Taylor & Maclaurin polynomials intro
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Worked example: Taylor polynomial of derivative function
- So, let's say we've been given all this information about the function g and it's derivative evaluated at x equals two. We know g of two is equal to three. G prime of two is equal to one. The second derivative of g evaluated two is negative one. The third derivative of g evaluated at two is two. Given that, what we're being tasked with is we want to use the second degree Taylor polynomial centered at x equals two to approximate g prime of one. Not g of one, g prime of one and so I encourage you to pause this video and try to think about it on your own. I'm assuming you've had a go at it. Let's just remind ourselves what a second degree Taylor polynomial centered at x equals two would look like for a general function f of x. F of x would approximately be equal to, it would be f of two plus f prime of two times x minus two plus f prime prime of two times x minus two squared, all of that over two factorial. That would get us to a second degree place because it's x minus two squared. This is gonna give us a second degree polynomial. This is the general case. If we want to find the approximation for f centered at x equals but we're gonna do it for g prime. Let me write this down. Alright, so I'll do it in blue. So, we have g prime of x is what we're going to try and approximate and then we're going to evaluate it at x equals 1. G prime of x is going to be approximately equal to, well same thing, it's going to be the function that I'm going to try to approximate evaluated at two so g prime of two. Notice, so I'm approximate f of x, it's that function evaluated at two. If I'm approximating g prime of x, it's that function evaluated at two. Then, plus the first derivative of this thing which is the second derivative of g. G prime prime of two times x minus two and then plus the second derivative of the function that I'm trying to approximate but the second derivative of g prime is going to be the third derivative of g. It's going to be g prime prime prime of two times x minus two squared, all of that over two factorial. Now, they tell us what these things are. Let me use some new colors here. They tell us g of two is equal to three. This right here, oh, actually no, that's not what we're gonna wanna use. They tell us we're using g of two. They're telling us g prime of two is equal to one. G prime of two, so this right over here is equal to one. G prime prime of two is equal to negative one. This is negative one right over here and then finally the third derivative of g evaluated at two is two. Two over two factorial, two factorials is two times one or it's just two. So that and that cancel out. What are we left with for our approximation our second degree approximation of g prime of x centered at x equals two? We are left with g prime of x is approximately equal to one minus x minus two. One minus x minus two, I guess I could write that as plus two minus x so plus two minus x. The negative of x minus two is two minus x. Plus x minus two squared and obviously I could simplify this even more. This is three minus x plus x minus two squared and now I can evaluate it. If I wanna approximate g prime of one, I could say g prime of one is going to be approximately equal to, wherever I see the x is here I put in a one there, so it's going to be three minus one plus one minus two squared. Well, this is going to be two and then one minus two is negative one but then if I square it I get a positive one. So this whole thing is one. Two plus one is equal to three. Once again, this is an approximation for g prime of one. What did we do here? We found the Taylor series. The second degree Taylor series approximation for g prime of x centered around x equals two and then we evaluated that approximation at x equals one to approximate g prime of one. Anyway, hopefully, you found that fun.