Answered: Use Taylor's residual formula to give…

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

Use Taylor's residual formula to give an estimate of π / 4 = tan − 1
(1) (without to use the π button on the calculator): Show that the residual term E1 (1) is in between
−1 and 0 and that π / 4 is equal to P1 (1) - 1/4 = 3/4 with an error less than 1/4

Expert Solution

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$f (t) = \tan^{- 1} (t) f (1) = \tan^{- 1} (1) = \frac{π}{4} . . (1) N o w, f (t) = \tan^{- 1} (t) d i f f e r e n t i a t e w i t h r e s p e c t t o t, w e h a v e f^{1} (t) = \frac{1}{t^{2} + 1} . . . (2) a g a i n d i f f e r e n t i a t e, w e h a v e f^{2} (t) = - \frac{2 t}{{(t^{2} + 1)}^{2}} . . . (3)$

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