12. Find a Taylor series representation for the function f(x) = sin(x) about a = 2 (a) 1 - (x-7² 2! + 2! 1! (1) (b) (1) (15) (1) + 3! 5! it (c) —1+ (d) (1) (1) + 1! 3! (e) 1 + ( +... 4! + 5! + ... ... (x-7)(x-7)²(x-7) ³ 3! ...+ 5! + ... Fin 2
12. Find a Taylor series representation for the function f(x) = sin(x) about a = 2 (a) 1 - (x-7² 2! + 2! 1! (1) (b) (1) (15) (1) + 3! 5! it (c) —1+ (d) (1) (1) + 1! 3! (e) 1 + ( +... 4! + 5! + ... ... (x-7)(x-7)²(x-7) ³ 3! ...+ 5! + ... Fin 2
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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Transcribed Image Text:**Problem Statement:**
Find a Taylor series representation for the function \(f(x) = \sin(x)\) about \(a = \frac{\pi}{2}\).
**Answer Choices:**
(a) \(1 - \frac{\left(x - \frac{\pi}{2}\right)^2}{2!} + \frac{\left(x - \frac{\pi}{2}\right)^4}{4!} + \cdots\)
(b) \(\frac{\left(x - \frac{\pi}{2}\right)}{1!} - \frac{\left(x - \frac{\pi}{2}\right)^3}{3!} + \frac{\left(x - \frac{\pi}{2}\right)^5}{5!} + \cdots\)
(c) \(-1 + \frac{\left(x - \frac{\pi}{2}\right)^2}{2!} - \frac{\left(x - \frac{\pi}{2}\right)^4}{4!} + \cdots\)
(d) \(\frac{\left(x - \frac{\pi}{2}\right)}{1!} + \frac{\left(x - \frac{\pi}{2}\right)^3}{3!} - \frac{\left(x - \frac{\pi}{2}\right)^5}{5!} + \cdots\)
(e) \(1 - \frac{\left(x - \frac{\pi}{2}\right)}{1!} + \frac{\left(x - \frac{\pi}{2}\right)^3}{3!} - \frac{\left(x - \frac{\pi}{2}\right)^5}{5!} + \cdots\)
**Explanation:**
The problem presents multiple choice answers for the Taylor series representation of \(\sin(x)\) expanded about the point \(a = \frac{\pi}{2}\). Each option gives a different potential series expansion. The correct series for the function \(\sin(x)\) about \(x = \frac{\pi}{2}\) needs to be identified from these choices.
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