Certamen 1 - Matemáticas III (2008) - Forma B.pdf - Certamen 1 MAT-023 Primer Semestre 2008 Departamento de Matem´ aticas UTFSM Tiempo 100 minutos

Certamen 1 - Matemáticas III (2008) - Forma B.pdf -...

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Certamen 1, MAT-023, Primer Semestre 2008 Fila B Departamento de Matem´ aticas, UTFSM Tiempo: 100 minutos. Cada ejercicio vale 25 puntos. No se puede utilizar calculadora. 1.) Considere la ecuaci´ on diferencial que modela el crecimiento de una poblaci´ on dy dt = y ( y - 1)( y - 3) 2 . Advertencia: ¡No tratar de resolverla! a) Identifique los puntos de equilibrio como sumideros, fuentes o nodos. Indique para qu´ e valores de y est´ a creciendo o decreciendo la soluci´ on. b) Considere la condici´ on inicial y (0) = y 0 . Suponga que y 0 0. ¿Cu´ al es la m´ ınima poblaci´ on inicial y 0 que debe existir de tal forma que la poblaci´ on no se extinga? Justifique su respuesta explicando qu´ e ocurre para este valor de y 0 y qu´ e ocurre para valores menores de y 0 . c) Considere la soluci´ on que satisface la condici´ on inicial y (0) = 4. Encuentre los dominios posibles de la soluci´ on. Determine los l´ ımites de la soluci´ on a medida que t crece o decrece para cada caso.
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