2. (a) Show that if x > 0, then 1 + x - x² < √1 + x < 1 + ½x.(b) Use part (a) to approximate √1.2 and √2. What is the best accuracy you can be sure of,using this inequality?

Answered: Image /qna-images/answer/69bfff6c-67cd-42f6-9b07-78d8f67b0cd4.jpg

Answered: 2. (a) Show that if a > 0, then 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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Question

2. (a) Show that if x > 0, then 1 + x - x² < √1 + x < 1 + ½x.
(b) Use part (a) to approximate √1.2 and √2. What is the best accuracy you can be sure of,
using this inequality?

Expert Solution

Step 1: To build up the inequality:

$left parenthesis a right parenthesis O n e space m a y space f i n d space t h a t space square root of 1 plus x end root almost equal to 1 plus x over 2 minus x squared over 8 plus x cubed over 16 minus fraction numerator 5 x to the power of 4 over denominator 128 end fraction plus fraction numerator 7 x to the power of 5 over denominator 256 end fraction plus O left parenthesis x to the power of 6 right parenthesis C l e a r l y space o n e space m a y space f i n d space t h a t square root of 1 plus x end root less or equal than 1 plus x over 2 space a n d 1 plus x over 2 minus x squared over 8 less or equal than square root of 1 plus x end root C o m b i n i n g space o n e space m a y space g e t 1 plus x over 2 minus x squared over 8 less or equal than square root of 1 plus x end root less or equal than 1 plus x over 2.$