Certamen 2 - Matemáticas III (2010-2).pdf - PAUTA del SEGUNDO CERTAMEN de Matem´ atica III(MAT023 Mi´ercoles 27 de Octubre Nombre del alumno Rol

Certamen 2 - Matemáticas III (2010-2).pdf - PAUTA del...

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PAUTA del SEGUNDO CERTAMEN de Matem´ atica III (MAT023) Mi´ ercoles 27 de Octubre Nombre del alumno: Rol: Paralelo o Profesor: ——————————– ————— ————————— 1. Considere la funci´ on f definida como sigue: f ( x, y ) = 2 yx 3 x 2 + y 2 si ( x, y ) 6 = (0 , 0) 0 si ( x, y ) = (0 , 0) a) Determine si la funci´ on f ( x, y ) es continua en todo R 2 . b) Determine si la funci´ on f ( x, y ) es diferenciable en todo R 2 . c) Determine el valor de f xy (0 , 0) y f yx (0 , 0) Soluci´ on: (a) La funci´ on es continua en todos los puntos donde el denominador no se anule, por lo tanto f ( x, y ) es continua ( x, y ) R 2 - { (0 , 0) } ( 2 puntos ) Falta analizar continuidad en el punto (0 , 0). Estudiamos al l´ ımite: lim ( x,y ) (0 , 0) 2 yx 3 x 2 + y 2 . Vemos que: 2 yx 3 x 2 + y 2 ≤ | 2 yx | y | 2 yx | → 0 cuando ( x, y ) (0 , 0). Y luego lim ( x,y ) (0 , 0) 2 yx 3 x 2 + y 2 = 0 = f (0 , 0). Por lo tanto, f es continua en (0 , 0) . ( 6 puntos ) Se concluye que f ( x, y ) es continua ( x, y ) R 2 . ( 1 punto ) (b) Se ver´ a la diferenciabilidad en el origen f x (0 , 0) = lim h 0 f ( h, 0) - 0 h = lim h 0 2 h 3 0 h 2 = lim h 0 ( 0 h 2 ) = lim h 0 (0) = 0 ( 1 punto )
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