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I have talked a lot about using polynomials to approximate

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functions. but I want to do is show you the

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approximation is actually happening.

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So right over here and I'm using Wolfram Alpha

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for this. It is a very cool website

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You can do all kinds of crazy mathamatical things

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on it. So wolframalpha.com

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and I got this copy paste. I met Steven Wolfram

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at a conference not too long ago.

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He said, yes, definitely use wolfram alpha in your

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videos and I said, Great ! I will, and so that is what I'm

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using right here and its super useful because what it

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does is - and we could have calculated all this on our own

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#
or even done it on a graphing calculator or we can

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do it with just one step on wolfram alpha -

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is see how well we can approximate

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sin of x using a Maclaurin series expansion

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or we can call it a Taylor series expansion at x = 0

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using more and more terms,

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having a good feel for the fact that

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the more terms we add

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the better it hugs the sine curve.

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So this over here in orange is sin of x.

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It should look fairly familiar to you

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and in previous videos we figured out

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what the Maclaurin expansion for sin of x is

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and wolfram alpha does it for us as well.

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They actually calculate the factorials for us.

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3 factorial is 6, 5 factorial is 120, and so forth.

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What's interesting here is, you can pick

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how many approximations you want to graph.

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So what they did is, if you want

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just one term of the approximation,

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if we just said that the whole polynomial is equal to x

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what does that look like?

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Well, that's going to be this graph right here.

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They tell us how many terms we use

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by how many dots there are,

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which I think is pretty clever.

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So this right here, that is a function p(x)

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p(x) = x