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akl3644 Group Title

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?

  • 2 years ago
  • 2 years ago

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  1. akl3644 Group Title
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    @KingGeorge can you help me on this question?

    • 2 years ago
  2. KingGeorge Group Title
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    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.

    • 2 years ago
  3. akl3644 Group Title
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    i am really confuse abt this question. i don't know what is ask for

    • 2 years ago
  4. KingGeorge Group Title
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    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.

    • 2 years ago
  5. akl3644 Group Title
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    what if the curve is centered at 0?

    • 2 years ago
  6. KingGeorge Group Title
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    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.

    • 2 years ago
  7. akl3644 Group Title
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    is could apply every function if the function is centered at 0?

    • 2 years ago
  8. KingGeorge Group Title
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    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.

    • 2 years ago
  9. KingGeorge Group Title
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    Of course, if you use the whole thing, it shouldn't be an approximation anymore. It would be exactly the same.

    • 2 years ago
  10. akl3644 Group Title
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    ok! thank you very much!!

    • 2 years ago
  11. KingGeorge Group Title
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    You're welcome.

    • 2 years ago
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