Taylor Series Question...

Question

agent_a · Answer

attachment

agent_a · Answer

Is my answer to letter c correct or is there a way to simplify this manually without much work? (Just to clarify, does the "!" mean that it is a factorial?)

amistre64 · Answer

the answer jpg should come with a warning label ...

amistre64 · Answer

if you found the results to B correctly, then its just a matter of inserting them into the parts for C

amistre64 · Answer

your derivatives look fine .... i wouldnt approximate them tho

amistre64 · Answer

f0 4^x

f1 4^x log4

f2 4^x (log4)^2

f3 4^x (log4)^3

fn  4^x (log4)^n

and at x=0 thats just (log4)^n

agent_a · Answer

@amistre64 is there a way to simplify the last line (with the factorials)? Or should I just leave it at that?

amistre64 · Answer

id leave it like that but i aint grading it :/

agent_a · Answer

@amistre64, Haha okay. Thanks! Yeah I don't think I'd be required to simplify it further without having to resort to Wolfram-Alpha :P .

amistre64 · Answer

id concur :)

amistre64 · Answer

log(x) is itself an infinite taylor poly so trying to square and simplify and such is just not something i see as feasible

agent_a · Answer

@amistre64 Ahhhh interesting. This is the first time I've encountered Taylor Series, actually. My teacher has not covered it yet.

amistre64 · Answer

they are an interesting idea;  given that 2 objects are equal if they share the same properties ... then taking finding a polynomial that acts like a particular function means that all their derivatives are equal.

amistre64 · Answer

in some cases the interval on which they are identical is only at 1 point, and sometimes at all points .... so an interval of convergence is a concept to look into as well

agent_a · Answer

And is that why you can always approximate the tangent line (because all their derivatives are equivalent)?

agent_a · Answer

Okay I'll look into that (Interval of Convergence).

amistre64 · Answer

well, a tangent line is itself an approximation that has a degree of error associated with it.

agent_a · Answer

Oh that's true...

amistre64 · Answer

the tangent line shares some properties in that it is equal to the point, and it is equal to the slope of the line at that point :)  as we go deeper into how the functions move (rates of change are all about derivatives) we can better form a finite polynomial

agent_a · Answer

Oh I see.

amistre64 · Answer

good luck ;)

agent_a · Answer

Thank You! :-)