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f(x) = { ₁ Sketch the odd periodic extension of f. 1 0

f(x) = { ₁ Sketch the odd periodic extension of f. 1 0

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.2: Arithmetic Sequences
Problem 67E
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#4 Just part D
**Mathematics and Fourier Series: Periodic Extensions and Fourier Series** Consider the function \( f(x) \) defined by: \[ f(x) = \begin{cases} 1 & 0 < x < \pi/2 \\ 2 & \pi/2 < x < \pi \end{cases} \] **Tasks:** 1. (a)-(c) [redacted] 2. (d) Sketch the odd periodic extension of \( f \). **Fourier Series Analysis:** (e) **Find the Fourier sine series of \( f \)**. (f) **Determine the convergence of the Fourier sine series**: - At \( x = 0 \) - At \( x = \pi/2 \) - At \( x = \pi \) - At \( x = 3\pi/2 \) - At \( x = 2\pi \) (g) **Even Periodic Extensions**: - Denote by \( f_{ep}(x) \) the even periodic extension of \( f(x) \). - Consider when we use periodic functions of the following form to approximate \( f_{ep}(x) \): \[ T(x) = A_0 + A_1 \cos x + B_1 \sin x + A_2 \cos(2x) + B_2 \sin(2x) \] - The error in mean is defined by: \[ \int_{-\pi}^{\pi} |f_{ep}(x) - T(x)|^2 \, dx \] - Determine the coefficients \( A_0, A_1, B_1, A_2, B_2 \) that minimize the error in mean. This transcription provides detailed instructions for students to perform tasks related to Fourier series, focusing on periodic extensions, convergence, and approximation.
Transcribed Image Text:**Mathematics and Fourier Series: Periodic Extensions and Fourier Series** Consider the function \( f(x) \) defined by: \[ f(x) = \begin{cases} 1 & 0 < x < \pi/2 \\ 2 & \pi/2 < x < \pi \end{cases} \] **Tasks:** 1. (a)-(c) [redacted] 2. (d) Sketch the odd periodic extension of \( f \). **Fourier Series Analysis:** (e) **Find the Fourier sine series of \( f \)**. (f) **Determine the convergence of the Fourier sine series**: - At \( x = 0 \) - At \( x = \pi/2 \) - At \( x = \pi \) - At \( x = 3\pi/2 \) - At \( x = 2\pi \) (g) **Even Periodic Extensions**: - Denote by \( f_{ep}(x) \) the even periodic extension of \( f(x) \). - Consider when we use periodic functions of the following form to approximate \( f_{ep}(x) \): \[ T(x) = A_0 + A_1 \cos x + B_1 \sin x + A_2 \cos(2x) + B_2 \sin(2x) \] - The error in mean is defined by: \[ \int_{-\pi}^{\pi} |f_{ep}(x) - T(x)|^2 \, dx \] - Determine the coefficients \( A_0, A_1, B_1, A_2, B_2 \) that minimize the error in mean. This transcription provides detailed instructions for students to perform tasks related to Fourier series, focusing on periodic extensions, convergence, and approximation.
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