(a) Give the second-order Taylor polynomial T₂ (x) for the function ƒ (x) = x² ln (x), about a = 1. (x-1)+(3/2)*(x-1)^2 (b) Use Taylor's Theorem to give the error term E2 (x) = f (x) – T₂ (x), as a function of x and some z between 1 and î. 2*(x-1)^3/(6*z) (c) Use the fact that 1 < < < x, and the fact that 0 < ln (2) < 1 for all 1 < z < 2, to find an upper bound on E₂ (x) 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) |< 0.001 for all 1 < x
(a) Give the second-order Taylor polynomial T₂ (x) for the function ƒ (x) = x² ln (x), about a = 1. (x-1)+(3/2)*(x-1)^2 (b) Use Taylor's Theorem to give the error term E2 (x) = f (x) – T₂ (x), as a function of x and some z between 1 and î. 2*(x-1)^3/(6*z) (c) Use the fact that 1 < < < x, and the fact that 0 < ln (2) < 1 for all 1 < z < 2, to find an upper bound on E₂ (x) 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) |< 0.001 for all 1 < x
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:(a) Give the second-order Taylor polynomial T₂ (x) for the function ƒ (x) = x² ln (x), about a = 1.
(x-1)+(3/2)*(x-1)^2
(b) Use Taylor's Theorem to give the error term E2 (x) = f (x) – T₂ (x), as a function of x and some z between 1 and î.
2*(x-1)^3/(6*z)
(c) Use the fact that 1 < < < x, and the fact that 0 < ln (2) < 1 for all 1 < z < 2, to find an upper bound on E₂ (x) 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) |< 0.001 for all 1 < x <p.
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