calculo2_a
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calculo2_a


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\uf0e6 \uf070
3
2,E , \uf0f7
\uf0f8
\uf0f6
\uf0e7
\uf0e8
\uf0e6 \uf070
6
5,3F e \uf0f7
\uf0f8
\uf0f6
\uf0e7
\uf0e8
\uf0e6 \uf070\uf02d
3
8,3G . 
Resolução: 
2
\uf070
3
\uf070
4
\uf070
\uf070
6
\uf070
3
\uf0702
4
\uf0703
6
\uf0705
6
\uf0707
4
\uf0705
3
\uf0704
3
\uf0705 4
\uf0707
6
\uf07011
2
\uf0703
\uf070
0
2
 
Resposta: 
 
 
 
 
 Cálculo II \u2013 (Lauro / Nunes) 2-4
2. Represente no plano os pontos ),( \uf071\uf072 onde: 
)
2
,1( \uf070\uf02d\uf02dA , )3,3( \uf070B , \uf0f7
\uf0f8
\uf0f6
\uf0e7
\uf0e8
\uf0e6 \uf070
4
7,2C , \uf0f7
\uf0f8
\uf0f6
\uf0e7
\uf0e8
\uf0e6 \uf070\uf02d\uf02d
4
3,
2
3D , \uf0f7
\uf0f8
\uf0f6
\uf0e7
\uf0e8
\uf0e6 \uf070\uf02d
6
,2E , \uf0f7
\uf0f8
\uf0f6
\uf0e7
\uf0e8
\uf0e6 \uf070\uf02d
6
31,3F e \uf0f7
\uf0f8
\uf0f6
\uf0e7
\uf0e8
\uf0e6 \uf070\uf02d
4
5,2G . 
Resolução: 
2
\uf070
3
\uf070
4
\uf070
\uf070
6
\uf070
3
\uf0702
4
\uf0703
6
\uf0705
6
\uf0707
4
\uf0705
3
\uf0704
3
\uf0705 4
\uf0707
6
\uf07011
2
\uf0703
\uf070
0
2
 
Resposta: 
2.2.1 Relações entre Coordenadas Cartesianas e Polares 
Para a representação do mesmo ponto em coordenadas cartesianas e coordenadas 
polares vamos tomar o ponto O como origem dos dois sistemas. Tome também o eixo polar 
coincidindo com o eixo \u201cOx\u201d. Se P não coincidir com o pólo (origem), temos: 
\uf071
\uf072
x
P
O
y
 
 
\uf0ee
\uf0ed
\uf0ec
\uf071\uf072\uf03d
\uf071\uf072\uf03d
sin
cos
y
x
 \uf0db 
\uf0ef
\uf0ef
\uf0ef
\uf0ee
\uf0ef
\uf0ef
\uf0ef
\uf0ed
\uf0ec
\uf0f7
\uf0f8
\uf0f6
\uf0e7
\uf0e8
\uf0e6\uf03d\uf071
\uf0ef
\uf0ef
\uf0fe
\uf0ef
\uf0ef
\uf0fd
\uf0fc
\uf02b
\uf03d\uf071
\uf02b
\uf03d\uf071
\uf02b\uf03d\uf072
x
y
yx
y
yx
x
yx
arctan
sin
cos
22
22
22
 
 
),( \uf071\uf072 é o ponto em coordenadas polares. 
),( yx é o ponto em coordenadas cartesianas. 
 
 
 
 Cálculo II \u2013 (Lauro / Nunes) 2-5
Definição 
Uma função em coordenadas polares é uma relação que associa a cada ângulo \uf071 
(medido em radianos) um único real \uf072 (que pode ser negativo). Representa-se por: 
 \uf072 \uf03d )(\uf071f 
Existem alguns casos especiais de funções em coordenadas polares que serão tratados 
a seguir. 
2.2.2 Caso Geral da Espiral de Arquimedes 
 \uf072 \uf03da \uf071 (\uf022 a \uf0b90; a \uf0ce\uf0c2) 
3. Construir o gráfico da função: 
\uf072 \uf03d \uf071, para 0 \uf0a3 \uf071 \uf0a3 2\uf070. 
\uf071 0 
4
\uf070 
2
\uf070 
3
2\uf070 \uf070 
4
5\uf070 
2
3\uf070 
4
7\uf070 2\uf070 
\uf072 0 
4
\uf070 
2
\uf070 
3
2\uf070 \uf070 
4
5\uf070 
2
3\uf070 
4
7\uf070 2\uf070 
\uf072~ 0 0,8 1,6 2,1 3,1 3,9 4,7 5,5 6,3 
Resolução: 
2
\uf070
3
\uf070
4
\uf070
\uf070
6
\uf070
3
\uf0702
4
\uf0703
6
\uf0705
6
\uf0707
4
\uf0705
3
\uf0704
3
\uf0705 4
\uf0707
6
\uf07011
2
\uf0703
\uf070
0
2
 
Resposta: 
2.2.3 Constante 
\uf072 \uf03d R (constante) é um círculo de raio R . 
 
R
 
 
 Cálculo II \u2013 (Lauro / Nunes) 2-6
2.2.4 Caso Geral da Cardióide 
O gráfico de qualquer uma das equações polares seguintes, com a \uf0b90, é uma 
CARDIÓIDE: 
 \uf072 \uf03d a (1\uf02b \uf071cos ) 
 \uf072 \uf03d a (1\uf02d \uf071cos ) 
 \uf072 \uf03da (1\uf02b \uf071sin ) 
 \uf072 \uf03da (1\uf02d \uf071sin ) 
4. Construir o gráfico da função: 
\uf072 \uf03d 2 \uf02b 2 \uf071cos (cardióide). 
Resolução: 
\uf071 0 
6
\uf070 
4
\uf070 
3
\uf070 
2
\uf070 
3
2\uf070 
4
3\uf070 
6
5\uf070 \uf070 
\uf072 
\uf072~ 
2
\uf070
3
\uf070
4
\uf070
\uf070
6
\uf070
3
\uf0702
4
\uf0703
6
\uf0705
6
\uf0707
4
\uf0705
3
\uf0704
3
\uf0705 4
\uf0707
6
\uf07011
2
\uf0703
\uf070
0
2
 
Resposta: 
2.2.5 Caso Geral do Caracol 
Se a e b não são nulos, então os gráficos das equações polares seguintes são 
CARACÓIS. 
 \uf072 \uf03d a \uf02b \uf071cosb , 
 \uf072 \uf03d a \uf02b \uf071sinb . 
 
 
 
 
 
 Cálculo II \u2013 (Lauro / Nunes) 2-7
5. Construir o gráfico da função: 
\uf072 \uf03d 2 \uf02b 4 \uf071cos (caracol). 
Resolução: 
\uf071 0 
6
\uf070 
4
\uf070 
3
\uf070 
2
\uf070 
3
2\uf070 
4
3\uf070 
6
5\uf070 \uf070 
\uf072 
\uf072~ 
2
\uf070
3
\uf070
4
\uf070
\uf070
6
\uf070
3
\uf0702
4
\uf0703
6
\uf0705
6
\uf0707
4
\uf0705
3
\uf0704
3
\uf0705 4
\uf0707
6
\uf07011
2
\uf0703
\uf070
0
2
 
Resposta: 
2.2.6 Caso Geral da Rosácea 
Qualquer uma das equações abaixo representa uma rosácea, considerando as 
condições seguintes: 
\uf022 a \uf0b90; a\uf0ce\uf0c2 e 
\uf022 n \uf03e1; n\uf0ce N 
 \uf072 \uf03d \uf071na sin 
 \uf072 \uf03d \uf071na cos 
O gráfico consiste em um certo número de laços pela origem. 
\uf0b7 Se n é par, há 2 n laços; 
\uf0b7 Se n é ímpar, há n laços. 
 
 
 
 
 
 
 
 
 
 
 
 
 Cálculo II \u2013 (Lauro / Nunes) 2-8
6. Construir os gráficos das rosáceas nos itens a) e b). 
Rosáceas de quatro pétalas (folhas): 
a) \uf072 \uf03d 3 \uf0712sin 
Resolução: 
\uf071 0 
6
\uf070 
4
\uf070 
3
\uf070 
2
\uf070 
3
2\uf070 
4
3\uf070 
6
5\uf070 \uf070 
\uf072~ 
2
\uf070
3
\uf070
4
\uf070
\uf070
6
\uf070
3
\uf0702
4
\uf0703
6
\uf0705
6
\uf0707
4
\uf0705
3
\uf0704
3
\uf0705 4
\uf0707
6
\uf07011
2
\uf0703
\uf070
0
2
 
Resposta: 
b) \uf072 \uf03d 3 \uf0712cos 
Resolução: 
\uf071 0 
6
\uf070 
4
\uf070 
3
\uf070 
2
\uf070 
3
2\uf070 
4
3\uf070 
6
5\uf070 \uf070 
\uf072~ 
2
\uf070
3
\uf070
4
\uf070
\uf070
6
\uf070
3
\uf0702
4
\uf0703
6
\uf0705
6
\uf0707
4
\uf0705
3
\uf0704
3
\uf0705 4
\uf0707
6
\uf07011
2
\uf0703
\uf070
0
2
 
Resposta: 
 
 
 
 Cálculo II \u2013 (Lauro / Nunes) 2-9
7. Se considerarmos o quadrado do primeiro termo na rosácea seguinte, temos: 
\uf0722 \uf03d 4 \uf0712cos (Lemniscata de Bernoulli). 
Dicas para fazer o gráfico: 
\uf072 \uf03d 2 \uf0712cos 0 \uf0a3 \uf0712cos \uf0a3 1 
Tome D\uf072 como o domínio de \uf072 tal que: 
D\uf072 \uf03d {\uf071\uf0ceR; \uf02d
2
\uf070 \uf02b 2n\uf070 \uf0a3 2\uf071 \uf0a3 
2
\uf070 \uf02b 2n\uf070, com n\uf0ceZ} 
D\uf072 \uf03d {\uf071\uf0ceR; \uf02d
4
\uf070 \uf02b n\uf070 \uf0a3 \uf071 \uf0a3 
4
\uf070 \uf02b n\uf070, com n\uf0ceZ} 
Resolução: 
\uf071 0 
6
\uf070 
4
\uf070 
3
\uf070 
2
\uf070 
3
2\uf070 
4
3\uf070 
6
5\uf070 \uf070 
\uf072~ 
2
\uf070
3
\uf070
4
\uf070
\uf070
6
\uf070
3
\uf0702
4
\uf0703
6
\uf0705
6
\uf0707
4
\uf0705
3
\uf0704
3
\uf0705 4
\uf0707
6
\uf07011
2
\uf0703
\uf070
0
2
 
Resposta: 
2.3 Gráficos diversos em coordenadas polares 
2.3.1 Equação do pólo (origem) 2.3.2 Equação que passa pela origem 
\uf072 \uf03d 0 
\uf071 \uf03d r (r constante) 
\uf071 \uf03d 
6
\uf070 ou \uf071 \uf03d 
6
7\uf070 
 
2
\uf070
3
\uf070
4
\uf070
\uf070
6
\uf070
3
\uf0702
4
\uf0703
6
\uf0705
6
\uf0707
4
\uf0705
3
\uf0704
3
\uf0705 4
\uf0707 6
\uf07011
2
\uf0703
\uf070
0
2
 
 
 
2
\uf070
3
\uf070
4
\uf070
\uf070
6
\uf070
3
\uf0702
4
\uf0703
6
\uf0705
6
\uf0707
4
\uf0705
3
\uf0704
3
\uf0705 4
\uf0707 6
\uf07011
2
\uf0703
\uf070
0
2
 
 
 Cálculo II \u2013 (Lauro / Nunes) 2-10
2.3.3 Retas paralelas e perpendiculares ao eixo polar 
a) \uf072\uf0d7sen\uf071 \uf03d b 
\uf072\uf0d7sin\uf071 \uf03d 3 ou \uf072 \uf03d 
\uf071sin
3 \uf072\uf0d7sin\uf071 \uf03d \uf02d3 ou \uf072 \uf03d \uf02d
\uf071sin
3 
2
\uf070
3
\uf070
4
\uf070
\uf070
6
\uf070
3
\uf0702
4
\uf0703
6
\uf0705
6
\uf0707
4
\uf0705
3
\uf0704
3
\uf0705 4
\uf0707 6
\uf07011
2
\uf0703
\uf070
0
2
 
2
\uf070
3
\uf070
4
\uf070
\uf070
6
\uf070
3
\uf0702
4
\uf0703
6
\uf0705
6
\uf0707
4
\uf0705
3
\uf0704
3
\uf0705 4
\uf0707 6
\uf07011
2
\uf0703
\uf070
0
2
 
 
b) \uf072\uf0d7cos\uf071 \uf03d a 
\uf072\uf0d7cos\uf071 \uf03d 3 ou \uf072 \uf03d 
\uf071cos
3 \uf072\uf0d7cos\uf071 \uf03d \uf02d3 ou \uf072 \uf03d \uf02d
\uf071cos
3 
2
\uf070
3
\uf070
4
\uf070
\uf070
6
\uf070
3
\uf0702
4
\uf0703
6
\uf0705
6
\uf0707
4
\uf0705
3
\uf0704
3
\uf0705 4
\uf0707 6
\uf07011
2
\uf0703
\uf070
0
2
 
2
\uf070
3
\uf070
4
\uf070
\uf070
6
\uf070
3
\uf0702
4
\uf0703
6
\uf0705
6
\uf0707
4
\uf0705
3
\uf0704
3
\uf0705 4
\uf0707 6
\uf07011
2
\uf0703
\uf070
0
2
 
2.3.4 Algumas circunferências 
a) \uf072 \uf03d r (constante) 
\uf072 \uf03d 2 
2
\uf070
3
\uf070
4
\uf070
\uf070
6
\uf070
3
\uf0702
4
\uf0703
6
\uf0705
6
\uf0707
4
\uf0705
3
\uf0704
3
\uf0705 4
\uf0707 6
\uf07011
2
\uf0703
\uf070
0
2
 
 
b) \uf072 \uf03d 2a\uf0d7cos\uf071 
\uf072 \uf03d 4cos\uf071 \uf0de (a \uf03e 0) \uf072 \uf03d \uf02d4cos\uf071 \uf0de (a \uf03c 0) 
2
\uf070
3
\uf070
4
\uf070
\uf070
6
\uf070
3
\uf0702
4
\uf0703
6
\uf0705
6
\uf0707
4
\uf0705
3
\uf0704
3
\uf0705 4
\uf0707 6
\uf07011
2
\uf0703
\uf070
0
2
 
2
\uf070
3
\uf070
4
\uf070
\uf070
6
\uf070
3
\uf0702
4
\uf0703
6
\uf0705
6
\uf0707
4
\uf0705
3
\uf0704
3
\uf0705 4
\uf0707 6
\uf07011
2
\uf0703
\uf070
0
2
 
 
 
 Cálculo II \u2013 (Lauro / Nunes) 2-11
c) \uf072 \uf03d 2b\uf0d7sin\uf071 
\uf072 \uf03d 4sin\uf071 \uf0de (b \uf03e 0) \uf072 \uf03d \uf02d4sin\uf071 \uf0de (b \uf03c 0) 
2
\uf070
3
\uf070
4
\uf070
\uf070
6
\uf070
3
\uf0702
4
\uf0703
6
\uf0705
6
\uf0707
4
\uf0705
3
\uf0704
3
\uf0705 4
\uf0707 6
\uf07011
2
\uf0703
\uf070
0
2
 
2
\uf070
3
\uf070
4
\uf070
\uf070
6
\uf070
3
\uf0702
4
\uf0703
6
\uf0705
6
\uf0707
4
\uf0705
3
\uf0704
3
\uf0705 4
\uf0707 6
\uf07011
2
\uf0703
\uf070
0
2
 
2.3.5 Limaçons 
\uf072 \uf03d a \uf0b1 b\uf0d7cos\uf071 ou \uf072 \uf03d a \uf0b1 b\uf0d7sin\uf071, onde a, b \uf0ce R. 
a) Se b \uf03e a \uf0de a curva tem um laço 
\uf072 \uf03d 1 \uf02b 2cos\uf071 \uf072 \uf03d 1 \uf02d 2cos\uf071 
2
\uf070
3
\uf070
4
\uf070
\uf070
6
\uf070
3
\uf0702
4
\uf0703
6
\uf0705
6
\uf0707
4
\uf0705
3
\uf0704
3
\uf0705 4
\uf0707 6
\uf07011
2
\uf0703
\uf070
0
2
 
2
\uf070
3
\uf070
4
\uf070
\uf070
6
\uf070
3
\uf0702
4
\uf0703
6
\uf0705
6
\uf0707
4
\uf0705
3
\uf0704
3
\uf0705 4
\uf0707 6
\uf07011
2
\uf0703
\uf070
0
2
 
 
\uf072 \uf03d 1 \uf02b 2sin\uf071