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# Visualizing Taylor series approximations

The larger the degree of a Taylor polynomial, the better it approximates the function. See that in action with sin(x) and its Taylor polynomials. Created by Sal Khan.

## Want to join the conversation?

• So we can approximate any function using Taylor expanison and derive it's polynomial representation.

Is it possible to approximate any function using a formula of sine curves? •   yes. Look into the Fourier series. it has a lot of applications like electrical engineering. the wikipedia page has snapshots of how the sum of sine functions can add up to a sawtooth wave, square wave or whatever...
• I think I get how the Taylor/Maclaurin series works, so can somebody tell me if my thinking is correct:
The infinite series works because we build into the polynomial an infinite chain of derivatives of the original function. It gives the value of the function at a certain point, and also the rate the function is changing at that point (1st der.), and also the way that change is changing (2nd der.), and the way that the change that changes the rate of change is changing (3rd der.), ad infinitum, so that we essentially have a "seed" that gives us the entire function just from that point. Am I thinking along the right lines? • I found something quite interesting : If you take the Maclauren series of sin(x) with a finite polynomial, then whatever how small the coefficients are, for a very big x, the biggest power will overcome the others, and p(x) begin to be very big and is going really far away from sin(x). However, if we take only the first approximation, we get p(x)= 0, which is never further than 1 from sin(x) for any x. Isn't it counter-intuitive? • Maybe we can think of this as the cost of being more precise in the center of the function.

For example, imagine you are trying to get a wire to fit along ripples on a beach. Each time you put a bend in the wire (add a term to the polynomial) you cause the end of the wire to move further away from the surface of the sand ...

I'll stop this analogy now, before everyone gets too bent out of shape ...
• Why is it that by using the values of the function and its derivatives at zero, we can obtain an approximation of the function at values other than zero? Are the other values of the function somehow "encoded" in the behavior of the curve at x = 0? I can see how that would be the case for simple functions like lines and parabolas, but does that continue to hold for more complex functions? For arbitrary functions?

In a piecewise function, I can see that this method will fail (I imagine a function that is equal to sin(x) at zero but is something entirely different at other values of x), so there are clearly restrictions on the types of functions for which Taylor/MacLaurin series can be taken. Is it simply a requirement that the function be continuous and differentiable? • How come I cant find any videos about taylor series and taylor polynomials involving 2 variables f(x,y)? Im sure this has been covered, and some help to find these videos would be highly appreciated. If that is not the case, maybe requesting this is not such a bad idea. Thank you. • Look for information involving applying "Taylor's theorem" to "multivariate functions". I don't know if Sal has done any videos there but it would involve partial derivitives and gradients to approximate the surface. I'm not familair with this, but it's not immediately clear to me that you could approximate a function of N variables using polynomials (or some other N-dimensional building block --what?) but those are some search terms for you. Hope it helps.
• Can I have the link to the tools you used of this taylor series from Wolfram.com??? I found the site, but I couldn't find the link to the tools. Thank you in advance. • do i assume thst since i know Mac's series i now Taylor's series? do i just take my derivative at any point and i will arrive at the Taylor's approximate? • No, you just know the Taylor series at a specific point (also the Maclaurin series) or, to be more clear, each succeeding polynomial in the series will hug more and more of the function with the specified point that x equals being the one point that every single function touches (in the video above, x equals 0). The Taylor series is generalized to x equaling every single possible point in the function's domain. You can take this to mean a Maclaurin series that is applicable to every single point; sort of like having a general derivative of a function that you can use to find the derivative of any specific point you want. Check out "Generalized Taylor Series Approximation" for a better explanation.   