2008-PF(calc2) - d d d d d d d d d d d d Universidade...

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¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ @ @ @ @ @@ @ @ @ @ @@ @ @ @ @ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ @ @ @ @ @@ @ @ @ @ @@ @ @ @ @ Universidade Federal do Rio de Janeiro Instituto de Matemática Departamento de Métodos Matemáticos Cálculo II Prova Final - 25/06/2008 Prof. Samuel Senti Respostas não justi cadas não valem pontos! Só são permitidos nas mesas identidade, lapis, caneta, borracha e caderno da prova. O que não é permitido é proibido. 1 a QUESTÃO : Seja a equação diferencial ordinária y 0 ( t ) = 2 y ( t )( y ( t ) - 1) . (a) Obtenha a solução geral da EDO acima de forma explícita: y ( t ) = ... (b) Seja y 0 = y (0) . Encontre a expressão para y ( t ) envolvendo y 0 . Suponha que 0 < y 0 < 1 , calcule o limite da solução deste problema quando t tende a in nito. 2 a QUESTÃO : Um projétil é disparado com uma rapidez de 400 m/s e ângulo π 4 . Determine uma equação da trajetória. Determine o alcance do projétil, a altura máxima atingida e a rapidez do impacto (assume que o módulo do campo gravitacional | ~g | = 10 m/s 2 ). 3 a QUESTÃO : Seja a quádrica cuja equação cartesiana é dada por
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