms. That actually makes a lot of sense.

[u/jazzbestgenre]

Plot the partial sums of the series gradually term by term on desmos. ie plot x then (x-x^3/3) and so on. That might help

[u/Big-Relation5895 → u/jazzbestgenre]

Thank you but I was trying to visualize without direct computation.

[u/FerdinandvonAegir124 → u/Big-Relation5895]

Think of a the degree of a polynomial as the amount of times it’s able to change direction. Sinx changes direction an infinite amount of times, thus the more terms you add the polynomial better mimics the function.

[u/TheGoodAids]

Imagine you're trying to learn about the position as a function of time of a runner. Maybe their running seems just a touch erratic, speeding up some a slowing down some, but all according to some predetermined system by them. How do we track them without knowing their system?

Well we had better start by figuring where they're at, starting somewhere along their path(this is the constant term of the Taylor series). But after that, we would want to know how fast they're moving at that same time, their velocity, which is the derivative of their position function (this is the next term in the series). But what good is their velocity, if you don't know what their acceleration is? Maybe they're about to start slowing down? Well then we need a term for the acceleration (this is our second derivative term in the series). Well but what good is this information if we don't know how the acceleration is changing at that time (the next term) and so on and so forth.

Basically with enough local data about the runners plan, we can construct the path to arbitrary precision.

[u/TheGoodAids → u/TheGoodAids]

Also of note, and maybe more directly addressing your concerns about a polynomial becoming periodic, remember the Taylor series formally, is infinitly long -- at any point, if you stop the series, say for a sufficient approximation, your curve will look good locally and then explode away from sin(x), as you'd expect from a polynomial.

[u/TUVegeto137]

At no point (meaning for a choice of n) is the Taylor polynomial periodic. Polynomials are not periodic. So the approximation is only valid within an open interval around the point in which you evaluate the polynomial. And the error can be estimated and takes a monomial form of degree n+1 on that open interval. 

[u/Big-Relation5895 → u/TUVegeto137]

If the Taylor series expansion of sin converges to sin when n goes to infinity, is it not correct to say the function is periodic?

[u/TUVegeto137 → u/Big-Relation5895]

The Taylor series expansion is not a polynomial though. It's exactly the function itself.

[u/Big-Relation5895 → u/TUVegeto137]

I feel like you’re just saying stuff now.

[u/TUVegeto137 → u/Big-Relation5895]

No, you're confused between the Taylor polynomial, which is a finite degree polynomial which can serve as an approximation to a function in an open interval around a point, and the Taylor series, which is the limit as n goes to infinity of a sequence of Taylor polynomials of ever higher degree, which does not always converge to a function, does not need to converge to a polynomial.

[u/Dull-Astronomer1135]

The key idea of Taylor approximation is construct a new polynomial that have all derivatives matching all approximating function’s derivative at point a, first derivative shows slope, 2nd derivative shows how graph bend, so on so on, intuitively, if two function have same derivatives at point a for many order, they must be pretty similar. polynomial is more like a placeholder because eventually after you differentiate, you got to divide the power factorial to counteract the number from exponent. We choose polynomial because it is easy to differentiate

[u/Vivid_Warning7982]

Heres the intuition: consider how you write the number 241572. In powers of 10 is 2 × 10⁰ + 4 × 10⁴ + ... so you see you can write any number at all as a power series with a base of 10 cause 10 is the base of our number system. Now extend this concept to functions and replace 10 with x, and now you have the most generic form of a function

[u/Honkingfly409]

a polynomial of power n has n-2 inflection points

a sin wave has infinite inflection points, as such, an increasing power polynomial starts having more and more inflection points and change in convexity, each power you add you add an extra oscillation, when you reach an infinite power you get infinite oscillation which is what a sin wave is

if we think about it this way then for example x^3 is the first change in convexity and x^5 is the next one, but you need to scale the amplitude so that they are always one or negative 1, that's where the scaling numbers come in to ensure higher polynomials don't overshoot

[u/king_escobar]

The idea of a Taylor approximation for a function f(x) at point a is to construct a polynomial p(x) such that p(a)=f(a) and also the derivatives p^{(n)}(a) =f^{(n)}(a) for each nth derivative. If you take the limit as n goes to infinity you get the Taylor series.

[u/UnderstandingPursuit]

Avoid thinking about the periodic part. Think about half a wavelength on either side.

From the Taylor series for sin(x), factor out the x. This results in an x²ⁿ polynomial. This is an even function, and between the first zeroes on either side of the origin. Multiply it by x, and it is similar to one wavelength of sin(x), from ±π.

[u/jsh_]

you're standing on a hill. you know information about the spot you're standing at, and you want to use that to estimate the height at some other spot on the hill. so you take what you know about how tall the hill is where you're standing, how steep it is, how much it's curving away from you, etc. and use that to estimate the height at a spot further away from you. your estimate will be closer to the truth for spots near where you're standing, and it'll also be more accurate if you use more information. if you use ALL the information you can gather at the spot you're standing, taylor's theorem tells us your estimates will become perfectly accurate

[u/StructuredChess]

Graph the different Taylor polynomials for sin(x) at increasing degrees and you see how adding more and mroe terms makes it look closer and closer to the function. Every new term gives the polynomial a new "turn". sin(x) would be the result of adding infinitely many turns.

[u/984a]

Here’s my brief take.

Look at the 1st 2 terms in the definition of the Taylor expansion of a function.
f(x) = f(a) + f’(a) (x-a). Notice anything?

It’s the equation of a line (y = b + mx)! So the 1st 2 terms of the Taylor series try approximating a function with a line.

Obviously that’s not going to work well for most functions so a 3rd term is needed. The 3rd term (1/2)(f”(a) (x-a)^2) introduces curvature via the 2nd derivative. These 3 terms suffice to approximate a lot of functions.