Transcribed Image Text: In this question, we investigate some ideas surrounding Taylors Theorem. That is, how to
evaluate functions as series. The partial sums of these series give approximations to the value
of the function. Taylors Theorem gives an estimate of how good this approximation is.
Let f: R ? R be a function. Assume that the kth derivative of f, denoted by f(k), exists for
all k 2 0. In particular, f(k) is continuous for all k > 0.
(a) If x E R, show that
= f(0) + s'(t) dt
State the name of the theorem that allows us to calculate the definite integral.
(b) Let n > 0 be an integer. Define Rn: R ? R to be the function
Rn(x) = | *-) f(n+1)(t) dt.
Using Integration by Parts, show that if x E R, then
f(n+1) (0)
(n + 1)!
Ra(x) :
+ Rn+1(x).
