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In the last video, we hopefully set up some of the intuition for why - or I should say what - the Maclaurin

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series is all about, and I said at the end of the videos that a Maclaurin series is just a special case

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of a Taylor series. In the case of a Maclaurin series, we're approximating

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this function around x is equal to 0, and a Taylor series, and we'll talk about that in a future video,

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you can pick an arbitrary x value - or f(x) value, we should say, around which to approximate the function.

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But with that said, let's just focus on Maclaurin, becuase to some degree it's a little bit simpler,

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and that by itself can lead us to some pretty profound conclusions about mathematics,

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and that's actually where I'm trying to get to.

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So let's take the Maclurin series of some interesting functions

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and I'm gonna do functions where it's pretty easy to take the derivatives, and you can

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/keep/ taking their derivatives over and over and over and over and over again.

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So let's take the Maclaurin series of cosine of x, so if f(x)=cos(x),

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then - before I even apply this formula, that we somewhat derived in the last video,

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or at least got the intuitive for in the last video -

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let's take a bunch of derivatives of f(x), just so we have a good sense of it.

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So, if we take the first derivative, if we take the first derivative, derivative of cos(x) = -sin(x)

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if we take the derivative of that, if we take the derivative of that,

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derivative of sin(x) is cos(x), and we have the negative there, so it's -cos(x)

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so if we take the derivative of that, so this is the third derivative of cos(x),

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now it's just going to be positive sine of x, and if we take the derivative of that,

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we get cos(x) again.

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We get cosine of x again. So if we take the derivative of that,

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this is the fourth derivative, I should, I should use this notation

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but you get the idea, we'll get cos(x) again.

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And if you look at what we talked about in the last video, we want the difference - we want

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the function, and we want it's various derivatives evaluated at 0,

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so let's evaluate it at 0. So f(0), cos(0) is 1, cosine of zero is one.

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Whether you're talking about zero radians or zero degrees, doesn't matter,

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sine of zero is zero, so this is f prime of - f prime of zero, is zero. And then cos(0)

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is, once again, one, but we have the negative out there, so it becomes negative one.

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So f - the second derivative evaluated at zero is negative one.

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Let's take the third derivative, the third derivative evaluated at zero

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well, sine of zero is just zero, and then the fourth derivative evaluated at zero,

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cosine of zero is one. So f prime prime prime at zero is now equal to one.

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So you see an interesting pattern here - one, zero, negative one, zero, one,

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then you go to zero, then you go to negative one, zero.

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So if we were to apply this to find it's Maclaurin representation, what would we get?

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Let me do my best attempt at this. So we would get, our polynomial would be -

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so our polynomial approximation of cosine of x is going to be f(0),

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f(0) is one, and then we have one plus f'(0) times x.

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But f'(0) is just zero, so we're not going to have this term over there, it's going to be

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zero times x, I won't even take the trouble of writing it down, it would be this zero

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time x, then plus f prime prime or second derivative, which is negative one,

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so I'll write negative - negative, this is a negative one right here,

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this is a negative one, times x squared, times x squared,

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over 2 factorial - over two factorial, which in this case is just going to be two.

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But I'll just write it down here as two factorial, to make the pattern a little bit more obvious,

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and then we go to the next term, the third derivative evaluated at zero

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but the third derivative evaluated at zero is just zero, so this term won't be there as well,

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then you go to the fourth derivative, the fourth derivative evaluated at zero is positive one,

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so this coefficient right here is going to be a one, and so you're going to have

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one times x to the fourth over four factorial, so plus x to the fourth over four factorial,

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and I think you start seeing a pattern now.

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You have sign switches - and you would see this if we kept going, so

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you can verify it for yourself if you don't believe me -

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so you have a positive sign, a negative sign, a positive sign, and then a

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negative sign, so on and so forth, and this is, uh, one times x to the zeroth power,

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then you jump two to x to the squared, jump two to x to the fourth, and

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so if we kept that up, we'd have a positive sign, now we have a negative sign,

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it would be x to the sixth over six factorial, then you have a positive sign,

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x to the eighth over eight factorial, and then you'd have a negative sign,

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x to the tenth over ten factorial, and you can just keep going that way.

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And if you kept going with this series, this would be the polynomial representation of

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cosine of x. And it's frankly just kind of cool that if can be represented this way.

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It's a pretty simple pattern here for a trigonometric function.

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Once again, it kind of tells you that all of this math is connected. And we'll see,

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two or three videos from now, it's connected in far more profound ways then you can possibly imagine.