(c)(1) Find the Taylor polynomial p2(x) for the function f(x)=/x at X = 9.(ii) Use it to approximate v 10, writing your answer as a single fraction.(iii) Use Taylor's Theorem to find an upper bound for the error in this approximation. (Write your answer as either a fraction or as a decimal to 1 significant figure.)

Answered: (c) (1) Find the Taylor polynomial…

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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(c)
(1) Find the Taylor polynomial p2(x) for the function f(x)=/x at X = 9.
(ii) Use it to approximate v 10, writing your answer as a single fraction.
(iii) Use Taylor's Theorem to find an upper bound for the error in this approximation. (Write your answer as either a fraction or as a decimal to 1 significant figure.)

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$(i) f (x) = \sqrt{x} T a y l o r s e r i e s e x p a n s i o n o r T a y l o r' s p o l y n o m i a l f (x) = f (9) + f' (9) (x - 9) + \frac{f'' (9)}{2!} {(x - 9)}^{2} + R_{3} (1) f (9) = \sqrt{9} = 3 (2) f' (x) = \frac{1}{2} x^{\frac{- 1}{2}} f' (9) = \frac{1}{2} {(9)}^{\frac{- 1}{2}} = \frac{1}{6} (3) f'' (x) = - \frac{1}{4} x^{\frac{- 3}{2}} f'' (9) = - \frac{1}{4} {(9)}^{\frac{- 3}{2}} = - \frac{1}{108} (4) f''' (x) = \frac{3}{8} x^{\frac{- 5}{2}} (5) P u t (2), (3), (4) i n (1) f (x) = 3 + \frac{1}{6} (x - 9) - \frac{1}{108} \frac{{(x - 9)}^{2}}{2!} + R_{3} f (x) = 3 + \frac{1}{6} (x - 9) - \frac{1}{216} {(x - 9)}^{2} + R_{3} (6)$

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