Consider f, a function such as f(2) = 1, f'(2) = 2, f''(2) = 3 and f'''(2) = 4 and that verifies:-3 < f(x) < 340 < f'(x) < 12-8 < f''(x) < 9-6 < f'''(x) < 5 for all x ∈ [0,4]. Find the second degree Taylor polynomial, T2(x), of f at a = 2, then use T2(x) to approximate f(1) and determine a bound on the approximation error. Give a bound on the approximation error f(x) ≈ T2(x) that is valid for all x in the interval [0,4]. -------I was able to find T2(x) = 1 + 2(x-2) + 3/2(x-2)2 and the approximation f(1) ≈ 1/2, but I don't understand how to find the approximation error bounds? Are we supposed to use Taylor's inequality definition? 33.) Soit f une fonction telle que f(2) = 1, f'(2)= 2, f"(2)=3et f"(2)= 4, et qui vérifie–3< f(x)<34 0

Given: f(2)=1, f'(2)=2, f''(2)=3 and f'''(2)=4 To find: Error bound in the approximation of f1.

Answered: Consider f, a function such as f(2) =…

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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Question

Consider f, a function such as f(2) = 1, f'(2) = 2, f''(2) = 3 and f'''(2) = 4 and that verifies:

-3 < f(x) < 34

0 < f'(x) < 12

-8 < f''(x) < 9

-6 < f'''(x) < 5 for all x ∈ [0,4].

Find the second degree Taylor polynomial, T₂(x), of f at a = 2, then use T₂(x) to approximate f(1) and determine a bound on the approximation error.

Give a bound on the approximation error f(x) ≈ T₂(x) that is valid for all x in the interval [0,4].

-------

I was able to find T₂(x) = 1 + 2(x-2) + 3/2(x-2)² and the approximation f(1) ≈ 1/2, but I don't understand how to find the approximation error bounds? Are we supposed to use Taylor's inequality definition?