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?

Question

anonymous · Answer

@KingGeorge can you help me on this question?

kinggeorge · Answer

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 · Answer

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

kinggeorge · Answer

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 · Answer

what if the curve is centered at 0?

kinggeorge · Answer

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 · Answer

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

kinggeorge · Answer

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 · Answer

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

anonymous · Answer

ok! thank you very much!!

kinggeorge · Answer

You're welcome.