Question

Outra forma de representar as séries de Fourier são através das equações trigonométricas formadas por senos, cossenos e exponenciais que pode apresentar a seguinte equação característica:

x left parenthesis space t thin space right parenthesis space equals space a subscript 0 over 2 plus space sum from k equals 1 to infinity of open parentheses a subscript k times space cos left parenthesis k space times omega subscript 0 times space t right parenthesis space plus space b subscript k times space s e n space left parenthesis k times space omega subscript 0 space times t right parenthesis space close parenthesese m space q u ea subscript k space equals space 2 over T subscript 0 times integral subscript T subscript 0 end subscript space x left parenthesis space t space right parenthesis space times cos left parenthesis k space times omega subscript 0 times space t right parenthesis space times d tb subscript k space equals space 2 over T subscript 0 integral subscript T subscript 0 end subscript space x left parenthesis space t space right parenthesis space times s e n left parenthesis k times space omega subscript 0 times end subscript space t right parenthesis space times d t



Considere agora a função f space left parenthesis space x space right parenthesis space equals space x cubed, que apresenta o Gráfico a seguir.



Gráfico 1 - Função f space left parenthesis space x space right parenthesis space equals space x cubed.

f(x) = x³

Fonte: Mello, 2017.

Dada a função cúbica e seu gráfico (Gráfico 1), assinale a alternativa que apresenta corretamente a expansão da Série de Fourier no intervalo negative pi space less than space x space less than space plus pi:

Escolha uma:
a.
f left parenthesis space x space right parenthesis space equals space fraction numerator partial differential space x cubed over denominator partial differential space x end fraction space equals space 3 space times x squared

b.
f left parenthesis space x space right parenthesis space equals integral x times space cos space left parenthesis k space times x right parenthesis times space d x space equals space 1 over k squared times space cos left parenthesis k times space x right parenthesis space plus space x over k space times s e n left parenthesis k times space x right parenthesis

c.
f left parenthesis x right parenthesis space equals space sum from n equals 1 to infinity of space open square brackets 2 space times space open parentheses fraction numerator negative pi squared over denominator n end fraction plus 6 over n cubed close parentheses times space cos open parentheses n times pi close parentheses close square brackets space times s e n space left parenthesis n times x right parenthesis Correto

d. f left parenthesis space x space right parenthesis space equals space integral x cubed space times d x space equals space x to the power of 4 over 4 space plus space C
e.
f left parenthesis space x space right parenthesis space equals space 3 over 2 space plus space sum from n equals 1 to infinity of open square brackets fraction numerator 6 over denominator n squared times pi squared end fraction times open parentheses 1 space minus space cos left parenthesis n times pi right parenthesis close parentheses close square brackets times cos open parentheses fraction numerator n times pi times x over denominator 3 end fraction close parentheses

Answer 1

essa está errada. Veja na imagem qual é errada.

Answer 2

começa com somatorio de n=1 tendendo ao infinito[ 2*(-pi^2/n+6/n^3) cos(n*pi) ] * sen (n*x)

corrigida pelo ava