Question 6 Help

[JebusMeany]

I don't know what this section involves. Joel said in lecture that a) and b) do not involve Taylor Polynom's at all, and that is should be obvious.

Well, it isn't.

Any ideas on what is being asked?

Thanks

[Rogue_Knight]

a) I used Taylor's polynomial expansion for 1/(1-t), which works nicely as once you find f^(k)(t) and plug it into the equation, things start cancelling.
That extra bit at the end is the Lagrange remainder.

b) I cheated for this by using what we know in part a, and then evaluating the polynomial we get by setting t=-u^2.

c) Integrate.  Everything except the remainder is easy to integrate, so do what they say and leave it as an integral named I.

d) The Taylor polynomial of nth degree for f(x), call it P_n(x) is the unique polynomial of a degree n S.T.
lim x->0 (f(x) - P_n(x))/x^n =0
Note that we already know f(x) and P_n(x) because we know the remainder, namely, I, which is exactly what we want.  With some slight modification of the limit, this question is easy.

Everything else is pretty self-explanatory calculation based on what we did.