Explanation Given: The given function is f ( x ) = { 1 − 2 < x < − 1 − x − 1 ≤ x < 0 x 0 ≤ x < 1 1 1 ≤ x < 2 . Calculation: First determine if the given function is odd or even then accordingly determine the sine or cosine series. If the function is odd then the sine series is evaluated and if it is even then the cosine series is evaluated. f ( x ) = { 1 − 2 < x < − 1 − x − 1 ≤ x < 0 x 0 ≤ x < 1 1 1 ≤ x < 2 . If − 2 < x < − 1 then f ( − x ) = 1 . It can be seen that f ( − x ) = f ( x ) Thus, the function is even in this interval. If − 1 ≤ x < 0 then f ( − x ) = − ( x ) = − x It can be seen that f ( − x ) = f ( x ) Thus, the function is even in this interval. If 0 ≤ x < 1 then f ( − x ) = − ( − x ) = x It can be seen that f ( − x ) = f ( x ) Thus, the function is even in this interval. If 1 ≤ x < 2 then f ( − x ) = 1 It can be seen that f ( − x ) = f ( x ) Thus, the function is even in this interval also. This means that the given function is even. Thus, the cosine series has to be evaluated. Now, the cosine series of an even function is given by f ( x ) = a 0 2 + ∑ n = 1 ∞ a n cos n π x L . (1) Where a n is the coefficient given by a n = 2 L ∫ 0 L f ( x ) cos n π x L d x , n ∈ N

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Chapter 12.3, Problem 21E

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To expand the function in sine or cosine series accordingly.

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Chapter 12.3, Problem 20E

Chapter 12.3, Problem 22E

Chapter 12 Solutions

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Ch. 12.1 - Prob. 1E Ch. 12.1 - Prob. 2E Ch. 12.1 - Prob. 3E Ch. 12.1 - Prob. 4E Ch. 12.1 - Prob. 5E Ch. 12.1 - Prob. 6E Ch. 12.1 - Prob. 7E Ch. 12.1 - Prob. 8E Ch. 12.1 - Prob. 9E Ch. 12.1 - Prob. 10E

Ch. 12.1 - Prob. 11E Ch. 12.1 - Prob. 12E Ch. 12.1 - Prob. 13E Ch. 12.1 - Prob. 14E Ch. 12.1 - Prob. 15E Ch. 12.1 - Prob. 16E Ch. 12.1 - Prob. 17E Ch. 12.1 - Prob. 18E Ch. 12.1 - Prob. 21E Ch. 12.1 - Prob. 22E Ch. 12.1 - Prob. 23E Ch. 12.1 - Prob. 24E Ch. 12.1 - Prob. 25E Ch. 12.1 - Prob. 26E Ch. 12.1 - Prob. 27E Ch. 12.2 - Prob. 1E Ch. 12.2 - Prob. 2E Ch. 12.2 - Prob. 3E Ch. 12.2 - Prob. 4E Ch. 12.2 - Prob. 5E Ch. 12.2 - Prob. 6E Ch. 12.2 - Prob. 7E Ch. 12.2 - Prob. 8E Ch. 12.2 - Prob. 9E Ch. 12.2 - Prob. 10E Ch. 12.2 - Prob. 11E Ch. 12.2 - Prob. 12E Ch. 12.2 - Prob. 13E Ch. 12.2 - Prob. 14E Ch. 12.2 - Prob. 15E Ch. 12.2 - Prob. 16E Ch. 12.2 - Prob. 17E Ch. 12.2 - Prob. 18E Ch. 12.2 - Prob. 19E Ch. 12.2 - Prob. 20E Ch. 12.2 - Prob. 21E Ch. 12.2 - Prob. 22E Ch. 12.3 - Prob. 1E Ch. 12.3 - Prob. 2E Ch. 12.3 - Prob. 3E Ch. 12.3 - Prob. 4E Ch. 12.3 - Prob. 5E Ch. 12.3 - Prob. 6E Ch. 12.3 - Prob. 7E Ch. 12.3 - Prob. 8E Ch. 12.3 - Prob. 9E Ch. 12.3 - Prob. 10E Ch. 12.3 - Prob. 12E Ch. 12.3 - Prob. 13E Ch. 12.3 - Prob. 14E Ch. 12.3 - Prob. 15E Ch. 12.3 - Prob. 16E Ch. 12.3 - Prob. 17E Ch. 12.3 - Prob. 18E Ch. 12.3 - Prob. 19E Ch. 12.3 - Prob. 20E Ch. 12.3 - Prob. 21E Ch. 12.3 - Prob. 22E Ch. 12.3 - Prob. 23E Ch. 12.3 - Prob. 24E Ch. 12.3 - Prob. 25E Ch. 12.3 - Prob. 26E Ch. 12.3 - Prob. 27E Ch. 12.3 - Prob. 28E Ch. 12.3 - Prob. 29E Ch. 12.3 - Prob. 30E Ch. 12.3 - Prob. 31E Ch. 12.3 - Prob. 32E Ch. 12.3 - Prob. 33E Ch. 12.3 - Prob. 34E Ch. 12.3 - Prob. 35E Ch. 12.3 - Prob. 36E Ch. 12.3 - Prob. 37E Ch. 12.3 - Prob. 38E Ch. 12.3 - Prob. 39E Ch. 12.3 - Prob. 40E Ch. 12.3 - Prob. 41E Ch. 12.3 - Prob. 42E Ch. 12.3 - Prob. 43E Ch. 12.3 - Prob. 44E Ch. 12.3 - Prob. 45E Ch. 12.3 - Prob. 46E Ch. 12.3 - Prob. 50E Ch. 12.5 - Prob. 3E Ch. 12.5 - Prob. 4E Ch. 12.5 - Prob. 5E Ch. 12.5 - Prob. 6E Ch. 12.5 - Prob. 7E Ch. 12.5 - Prob. 8E Ch. 12.5 - Prob. 9E Ch. 12.5 - Prob. 10E Ch. 12.5 - Prob. 11E Ch. 12.5 - Prob. 13E Ch. 12.6 - Prob. 3E Ch. 12.6 - Prob. 4E Ch. 12.6 - Prob. 5E Ch. 12.6 - Prob. 6E Ch. 12.6 - Prob. 7E Ch. 12.6 - Prob. 8E Ch. 12.6 - Prob. 9E Ch. 12.6 - Prob. 10E Ch. 12.6 - Prob. 13E Ch. 12.6 - Prob. 14E Ch. 12.6 - Prob. 15E Ch. 12.6 - Prob. 16E Ch. 12.6 - Prob. 17E Ch. 12.6 - Prob. 18E Ch. 12.6 - Prob. 19E Ch. 12.6 - Prob. 20E Ch. 12.6 - Prob. 21E Ch. 12.6 - Prob. 22E Ch. 12.6 - Prob. 23E Ch. 12.6 - Prob. 24E Ch. 12 - Prob. 1CR Ch. 12 - Prob. 2CR Ch. 12 - Prob. 3CR Ch. 12 - Prob. 4CR Ch. 12 - Prob. 5CR Ch. 12 - Prob. 6CR Ch. 12 - Prob. 11CR Ch. 12 - Prob. 12CR Ch. 12 - Prob. 13CR Ch. 12 - Prob. 14CR Ch. 12 - Prob. 16CR Ch. 12 - Prob. 17CR Ch. 12 - Prob. 19CR Ch. 12 - Prob. 20CR Ch. 12 - Prob. 21CR Ch. 12 - Prob. 22CR Ch. 12 - Prob. 23CR Ch. 12 - Prob. 24CR Ch. 12 - Prob. 25CR

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Derivatives of Trigonometric Functions - Product Rule Quotient & Chain Rule - Calculus Tutorial; Author: The Organic Chemistry Tutor;https://www.youtube.com/watch?v=_niP0JaOgHY;License: Standard YouTube License, CC-BY

To expand the function in sine or cosine series accordingly.

Solution Summary: The author explains how the given function can be expanded in the sine or cosine series as f(x)=l1.

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To expand the function in sine or cosine series accordingly.

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Chapter 12 Solutions