Aula 9 - Séries de Taylor, Maclaurin e Fourier - Exercícios
1 pág.

Aula 9 - Séries de Taylor, Maclaurin e Fourier - Exercícios


DisciplinaEquações Diferenciais I6.065 materiais35.000 seguidores
Pré-visualização1 página
Equações Diferenciais e Séries 
Professor Hans 
Aula 9: Séries de Taylor, Maclaurin e Fourier - Exercícios 
 
1) Faça a expansão em série de Taylor solicitada: 
 
a) 
( ) xf x e\uf03d
 até terceira ordem com 
0a \uf03d
. 
b) 
\uf028 \uf029
2
( ) 2f x x
\uf02d
\uf03d \uf02b
 até terceira ordem com 
0a \uf03d
. 
c) 
3( )f x x\uf03d
 até segunda ordem com 
8a \uf03d
. 
d) 
( )f x senx\uf03d
 até quinta ordem com 
0a \uf03d
. 
e) 
\uf028 \uf029
1
2( ) 1f x x
\uf02d
\uf03d \uf02b
 até terceira ordem com 
0a \uf03d
. 
 
2) Mostre que a equação 
 
 
2 2 2
( ) 1
2
Q Z
E r
R Z R\uf070 \uf065
\uf0e6 \uf0f6
\uf03d \uf02d\uf0e7 \uf0f7
\uf02b\uf0e8 \uf0f8
 
 
 
para o campo de um disco carregado, em pontos sobre 
seu eixo, reduz-se ao campo de uma carga pontual 
2
( )
4
Q
E r
Z\uf070\uf065
\uf03d
 para 
Z R
. 
Sugestão: use a expansão 
1
2 2
2
1
R
Z
\uf02d
\uf0e6 \uf0f6
\uf02b\uf0e7 \uf0f7
\uf0e8 \uf0f8
 até a primeira 
ordem. 
 
 
 
3) Calcule a série de Fourier de f (x) no intervalo 
dado: 
 
a) 0, 0
( )
1,0
x
f x
x
\uf070
\uf070
\uf02d \uf0a3 \uf03c\uf0ec
\uf03d \uf0ed
\uf0a3 \uf03c\uf0ee
 
 
b) 1, 1 0
( )
,0 1
x
f x
x x
\uf02d \uf03c \uf03c\uf0ec
\uf03d \uf0ed
\uf0a3 \uf03c\uf0ee
 
 
c)
( )f x x \uf070\uf03d \uf02b
, 
x\uf070 \uf070\uf02d \uf03c \uf03c
 
 
 
 
 
 
 
 
 
4) (Onda Quadrada) Escreva a série de Fourier que 
descreve a onda quadrada abaixo com 
x\uf070 \uf070\uf02d \uf0a3 \uf0a3
 até 
n = 7. 
 
1,
2 2
( )
0,
2
se x
f x
se x
\uf070 \uf070
\uf070
\uf0ec
\uf02d \uf0a3 \uf0a3\uf0ef\uf0ef
\uf03d \uf0ed
\uf0ef \uf03e
\uf0ef\uf0ee
 
 
Gabarito 
1) 
a) 2 3
( ) 1
2 6
x x
f x x\uf03d \uf02b \uf02b \uf02b
 b) 2 31 3
( )
4 4 16 8
x x x
f x \uf03d \uf02d \uf02b \uf02d
 
c) \uf028 \uf029 \uf028 \uf02928 8
( ) 2
12 288
x x
f x
\uf02d \uf02d
\uf03d \uf02b \uf02d
d) 3 5
( )
6 120
x x
f x x\uf03d \uf02d \uf02b
 
e) 2 33 5
( ) 1
2 8 16
x x x
f x \uf03d \uf02d \uf02b \uf02d
 
 
2) Demonstração 
 
3) 
a) \uf028 \uf029
1
1 11 1
( ) n
2
n
n
f x sen x
n\uf070
\uf0a5
\uf03d
\uf02d \uf02d
\uf03d \uf02b \uf0e5
 
b) \uf028 \uf029
2 2
1
1 13 1
( ) cos n
4
n
n
f x n x sen x
n n
\uf070 \uf070\uf070 \uf070
\uf0a5
\uf03d
\uf0ec \uf0fc\uf02d \uf02d\uf0ef \uf0ef
\uf03d \uf02b \uf02d\uf0ed \uf0fd
\uf0ef \uf0ef\uf0ee \uf0fe
\uf0e5
 
c) \uf028 \uf029 1
1
1
( ) 2 n
n
n
f x sen x
n
\uf070
\uf02b
\uf0a5
\uf03d
\uf02d
\uf03d \uf02b \uf0e5
 
4)
1 2 2 2 2
( ) cos cos3 cos5 cos7
2 3 5 7
f x x x x x\uf070 \uf070 \uf070 \uf070\uf03d \uf02b \uf02d \uf02b \uf02d