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Questão de Matemática — Transformadas e séries — VUNESP 2022

MatemáticaTransformadas e séries
Código
vu149835
Banca
VUNESP
Órgão
EsFCEx
Ano
2022
Cargo
CFO/QC ( )
Leia o texto para responder à questão.   Considere a função f : ] -2, 2 [ →\rightarrow \mathbb{R}   f(x) = { {1,−2<x<0x,0<x<2\begin{cases} 1, -2 < x < 0\\x, 0 < x < 2 \end{cases}} A série de Fourier de f(x) é igual a
  1. A1 + \displaystyle \sum_{n=1}^∞\infty {\begin{Bmatrix} { \large 2 \over nπn \pi ^2} { [−1+(−1)n]\begin{bmatrix} -1 + ( -1)^n \end{bmatrix}} cos { \large n π\pi x \over 2} + { \large 1 \over n π\pi} { [−1+(−1)n+1]\begin{bmatrix} -1 + (-1)^{n+1} \end{bmatrix}} sen { \large n π\pi x \over 2} \end{Bmatrix}}
  2. B1 + \displaystyle \sum_{n=1}^∞\infty {\begin{Bmatrix} { \large 1 \over nπn \pi ^2} { [−1+(−1)n]\begin{bmatrix} -1 + ( -1)^n \end{bmatrix}} cos { \large n π\pi x \over 2} + { \large 1 \over n π\pi} { [−1+(−1)n+1]\begin{bmatrix} -1 + (-1)^{n+1} \end{bmatrix}} sen { \large n π\pi x \over 2} \end{Bmatrix}}
  3. C1 + \displaystyle \sum_{n=1}^∞\infty {\begin{Bmatrix} { \large 2 \over n π\pi} { [−1+(−1)n]\begin{bmatrix} -1 + ( -1)^n \end{bmatrix}} cos { \large n π\pi x \over 2} + { \large 1 \over nπn \pi^2} { [−1+(−1)n+1]\begin{bmatrix} -1 + (-1)^{n+1} \end{bmatrix}} sen { \large n π\pi x \over 2} \end{Bmatrix}}
  4. D1 + \displaystyle \sum_{n=1}^∞\infty {\begin{Bmatrix} { \large 1 \over nπn \pi} { [−1+(−1)n]\begin{bmatrix} -1 + ( -1)^n \end{bmatrix}} cos { \large n π\pi x \over 2} + { \large 1 \over nπn \pi^2} { [−1+(−1)n+1]\begin{bmatrix} -1 + (-1)^{n+1} \end{bmatrix}} sen { \large n π\pi x \over 2} \end{Bmatrix}}
  5. E1 + \displaystyle \sum_{n=1}^∞\infty {\begin{Bmatrix} { \large 2 \over nπn \pi ^2} { [−1+(−1)n]\begin{bmatrix} -1 + ( -1)^n \end{bmatrix}} sen { \large n π\pi x \over 2} + { \large 1 \over n π\pi} { [−1+(−1)n+1]\begin{bmatrix} -1 + (-1)^{n+1} \end{bmatrix}} cos { \large n π\pi x \over 2} \end{Bmatrix}}
Revelar gabarito e comentário

GabaritoA — 1 + \displaystyle \sum_{n=1}^\infty {\begin{Bmatrix} { \large 2 \over ( n \pi) ^2} { \begin{bmatrix} -1 + ( -1)^n \end{bmatrix}} cos { \large n \pi x \over 2} + { \large 1 \over n \pi} { \begin{bmatrix} -1 + (-1)^{n+1} \end{bmatrix}} sen { \large n \pi x \over 2} \end{Bmatrix}}

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