5 Ejercicios Deber.docx

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ESCUELA SUPERIOR POLITECNICA DE CHIMBORAZO FACULTAD DE INFORMÁTICA Y ELECTRÓNICA CARRERA DE ELECTRÓNICA Y AUTOMATIZACIÓN NOMBRE: Gómez Regina CÓDIGO: 290 CURSO: 3° “B” FECHA: 30/11/2018 EJERCICIO 1 𝑓(𝑥) = { 𝑎0 =

0 1

2𝜋 1 𝜋 [∫ 0 𝑑𝑡 + ∫ 1 𝑑𝑡] 𝜋 0 𝜋

𝒂𝟎 = 𝟏 𝑎𝑛 =

2𝜋 1 𝜋 [∫ 0 cos(𝑛𝑡)𝑑𝑡 + ∫ 1 ∗ cos(𝑛𝑡)𝑑𝑡] 𝜋 0 𝜋

𝒂𝒏 = 𝟎 𝑏𝑛 =

2𝜋 1 𝜋 [∫ 0 sin(𝑛𝑡)𝑑𝑡 + ∫ 1 ∗ sin(𝑛𝑡)𝑑𝑡] 𝜋 0 𝜋

𝒃𝒏 = −

(−𝟏)𝒏 − 𝟏 𝝅𝒏 ∞

𝟏 (−𝟏)𝒏 − 𝟏 𝒇(𝒕) = + ∑ − ∗ 𝒔𝒊𝒏(𝒏𝒕) 𝟐 𝝅𝒏 𝒏=𝟏

0<𝑡<𝜋 𝜋 < 𝑡 < 2𝜋

EJERCICIO 2 𝑡 −𝜋 <𝑡 <0 𝑓(𝑥) = { 𝑡 0<𝑡<𝜋 𝑎0 =

𝜋 1 0 [∫ 𝑡 𝑑𝑡 + ∫ 𝑑𝑡 ] 𝜋 −𝜋 0

𝒂𝟎 = 𝟎 𝑎𝑛 =

𝜋 1 0 [∫ 𝑡 𝑐𝑜𝑠(𝑛𝑡) 𝑑𝑡 + ∫ 𝑡𝑐𝑜𝑠(𝑛𝑡)𝑑𝑡] 𝜋 −𝜋 0

𝒂𝒏 = 𝟎 𝑏𝑛 =

2𝜋 1 𝜋 [∫ 0 sin(𝑛𝑡)𝑑𝑡 + ∫ 1 ∗ sin(𝑛𝑡)𝑑𝑡] 𝜋 0 𝜋

𝒂𝒏 =

𝟐 ∗ (−𝒄𝒐𝒔(𝒏𝝅) − 𝒄𝒐𝒔(−𝝅𝒏)) 𝒏 ∞

𝒇(𝒕) = ∑ 𝒏=𝟏

𝟐 ∗ (−𝒄𝒐𝒔(𝒏𝝅) − 𝒄𝒐𝒔(−𝝅𝒏)) ∗ 𝒔𝒊𝒏(𝒏𝒕) 𝒏

EJERCICIO 3 2 𝑓(𝑥) = {𝐴𝑡

𝑎0 =

𝐴 𝜋 2 [∫ 𝑡 𝑑𝑡] 𝜋 −𝜋

𝒂𝟎 =

𝟐𝑨𝝅𝟐 𝟑

−𝜋<𝑡 <𝜋 −

𝐴 𝜋 2 𝑎𝑛 = [∫ 𝑡 cos(𝑛𝑡) 𝑑𝑡] 𝜋 −𝜋 𝒂𝒏 =

𝟐𝑨 (𝒄𝒐𝒔(𝒏𝝅) + 𝒄𝒐𝒔(−𝒏𝝅)) 𝒏𝟐

𝑏𝑛 =

𝐴 𝜋 2 [∫ 𝑡 sin(𝑛𝑡) 𝑑𝑡] 𝜋 −𝜋

𝒃𝒏 = 𝟎 ∞

𝟐𝑨𝝅𝟐 𝟐𝑨 𝒇(𝒕) = + ∑ 𝟐 (𝒄𝒐𝒔(𝒏𝝅) + 𝒄𝒐𝒔(−𝒏𝝅)) ∗ 𝒄𝒐𝒔(𝒏𝒕) 𝟑 𝒏 𝒏=𝟏

EJERCICIO 4 𝑡 𝑓(𝑥) = {𝐴𝑒

𝑎0 =

𝐴 𝜋 𝑡 [∫ 𝑒 𝑑𝑡] 𝜋 −𝜋

𝒂𝟎 =

(𝒆𝝅 − 𝒆−𝝅 ) 𝝅

𝑎𝑛 =

𝐴 𝜋 𝑡 [∫ 𝑒 𝑑𝑡] 𝜋 −𝜋

−𝜋 <𝑡 <𝜋 −

(−𝟏)𝒏 (𝒆𝝅 − 𝒆−𝝅 ) 𝒂𝒏 = (𝒏𝟐 + 𝟏)𝝅 𝑏𝑛 =

𝐴 𝜋 𝑡 [∫ 𝑒 𝑑𝑡] 𝜋 −𝜋

𝒃𝒏 =

−(−𝟏)𝒏 (𝒆𝝅 − 𝒆−𝝅 )𝒏 (𝒏𝟐 + 𝟏)𝝅 ∞

(−𝟏)𝒏 (𝒆𝝅 − 𝒆−𝝅 )𝒏 (𝒆𝝅 − 𝒆−𝝅 ) (−𝟏)𝒏 (𝒆𝝅 − 𝒆−𝝅 ) 𝒇(𝒕) = +∑ ∗ 𝒄𝒐𝒔(𝒏𝒕) − ∗ 𝒔𝒊𝒏(𝒏𝒕) (𝒏𝟐 + 𝟏)𝝅 𝟐𝝅 (𝒏𝟐 + 𝟏)𝝅 𝒏=𝟏

EJERCICIO 4 1 𝑓(𝑥) = {𝐴𝑠𝑖𝑛(𝑤𝑡) 0 < 𝑡 < 2 − 1

𝐴 2 𝑎0 = [∫ 𝑠𝑖𝑛(𝑤𝑡) 𝑑𝑡] 3 0 𝒂𝟎 =

𝟐𝑨 𝝅 1

𝐴 2 𝑎𝑛 = [∫ 𝑠𝑖𝑛(𝑤𝑡) 𝑑𝑡] 3 0 𝒂𝒏 =

𝟗𝑨 𝟐(𝟏 − 𝟒𝝅𝒏𝟐 )

𝑏𝑛 =

𝐴 𝜋 𝑡 [∫ 𝑒 𝑑𝑡] 𝜋 −𝜋

𝒃𝒏 = 𝟎 ∞

𝟐𝑨 𝟗𝑨 𝒇(𝒕) = +∑ ∗ 𝒄𝒐𝒔(𝒏𝒕) 𝝅 𝟐(𝟏 − 𝟒𝝅𝒏𝟐 ) 𝒏=𝟏

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