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right the first four nonzero terms of the Maclaurin series for e to the X use the Maclaurin series for e to the X to write the third degree Taylor polynomial for G to the X is equal to e to the x times f of X about x equals zero so Maclaurin series if that looks familiar to you watch the videos on Khan Academy on Maclaurin series I'll give a little bit of a primer if we're taking the Maclaurin series of f of X that's going to be equal to f of 0 we could view it as f of 0 times X to the zero over zero factorial if you assume zero factorial is equal to 1 and then plus F prime of 0 times X to the first over 1 factorial plus F prime prime the second derivative evaluated at 0 times x squared over 2 factorial I think you see the pattern here plus F prime prime prime the third derivative evaluated at 0 of X to the third power over 3 factorial I think you see where this is going and of course this first one X to the 0 over 0 factorial that's just 1 so it oftentimes will just be written F of 0 and this term 1 factorial is just 1 so oftentimes just to see it written this F prime of 0 times X so on and so forth so I'll actually let me just write approximately right over there and so let's do it for e to the X so e to the X is approximately equal to well it's going to be e to the 0 which is 1 plus you might already know that F f of X if f of X is equal to e to the X then F prime of X is also equal to e to the X that's one of the magical things about e to the X the derivative the slope of the tangent line at any point is well equal to the value is equal to the x value there and if you take the third derivative or a second derivative it's also you could take as many derivatives you want you still get e to the X that's one of the special things about e so the first term that evaluated 0 well that's still e to the 0 power times X to the first so plus X and then we have the sec evaluated at zero well that's still one so it's going to be times or so it's one times x squared over two factorial so we could say plus x squared over two plus x squared over two two factorial is just two times one and then Plus once again the third derivative evaluated zero that's just e to the X evaluated zero R which is e to the 0 which is 1 so X to the third over three factorial so plus X to the third over three factorial we could write it as three factorial or that's three times two times one that's equal to six and then we keep going so we just wrote the first four nonzero terms of the Maclaurin series for e to the X that's one two three four nonzero terms now we want to use so let me just underline that that's part of what they're asking us to do then they say use the Maclaurin series for e to the X to write the third degree Taylor polynomial for G of X which is equal to the product of e to the X and f of X so what I'm gonna do is I'm gonna write our original f of X let's write down the first few terms of it and then what we can do is think about well how can we multiply those two polynomials and we just have to know enough about the multiplication of those two polynomials to just get us our our terms that are no higher than third degree so f of X is approximately equal to let's see it's X I have a bad memory it is X minus three halves x squared X X minus three has x squared and then plus three X to the third power plus three X to the third power plus three X to the third power plus and actually you could say minus if you like cuz then it's gonna be you're gonna have it's plus minus plus minus however you want us to do it and that's enough and why do I feel confident that it's enough well we only want we are only we only want to write the third degree Taylor polynomial so if we multiply these and we involve terms that are higher than third degree well we're gonna we're that's going to give us a high the the the terms in our polynomial that are higher than third degree so let's just think about it what's going to be the product so e to the X e to the x times f of X that's going to be approximately equal to well let's see we are going to multiply this infinite polynomial times this infinite polynomial and that might seem intimidating to you at first but what you could do is you could go for each of these terms I'll start multiplying it times each of these terms out here essentially you know when you're multiplying polynomials you're just repeatedly doing the distributive property so you take this and distribute it onto that but we should only worry about the terms up to third degree because anything beyond that well that's going to be that's going to add up to a higher than third degree term so x times one is x x times x squared or sorry x times X is x squared so plus x squared x times x squared over two is X to the X to the third over two so I'll just write plus one half X to the third power and I'm gonna stop there because if I do x times that that's gonna give me a fourth degree term and I don't want to worry about that we are writing the third degree Taylor polynomial so X so that's X x squared X to the third over two so now let's distribute the negative three halves x squared and I'll use another color here just to help explain it a little bit so if you multiply that times one that's going to be minus three halves x squared I'm just going to a second line here so I can line things up and add them nicely and then this times this is negative three halves X to the third negative three halves X to the third and once again I'm gonna stop there if I multiply these two I'm gonna get a fourth degree term and I don't care about the fourth degree terms and then and then let's do let's worry about this guy and so let's start distributing so if I multiply it times this one I'm gonna get 3x to the third power 3x to the third power and I'm gonna stop there because then if I start multiplying it times that guy that's giving me a fourth degree term and then a fifth degree term and then a sixth degree term which I don't need to worry about so these are all the pieces that are going to make up that third degree polynomial and so what is that going to be equal to what is that going to be equal to this is a little bit this was a little bit of you know getting your math intuition for multiplying infinite polynomials let's see you're gonna have X and then you're going to have this is 1x squared minus 1 and 1/2 x squared so that's going to be negative 1/2 x squared and let's see here you have plus 1/2 minus 1 and 1/2 which would give you negative 2 and then plus 3 plus oh no sorry 1/2 minus 1 and 1/2 which would be negative 1 plus 3 is positive 2 plus 2 X to the third power so we could say e to the x times f of X the third degree Taylor polynomial for this is X minus 1/2 x squared plus 2 X to the third power and we are done so this was a little bit tricky you had to appreciate how to end but it's not calculus it's a little bit of just algebra of just appreciating Gallina third degree I don't have to distribute this times an infinite number of terms and just because at first you might say well that's that's super hard how do I multiply two infinite polynomials the key is we only worry about the third degree up to the third degree
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