tag:blogger.com,1999:blog-34232393374266316852024-03-08T02:03:18.579-03:00Matemática na PráticaAndrei FLhttp://www.blogger.com/profile/12516487634069169995noreply@blogger.comBlogger2125tag:blogger.com,1999:blog-3423239337426631685.post-26176549312367554352012-05-26T21:30:00.001-03:002012-05-26T21:51:01.781-03:00Matemática Financeira<br />
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<b><span style="color: #990000; font-family: "Verdana","sans-serif"; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-bidi-font-size: 11.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;">MATEMÁTICA FINANCEIRA</span></b><span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"><o:p></o:p></span></div>
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<b><span style="color: #990000; font-family: "Arial","sans-serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;">Conceitos básicos</span></b><span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"><o:p></o:p></span></div>
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<span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"> </span><span style="font-family: Arial, sans-serif; font-size: 10pt;">A Matemática
Financeira é uma ferramenta útil na análise de algumas alternativas de
investimentos ou financiamentos de bens de consumo. Consiste em empregar
procedimentos matemáticos para simplificar a operação financeira a um Fluxo de
Caixa.</span><span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"><o:p></o:p></span></div>
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<b><span style="color: #990000; font-family: "Arial","sans-serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;"> Capital</span></b><span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"><o:p></o:p></span></div>
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<span style="font-family: Arial, sans-serif; font-size: 10pt;">O Capital é o valor
aplicado através de alguma operação financeira. Também conhecido como:
Principal, Valor Atual, Valor Presente ou Valor Aplicado. Em inglês usa-se
Present Value (indicado pela tecla PV nas calculadoras financeiras).<br />
</span><span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"><o:p></o:p></span></div>
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<span style="font-family: Arial, sans-serif; font-size: 12pt;"> </span><span style="color: #990000; font-family: "Arial","sans-serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;"> <b>Juros</b></span><span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"><o:p></o:p></span></div>
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<span style="font-family: Arial, sans-serif; font-size: 10pt;">Juros representam a
remuneração do Capital empregado em alguma atividade produtiva. Os juros podem
ser capitalizados segundo dois regimes: <u>simples</u> ou <u>compostos</u>.<br />
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<b><span style="color: #990000; font-family: "Arial","sans-serif"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;">JUROS SIMPLES</span></b><span style="color: #990000; font-family: "Arial","sans-serif"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;">:</span><span style="font-family: "Arial","sans-serif"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;"> o juro de cada intervalo de tempo sempre é calculado sobre o
capital inicial emprestado ou aplicado.</span><span style="font-family: "Times New Roman","serif"; font-size: 8.5pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;"><o:p></o:p></span></div>
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<b><span style="color: #990000; font-family: "Arial","sans-serif"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;">JUROS COMPOSTOS</span></b><span style="color: #990000; font-family: "Arial","sans-serif"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;">:</span><span style="font-family: "Arial","sans-serif"; font-size: 10.0pt; mso-bidi-font-size: 11.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;"> </span><span style="font-family: "Arial","sans-serif"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;">o juro de cada intervalo de tempo é calculado a
partir do saldo no início de correspondente intervalo. Ou seja: o juro de
cada intervalo de tempo é incorporado ao capital inicial e passa a render
juros também.</span><span style="font-family: "Times New Roman","serif"; font-size: 8.5pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;"><o:p></o:p></span></div>
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<span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"> <br />
</span><span style="font-family: Arial, sans-serif; font-size: 10pt;">O<b> juro </b>é a remuneração pelo empréstimo do dinheiro. Ele
existe porque a maioria das pessoas prefere o consumo imediato, e está disposta
a pagar um preço por isto. Por outro lado, quem for capaz de esperar até
possuir a quantia suficiente para adquirir seu desejo, e neste ínterim estiver
disposta a emprestar esta quantia a alguém, menos paciente, deve ser
recompensado por esta abstinência na proporção do <b>tempo</b> e <b>risco</b>,
que a operação envolver. O tempo, o risco e a quantidade de dinheiro disponível
no mercado para empréstimos definem qual deverá ser a remuneração, mais
conhecida como<b>taxa de juros</b>.</span><span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"><o:p></o:p></span></div>
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<span style="font-family: Verdana, sans-serif; font-size: 12pt;"> </span><b><span style="color: navy; font-family: "Arial","sans-serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;"> </span></b><b><span style="color: #990000; font-family: "Arial","sans-serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;">Quando usamos juros
simples e juros compostos?</span></b><span style="font-family: Verdana, sans-serif; font-size: 12pt;"><br />
</span><span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"><br />
</span><span style="font-family: Arial, sans-serif; font-size: 10pt;">A maioria das operações envolvendo dinheiro utiliza <b>juros
compostos</b>. Estão incluídas: compras a médio e longo prazo, compras com
cartão de crédito, empréstimos bancários, as aplicações financeiras usuais como
Caderneta de Poupança e aplicações em fundos de renda fixa, etc. Raramente
encontramos uso para o regime de juros simples: é o caso das operações de
curtíssimo prazo, e do processo de desconto simples de duplicatas.</span><span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"><o:p></o:p></span></div>
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<span style="font-family: Verdana, sans-serif; font-size: 12pt;"> </span><b><span style="color: navy; font-family: "Arial","sans-serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;"> </span></b><b><span style="color: #990000; font-family: "Arial","sans-serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;">Taxa de juros</span></b><span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"><o:p></o:p></span></div>
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<span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"> </span><span style="font-family: Arial, sans-serif; font-size: 10pt;"> A taxa de juros
indica qual remuneração será paga ao dinheiro emprestado, para um determinado
período. Ela vem normalmente expressa da forma percentual, em seguida da
especificação do período de tempo a que se refere:</span><span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"><o:p></o:p></span></div>
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<span style="font-family: Arial, sans-serif; font-size: 10pt;">8 % a.a. - (a.a.
significa ao ano).<br />
10 % a.t. - (a.t. significa ao trimestre).</span><span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"><o:p></o:p></span></div>
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<span style="font-family: Arial, sans-serif; font-size: 10pt;"> Outra forma de apresentação da taxa de juros é a
unitária, que é igual a taxa percentual dividida por 100, sem o símbolo %:</span><span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"><o:p></o:p></span></div>
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<span style="font-family: Arial, sans-serif; font-size: 10pt;">0,15 a.m. - (a.m.
significa ao mês).<br />
0,10 a.q. - (a.q. significa ao quadrimestre)</span><span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"><o:p></o:p></span></div>
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<b><span style="color: #990000; font-family: "Verdana","sans-serif"; font-size: 13.5pt; mso-bidi-font-family: Arial; mso-bidi-font-size: 11.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;">JUROS SIMPLES</span></b><span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"><o:p></o:p></span></div>
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<span style="font-family: Arial, sans-serif; font-size: 10pt;"> O
regime de juros será simples quando o percentual de juros incidir apenas sobre
o valor principal. Sobre os juros gerados a cada período não incidirão novos
juros.</span><span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"> </span><span style="font-family: Arial, sans-serif; font-size: 10pt;">Valor Principal ou simplesmente
principal é o valor inicial emprestado ou aplicado, antes de somarmos os juros.
Transformando em fórmula temos:</span><span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"><o:p></o:p></span></div>
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<b><span style="color: #990000; font-family: "Times New Roman","serif"; font-size: 18.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;">J = P . i . n</span></b><span style="font-family: "Times New Roman","serif"; font-size: 8.5pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;"><o:p></o:p></span></div>
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<span style="font-family: Arial, sans-serif; font-size: 10pt;">Onde:</span><span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"><o:p></o:p></span></div>
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<b><span style="font-family: "Arial","sans-serif"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;">J</span></b><span style="font-family: "Arial","sans-serif"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;"> = juros<br />
<b>P</b> = principal (capital)<br />
<b>i</b> = taxa de juros<br />
<b>n</b> = número de períodos</span><span style="font-family: "Times New Roman","serif"; font-size: 8.5pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;"><o:p></o:p></span></div>
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<span style="font-family: Arial, sans-serif; font-size: 10pt;"> </span><span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"><o:p></o:p></span></div>
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<span style="font-family: Arial, sans-serif; font-size: 10pt;"> <u>Exemplo</u>:
Temos uma dívida de R$ 1000,00 que deve ser paga com juros de 8% a.m. pelo
regime de juros simples e devemos pagá-la em 2 meses. Os juros que pagarei
serão:</span><span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"><o:p></o:p></span></div>
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<span style="font-family: Arial, sans-serif; font-size: 10pt;">J = 1000 x 0.08 x 2 = <b>160</b></span><span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"><o:p></o:p></span></div>
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<span style="font-family: Arial, sans-serif; font-size: 10pt;"> Ao
somarmos os juros ao valor principal temos o <b>montante</b>.</span><span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"><o:p></o:p></span></div>
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<span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"> </span><span style="font-family: Arial, sans-serif; font-size: 10pt;">Montante = Principal + Juros<br />
Montante = Principal + ( Principal x Taxa de juros x Número
de períodos )</span><span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"><o:p></o:p></span></div>
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<b><span style="color: #990000; font-family: "Times New Roman","serif"; font-size: 18.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;">M = P . ( 1 + ( i . n ) )</span></b><span style="font-family: "Times New Roman","serif"; font-size: 8.5pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;"><o:p></o:p></span></div>
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<span style="font-family: Arial, sans-serif; font-size: 10pt;"> <u>Exemplo</u>:
Calcule o montante resultante da aplicação de R$70.000,00 à taxa de 10,5% a.a.
durante 145 dias.</span><span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"><o:p></o:p></span></div>
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SOLUÇÃO:<br />
M = P . ( 1 + (i.n) )<br />
M = 70000 [1 + (10,5/100).(145/360)] = R$72.960,42</span><span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"><o:p></o:p></span></div>
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<span style="font-family: Arial, sans-serif; font-size: 10pt;">
Observe que expressamos a taxa <b>i</b> e o período <b>n</b>, na
mesma unidade de tempo, ou seja, anos. Daí ter dividido 145 dias por 360, para
obter o valor equivalente em anos, já que um ano comercial possui 360 dias.</span><span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"><o:p></o:p></span></div>
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<span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"> </span><b><i><u><span style="font-family: Arial, sans-serif; font-size: 10pt;">Exercícios sobre juros simples:</span></u></i></b><span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"><o:p></o:p></span></div>
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<b><span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"> </span></b><b><span style="color: #990000; font-family: "Arial","sans-serif"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;">1)</span></b><b><span style="color: #990000; font-family: "Verdana","sans-serif"; font-size: 8.5pt; mso-bidi-font-family: Arial; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;"> </span></b><b><span style="color: #990000; font-family: "Arial","sans-serif"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;">Calcular os juros simples de R$ 1200,00 a 13 % a.t. por 4 meses e 15
dias.</span></b><span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"><o:p></o:p></span></div>
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<span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"> </span><span style="font-family: Arial, sans-serif; font-size: 10pt;">0.13 / 6 = 0.02167<br />
logo, 4m15d = 0.02167 x 9 = 0.195</span><span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"><o:p></o:p></span></div>
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<span style="font-family: Arial, sans-serif; font-size: 10pt;"> j
= 1200 x 0.195 = 234</span><span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"><o:p></o:p></span></div>
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<br /></div>
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<b><span style="color: navy; font-family: "Arial","sans-serif"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;"> </span></b><b><span style="color: #990000; font-family: "Arial","sans-serif"; font-size: 10.0pt; mso-bidi-font-size: 11.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;"> </span></b><b><span style="color: #990000; font-family: "Arial","sans-serif"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;">2 - Calcular os juros simples produzidos por R$40.000,00, aplicados à
taxa de 36% a.a., durante 125 dias.</span></b><span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"><o:p></o:p></span></div>
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<span style="font-family: Arial, sans-serif; font-size: 10pt;">
Temos: J = P.i.n<br />
A taxa de 36% a.a. equivale a 0,36/360 dias = 0,001 a.d.<br />
Agora, como a taxa e o período estão referidos à mesma
unidade de tempo, ou seja, dias, poderemos calcular diretamente:<br />
J = 40000.0,001.125 = R$5000,00</span><span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"><o:p></o:p></span></div>
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<br /></div>
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<b><span style="color: navy; font-family: "Arial","sans-serif"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;"> </span></b><b><span style="color: #990000; font-family: "Arial","sans-serif"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;"> 3 - Qual o
capital que aplicado a juros simples de 1,2% a.m. rende R$3.500,00 de juros em
75 dias?</span></b><span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"><o:p></o:p></span></div>
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<span style="font-family: Arial, sans-serif; font-size: 10pt;">
Temos imediatamente: J = P.i.n ou seja: 3500 = P.(1,2/100).(75/30)<br />
Observe que expressamos a taxa <b>i</b> e o
período <b>n</b> em relação à mesma unidade de tempo, ou seja, meses.
Logo,<br />
3500 = P. 0,012 . 2,5 = P . 0,030; Daí, vem:<br />
P = 3500 / 0,030 = R$116.666,67</span><span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"><o:p></o:p></span></div>
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<br /></div>
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<span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"> </span><span style="color: #990000; font-family: "Verdana","sans-serif"; font-size: 8.5pt; mso-bidi-font-family: Arial; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;"> </span><b><span style="color: #990000; font-family: "Arial","sans-serif"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;">4 - Se a taxa de
uma aplicação é de 150% ao ano, quantos meses serão necessários para dobrar um
capital aplicado através de capitalização simples?</span></b><span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"><br />
<br />
</span><span style="font-family: Arial, sans-serif; font-size: 10pt;"> Objetivo: M = 2.P<br />
Dados: i = 150/100 = 1,5<br />
Fórmula: M = P (1 + i.n)<br />
Desenvolvimento<o:p></o:p></span></div>
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<span style="font-family: Arial, sans-serif; font-size: 10pt;">2P = P (1 + 1,5 n)</span><span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"><o:p></o:p></span></div>
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<span style="font-family: Arial, sans-serif; font-size: 10pt;">2 = 1 + 1,5 n<o:p></o:p></span></div>
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<span style="font-family: Arial, sans-serif; font-size: 10pt;">n = 2/3 ano = 8 meses<o:p></o:p></span></div>
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<b><span style="color: #c00000; font-family: "Verdana","sans-serif"; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-bidi-font-size: 11.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;">JUROS COMPOSTOS</span></b><span style="color: #c00000; font-family: "Verdana","sans-serif"; font-size: 8.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;"><o:p></o:p></span></div>
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<span style="font-family: Arial, sans-serif; font-size: 10pt;"> O regime de juros compostos é o mais comum no sistema
financeiro e portanto, o mais útil para cálculos de problemas do dia-a-dia. Os
juros gerados a cada período são incorporados ao principal para o cálculo dos
juros do período seguinte.</span><span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"><o:p></o:p></span></div>
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<br /></div>
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<span style="font-family: Arial, sans-serif; font-size: 10pt;"> Chamamos de capitalização o momento em que os juros
são incorporados ao principal.</span><span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"><o:p></o:p></span></div>
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<br /></div>
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<span style="font-family: Arial, sans-serif; font-size: 10pt;">Após três meses de capitalização, temos:</span><span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"><o:p></o:p></span></div>
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<span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"> 1º</span><span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"> </span><span style="font-family: Arial, sans-serif; font-size: 10pt;">mês:</span><span style="font-family: Arial, sans-serif; font-size: 10pt;"> </span><b><span style="font-family: Arial, sans-serif; font-size: 10pt;">M =P.(1 + i)</span></b><span style="font-family: Arial, sans-serif; font-size: 10pt;"><br />
2º mês: o principal é igual ao montante do mês anterior:</span><span style="font-family: Arial, sans-serif; font-size: 10pt;"> </span><b><span style="font-family: Arial, sans-serif; font-size: 10pt;">M = P x (1 + i) x (1 + i)</span></b><b><span style="font-family: Arial, sans-serif; font-size: 10pt;"> </span></b><span style="font-family: Arial, sans-serif; font-size: 10pt;"><br />
3º mês: o principal é igual ao montante do mês anterior:</span><span style="font-family: Arial, sans-serif; font-size: 10pt;"> </span><b><span style="font-family: Arial, sans-serif; font-size: 10pt;">M = P x (1 + i) x (1 + i) x (1 + i)</span></b><span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"><o:p></o:p></span></div>
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<span style="font-family: Arial, sans-serif; font-size: 10pt;"> Simplificando, obtemos a fórmula:<br />
</span><span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"><o:p></o:p></span></div>
<div align="center">
<table border="1" cellpadding="0" cellspacing="0" class="MsoNormalTable" style="border: outset silver 1.0pt; mso-border-alt: outset silver .75pt; mso-cellspacing: 0cm; mso-padding-alt: 3.75pt 3.75pt 3.75pt 3.75pt; mso-yfti-tbllook: 1184; width: 50.0%;">
<tbody>
<tr>
<td style="background: whitesmoke; border: inset silver 1.0pt; mso-border-alt: inset silver .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt; width: 100.0%;" width="100%">
<div align="center" class="MsoNormal" style="margin-bottom: 0.0001pt; text-align: center;">
<b><span style="color: #c00000; font-family: "Verdana","sans-serif"; font-size: 18.0pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;">M = P . (1 + i)<sup>n</sup></span></b><span style="color: #c00000; font-family: "Verdana","sans-serif"; font-size: 8.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;"><o:p></o:p></span></div>
</td>
</tr>
</tbody></table>
</div>
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<br /></div>
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<span style="font-family: Arial, sans-serif; font-size: 10pt;"> </span><span style="font-family: Arial, sans-serif; font-size: 10pt;"> </span><u><span style="font-family: Arial, sans-serif; font-size: 10pt;">Importante:</span></u><span style="font-family: Arial, sans-serif; font-size: 10pt;"> </span><span style="font-family: Arial, sans-serif; font-size: 10pt;">a taxa</span><span style="font-family: Arial, sans-serif; font-size: 10pt;"> </span><b><span style="font-family: Arial, sans-serif; font-size: 10pt;">i</span></b><span style="font-family: Arial, sans-serif; font-size: 10pt;"> </span><span style="font-family: Arial, sans-serif; font-size: 10pt;">tem que ser expressa
na mesma medida de tempo de</span><span style="font-family: Arial, sans-serif; font-size: 10pt;"> </span><b><span style="font-family: Arial, sans-serif; font-size: 10pt;">n</span></b><span style="font-family: Arial, sans-serif; font-size: 10pt;">, ou seja, taxa de juros ao mês para n
meses.</span><span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"><o:p></o:p></span></div>
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<span style="font-family: Arial, sans-serif; font-size: 10pt;"> Para calcularmos apenas os juros basta diminuir o
principal do montante ao final do período:<br />
</span><span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"><o:p></o:p></span></div>
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<table border="1" cellpadding="0" cellspacing="0" class="MsoNormalTable" style="border: outset silver 1.0pt; mso-border-alt: outset silver .75pt; mso-cellspacing: 0cm; mso-padding-alt: 3.75pt 3.75pt 3.75pt 3.75pt; mso-yfti-tbllook: 1184; width: 30.0%;">
<tbody>
<tr>
<td style="background: whitesmoke; border: inset silver 1.0pt; mso-border-alt: inset silver .75pt; padding: 3.75pt 3.75pt 3.75pt 3.75pt; width: 100.0%;" width="100%">
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<b><span style="color: #c00000; font-family: "Verdana","sans-serif"; font-size: 18.0pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;">J = M - P</span></b><span style="color: #c00000; font-family: "Verdana","sans-serif"; font-size: 8.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;"><o:p></o:p></span></div>
</td>
</tr>
</tbody></table>
</div>
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<br /></div>
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<span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"> </span><span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"> </span><b><i><span style="font-family: Arial, sans-serif; font-size: 10pt;">Exemplo:</span></i></b><span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"><o:p></o:p></span></div>
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<span style="font-family: Verdana, sans-serif; font-size: 10pt;"> </span><span style="font-family: Verdana, sans-serif; font-size: 10pt;"> </span><span style="font-family: Arial, sans-serif; font-size: 10pt;">Calcule o montante de um capital de
R$6.000,00, aplicado a juros compostos, durante 1 ano, à taxa de 3,5% ao mês.<br />
(use</span><span style="font-family: Arial, sans-serif; font-size: 10pt;"> </span><i><span style="font-family: Arial, sans-serif; font-size: 10pt;">log 1,035=0,0149</span></i><span style="font-family: Arial, sans-serif; font-size: 10pt;"> </span><span style="font-family: Arial, sans-serif; font-size: 10pt;">e</span><span style="font-family: Arial, sans-serif; font-size: 10pt;"> </span><i><span style="font-family: Arial, sans-serif; font-size: 10pt;">log 1,509=0,1788</span></i><span style="font-family: Arial, sans-serif; font-size: 10pt;">)</span><span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"><o:p></o:p></span></div>
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<span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"> </span><span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"> </span><i><span style="font-family: Arial, sans-serif; font-size: 10pt;">Resolução:</span></i><span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"><o:p></o:p></span></div>
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<span style="color: #c00000; font-family: "Verdana","sans-serif"; font-size: 8.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;"> </span><span style="color: #c00000; font-family: "Verdana","sans-serif"; font-size: 8.5pt; mso-bidi-font-family: "Times New Roman"; mso-bidi-font-size: 11.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;"> </span><span style="color: #c00000; font-family: "Arial","sans-serif"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;">P = R$6.000,00<br />
t = 1 ano = 12 meses<br />
i = 3,5 % a.m. = 0,035<br />
M = ?</span><span style="color: #c00000; font-family: "Verdana","sans-serif"; font-size: 8.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;"><o:p></o:p></span></div>
<div class="MsoNormal" style="background-color: white; background-position: initial initial; background-repeat: initial initial; margin-bottom: 0.0001pt;">
<span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"> <o:p></o:p></span></div>
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<span style="font-family: Arial, sans-serif; font-size: 10pt;"> Usando a fórmula</span><span style="font-family: Arial, sans-serif; font-size: 10pt;"> </span><b><span style="font-family: Arial, sans-serif; font-size: 10pt;">M=P.(1+i)<sup>n</sup></span></b><span style="font-family: Arial, sans-serif; font-size: 10pt;">, obtemos:</span><span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"><o:p></o:p></span></div>
<div class="MsoNormal" style="background-color: white; background-position: initial initial; background-repeat: initial initial; margin-bottom: 0.0001pt;">
<span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"> </span><span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"> </span><span style="font-family: Arial, sans-serif; font-size: 10pt;">M = 6000.(1+0,035)<sup>12</sup>
= 6000. (1,035)<sup>12</sup><br />
Fazendo x = 1,035<sup>12</sup></span><span style="font-family: Arial, sans-serif; font-size: 10pt;"> </span><span style="font-family: Arial, sans-serif; font-size: 10pt;">e aplicando logaritmos,</span><span style="font-family: Arial, sans-serif; font-size: 10pt;"> </span><span style="font-family: Arial, sans-serif; font-size: 10pt;">encontramos:</span><span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"><o:p></o:p></span></div>
<div class="MsoNormal" style="background-color: white; background-position: initial initial; background-repeat: initial initial; margin-bottom: 0.0001pt;">
<span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"> </span><span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"> </span><span style="font-family: Arial, sans-serif; font-size: 10pt;">log x = log</span><span style="font-family: Arial, sans-serif; font-size: 10pt;"> </span><span style="font-family: Arial, sans-serif; font-size: 10pt;">1,035<sup>12</sup> => log x = 12 log
1,035 => log x = 0,1788
=> x = 1,509</span><span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"><o:p></o:p></span></div>
<div class="MsoNormal" style="background-color: white; background-position: initial initial; background-repeat: initial initial; margin-bottom: 0.0001pt;">
<span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"> </span><span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"> </span><span style="font-family: Arial, sans-serif; font-size: 10pt;">Então M = 6000.1,509 = 9054.</span><span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"><o:p></o:p></span></div>
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<span style="font-family: Arial, sans-serif; font-size: 10pt;"> Portanto o montante é R$9.054,00</span><span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"><o:p></o:p></span></div>
<div align="center" class="MsoNormal" style="background-color: white; background-position: initial initial; background-repeat: initial initial; margin-bottom: 0.0001pt; text-align: center;">
<b><span style="color: #c00000; font-family: "Verdana","sans-serif"; font-size: 13.5pt; mso-bidi-font-family: "Times New Roman"; mso-bidi-font-size: 11.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;">Relação entre juros e progressões</span></b><span style="color: #c00000; font-family: "Verdana","sans-serif"; font-size: 8.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;"><o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span style="background-color: white; background-position: initial initial; background-repeat: initial initial; font-family: Verdana, sans-serif; font-size: 8.5pt;"> </span><span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"> </span><span style="background-color: white; background-position: initial initial; background-repeat: initial initial; font-family: Arial, sans-serif; font-size: 10pt;">No regime de juros
simples:<br />
<b> </b></span><span style="font-family: Arial, sans-serif; font-size: 10pt;"> </span><b><span style="background-color: white; background-position: initial initial; background-repeat: initial initial; font-family: Arial, sans-serif; font-size: 10pt;">M( n ) = P + n r P</span></b><span style="background-color: white; background-position: initial initial; background-repeat: initial initial; font-family: Arial, sans-serif; font-size: 10pt;"><br />
<br />
No regime de juros compostos:<br />
<b> </b></span><span style="font-family: Arial, sans-serif; font-size: 10pt;"> </span><b><span style="background-color: white; background-position: initial initial; background-repeat: initial initial; font-family: Arial, sans-serif; font-size: 10pt;">M( n ) = P . ( 1 + r )</span></b><b><sup><span style="font-family: Arial, sans-serif; font-size: 10pt;"> </span></sup></b><b><sup><span style="background-color: white; background-position: initial initial; background-repeat: initial initial; font-family: Arial, sans-serif; font-size: 10pt;">n</span></sup></b><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;"><o:p></o:p></span></div>
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<span style="font-family: Arial, sans-serif; font-size: 10pt;"> Portanto:</span><span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"><o:p></o:p></span></div>
<ul type="disc">
<li class="MsoNormal" style="background-color: white; background-position: initial initial; background-repeat: initial initial;"><span style="font-family: "Arial","sans-serif"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;">num
regime de capitalização</span><span style="font-family: "Arial","sans-serif"; font-size: 10.0pt; mso-bidi-font-size: 11.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;"> </span><b><span style="font-family: "Arial","sans-serif"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;">a juros simples</span></b><span style="font-family: "Arial","sans-serif"; font-size: 10.0pt; mso-bidi-font-size: 11.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;"> </span><span style="font-family: "Arial","sans-serif"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;">o saldo cresce em</span><span style="font-family: "Arial","sans-serif"; font-size: 10.0pt; mso-bidi-font-size: 11.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;"> </span><b><span style="font-family: "Arial","sans-serif"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;">progressão aritmética</span></b><span style="font-family: "Verdana","sans-serif"; font-size: 8.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;"><o:p></o:p></span></li>
<li class="MsoNormal" style="background-color: white; background-position: initial initial; background-repeat: initial initial;"><span style="font-family: "Arial","sans-serif"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;">num
regime de capitalização</span><span style="font-family: "Arial","sans-serif"; font-size: 10.0pt; mso-bidi-font-size: 11.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;"> </span><b><span style="font-family: "Arial","sans-serif"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;">a juros compostos</span></b><span style="font-family: "Arial","sans-serif"; font-size: 10.0pt; mso-bidi-font-size: 11.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;"> </span><span style="font-family: "Arial","sans-serif"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;">o saldo cresce em</span><span style="font-family: "Arial","sans-serif"; font-size: 10.0pt; mso-bidi-font-size: 11.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;"> </span><b><span style="font-family: "Arial","sans-serif"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;">progressão geométrica</span></b><span style="font-family: "Verdana","sans-serif"; font-size: 8.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;"><o:p></o:p></span></li>
</ul>
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<br /></div>
<div align="center" class="MsoNormal" style="background-color: white; background-position: initial initial; background-repeat: initial initial; margin-bottom: 0.0001pt; text-align: center;">
<b><span style="color: #c00000; font-family: "Verdana","sans-serif"; font-size: 18.0pt; mso-bidi-font-family: "Times New Roman"; mso-bidi-font-size: 11.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;">TAXAS EQUIVALENTES</span></b><span style="color: #c00000; font-family: "Verdana","sans-serif"; font-size: 8.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;"><o:p></o:p></span></div>
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<span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"> </span><span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"> </span><span style="font-family: Arial, sans-serif; font-size: 10pt;">Duas taxas i<sub>1</sub></span><span style="font-family: Arial, sans-serif; font-size: 10pt;"> </span><span style="font-family: Arial, sans-serif; font-size: 10pt;">e i<sub>2</sub></span><span style="font-family: Arial, sans-serif; font-size: 10pt;"> </span><span style="font-family: Arial, sans-serif; font-size: 10pt;">são equivalentes, se aplicadas ao mesmo
Capital</span><span style="font-family: Arial, sans-serif; font-size: 10pt;"> </span><b><span style="font-family: Arial, sans-serif; font-size: 10pt;">P</span></b><span style="font-family: Arial, sans-serif; font-size: 10pt;"> </span><span style="font-family: Arial, sans-serif; font-size: 10pt;">durante o mesmo
período de tempo, através de diferentes períodos de capitalização, produzem o
mesmo montante final.</span><span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"><o:p></o:p></span></div>
<ul type="disc">
<li class="MsoNormal" style="background-color: white; background-position: initial initial; background-repeat: initial initial;"><span style="font-family: "Arial","sans-serif"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;">Seja
o capital</span><span style="font-family: "Arial","sans-serif"; font-size: 10.0pt; mso-bidi-font-size: 11.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;"> </span><b><span style="font-family: "Arial","sans-serif"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;">P</span></b><span style="font-family: "Arial","sans-serif"; font-size: 10.0pt; mso-bidi-font-size: 11.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;"> </span><span style="font-family: "Arial","sans-serif"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;">aplicado por um ano a uma
taxa anual i<sub>a</sub></span><span style="font-family: "Arial","sans-serif"; font-size: 10.0pt; mso-bidi-font-size: 11.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;"> </span><span style="font-family: "Arial","sans-serif"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;">.</span><span style="font-family: "Verdana","sans-serif"; font-size: 8.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;"><o:p></o:p></span></li>
<li class="MsoNormal" style="background-color: white; background-position: initial initial; background-repeat: initial initial;"><span style="font-family: "Arial","sans-serif"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;">O
montante</span><span style="font-family: "Arial","sans-serif"; font-size: 10.0pt; mso-bidi-font-size: 11.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;"> </span><b><span style="font-family: "Arial","sans-serif"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;">M</span></b><span style="font-family: "Arial","sans-serif"; font-size: 10.0pt; mso-bidi-font-size: 11.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;"> </span><span style="font-family: "Arial","sans-serif"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;">ao final do período de 1 ano
será igual a M = P(1 + i</span><span style="font-family: "Arial","sans-serif"; font-size: 10.0pt; mso-bidi-font-size: 11.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;"> </span><sub><span style="font-family: "Arial","sans-serif"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;">a</span></sub><span style="font-family: "Arial","sans-serif"; font-size: 10.0pt; mso-bidi-font-size: 11.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;"> </span><span style="font-family: "Arial","sans-serif"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;">)</span><span style="font-family: "Verdana","sans-serif"; font-size: 8.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;"><o:p></o:p></span></li>
<li class="MsoNormal" style="background-color: white; background-position: initial initial; background-repeat: initial initial;"><span style="font-family: "Arial","sans-serif"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;">Consideremos
agora, o mesmo capital</span><span style="font-family: "Arial","sans-serif"; font-size: 10.0pt; mso-bidi-font-size: 11.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;"> </span><b><span style="font-family: "Arial","sans-serif"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;">P</span></b><span style="font-family: "Arial","sans-serif"; font-size: 10.0pt; mso-bidi-font-size: 11.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;"> </span><span style="font-family: "Arial","sans-serif"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;">aplicado por 12 meses a uma
taxa mensal i<sub>m</sub></span><span style="font-family: "Arial","sans-serif"; font-size: 10.0pt; mso-bidi-font-size: 11.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;"> </span><span style="font-family: "Arial","sans-serif"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;">.</span><span style="font-family: "Verdana","sans-serif"; font-size: 8.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;"><o:p></o:p></span></li>
<li class="MsoNormal" style="background-color: white; background-position: initial initial; background-repeat: initial initial;"><span style="font-family: "Arial","sans-serif"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;">O
montante</span><span style="font-family: "Arial","sans-serif"; font-size: 10.0pt; mso-bidi-font-size: 11.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;"> </span><b><span style="font-family: "Arial","sans-serif"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;">M’</span></b><span style="font-family: "Arial","sans-serif"; font-size: 10.0pt; mso-bidi-font-size: 11.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;"> </span><span style="font-family: "Arial","sans-serif"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;">ao final do período de 12
meses será igual a M’ = P(1 + i<sub>m</sub>)<sup>12</sup></span><span style="font-family: "Arial","sans-serif"; font-size: 10.0pt; mso-bidi-font-size: 11.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;"> </span><span style="font-family: "Arial","sans-serif"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;">.</span><span style="font-family: "Verdana","sans-serif"; font-size: 8.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;"><o:p></o:p></span></li>
</ul>
<div class="MsoNormal" style="background-color: white; background-position: initial initial; background-repeat: initial initial; margin-bottom: 0.0001pt;">
<span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"> </span><span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"> </span><span style="font-family: Arial, sans-serif; font-size: 10pt;">Pela definição de taxas equivalentes
vista acima, deveremos ter</span><span style="font-family: Arial, sans-serif; font-size: 10pt;"> </span><b><span style="font-family: Arial, sans-serif; font-size: 10pt;">M = M’</span></b><span style="font-family: Arial, sans-serif; font-size: 10pt;">.</span><span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"><o:p></o:p></span></div>
<div class="MsoNormal" style="background-color: white; background-position: initial initial; background-repeat: initial initial; margin-bottom: 0.0001pt;">
<span style="font-family: Arial, sans-serif; font-size: 10pt;"> Portanto, P(1 + i<sub>a</sub>) = P(1 + i<sub>m</sub>)<sup>12</sup><br />
Daí concluímos que</span><span style="font-family: Arial, sans-serif; font-size: 10pt;"> </span><b><span style="color: #c00000; font-family: "Arial","sans-serif"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;">1 + i<sub>a</sub></span></b><b><span style="color: #c00000; font-family: "Arial","sans-serif"; font-size: 10.0pt; mso-bidi-font-size: 11.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;"> </span></b><b><span style="color: #c00000; font-family: "Arial","sans-serif"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;">= (1 + i<sub>m</sub>)<sup>12</sup></span></b><span style="font-family: Arial, sans-serif; font-size: 10pt;"><br />
Com esta fórmula podemos calcular a taxa anual equivalente a
uma taxa mensal conhecida.</span><span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"><o:p></o:p></span></div>
<div class="MsoNormal" style="background-color: white; background-position: initial initial; background-repeat: initial initial; margin-bottom: 0.0001pt;">
<span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"> </span><span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"> </span><b><i><span style="font-family: Arial, sans-serif; font-size: 10pt;">Exemplos:</span></i></b><span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"><o:p></o:p></span></div>
<div class="MsoNormal" style="background-color: white; background-position: initial initial; background-repeat: initial initial; margin-bottom: 0.0001pt;">
<b><span style="color: #c00000; font-family: "Arial","sans-serif"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;"> 1 - Qual a taxa anual equivalente a 8% ao semestre?</span></b><span style="color: #c00000; font-family: "Verdana","sans-serif"; font-size: 8.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;"><o:p></o:p></span></div>
<div class="MsoNormal" style="background-color: white; background-position: initial initial; background-repeat: initial initial; margin-bottom: 0.0001pt;">
<span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"> </span><span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"> </span><span style="font-family: Arial, sans-serif; font-size: 10pt;">Em um ano temos dois semestres, então
teremos: 1 + i<sub>a</sub></span><span style="font-family: Arial, sans-serif; font-size: 10pt;"> </span><span style="font-family: Arial, sans-serif; font-size: 10pt;">= (1 + i<sub>s</sub>)<sup>2</sup><br />
1 + i<sub>a</sub></span><span style="font-family: Arial, sans-serif; font-size: 10pt;"> </span><span style="font-family: Arial, sans-serif; font-size: 10pt;">= 1,08<sup>2</sup><br />
i<sub>a</sub></span><span style="font-family: Arial, sans-serif; font-size: 10pt;"> </span><span style="font-family: Arial, sans-serif; font-size: 10pt;">= 0,1664 = 16,64%
a.a.<br />
</span><span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"><o:p></o:p></span></div>
<div class="MsoNormal" style="background-color: white; background-position: initial initial; background-repeat: initial initial; margin-bottom: 0.0001pt;">
<b><span style="color: navy; font-family: "Arial","sans-serif"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;">
</span></b><b><span style="color: #c00000; font-family: "Arial","sans-serif"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;">2 - Qual a taxa anual equivalente a 0,5% ao mês?</span></b><span style="color: #c00000; font-family: "Verdana","sans-serif"; font-size: 8.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;"><o:p></o:p></span></div>
<div class="MsoNormal" style="background-color: white; background-position: initial initial; background-repeat: initial initial; margin-bottom: 0.0001pt;">
<span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"> </span><span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"> </span><span style="font-family: Arial, sans-serif; font-size: 10pt;">1 + i<sub>a</sub></span><span style="font-family: Arial, sans-serif; font-size: 10pt;"> </span><span style="font-family: Arial, sans-serif; font-size: 10pt;">= (1 + i<sub>m</sub>)<sup>12</sup><br />
1 + i<sub>a</sub></span><span style="font-family: Arial, sans-serif; font-size: 10pt;"> </span><span style="font-family: Arial, sans-serif; font-size: 10pt;">= (1,005)<sup>12</sup><br />
i<sub>a</sub></span><span style="font-family: Arial, sans-serif; font-size: 10pt;"> </span><span style="font-family: Arial, sans-serif; font-size: 10pt;">= 0,0617 = 6,17% a.a.</span><span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"><o:p></o:p></span></div>
<div align="center" class="MsoNormal" style="background-color: white; background-position: initial initial; background-repeat: initial initial; margin-bottom: 0.0001pt; text-align: center;">
<span style="font-family: Arial, sans-serif; font-size: 10pt;"><br />
</span><b><span style="color: #c00000; font-family: "Verdana","sans-serif"; font-size: 18.0pt; mso-bidi-font-family: "Times New Roman"; mso-bidi-font-size: 11.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;">TAXAS NOMINAIS</span></b><span style="color: #c00000; font-family: "Verdana","sans-serif"; font-size: 8.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;"><o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span style="background-color: white; background-position: initial initial; background-repeat: initial initial; font-family: Arial, sans-serif; font-size: 10pt;"> A taxa nominal é quando o
período de formação e incorporação dos juros ao Capital não coincide com aquele
a que a taxa está referida. Alguns exemplos:<br />
- 340% ao semestre com capitalização mensal.<br />
- 1150% ao ano com capitalização mensal.<br />
- 300% ao ano com capitalização trimestral.</span><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;"><o:p></o:p></span></div>
<div class="MsoNormal" style="background-color: white; background-position: initial initial; background-repeat: initial initial; margin-bottom: 0.0001pt;">
<span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"> </span><span style="font-family: Arial, sans-serif; font-size: 10pt;"> </span><b><i><span style="font-family: Arial, sans-serif; font-size: 10pt;">Exemplo:</span></i></b><span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"><o:p></o:p></span></div>
<div class="MsoNormal" style="background-color: white; background-position: initial initial; background-repeat: initial initial; margin-bottom: 0.0001pt;">
<span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"> </span><span style="font-family: Arial, sans-serif; font-size: 10pt;"> </span><span style="font-family: Arial, sans-serif; font-size: 10pt;">Uma taxa de 15 % a.a., capitalização mensal, terá 16.08 % a.a. como taxa
efetiva:</span><span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"><o:p></o:p></span></div>
<div class="MsoNormal" style="background-color: white; background-position: initial initial; background-repeat: initial initial; margin-bottom: 0.0001pt;">
<span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"> </span><span style="font-family: Arial, sans-serif; font-size: 10pt;"> </span><span style="font-family: Arial, sans-serif; font-size: 10pt;">15/12 = 1,25
1,0125<sup>12</sup></span><sup><span style="font-family: Arial, sans-serif; font-size: 10pt;"> </span></sup><span style="font-family: Arial, sans-serif; font-size: 10pt;">= 1,1608</span><span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"><o:p></o:p></span></div>
<div class="MsoNormal" style="background-color: white; background-position: initial initial; background-repeat: initial initial; margin-bottom: 0.0001pt;">
<br /></div>
<div align="center" class="MsoNormal" style="background-color: white; background-position: initial initial; background-repeat: initial initial; margin-bottom: 0.0001pt; text-align: center;">
<span style="font-family: Arial, sans-serif; font-size: 10pt;"><br />
</span><b><span style="color: #c00000; font-family: "Verdana","sans-serif"; font-size: 18.0pt; mso-bidi-font-family: "Times New Roman"; mso-bidi-font-size: 11.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;">TAXAS EFETIVAS</span></b><span style="color: #c00000; font-family: "Verdana","sans-serif"; font-size: 8.5pt; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;"><o:p></o:p></span></div>
<div class="MsoNormal" style="background-color: white; background-position: initial initial; background-repeat: initial initial; margin-bottom: 0.0001pt;">
<span style="font-family: Arial, sans-serif; font-size: 10pt;"> A taxa Efetiva é quando o período de formação e
incorporação dos juros ao Capital coincide com aquele a que a taxa está
referida. Alguns exemplos:<br />
- 140% ao mês com capitalização mensal.<br />
- 250% ao semestre com capitalização semestral.<br />
- 1250% ao ano com capitalização anual.</span><span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"><o:p></o:p></span></div>
<div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<b><span style="background: white; color: navy; font-family: "Arial","sans-serif"; font-size: 10.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;"> Taxa Real:</span></b><span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"> </span><span style="background-color: white; background-position: initial initial; background-repeat: initial initial; font-family: Arial, sans-serif; font-size: 10pt;">é a taxa efetiva
corrigida pela taxa inflacionária do período da operação.</span><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: PT-BR;"><o:p></o:p></span></div>
<div class="MsoNormal" style="background-color: white; background-position: initial initial; background-repeat: initial initial; margin-bottom: 0.0001pt;">
<br /></div>
<div align="center" style="background: white; margin-bottom: .0001pt; margin: 0cm; text-align: center;">
<strong><span style="color: #c00000; font-family: "Verdana","sans-serif"; font-size: 18.0pt;">FLUXO DE CAIXA</span></strong><span style="color: #c00000; font-family: "Verdana","sans-serif"; font-size: 8.5pt;"><o:p></o:p></span></div>
<div style="background-color: white; background-position: initial initial; background-repeat: initial initial; margin: 0cm 0cm 0.0001pt;">
<span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"> <span class="apple-converted-space"> </span></span><span style="font-family: Arial, sans-serif; font-size: 10pt;">O fluxo
de caixa serve para demonstrar graficamente as transações financeiras em um
período de tempo. O tempo é representado na horizontal dividido pelo número de
períodos relevantes para análise. As</span><b><span style="color: green; font-family: "Arial","sans-serif"; font-size: 10.0pt;">entradas</span></b><span class="apple-converted-space"><span style="font-family: Arial, sans-serif; font-size: 10pt;"> </span></span><span style="font-family: Arial, sans-serif; font-size: 10pt;">ou<span class="apple-converted-space"> </span></span><b><span style="color: green; font-family: "Arial","sans-serif"; font-size: 10.0pt;">recebimentos</span></b><span class="apple-converted-space"><span style="font-family: Arial, sans-serif; font-size: 10pt;"> </span></span><span style="font-family: Arial, sans-serif; font-size: 10pt;">são representados por setas verticais
apontadas para<span class="apple-converted-space"> </span><b>cima</b><span class="apple-converted-space"> </span>e as<span class="apple-converted-space"> </span></span><b><span style="color: green; font-family: "Arial","sans-serif"; font-size: 10.0pt;">saídas</span></b><span class="apple-converted-space"><span style="font-family: Arial, sans-serif; font-size: 10pt;"> </span></span><span style="font-family: Arial, sans-serif; font-size: 10pt;">ou</span><b><span style="color: green; font-family: "Arial","sans-serif"; font-size: 10.0pt;">pagamentos</span></b><span class="apple-converted-space"><span style="font-family: Arial, sans-serif; font-size: 10pt;"> </span></span><span style="font-family: Arial, sans-serif; font-size: 10pt;">são representados por setas verticais
apontadas para<span class="apple-converted-space"> </span><b>baixo</b>.
Observe o gráfico abaixo:</span><span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"><o:p></o:p></span></div>
<div style="background-color: white; background-position: initial initial; background-repeat: initial initial; margin: 0cm 0cm 0.0001pt;">
<br /></div>
<div align="center" style="background-color: white; background-position: initial initial; background-repeat: initial initial; margin: 0cm 0cm 0.0001pt; text-align: center;">
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<div align="center" style="background-color: white; background-position: initial initial; background-repeat: initial initial; margin: 0cm 0cm 0.0001pt; text-align: center;">
<br /></div>
<div style="background-color: white; background-position: initial initial; background-repeat: initial initial; margin: 0cm 0cm 0.0001pt;">
<span style="font-family: Arial, sans-serif; font-size: 10pt;"> Chamamos de<span class="apple-converted-space"> </span></span><b><span style="color: navy; font-family: "Arial","sans-serif"; font-size: 10.0pt;">VP</span></b><span class="apple-converted-space"><span style="font-family: Arial, sans-serif; font-size: 10pt;"> </span></span><span style="font-family: Arial, sans-serif; font-size: 10pt;">o<span class="apple-converted-space"> </span><b>valor
presente</b>, que significa o valor que eu tenho na data 0;<span class="apple-converted-space"> </span></span><b><span style="color: navy; font-family: "Arial","sans-serif"; font-size: 10.0pt;">VF</span></b><span class="apple-converted-space"><span style="font-family: Arial, sans-serif; font-size: 10pt;"> </span></span><span style="font-family: Arial, sans-serif; font-size: 10pt;">é o<span class="apple-converted-space"><b> </b></span><b>valor
futuro</b>, que será igual ao valor que terei no final do fluxo, após juros,
entradas e saídas. </span><span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"><o:p></o:p></span></div>
<div style="background-color: white; background-position: initial initial; background-repeat: initial initial; margin: 0cm 0cm 0.0001pt;">
<br /></div>
<div style="background-color: white; background-position: initial initial; background-repeat: initial initial; margin: 0cm 0cm 0.0001pt;">
<br /></div>
<div align="center" style="background-color: white; background-position: initial initial; background-repeat: initial initial; margin: 0cm 0cm 0.0001pt; text-align: center;">
<strong><span style="color: #c00000; font-family: "Verdana","sans-serif"; font-size: 18.0pt;">VALOR
PRESENTE e VALOR FUTURO</span></strong><span style="color: #c00000; font-family: "Verdana","sans-serif"; font-size: 8.5pt;"><o:p></o:p></span></div>
<div style="background-color: white; background-position: initial initial; background-repeat: initial initial; margin: 0cm 0cm 0.0001pt;">
<span style="font-family: Arial, sans-serif; font-size: 10pt;"> Na fórmula<span class="apple-converted-space"> </span><b>M
= P . (1 + i)<sup>n</sup></b><span class="apple-converted-space"> </span>, o
principal P é também conhecido como<span class="apple-converted-space"> </span><b>Valor
Presente</b><span class="apple-converted-space"> </span>(</span><b><span style="color: navy; font-family: "Arial","sans-serif"; font-size: 10.0pt;">PV</span></b><span class="apple-converted-space"><span style="font-family: Arial, sans-serif; font-size: 10pt;"> </span></span><span style="font-family: Arial, sans-serif; font-size: 10pt;">= present value) e o montante M é também
conhecido como<span class="apple-converted-space"> </span><b>Valor Futuro</b><span class="apple-converted-space"> </span>(</span><b><span style="color: navy; font-family: "Arial","sans-serif"; font-size: 10.0pt;">FV</span></b><span class="apple-converted-space"><span style="font-family: Arial, sans-serif; font-size: 10pt;"> </span></span><span style="font-family: Arial, sans-serif; font-size: 10pt;">= future value). </span><span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"><o:p></o:p></span></div>
<div style="background-color: white; background-position: initial initial; background-repeat: initial initial; margin: 0cm 0cm 0.0001pt;">
<span style="font-family: Arial, sans-serif; font-size: 8.5pt;"> <span class="apple-converted-space"> </span></span><span style="font-family: Arial, sans-serif; font-size: 10pt;">Então
essa fórmula pode ser escrita como </span><span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"><o:p></o:p></span></div>
<div style="background-color: white; background-position: initial initial; background-repeat: initial initial; margin: 0cm 0cm 0.0001pt;">
<span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"> </span><span class="apple-converted-space"><span style="font-family: Arial, sans-serif; font-size: 10pt;"> </span></span><span style="font-family: Arial, sans-serif; font-size: 10pt;">FV = PV
(1 + i)<span class="apple-converted-space"> </span><sup>n</sup> </span><span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"><o:p></o:p></span></div>
<div style="background-color: white; background-position: initial initial; background-repeat: initial initial; margin: 0cm 0cm 0.0001pt;">
<span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"> </span><span class="apple-converted-space"><span style="font-family: Arial, sans-serif; font-size: 10pt;"> </span></span><span style="font-family: Arial, sans-serif; font-size: 10pt;">Isolando
PV na fórmula temos:</span><span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"><o:p></o:p></span></div>
<div style="background-color: white; background-position: initial initial; background-repeat: initial initial; margin: 0cm 0cm 0.0001pt;">
<span style="color: #c00000; font-family: "Verdana","sans-serif"; font-size: 8.5pt;"> </span><span class="apple-converted-space"><span style="color: #c00000; font-family: "Arial","sans-serif"; font-size: 10.0pt;"> </span></span><b><span style="color: #c00000; font-family: "Arial","sans-serif";">PV = FV / (1+i)<sup>n</sup></span></b><span style="color: #c00000; font-family: "Verdana","sans-serif"; font-size: 8.5pt;"><o:p></o:p></span></div>
<div style="background-color: white; background-position: initial initial; background-repeat: initial initial; margin: 0cm 0cm 0.0001pt;">
<span style="font-family: Arial, sans-serif; font-size: 10pt;"> Na HP-12C, o valor presente é representado pela
tecla PV.</span><span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"><o:p></o:p></span></div>
<div style="background-color: white; background-position: initial initial; background-repeat: initial initial; margin: 0cm 0cm 0.0001pt;">
<span style="font-family: Arial, sans-serif; font-size: 10pt;"> Com esta mesma fórmula podemos calcular o valor
futuro a partir do valor presente.</span><span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"><o:p></o:p></span></div>
<div style="background-color: white; background-position: initial initial; background-repeat: initial initial; margin: 0cm 0cm 0.0001pt;">
<b><i><span style="font-family: Arial, sans-serif; font-size: 10pt;"> <span class="apple-converted-space"> </span><u>Exemplo</u>:</span></i></b><span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"><o:p></o:p></span></div>
<div style="background-color: white; background-position: initial initial; background-repeat: initial initial; margin: 0cm 0cm 0.0001pt;">
<span style="font-family: Arial, sans-serif; font-size: 10pt;"> Quanto teremos daqui a 12 meses se aplicarmos
R$1.500,00 a 2% ao mês?<br />
<span class="apple-converted-space"> </span><i>Solução</i>:</span><span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"><o:p></o:p></span></div>
<div style="background-color: white; background-position: initial initial; background-repeat: initial initial; margin: 0cm 0cm 0.0001pt;">
<span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"> </span><span style="font-family: Arial, sans-serif; font-size: 10pt;"> FV = 1500 . (1
+ 0,02)<sup>12</sup><span class="apple-converted-space"> </span>= R$
1.902,36</span><span style="font-family: Verdana, sans-serif; font-size: 8.5pt;"><o:p></o:p></span></div>
<div class="MsoNormal">
<br /></div>Andrei FLhttp://www.blogger.com/profile/12516487634069169995noreply@blogger.com0tag:blogger.com,1999:blog-3423239337426631685.post-89954068718282947962012-05-26T20:27:00.002-03:002012-05-26T20:27:24.914-03:00Conjuntos Numéricos<br />
<h2 class="posttitle" style="background-color: #eeeae8; border-bottom-color: rgb(136, 134, 133); border-bottom-style: dotted; border-bottom-width: 1px; color: #4a4a49; font-family: Verdana, serif; font-size: 11px; margin: 0.6em 0px 0px; padding: 0px 0px 1px; position: relative; text-align: center; text-transform: uppercase;">
CONJUNTOS NUMÉRICOS</h2>
<h2 class="posttitle" style="background-color: #eeeae8; border-bottom-color: rgb(136, 134, 133); border-bottom-style: dotted; border-bottom-width: 1px; color: #4a4a49; font-family: Verdana, serif; font-size: 11px; margin: 0.6em 0px 0px; padding: 0px 0px 1px; position: relative; text-align: left; text-transform: uppercase;">
<br /><div style="font-family: Arial, sans-serif; font-size: 12px; font-weight: normal; line-height: 1.5em; margin-bottom: 1.2em; margin-top: 1.2em;">
<strong>CONJUNTO NUMÉRICOS – PARTE 1</strong></div>
<div style="font-family: Arial, sans-serif; font-size: 12px; font-weight: normal; line-height: 1.5em; margin-bottom: 1.2em; margin-top: 1.2em;">
<span class="embed-youtube" style="display: block; text-align: center;"><iframe class="youtube-player" frameborder="0" height="309" src="http://www.youtube.com/embed/ooZatdw2DY4?version=3&rel=1&fs=1&showsearch=0&showinfo=1&iv_load_policy=1&wmode=transparent" style="max-width: 100%;" type="text/html" width="495"></iframe></span></div>
<div style="font-family: Arial, sans-serif; font-size: 12px; font-weight: normal; line-height: 1.5em; margin-bottom: 1.2em; margin-top: 1.2em;">
<strong>CONJUNTO NUMÉRICOS – PARTE 2</strong></div>
<div style="font-family: Arial, sans-serif; font-size: 12px; font-weight: normal; line-height: 1.5em; margin-bottom: 1.2em; margin-top: 1.2em;">
<span class="embed-youtube" style="display: block; text-align: center;"><iframe class="youtube-player" frameborder="0" height="309" src="http://www.youtube.com/embed/62Aq3kUxbU0?version=3&rel=1&fs=1&showsearch=0&showinfo=1&iv_load_policy=1&wmode=transparent" style="max-width: 100%;" type="text/html" width="495"></iframe></span></div>
<div style="font-family: Arial, sans-serif; font-size: 12px; font-weight: normal; line-height: 1.5em; margin-bottom: 1.2em; margin-top: 1.2em;">
<strong>MATEMÁTICA – AULA 2 – CONJUNTOS NUMÉRICOS – PARTE 1</strong></div>
<div style="font-family: Arial, sans-serif; font-size: 12px; font-weight: normal; line-height: 1.5em; margin-bottom: 1.2em; margin-top: 1.2em;">
<span class="embed-youtube" style="display: block; text-align: center;"><iframe class="youtube-player" frameborder="0" height="309" src="http://www.youtube.com/embed/FL6WQG6Z5tQ?version=3&rel=1&fs=1&showsearch=0&showinfo=1&iv_load_policy=1&wmode=transparent" style="max-width: 100%;" type="text/html" width="495"></iframe></span></div>
<div style="font-family: Arial, sans-serif; font-size: 12px; font-weight: normal; line-height: 1.5em; margin-bottom: 1.2em; margin-top: 1.2em;">
<strong>MATEMÁTICA – AULA 2 – CONJUNTOS NUMÉRICOS – PARTE 2</strong></div>
<div style="font-family: Arial, sans-serif; font-size: 12px; font-weight: normal; line-height: 1.5em; margin-bottom: 1.2em; margin-top: 1.2em;">
<span class="embed-youtube" style="display: block; text-align: center;"><iframe class="youtube-player" frameborder="0" height="309" src="http://www.youtube.com/embed/19Gx6x5EKIg?version=3&rel=1&fs=1&showsearch=0&showinfo=1&iv_load_policy=1&wmode=transparent" style="max-width: 100%;" type="text/html" width="495"></iframe></span></div>
<div style="font-family: Arial, sans-serif; font-size: 12px; font-weight: normal; line-height: 1.5em; margin-bottom: 1.2em; margin-top: 1.2em;">
<strong>MATEMÁTICA – AULA 2 – CONJUNTOS NUMÉRICOS – PARTE 3 – FINAL</strong></div>
<div style="font-family: Arial, sans-serif; font-size: 12px; font-weight: normal; line-height: 1.5em; margin-bottom: 1.2em; margin-top: 1.2em;">
<span class="embed-youtube" style="display: block; text-align: center;"><iframe class="youtube-player" frameborder="0" height="309" src="http://www.youtube.com/embed/5luhLfmp6ig?version=3&rel=1&fs=1&showsearch=0&showinfo=1&iv_load_policy=1&wmode=transparent" style="max-width: 100%;" type="text/html" width="495"></iframe></span></div>
<div style="font-family: Arial, sans-serif; font-size: 12px; font-weight: normal; line-height: 1.5em; margin-bottom: 1.2em; margin-top: 1.2em;">
<strong>CONJUNTOS NUMÉRICOS</strong></div>
<div style="font-family: Arial, sans-serif; font-size: 12px; font-weight: normal; line-height: 1.5em; margin-bottom: 1.2em; margin-top: 1.2em;">
<span class="embed-youtube" style="display: block; text-align: center;"><iframe class="youtube-player" frameborder="0" height="309" src="http://www.youtube.com/embed/bV2S9sizNJ4?version=3&rel=1&fs=1&showsearch=0&showinfo=1&iv_load_policy=1&wmode=transparent" style="max-width: 100%;" type="text/html" width="495"></iframe></span></div>
<div style="font-family: Arial, sans-serif; font-size: 12px; font-weight: normal; line-height: 1.5em; margin-bottom: 1.2em; margin-top: 1.2em;">
<strong>1_NOÇÕES DE CONJUNTOS</strong></div>
<div style="font-family: Arial, sans-serif; font-size: 12px; font-weight: normal; line-height: 1.5em; margin-bottom: 1.2em; margin-top: 1.2em;">
<span class="embed-youtube" style="display: block; text-align: center;"><iframe class="youtube-player" frameborder="0" height="309" src="http://www.youtube.com/embed/eAZXi4GyzLQ?version=3&rel=1&fs=1&showsearch=0&showinfo=1&iv_load_policy=1&wmode=transparent" style="max-width: 100%;" type="text/html" width="495"></iframe></span></div>
<div style="font-family: Arial, sans-serif; font-size: 12px; font-weight: normal; line-height: 1.5em; margin-bottom: 1.2em; margin-top: 1.2em;">
<strong>2_CONJUNTOS DIAGRAMA DE VENN</strong></div>
<div style="font-family: Arial, sans-serif; font-size: 12px; font-weight: normal; line-height: 1.5em; margin-bottom: 1.2em; margin-top: 1.2em;">
<span class="embed-youtube" style="display: block; text-align: center;"><iframe class="youtube-player" frameborder="0" height="309" src="http://www.youtube.com/embed/HQJdSTbRA0w?version=3&rel=1&fs=1&showsearch=0&showinfo=1&iv_load_policy=1&wmode=transparent" style="max-width: 100%;" type="text/html" width="495"></iframe></span></div>
<div style="font-family: Arial, sans-serif; font-size: 12px; font-weight: normal; line-height: 1.5em; margin-bottom: 1.2em; margin-top: 1.2em;">
<strong>3_CONJUNTOS DIAGRAMA DE VENN PASSO A PASSO</strong></div>
<div style="font-family: Arial, sans-serif; font-size: 12px; font-weight: normal; line-height: 1.5em; margin-bottom: 1.2em; margin-top: 1.2em;">
<span class="embed-youtube" style="display: block; text-align: center;"><iframe class="youtube-player" frameborder="0" height="309" src="http://www.youtube.com/embed/_VKsIIWK7go?version=3&rel=1&fs=1&showsearch=0&showinfo=1&iv_load_policy=1&wmode=transparent" style="max-width: 100%;" type="text/html" width="495"></iframe></span></div>
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</h2>Andrei FLhttp://www.blogger.com/profile/12516487634069169995noreply@blogger.com0