In this question, we investigate some ideas surrounding Taylor's Theorem. That is, how to evaluate functions as series. The partial sums of these series give approximations to the value of the function. Taylor's 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 > 0. In particular, f(k) is continuous for all k > 0. (a) If x E R, show that f(x) = f(0) + / f' (t) dt 0, State the name of the theorem that allows us to calculate the definite integral.
In this question, we investigate some ideas surrounding Taylor's Theorem. That is, how to evaluate functions as series. The partial sums of these series give approximations to the value of the function. Taylor's 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 > 0. In particular, f(k) is continuous for all k > 0. (a) If x E R, show that f(x) = f(0) + / f' (t) dt 0, State the name of the theorem that allows us to calculate the definite integral.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:In this question, we investigate some ideas surrounding Taylor's Theorem. That is, how to
evaluate functions as series. The partial sums of these series give approximations to the value
of the function. Taylor's 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).
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