Answered: Let f: ℝ → (-π/2, π/2) be given by f(t)…

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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Let f: ℝ → (-π/2, π/2) be given by f(t) = tan^-1(t)

a) What is Taylor's polynomial P₁(t) of order 1 f if t = 0? What is the residual E₁(t) in Taylor's formula f(t) = P₁(t) + E₁(t)?

b) Use Taylor's residual formula to give an estimate of π/4 = tan^-1(t) (without using π on the calculator). Show that the residual term E₁(t) is between -1 and 0 and that π/4 is equal to P₁(t) - 1/4 = 3/4 with an error less than 1/4

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