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Main content
Current time:0:00Total duration:5:37
LIM‑8 (EU)
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Video transcript

in the last video we hopefully set up some of the intuition for why or I should say what a Maclaurin series is all about and I had said at the end of the videos that a Maclaurin series is just a special case of a Taylor series in the case of a Maclaurin series were approximating this function around X is equal to zero in a Taylor series and we'll talk about that in a future video you can pick an arbitrary x value or f of X value we should say around which to approximate the function but with that said let's just focus on the Maclaurin because to some degree it's a little bit simpler and it'll be that by itself can lead us to some pretty profound conclusions about mathematics and that's actually where I'm trying to get to so let's take the Maclaurin series of some interesting functions and I'm going to do functions where it's pretty easy to take the derivatives and you can keep taking their derivatives over and over and over and over and over again so let's take the Maclaurin series of cosine of X so if f of X is equal to cosine of X then before I even apply this formula that we somewhat derived in the last video or at least got the intuitive 4 in the last video let's take a bunch of derivatives of f of X just so that we have a good sense of it so if we take the first derivative if we take the first derivative derivative cosine of X is negative sine of X if we take the derivative of that if we take the derivative of that derivative of sine of X is cosine of X but we have that negative there so it's negative cosine of X if we take the derivative of that so this is the third derivative of cosine of X now it's going to be positive sine of X and if we take the derivative of that we get cosine of X again we get cosine of X again so if we take the derivative of that this is the fourth derivative I should you I shouldn't use this notation but you get the idea we'll get cosine of X again and if you look at what we talked about in the last video we want the different we want the function and we want its various derivatives evaluated at 0 so let's evaluate them let's evaluate them at 0 so f of 0 cosine of 0 is 1 cosine of 0 is 1 whether you're talking about 0 radians or is degrees doesn't matter sine of zero is zero so this is F prime of F prime of zero is zero and then cosine of zero is once again one but we have the negative out there so it becomes negative one so F the second derivative evaluated zero is negative one let's take the third derivative the third derivative evaluated at zero well sine of zero is just zero and then the fourth derivative evaluated at zero cosine of zero is one so f of prime prime prime at zero is now equal to one so you see an interesting pattern here 1 0 negative 1 0 1 then go to 0 then you go to negative 1 0 and so if we were to apply this to find its Maclaurin representation what would we get so let me do my best attempt at this so we would get our polynomial would be so our polynomial approximation of cosine of X is going to be f of 0 F of 0 is 1 is 1 and then we have 1 plus f prime of 0 times X but F prime of 0 is just 0 so we're not going to have this term over there it's going to be 0 times X I won't even take the trouble of writing it down it would be this 0 times X then plus F prime prime or second derivative which is negative 1 so I'll write negative negative this is negative 1 right here this is the negative 1 times x squared times x squared over 2 factorial over 2 factorial which in this case is just going to be 2 but I'll just write it down here as 2 factorial it'll make the pattern a little bit more obvious and then we go to the next term the third derivative valuated 0 but the third derivative evaluated at 0 is just 0 so this term won't be there as well then you go to the fourth derivative the fourth derivative evaluated 0 is positive 1 so this coefficient right here is going to be a 1 and so you're going to have 1 times X to the 4th over 4 factorial so plus X to the 4th over 4 factorial I think you start seeing a pattern now you have sign switches and you would see this if we kept going so you can verify it for yourself if you don't believe me so you have a positive sign a negative sign positive sign and then a positive sign you're gonna have a negative sign so on and so forth and this is 1 times X to the zeroth power then you jump to 2 X to the squared jump to 2 X to the 4th and so if we kept that up we had a positive sign now you have a negative sign it would be X to the sixth over 6 factorial then you have a positive sign X to the 8th over 8 factorial and then you'd have a negative sign X to the 10th over 10 factorial and you could just keep going that way and if you kept going with this series this would be the polynomial representation of cosine of X and it's frankly just kind of cool that it can be represented this way that it's a it's a pretty simple pattern here for a trigonometric function once again it kind of tells you that all of this math is connected and we'll see 2 or 3 videos from now it's connected in far more profound ways than you can possibly imagine