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## akl3644 3 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

@KingGeorge can you help me on this question?

2. 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.

3. akl3644

i am really confuse abt this question. i don't know what is ask for

4. 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.

5. akl3644

what if the curve is centered at 0?

6. 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.

7. akl3644

is could apply every function if the function is centered at 0?

8. 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.

9. KingGeorge

Of course, if you use the whole thing, it shouldn't be an approximation anymore. It would be exactly the same.

10. akl3644

ok! thank you very much!!

11. KingGeorge

You're welcome.

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