ejercicios_3 - UNIVERSIDAD AUTONOMA DE OCCIDENTE FACULTAD DE CIENCIAS BASICAS DEPARTAMENTO DE MATEMATICAS Curso Calculo II(131230 Profesor Victor Hugo

ejercicios_3 - UNIVERSIDAD AUTONOMA DE OCCIDENTE FACULTAD...

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UNIVERSIDAD AUT ´ ONOMA DE OCCIDENTE FACULTAD DE CIENCIAS B ´ ASICAS DEPARTAMENTO DE MATEM ´ ATICAS Curso: Calculo II (131230) Profesor: Victor Hugo Gil A. 20 / 02 / 2017 Ejercicios # 3 Instrucciones. Primero lea cuidadosa y detalladamente el taller, despu´ es, responda de manera clara y ordenada. Justifique todas las respuestas. 1. ( Derivadas direccionales ) En los siguientes problemas determine la derivada direccional de f en el punto p en la direcci´on de u . a) f ( x,y ) = x 2 y ; p = (1 , 2); u = 3 i - 4 j b) f ( ) = y 2 ln( x ); p = (1 , 4); u = i - j c) f ( ) = 2 x 2 + xy - y 2 ; p = (3 , - 2); u = i - j d) f ( ) = e x sin( y ); p = (0 ,π/ 4); u = i + 3 j e) f ( ) = e - xy ; p = (1 , - 1); u = - i + 3 j f) f ( x,y,z ) = x 2 + y 2 + z 2 ; p = (1 , - 1 , 2); u = 2 i - j - k g) f ( ) = x 2 + 2 y 2 - 3 z 2 ; p = (1 , 1 , 1); u = i + j + k h) f ( ) = cos( xy ) + e yz + ln( zx ); p = (1 , 0 , 1 / 2); u = i + 2 j + 2 k 2. ( Direcciones de incremento y decremento m´ as r´ apido ) En los siguientes problemas determine un vector unitario en la direcci´on en que f crece o decrece m´as r´apidamente en p . ¿Cu´al es la raz´on de cambio en esta direcci´on? p = (5 π/ 6 , 0) p = (2 , 0 , - 4) p = (1 , - 1 , 2) p = (1 , ln(2) , 1 / 2) p = (1 , 1 , 1) h) f ( ) = ln( x 2 + y 2 - 1) + y + 6 z ; p = (1 , 1 , 0) 3. ( Un poco de teor´ ıa ) a) ¿En qu´ e direcci´on se anula la derivada de f ( ) = xy + y 2 en p = (3 , 2)? b) ¿En qu´ e direcciones se anula la derivada de f ( ) = ( x 2 - y 2 ) / ( x 2 + y 2 ) en p = (1 , 1)? c) Bosqueje la curva de nivel de f ( ) = y/x 2 que pasa por p = (1 , 2). Calcule el vector gra- diente f ( p ) y trace ese vector, colocando su punto inicial en p . ¿ Que debe ocurrir con f ( p )? d) ¿ existe una direcci´on u en que la razon de cam- bio de f ( ) = x 2 - 3 xy + 4 y 2 en p = (1 , 2) sea igual a 14? Justifique su respuesta.
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