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(a) Give the second-order Taylor polynomial T₂ (2) for the function f(x) = x³ In (2), centered at x = 1. (b) Use Taylor's Theorem to give the error term E₂ (x) = f(x) – T₂ (2), as a function of and some z between 1 and 2. (c) Use the fact that 1 < z < x, and/or the fact that 0 < ln (z) < 1 for all 1

(a) Give the second-order Taylor polynomial T₂ (2) for the function f(x) = x³ In (2), centered at x = 1. (b) Use Taylor's Theorem to give the error term E₂ (x) = f(x) – T₂ (2), as a function of and some z between 1 and 2. (c) Use the fact that 1 < z < x, and/or the fact that 0 < ln (z) < 1 for all 1

Advanced Engineering Mathematics
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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(a) Give the second-order Taylor polynomial T₂ (x) for the function f(x) = x³ ln (x), centered at x = 1. (b) Use Taylor's Theorem to give the error term E₂ (x) = f(x) - T₂ (x), as a function of and some z between 1 and x. (c) Use the fact that 1 < z < x, and/or the fact that 0 < ln (z) < 1 for all 1 < z < 2, to find an upper bound on E₂ (2) when 1 < x < 2. Enter this bound as a function involving a (but not z). (d) Use the upper bound to enter a value p for which E₂ (x) < 0.01 for all 1<x<p.
Transcribed Image Text:(a) Give the second-order Taylor polynomial T₂ (x) for the function f(x) = x³ ln (x), centered at x = 1. (b) Use Taylor's Theorem to give the error term E₂ (x) = f(x) - T₂ (x), as a function of and some z between 1 and x. (c) Use the fact that 1 < z < x, and/or the fact that 0 < ln (z) < 1 for all 1 < z < 2, to find an upper bound on E₂ (2) when 1 < x < 2. Enter this bound as a function involving a (but not z). (d) Use the upper bound to enter a value p for which E₂ (x) < 0.01 for all 1<x<p.
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