to searchF2Problem 3Given f(x) = sin(x) and xo = 0a) Find the following Taylor polynomials P5(x), P7(x), and P⁹(x).b) Show that if x ≤ 1, then the E9(x) < 1/10!F3100F442)DILF5☀F6F7PrtScnF8HomeF9

The Taylor series expansion of fx is fx=∑n=0∞xnn!fnx0 Where the error term is…

Answered: Given f(x)=sin(x) and xo = 0 a) Find…

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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Suppose that g(x) is a function that is continuous for all real values of x. Let's say that you want to compute the third-degree Taylor polynomial T3(x) centered at 8 for g(x), and you know that g(8) = 2, g' (8) = -5, g "(8) = 18, and g "(8) = 48. (In case you are having trouble reading the notation, I just gave you the value of g at 8, followed by the values of the first, second, and third derivatives of g at 8, in that order.) What is T3(x)?
2. Find the fourth Taylor polynonmial for f (x) = In x at x = 1.
Given the Taylor polynomial for f(x) at x = 3, find f(³) (3): T₁(x) = 1 + x −3+2(x − 3)² + 2(x-3)³ - 3(x − 3)¹ f(3) (3) = HHH
The first four Taylor polynomials of the function f(x) = e* at a = 0 are given as follows: Tо (х) 1 Ti (х) 1+x T2(x) 1+x + T3(x) 1+x + %3D It is known that the Taylor polynomials of the function f(x) = e* at a = 0 satisfy the following property: If h is a very small positive number then, for any real number x and any positive integer n, T„(x + h) – T,(x) - Tn-1(x). h (x means approximately equal to, when h is very small) You are asked to prove this for specific values of x and n as follows: (a) For x = 1 and n = 1: Using the Taylor polynomials, show that if h is very small, then T;(1 + h) – T¡(1) z To(1). h (b) For x = 1 and n = 2: Using the Taylor polynomials, show that if h is very small, then T2(1+ h) – T2(1) - T¡(1). h
a) Use the Taylor polynomial for f(x) = sqrt(x) of degree 2 at point 100 to find an approximate value for sqrt(101). And then give an estimate of the inaccuracy of the value found. b) Justify whether the approximate value found in (a) is too large or too small compared to the real value of sqrt(101)
(b) Using generating function for Legendre polynomials, prove that (21+1)P(x)t' : 1- t2 (1 – 2xt + t2)3/2 %3D
Find the third Taylor polynomial for f(x): = 12 X expanded about c = 1. Select one: O a. P(x) = 12- 12(x-1)-72(x - 1)²- 12(x - 1)³ O b. P(x) = 12- 12(x-1)+24(x - 1)²-24(x-1)³ O c. P(x) = 12- 12(x+1)+12(x + 1)²- 12(x+1)³ 12(x-1)-24(x-1)²- 12(x-1)³ O d. P(x) = 12- O e. P3(x) = 12- 12(x-1)+12(x-1)² - 12(x-1)³

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Given f(x)=sin(x) and xo = 0 a) Find the following Taylor polynomials P5(x), P7(x), and P9(x). b) Show that if x ≤ 1, then the E9(x) < 1/10!