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Main content
Current time:0:00Total duration:6:10
AP.CALC:
LIM‑8 (EU)
,
LIM‑8.E (LO)
,
LIM‑8.E.1 (EK)
,
LIM‑8.F (LO)
,
LIM‑8.F.2 (EK)

Video transcript

now let's do something pretty interesting this will some degree one of the easiest functions to find the Maclaurin series representation of but let's try to approximate e to the X f of X is equal to e to the X and what makes this really simple is when you take the derivative this is frankly one of the amazing things about the number E is when you take the derivative of e to the X you get e to the X so this is equal to F prime of X this is equal to F the second derivative of x this is equal to the third derivative of x this is equal to the nth derivative of X it's always equal to e to the X that's what that's kind of the first mind-blowing thing about the number E that's it's just you can keep taking its derivative the slope the slope at any point on that curve is the same as the value of that point on that curve that's kind of crazy anyway with that said let's take its Maclaurin representation so we have to find f of 0 F prime of 0 the second derivative 0 when you take e to the 0 e to the 0 is just equal to 1 and so this is going to be equal to f of 0 this is going to be equal to F prime of 0 it's going to be equal to any of the derivatives any of the derivatives evaluated any of the derivatives evaluated at 0 the nth derivative evaluated 0 and that's why it takes that's why it makes applying the Maclaurin series formula fairly straightforward if I wanted to approximate e to the X using a Maclaurin series so e to the X and I'll put a little approximately over here and we'll get closer and closer to the real EDX as we keep adding more and more terms and especially if we add an infinite number of terms it would look like this F of 0 F of 0 let me do it in what colors did I use first cosine and sine so I used pink and I used green so let me use a non pink non green I'll use the yellow here so f of 0 is 1 plus F prime of 0 times X F prime of 0 is also 1 so plus X plus this is also 1 so it's going to be x squared over 2 factorial so plus x squared over 2 factorial this all of these things are going to be 1 this is 1 this is 1 when we're talking about a to the X so you go to the third term this is 1 just have X to the third over 3 factorial plus X to the third over 3 factorial I think you see the pattern here we just keep adding terms X to the fourth over 4 factorial plus X to the fifth over 5 factorial plus X to the sixth over 6 factorial and something pretty neat is starting to emerge is that e to the X 1 this is just really cool that e to the X can be approximated by 1 plus X plus x squared over 2 factorial plus X to the third over 3 factorial once again e to the X is starting to look like a pretty cool thing this also leads to other interesting results that if you wanted to approximate e you just evaluate this at X is equal to 1 this isn't so this is so if you want to approximate e you'd say e is approximate to eat well e is e to the first power and that's going to be approximately equal to this polynomial evaluated at 1 if X is 1 here we make X 1 over here so it'll be 1 plus 1 so it'll be 1 plus 1 plus 1 over 2 factorial plus 1 over 3 factorial plus 1 over 4 factorial so on and so forth all the way into infinity and you could view this as you could also view this as 1 over 1 factorial as well 1 over 1 factorial but what's really cool is it's this is another really neat way to represent e it shows that e once again shows up in this kind of neat thing it's kind of 2 plus 1/2 plus 1/6 plus if you just keep doing this you get close to the number E but that by itself isn't the only fascinating thing if we look back at our Maclaurin representations of these other of these other functions cosine of X let me copy and paste cosine of X so cosine of X right up here so let me do my best to a copy and paste the whole thing so copy and paste copy and paste so that is cosine of X and let's do the sahte the same thing for the sine of X that we did last video so the sine of X so sine of X let me copy and paste that copy and then let me paste that and it pasted so do we see any relationship between these approximations so before you know you probably would have guessed maybe there's some relationship between cosine and sine but what about e to the X and what you see here is that cosine of X looks a lot like this term Plus this term although we would want to put a negative out front here so it's a negative version of this term right here plus this term right here plus a negative version of this term right over here and sine of X sine of X looks just like this term plus a negative version of this term Plus this term plus a negative version of the next term so if we can somehow reconcile the negatives in some interesting way it looks like e to the X is somehow or at least it's polynomial representation of e to the X is somehow related to a combination of the polynomial representations of cosine of X and sine of X so this is starting to get really really really cool we're starting to see a connection between something related to compound interest or a function whose derivative is always equal to that function and these things that come out of the unit circle an oscillatory motion and all of those things there starts to see seem some type of pure connectedness here but I'll leave you there in that video and in the next video I'll show you how you we can actually reconcile these these three fascinating functions