Finding Taylor or Maclaurin series for a function
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Visualizing Taylor series approximations
I've talked a lot about using polynomials to approximate functions, but what I want to do in this video is actually show you that the approximation is actually happening. So right over here-- and I'm using WolframAlpha for this. It's a very cool website. You can do all sorts of crazy mathematical things on it. So WolframAlpha.com-- I got this copied and pasted from them. I met Steven Wolfram at a conference not too long ago. He said yes, you should definitely use WolframAlpha in your videos. And I said, great. I will. And so that's what I'm doing right here. And it's super useful, because what it does is-- and we could have calculated a lot of this on our own, or even done it on a graphic calculator. But you usually can do it just with one step on WolframAlpha-- is see how well we can approximate sine of x using-- you could call it a Maclaurin series expansion, or you could call it a Taylor series expansion-- at x is equal to 0 using more and more terms. And having a good feel for the fact that the more terms we add, the better it hugs the sine curve. So this over here in orange is sine of x. That should hopefully look fairly familiar to you. And in previous videos, we figured out what that Maclaurin expansion for sine of x is. And WolframAlpha does it for us as well. They actually calculate the factorials. 3 factorial is 6, 5 factorial is 120, so on and so forth. But what's interesting here is you can pick how many of the approximations you want to graph. And so what they did is if you want just one term of the approximation-- so if we didn't have this whole thing. If we just said that our polynomial is equal to x, what does that look like? Well, that's going to be this graph right over here. They tell us which term-- how many terms we used by how many dots there are right over here, which I think is pretty clever. So this right here, that is the function p of x is equal to x. And so it's a very rough approximation, although for sine of x, it doesn't do a bad job. It hugs the sine curve right over there. And then it starts to veer away from the sine curve again. You add another term. So if you have the x minus x to the third over 6. So now you have two terms in the expansion. Or I guess we should say we were up to the third-order term, because that's how their numbering the dots. Because they're not talking about the number of terms. They're talking about the order of the terms. So they have one dot here, because we have only one first-degree term. When we have two terms here, since we-- when you do the expansion for sine of x, it doesn't have a second-degree term. We now have a third-degree polynomial approximation. And so let's look at the third-degree. We should look for three dots. That's this curve right over here. So if you just have that first term, you just get that straight line. You add the negative x to the third over 6 to that x. You now get a curve that looks like this. And notice it starts hugging sine a little bit earlier. And it keeps hugging it a little bit later. So once again, just adding that second term does a pretty good job. It hugs the sine curve pretty well, especially around smaller numbers. You add another term. And now we're at an order five polynomial, right over here. So x minus x to the third over 6 plus x to the fifth over 120. So let's look for the five dots. So that's this one right over here-- one, two, three, four, five. So that's this curve right over here. And notice it begins hugging the line a little bit earlier than the magenta version, and it keeps hugging it a little bit longer. Then it flips back up like this. So it hugged it a little bit longer. And you can see I'll keep going. If you have all these first four terms, it gives us a seventh degree polynomial. So let's look for the seven dots over here. So they come in just like this. And then once again, it hugs the curve sooner than when we just had the first three terms. And it keeps hugging the curve all the way until here. And then the last one. If you have all of these terms up to x to the ninth, it does it even more. You start here. It hugs the curve longer than the others. And goes out. And if you think about it, it makes sense, because what's happening here is each successive term that we're adding to the expansion, they have a higher degree of x over a much, much, much, much larger number. So for small x value-- so when you're close to the origin for small x values, this denominator is going to overpower the numerator, especially when you're below 1. Because when you take something that has absolute value less than 1 to a power, you're actually shrinking it down. So when you're close to the origin, these latter terms don't matter much. So you're kind of not losing some of the precision of some of the earlier terms. When these tweaking terms come in, these come in when the numerator can start to overpower the denominator. So this last term, it starts to become relevant out here, where all of a sudden x to the ninth can overpower 362,880. And the same thing on the negative side. So hopefully this gives you a sense. We only have one, two, three, four, five terms here. Imagine what would happen if we had an infinite number of terms. I think you get a pretty good sense that it would kind of hug the sine curve out to infinity. So hopefully that makes you feel a little bit better about this. And for fun, you might want to go type in-- you can type in Taylor expansion at 0 and sine of x, or Maclaurin expansion or Maclaurin series for sine of x, cosine of x, e to the x, at WolframAlpha.com. And try it out for a bunch of different functions. And you can keep adding or taking away terms to see how well it hugs the curve.
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