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akl3644

  • 2 years ago

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?

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  1. akl3644
    • 2 years ago
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    @KingGeorge can you help me on this question?

  2. KingGeorge
    • 2 years ago
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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.

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

  4. KingGeorge
    • 2 years ago
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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.

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

  6. KingGeorge
    • 2 years ago
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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.

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

  8. KingGeorge
    • 2 years ago
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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.

  9. KingGeorge
    • 2 years ago
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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.

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

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

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