anonymous
  • anonymous
taylor expansion question: what part of the expansion of a function of f(x) in powers of x best reflects the behavior of the function for x's close to 0?
Mathematics
jamiebookeater
  • jamiebookeater
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anonymous
  • anonymous
@KingGeorge can you help me on this question?
KingGeorge
  • KingGeorge
Well, if we divide it up into parts where a "part" is the first n terms, I have an idea. However, I would like to see what you think before I start an explanation.
anonymous
  • anonymous
i am really confuse abt this question. i don't know what is ask for

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KingGeorge
  • KingGeorge
Well, it's talking about the powers of x. From that, I would make a guess that they want you to say that it's the all the terms up to the \(x^1\)th term. However, there's no real way to say for sure since this isn't necessarily a McLaurin Series. Thus, it's not necessarily centered at 0, so it really depends on the function.
anonymous
  • anonymous
what if the curve is centered at 0?
KingGeorge
  • KingGeorge
However, if we assume it is centered at 0, then let's throw away all the terms except the first two terms. So we have a function that looks like \(T_1(x)=ax+b\). It is precisely correct at \(x=0\) since it's centered at 0, and for very close points, it has nearly the same slope. So for points very close to \(x=0\), this is a good approximation.
anonymous
  • anonymous
is could apply every function if the function is centered at 0?
KingGeorge
  • KingGeorge
If it's centered at 0, I would say the first two terms. If it's not centered at 0, you really can't say anything. However, the best approximation, is the whole Taylor series.
KingGeorge
  • KingGeorge
Of course, if you use the whole thing, it shouldn't be an approximation anymore. It would be exactly the same.
anonymous
  • anonymous
ok! thank you very much!!
KingGeorge
  • KingGeorge
You're welcome.

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