apostila matematica colfundamental 5 8
DESCRIPTION
Apostila de Matemática BasicaTRANSCRIPT
APOSTILA DE MATEMTICA BSICA - Coleo Fundamental - 5/8
PAGE www.ResumosConcursos.hpg.com.br
Apostila: Matemtica Bsica vol. V por Prof. Paulo Cesar Pfaltzgraff Ferreira
Apostila de Matemtica Bsica
Assunto:
MATEMTICA BSICAColeo Fundamental - volume 5/8
Autor:
Prof. Paulo Cesar Pfaltzgraff Ferreiralogo
porm
de modo que
ou
Uma conseqncia imediata da desigualdade do tringulo que
(89)
que pode ser demonstrada a partir de
o que nos leva a
(90)
que a desigualdade (89) quando . Se tivermos , basta trocar e na desigualdade (90) para obter
que o resultado desejado.
A desigualdade (89) traduz o fato de que o comprimento de um todo de um tringulo no pode ser menor que a diferena dos comprimentos dos outros dois lados.
Podemos tambm obter formas alternativas teis para as desigualdades (88) e (89) trocando por :
(91)
(92)Exemplo 1.28
Verificar a desigualdade do tringulo para e .
Soluo:Temos que:
e est verificada a desigualdade.
1.14.6 Curvas e Regies no Plano Complexo
Vamos considerar agora alguns tipos importantes de curvas e regies no plano complexo e suas representaes por meio de equaes e desigualdades.
a) Circunferncia
Conforme j visto em 1.14.4.b, a distncia entre os pontos do plano definidos pelos complexos e . Segue-se ento que uma circunferncia C de raio ( com centro em um ponto a (fig. 1.31) pode ser representada sob a forma
(93)
Fig. 1.31
Onde z um ponto qualquer da circunferncia.
Exemplo 1.29
Identificar o lugar geomtrico representado por (a) ; (b) .
a)
Fig. 1.32
Trata-se pois de uma circunferncia centrada em e raio 3.
b)
Fig. 1.33
Temos ento uma circunferncia centrada em e raio 4.
b) Disco Fechado
Em conseqncia do que foi visto em (a), para um disco fechado de raio ( e centro em a temos:
(94)
Fig. 1.34
Exemplo 1.30
Identificar o lugar geomtrico representado por .
Fig. 1.35
c) Disco Aberto
Para o disco aberto temos:
(95)
Fig. 1.36
d) Exterior da Circunferncia
Semelhantemente a desigualdade
(96)Representa o exterior da circunferncia de raio ( centrada em a.
Fig. 1.37
e) Coroa Fechada
A regio entre duas circunferncias concntricas de raios e , sendo , pode ser representada por:
(97)
Fig. 1.38
f) Coroa Aberta
Temos ento:
(98)
Fig. 1.39
g) Circunferncia Unitria
A equao
(99)representa a chamada circunferncia unitria, ou seja, a circunferncia de raio 1 e centro na origem, que representa papel importante na seqncia do estudo de variveis complexas.
Fig. 1.40
h) Reta que une dois pontos
Fig. 1.41
Sejam e os complexos representando dois pontos quaisquer e do plano, conforme aparece na figura 1.41, e z o complexo representando um ponto P qualquer da reta que passa pelos dois pontos inicialmente mencionados.
Da figura em questo percebe-se que:
Sendo AP e AB segmentos orientados paralelos temos:
AP = kAB
o que nos permite escrever:
+ kABNo entanto,
AB
o que nos conduz a:
(100a)
(100a)Se queremos representar qualquer ponto da reta devemos ter mas, se queremos representar apenas os pontos do segmento que une os pontos e devemos ter .
1.15. Exerccios Propostos sobre Nmeros Complexos
1. Determine para que o nmero complexo seja real.
2. Determine os valores reais de a para os quais um nmero real.
3. Efetuar as seguintes potncias:
a)
b)
c)
d)
e)
f)
4. Calcular .
5. Calcule o mdulo e o argumento dos seguintes nmeros complexos:
a)
b)
c)
d)
e)
f)
6. Exprimir cada um dos seguintes nmeros complexos na forma polar:
a)
b)
c)
7. Passar os seguintes nmeros complexos da forma polar para a forma retangular:
a)
b)
c)
8. Escrever na forma trigonomtrica os seguintes nmeros complexos e represent-los no plano de Argand-Gauss:
a)
b)
c)
9. Determinar x e y ( R de modo que .
10. Determine x e y ( R tais que .
11. Calcule o valor de n sabendo-se que .
12. Calcular
a)
b)
c)
d)
e)
f)
g)
h)
13. Se e calcular .
14. Calcular os seguintes produtos:
a)
EMBED Equation.3
b)
15. Sendo calcular .
16. Sendo
calcular .
17. Expressar na forma polar os seguintes complexos:
a)
b)
18. Calcular:
a)
b)
c)
d)
e)
f)
g)
h)
19. Determinar o nmero real x tal que o produto , onde e , seja tambm um nmero real.
20. Determine o nmero complexo z tal que .
21. Determine a ( R para que seja imaginrio puro.
22. Determinar o resultado da expresso sendo e .
23. Sendo e dois nmeros complexos, resolver o sistema .
24. Calcule o argumento do complexo .
25. Sendo e calcular apresentando o resultado na forma trigonomtrica.
26. Dados e determine:
a)
b)
c)
d)
e)
f)
g)
27. Dados , e , calcular:
a)
b)
c)
d)
e)
28. Os ngulos agudos de um tringulo retngulo so ( e (. Calcule o valor do produto .
29. Calcular .
30. Dado o nmero complexo calcular .
31. Calcular .
32. Calcular .
33. Achar o conjugado do complexo onde , com e .
34. Calcular o menor valor natural n para o qual um imaginrio puro.
35. Calcule o valor da expresso .
36. Calcule o resultado de .
37. Dados os complexos e calcule o valor de .
38. Calcular as seguintes razes e represent-las no plano complexo:
a)
b)
c)
d)
e)
f)
g)
h)
i)
39. Determinar o conjunto-soluo em C para cada uma das seguintes equaes:
a)
b)
c)
d)
e)
f)
g)
h)
i)
j)
40. Demonstre, por induo matemtica, a desigualdade seguinte, e interprete o resultado graficamente.
41. Estabelecer as equaes cartesianas, identificar e traar os grficos dos lugares geomtricos representados por:
a)
b)
c)
d)
e)
f)
g)
h)
i)
j)
k)
l)
m)
n)
o)
p)
q)
r)
s)
t)
u)
v)
w)
x)
y)
z) , sendo e nmeros complexos quaisquer, ( e ( reais e no negativos e .
1.16. Resposta dos Exerccios Propostos sobre Nmeros Complexos:
1.
2. 0; 1 e
3. a) 1; b) 1; c) j; d) 1 para n par e para n mpar; e) ; f) j par n par e para n mpar.
4. j5. a) 1,414 e 45; b) 2 e 90; c) 3 e 0; d) 2 e 60; e) 3 e 90; f) 2 e 30.
6. a)
; b)
; c)
7. a) ; b) ; c)
8. a)
b)
c)
9. x = 7, y = 2
10. x = 0, y = 2
11. n = 3
12. a) ; b) 7; c) ; d) ; e) 16; f) 5; g) 5 + j3; h) 15 j13.
14. a)
; b)
15. 6,25
16. 100
17. a)
; b)
18. a) 1,5 j0,5; b) 1 j; c) j; d) 1 + j1,414;
e) 1,176 j0,117; f) 1,5 j1,5; g) 3,9 + j1,3 ; h) 2,5 j0,5.
19. x = 8
20. 0 e j2
21. a = 6
22.
23. e
24.
25.
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EMBED Equation.3
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_1066123875.doc
EMBED Equation.3
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_1066124011.doc
EMBED Equation.3
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EMBED Equation.3
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EMBED Equation.3
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_1061877972.unknown
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EMBED Equation.3
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_1061877972.unknown
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_1066017040.doc
EMBED Equation.3
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_1061876463.unknown
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_1066017104.doc
EMBED Equation.3
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_1061876463.unknown
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_1066017219.doc
EMBED Equation.3
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_1061876463.unknown
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EMBED Equation.3
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_1061876463.unknown
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EMBED Equation.3
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_1061876463.unknown
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EMBED Equation.3
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