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I have talked a lot about using polynomials to approximate
functions. but I want to do is show you the
approximation is actually happening.
So right over here and I'm using Wolfram Alpha
for this. It is a very cool website
You can do all kinds of crazy mathamatical things
on it. So wolframalpha.com
and I got this copy paste. I met Steven Wolfram
at a conference not too long ago.
He said, yes, definitely use wolfram alpha in your
videos and I said, Great ! I will, and so that is what I'm
using right here and its super useful because what it
does is - and we could have calculated all this on our own
# or even done it on a graphing calculator or we can
do it with just one step on wolfram alpha -
is see how well we can approximate
sin of x using a Maclaurin series expansion
or we can call it a Taylor series expansion at x = 0
using more and more terms,
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 sin of x.
It should look fairly familiar to you
and in previous videos we figured out
what the Maclaurin expansion for sin of x is
and wolfram alpha does it for us as well.
They actually calculate the factorials for us.
3 factorial is 6, 5 factorial is 120, and so forth.
What's interesting here is, you can pick
how many approximations you want to graph.
So what they did is, if you want
just one term of the approximation,
if we just said that the whole polynomial is equal to x
what does that look like?
Well, that's going to be this graph right here.
They tell us how many terms we use
by how many dots there are,
which I think is pretty clever.
So this right here, that is a function p(x)
p(x) = x