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I have a kind of general question. So I understand that the goal of a Taylor function is to approximate a transcendental function using a polynomial function. This makes things easier to deal with sometimes. I understand that this works by chosing a polynomial function that seems to behave like the transcendental function does, at least in a localized area. For example f(c)=P(c) where f is our transcendental function and P is our polynomial function. As we add additional constrantaints, like f'(c)=P'(c) and f''(c)=P''(c) this approximation becomes better and better until eventually we can come up with a power series that is exactly the function.

My question is, WHY is it that as the higher order derivatives match, the function becomes a better and better approximation? I mean, intuitively it makes sense: there are more things that the two functions have in common so it seems natural that a function that has more in common would be a better approximation than one that has less in common. I just don't understand what it is that is making it a better approximation. Why are the higher order derivatives so important?

I'm not sure if this makes sense. I feel like I understand how it works, I just don't understand why.