Optimizaci\303\263n: Una introducci\303\263n intuitiva-gr\303\241fica al problema Dr. Ranfer\303\255 Guti\303\251rrez matematicaurl@gmail.com

Uno de los temas que presenta mayor dificultad para el estudiante novato de C\303\241lculo diferencial es el de optimizaci\303\263n. Las dificultades van desde la construcci\303\263n de la funci\303\263n que modela un problema aplicado particular, hasta la comprensi\303\263n y aplicaci\303\263n del proceso anal\303\255tico para encontrar el valor que maximiza o minimiza la misma, seg\303\272n corresponda. En esta gu\303\255a electr\303\263nica se presenta el tema de la optimizaci\303\263n de una funci\303\263n desde un punto de vista intuitivo, apoy\303\241ndose en las capacidades de Maple para crear gr\303\241ficas interactivas, con el objetivo de que el estudiante adquiera una mejor comprensi\303\263n de lo que significa optimizar una funci\303\263n. Se deja para el desarrollo en el aula el analizar los problemas de optimizaci\303\263n de una funci\303\263n utilizando el concepto de derivada. Durante el desarrollo de esta gu\303\255a se estudian tres problemas aplicados, en los cuales se enfatizan los siguientes pasos para resolverlos: Construcci\303\263n de la funci\303\263n que modela el problema. Determinaci\303\263n del dominio de la funci\303\263n, de acuerdo a las condiciones del problema. B\303\272squeda de el valor m\303\241ximo o m\303\255nimo de la funci\303\263n en la gr\303\241fica de la misma. Los comentarios al final de cada problema ayudan al estudiante a reconocer los aspectos generales m\303\241s importantes de los problemas de optimizaci\303\263n y esto, aunado con el an\303\241lisis detallado de las gr\303\241ficas interactivas que encontrar\303\241 en esta gu\303\255a, lo preparar\303\241n para afrontar con mayor \303\251xito problemas de optimizaci\303\263n que requieran la aplicaci\303\263n del concepto de derivada, as\303\255 como comprender mejor los criterios para determinar si en un punto dado la funci\303\263n tiene un m\303\241ximo o un m\303\255nimo.

En esta secci\303\263n usted encontrar\303\241 un problema de juguete con el cual podr\303\241 comprender de manera intuitiva lo que significa resolver un problema de optimizaci\303\263n en c\303\241lculo, as\303\255 como familiarizarse con la terminolog\303\255a y procedimiento de an\303\241lisis que se realiza en este tipo de problemas.

Suponga que tiene una figura con base inferior y altura constantes y de igual longitud, pero con una base superior cuya dimensi\303\263n puede variarse, tal como se muestra en la figura de abajo. Mueva la barra deslizante para ver c\303\263mo cambia la forma de la figura al variar el tama\303\261o de la base superior. LSUlUExPVEc2JS0lKVBPTFlHT05TRzYkNyY3JCQhI10hIiIkIiIhISIiNyQkISoxZUEuJSEiKSQiIz8hIiI3JCQiKjFlQS4lISIpJCIjPyEiIjckJCIjXSEiIiQiIiEhIiItJSZDT0xPUkc2JiUkUkdCRyQiIiEhIiIkIiM1ISIiJCIjNSEiIi0lKkFYRVNTVFlMRUc2IyUlTk9ORUctJSVST09URzYnLSUpQk9VTkRTX1hHNiMkIiNxISIiLSUpQk9VTkRTX1lHNiMkIiNdISIiLSUtQk9VTkRTX1dJRFRIRzYjJCIlSVEhIiItJS5CT1VORFNfSEVJR0hURzYjJCIlSVEhIiItJSlDSElMRFJFTkc2Ig== use DocumentTools in # Enter Maple commands to be executed when the specified # action is carried out on the component. # Use: # Do( %component_name ); # and # Do( %component_name = value ); # to set and get properties of the component. # You can also use arbitrary expressions # involving components, e.g.: # Do( %target = %input1 + 2*%input2 ); # Note the %-prefix to each component name. # See ?CustomizingComponents for more information. Do(b=%Slider1); trapecio:=plottools[polygon]([[-5,0],[-b,2],[b,2],[5,0]], color="Aqua"); Do(%Plot2=plots[display](trapecio, axes=none)); end use; Note que la figura, en general, corresponde a un trapecio, el cual puede convertirse en un tri\303\241ngulo o en un cuadrado, como casos l\303\255mites al variar al m\303\255nimo o al m\303\241ximo, respectivamente, la dimensi\303\263n de la base superior. \302\277Cu\303\241l es el valor de \303\241rea m\303\241s peque\303\261a y de \303\241rea m\303\241s grande que puede obtenerse con la figura de arriba? Suponga que la base inferior y la altura tienen, cada una, un tama\303\261o de 10 u.

Es obvio que el \303\241rea m\303\241s peque\303\261a corresponde al caso extremo en que se forma un tri\303\241ngulo, mientras que el \303\241rea m\303\241s grande corresponde al caso extremo en que se forma un cuadrado. El \303\241rea m\303\241s peque\303\261a que se puede obtener con la figura de arriba tiene un valor 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1cEdGJDYlLUkjbWlHRiQ2JlEidUYnLyUlc2l6ZUdRIzE2RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNictSSNtbkdGJDYlUSIyRidGMi9GOVEnbm9ybWFsRidGMkY1LyUwZm9udF9zdHlsZV9uYW1lR1ElVGV4dEYnRjgvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRjJGQ0ZB, mientras que el \303\241rea m\303\241s grande tiene un valor 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.

Ahora daremos un paso m\303\241s y utilizaremos conceptos del curso de Pre-c\303\241lculo para obtener una funci\303\263n que modele el \303\241rea de la figura traviesa y luego encontrar en forma gr\303\241fica los valores LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiQUYnLyUlc2l6ZUdRIzE2RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXJvd0dGJDYlLUYsNiZRIm1GJ0YvL0YzUSZmYWxzZUYnL0Y2USdub3JtYWxGJy1JI21vR0YkNi9RI2luRidGLy8lJWJvbGRHRj9GQC8lJmZlbmNlR0Y/LyUqc2VwYXJhdG9yR0Y/LyUpc3RyZXRjaHlHRj8vJSpzeW1tZXRyaWNHRj8vJShsYXJnZW9wR0Y/LyUubW92YWJsZWxpbWl0c0dGPy8lJ2FjY2VudEdGPy8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlhGQC8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYn y LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiQUYnLyUlc2l6ZUdRIzE2RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXJvd0dGJDYkLUYsNiZRJG1heEYnRi8vRjNRJmZhbHNlRicvRjZRJ25vcm1hbEYnRkAvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJw== indicados en el p\303\241rrafo anterior: Comenzaremos introduciendo un sistema de coordenadas, como se ilustra en la figura de abajo. Observe que en la figura de abajo se ha denotado con LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JlEieEYnLyUlc2l6ZUdRIzE2RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0Y2USdub3JtYWxGJw== la coordenada del punto en el extremo superior derecho (el punto azul). 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 use DocumentTools in # Enter Maple commands to be executed when the specified # action is carried out on the component. # Use: # Do( %component_name ); # and # Do( %component_name = value ); # to set and get properties of the component. # You can also use arbitrary expressions # involving components, e.g.: # Do( %target = %input1 + 2*%input2 ); # Note the %-prefix to each component name. # See ?CustomizingComponents for more information. Do(b2=%Slider2); trapecio:=plottools[polygon]([[-5,0],[-b2,10],[b2,10],[5,0]], color="Aqua"); texto:=plots[textplot]([b2-0.7, 9.5, sprintf("x=%1.1f",b2)],color=blue,labelfont=["TIMES",16], font=[TIMES, BOLD, 16]): punto:=plottools[point]([b2,10],color=blue,symbol=solidcircle,symbolsize=15): Do(%Plot3=plots[display]({trapecio,texto,punto}, axes=normal,labels=["x","f(x)"])); end use; Por otro lado, en la figura de abajo se muestra un trapecio y la f\303\263rmula que permite calcular su \303\241rea: 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Note que si LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiYkYnLyUlc2l6ZUdRIzE2RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiPUYnRi8vRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZALyUpc3RyZXRjaHlHRkAvJSpzeW1tZXRyaWNHRkAvJShsYXJnZW9wR0ZALyUubW92YWJsZWxpbWl0c0dGQC8lJ2FjY2VudEdGQC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRk8tSSNtbkdGJDYlUSIwRidGL0Y8Rjw=, el trapecio se convierte en un tri\303\241ngulo, y en este caso la f\303\263rmula para calcular el \303\241rea de un trapecio se reduce a la f\303\263rmula para calcular el \303\241rea de un tri\303\241ngulo: 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. Por otro lado, si LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiYkYnLyUlc2l6ZUdRIzE2RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiPUYnRi8vRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZALyUpc3RyZXRjaHlHRkAvJSpzeW1tZXRyaWNHRkAvJShsYXJnZW9wR0ZALyUubW92YWJsZWxpbWl0c0dGQC8lJ2FjY2VudEdGQC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRk8tRiw2JlEiQkYnRi9GMkY1Rjw=, el trapecio se convierte en un rect\303\241ngulo, y en este caso la f\303\263rmula para calcular el \303\241rea de un trapecio se reduce a la f\303\263rmula para calcular el \303\241rea de un rect\303\241ngulo: 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. Esto justifica la utilizaci\303\263n de la f\303\263rmula del \303\241rea de un trapecio para nuestro problema de juguete. Ahora construiremos una funci\303\263n LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiQUYnLyUlc2l6ZUdRIzE2RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNiQtRiw2JlEieEYnRi9GMkY1L0Y2USdub3JtYWxGJ0YvRkBGQA== que permita calcular el \303\241rea de la figura traviesa. En nuestro caso, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiQkYnLyUlc2l6ZUdRIzE2RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiPUYnRi8vRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZALyUpc3RyZXRjaHlHRkAvJSpzeW1tZXRyaWNHRkAvJShsYXJnZW9wR0ZALyUubW92YWJsZWxpbWl0c0dGQC8lJ2FjY2VudEdGQC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRk8tSSNtbkdGJDYlUSMxMEYnRi9GPEY8 y LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiaEYnLyUlc2l6ZUdRIzE2RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiPUYnRi8vRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZALyUpc3RyZXRjaHlHRkAvJSpzeW1tZXRyaWNHRkAvJShsYXJnZW9wR0ZALyUubW92YWJsZWxpbWl0c0dGQC8lJ2FjY2VudEdGQC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRk8tSSNtbkdGJDYlUSMxMEYnRi9GPEY8; adem\303\241s, 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. La funci\303\263n que permite calcular el \303\241rea de la figura traviesa es 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 Ahora determinaremos el dominio de la funci\303\263n 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. Sabemos que el caso LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEieEYnLyUlc2l6ZUdRIzE2RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiPUYnRi8vRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZALyUpc3RyZXRjaHlHRkAvJSpzeW1tZXRyaWNHRkAvJShsYXJnZW9wR0ZALyUubW92YWJsZWxpbWl0c0dGQC8lJ2FjY2VudEdGQC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRk8tSSNtbkdGJDYlUSIwRidGL0Y8Rjw= corresponde al tri\303\241ngulo, mientras que el caso LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEieEYnLyUlc2l6ZUdRIzE2RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiPUYnRi8vRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZALyUpc3RyZXRjaHlHRkAvJSpzeW1tZXRyaWNHRkAvJShsYXJnZW9wR0ZALyUubW92YWJsZWxpbWl0c0dGQC8lJ2FjY2VudEdGQC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRk8tSSNtbkdGJDYlUSI1RidGL0Y8Rjw= corresponde al cuadrado. Abajo se muestra la gr\303\241fica de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiQUYnLyUlc2l6ZUdRIzE2RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNiQtRiw2JlEieEYnRi9GMkY1L0Y2USdub3JtYWxGJ0YvRkBGQA==, en el dominio admisible para nuestro problema de juguete: 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Ahora buscaremos en la gr\303\241fica los valores m\303\241ximo y m\303\255nimo de la funci\303\263n, dentro del dominio admisible para la misma: De la gr\303\241fica de arriba es claro que el valor m\303\255nimo del \303\241rea corresponde a LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEieEYnLyUlc2l6ZUdRIzE2RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiPUYnRi8vRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZALyUpc3RyZXRjaHlHRkAvJSpzeW1tZXRyaWNHRkAvJShsYXJnZW9wR0ZALyUubW92YWJsZWxpbWl0c0dGQC8lJ2FjY2VudEdGQC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRk8tSSNtbkdGJDYlUSIwRidGL0Y8Rjw=: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYrLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEiQUYnLyUlc2l6ZUdRIzE2RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtRi82JlEkbWluRidGMi9GNlEmZmFsc2VGJy9GOVEnbm9ybWFsRidGMkY1RjgvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1JI21vR0YkNi5RIj1GJ0YyRkIvJSZmZW5jZUdGQS8lKnNlcGFyYXRvckdGQS8lKXN0cmV0Y2h5R0ZBLyUqc3ltbWV0cmljR0ZBLyUobGFyZ2VvcEdGQS8lLm1vdmFibGVsaW1pdHNHRkEvJSdhY2NlbnRHRkEvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZlbkYuLUkobWZlbmNlZEdGJDYlLUYjNiQtSSNtbkdGJDYlRkZGMkZCRkJGMkZCRkctRl5vNiVRIzUwRidGMkZCLUZINi5RIn5GJ0YyRkJGS0ZNRk9GUUZTRlVGVy9GWlEmMC4wZW1GJy9GZ25GZ28tSSVtc3VwR0YkNiUtRi82JlEidUYnRjJGNUY4LUYjNiYtRl5vNiVRIjJGJ0YyRkJGMkY1RjgvJTFzdXBlcnNjcmlwdHNoaWZ0R0ZGRkI=, mientras que el valor m\303\241ximo del \303\241rea corresponde a 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. Comentarios: Este problema de juguete muestra la esencia de lo que significa en c\303\241lculo resolver un problema de optimizaci\303\263n: Buscar, restringidos a las condiciones del problema, el valor m\303\241ximo o m\303\255nimo de cierta cantidad de inter\303\251s. En los textos resaltados en color verde se muestran los pasos b\303\241sicos de an\303\241lisis que hay que realizar en los problemas de optimizaci\303\263n: Construir una funci\303\263n que permita calcular de forma general la cantidad que se quiere maximizar o minimizar. Con frecuencia esto necesitar\303\241 de introducir un sistema de coordenadas, aunque no es una regla general. En ocasiones se pueden utilizar f\303\263rmulas conocidas, como por ejemplo de geometr\303\255a; en otros casos usted deber\303\241 deducir la funci\303\263n. Determinar el dominio de la funci\303\263n que se contruy\303\263 en el paso anterior. Este paso es importante ya que el problema impondr\303\241 de forma impl\303\255cita, o expl\303\255cita, l\303\255mites que tengan sentido para la variable independiente. Por ejemplo, en el problema de juguete, la funci\303\263n 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 admite, matem\303\241ticamente hablando, valores 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, pero es claro que en este caso, los \303\272nicos valores que tienen sentido son aquellos que pertenecen al intervalo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYnLUYjNiYtSSNtbkdGJDYlUSIwRicvJSVzaXplR1EjMTZGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYuUSIsRidGNEY3LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR1EldHJ1ZUYnLyUpc3RyZXRjaHlHRkAvJSpzeW1tZXRyaWNHRkAvJShsYXJnZW9wR0ZALyUubW92YWJsZWxpbWl0c0dGQC8lJ2FjY2VudEdGQC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYxNiVRIjVGJ0Y0RjdGN0Y0RjcvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRidGNw==. Determinar el valor m\303\241ximo o m\303\255nimo de la funci\303\263n, seg\303\272n corresponda. En esta gu\303\255a, la determinaci\303\263n de los valores extremos de la funci\303\263n se realizar\303\241 a partir de la observaci\303\263n de su gr\303\241fica. En clase usted aprender\303\241 m\303\251todos anal\303\255ticos que permiten calcular en forma precisa los valores extremos de dicha funci\303\263n. JSFH

En este problema se trata de determinar la distancia m\303\241s corta que hay desde el origen hasta un punto que pertenece a la gr\303\241fica de una funci\303\263n dada. Para resolver este problema es importante recordar que la distancia LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JlEiZEYnLyUlc2l6ZUdRIzE2RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0Y2USdub3JtYWxGJw== entre dos puntos 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 y 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 est\303\241 dada por 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 (1)

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En la figura de abajo se ilustra las variables involucradas en (1). 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(a) Construya la funci\303\263n LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiZEYnLyUlc2l6ZUdRIzE2RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNiQtRiw2JlEieEYnRi9GMkY1L0Y2USdub3JtYWxGJ0YvRkBGQA== que permite calcular la distancia desde el origen hasta un punto de la gr\303\241fica de 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 Determine el dominio de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEiZEYnLyUlc2l6ZUdRIzE2RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNiYtRiw2JlEieEYnRi9GMkY1LyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJTBmb250X3N0eWxlX25hbWVHUSgyRH5NYXRoRicvRjZRJ25vcm1hbEYnRi9GRkZARkNGRg==, adecuado para este problema . (b) Construya la gr\303\241fica de la funci\303\263n LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiZEYnLyUlc2l6ZUdRIzE2RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNiQtRiw2JlEieEYnRi9GMkY1L0Y2USdub3JtYWxGJ0YvRkBGQA== y util\303\255cela para determinar la distancia m\303\241s corta desde el origen hasta un punto de la gr\303\241fica de 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

(a) En la figura de abajo se muestra la gr\303\241fica de 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, un punto arbitrario LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYlLUYjNiYtSSNtaUdGJDYmUSJ4RicvJSVzaXplR1EjMTZGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYuUSIsRidGNC9GO1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjkvJSlzdHJldGNoeUdGRS8lKnN5bW1ldHJpY0dGRS8lKGxhcmdlb3BHRkUvJS5tb3ZhYmxlbGltaXRzR0ZFLyUnYWNjZW50R0ZFLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjE2JlEieUYnRjRGN0Y6RkFGNEZBRkE= sobre la misma y un segmento que lo une al origen. 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Figura 1. Gr\303\241fica de 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, en color rojo, y de un segmento (en color azul) que une al or\303\255gen de coordenadas con un punto LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYkLUYjNiYtSSNtaUdGJDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiLEYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GMTYlUSJ5RidGNEY3Rj5GPkY+ de la par\303\241bola. Asignando los puntos 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 y 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 -recuerde que LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JlEieUYnLyUlc2l6ZUdRIzE2RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiPUYnRi8vRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZALyUpc3RyZXRjaHlHRkAvJSpzeW1tZXRyaWNHRkAvJShsYXJnZW9wR0ZALyUubW92YWJsZWxpbWl0c0dGQC8lJ2FjY2VudEdGQC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRk8tSSVtc3VwR0YkNiUtRiw2JlEieEYnRi9GMkY1LUYjNiQtSSNtbkdGJDYlUSIyRidGL0Y8RjwvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLUY5Ni5RKCZtaW51cztGJ0YvRjxGPkZBRkNGRUZHRklGSy9GTlEsMC4yMjIyMjIyZW1GJy9GUUZfby1GZW42JVEiMUYnRi9GPEY8- entonces la aplicacion de (1) da como resultado que la funci\303\263n buscada es LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEiZEYnLyUlc2l6ZUdRIzE2RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNiQtRiw2JlEieEYnRi9GMkY1L0Y2USdub3JtYWxGJ0YvRkAtSSNtb0dGJDYuUSI9RidGL0ZALyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZILyUpc3RyZXRjaHlHRkgvJSpzeW1tZXRyaWNHRkgvJShsYXJnZW9wR0ZILyUubW92YWJsZWxpbWl0c0dGSC8lJ2FjY2VudEdGSC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlctSSZtc3FydEdGJDYjLUYjNictSSVtc3VwR0YkNiVGPS1GIzYmLUkjbW5HRiQ2JVEiMkYnRi9GQEYvRjJGNS8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRictRkM2LlEiK0YnRi9GQEZGRklGS0ZNRk9GUUZTL0ZWUSwwLjIyMjIyMjJlbUYnL0ZZRmlvLUZqbjYlLUY5NiUtRiM2J0Zpbi1GQzYuUSgmbWludXM7RidGL0ZARkZGSUZLRk1GT0ZRRlNGaG9Gam8tRl9vNiVRIjFGJ0YvRkBGL0ZARi9GQEZcb0Zib0YvRkBGQA== la cual equivale a 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El dominio de la funci\303\263n (2) corresponde a todos los reales ya que LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYsLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEieUYnLyUlc2l6ZUdRIzE2RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtSSNtbkdGJDYlUSIxRidGMi9GOVEnbm9ybWFsRidGMkY1RjgvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1JI21vR0YkNi5RIj1GJ0YyRkEvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkwvJSlzdHJldGNoeUdGTC8lKnN5bW1ldHJpY0dGTC8lKGxhcmdlb3BHRkwvJS5tb3ZhYmxlbGltaXRzR0ZMLyUnYWNjZW50R0ZMLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGZW4tSSVtc3VwR0YkNiUtRi82JlEieEYnRjJGNUY4LUYjNiYtRj42JVEiNEYnRjJGQS8lK2V4ZWN1dGFibGVHRjcvJTBmb250X3N0eWxlX25hbWVHUSkyRH5JbnB1dEYnRkEvJTFzdXBlcnNjcmlwdHNoaWZ0R0ZFLUZHNi5RKCZtaW51cztGJ0YyRkFGSkZNRk9GUUZTRlVGVy9GWlEsMC4yMjIyMjIyZW1GJy9GZ25GXnAtRmluNiVGW28tRiM2Ji1GPjYlUSIyRidGMkZBRmNvRmVvRkFGaG8tRkc2LlEiK0YnRjJGQUZKRk1GT0ZRRlNGVUZXRl1wRl9wRj1GY29GZW9GQQ== siempre es positiva, tal como se ilustra en la figura de abajo: 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Figura 2. Gr\303\241fica de 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. Adem\303\241s, recuerde que el punto LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYlLUYjNiYtSSNtaUdGJDYmUSJ4RicvJSVzaXplR1EjMTZGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYuUSIsRidGNC9GO1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjkvJSlzdHJldGNoeUdGRS8lKnN5bW1ldHJpY0dGRS8lKGxhcmdlb3BHRkUvJS5tb3ZhYmxlbGltaXRzR0ZFLyUnYWNjZW50R0ZFLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjE2JlEieUYnRjRGN0Y6RkFGNEZBRkE= cuya distancia al origen se quiere minimizar pertenece a la par\303\241bola 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, la cual admite valores 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 (b) Para determinar el valor de la distancia mas corta desde el origen a cualquier punto LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYlLUYjNiYtSSNtaUdGJDYmUSJ4RicvJSVzaXplR1EjMTZGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYuUSIsRidGNC9GO1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjkvJSlzdHJldGNoeUdGRS8lKnN5bW1ldHJpY0dGRS8lKGxhcmdlb3BHRkUvJS5tb3ZhYmxlbGltaXRzR0ZFLyUnYWNjZW50R0ZFLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjE2JlEieUYnRjRGN0Y6RkFGNEZBRkE= que pertenece a 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 construiremos, tal como lo pide el problema, la gr\303\241fica de la funci\303\263n 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. Para lograr una mejor comprensi\303\263n de lo que est\303\241 realizando al buscar la distancia m\303\241s corta que pide el problema, abajo encontrar\303\241 dos gr\303\241ficas: La Figura 3 es similar a la Figura 1, salvo que ahora se puede modificar la posici\303\263n del punto LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYlLUYjNiYtSSNtaUdGJDYmUSJ4RicvJSVzaXplR1EjMTZGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYuUSIsRidGNC9GO1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjkvJSlzdHJldGNoeUdGRS8lKnN5bW1ldHJpY0dGRS8lKGxhcmdlb3BHRkUvJS5tb3ZhYmxlbGltaXRzR0ZFLyUnYWNjZW50R0ZFLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjE2JlEieUYnRjRGN0Y6RkFGNEZBRkE= mediante la barra deslizante abajo de esa figura. La Figura 4 muestra el valor de la distancia entre el punto LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYlLUYjNiYtSSNtaUdGJDYmUSJ4RicvJSVzaXplR1EjMTZGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYuUSIsRidGNC9GO1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjkvJSlzdHJldGNoeUdGRS8lKnN5bW1ldHJpY0dGRS8lKGxhcmdlb3BHRkUvJS5tb3ZhYmxlbGltaXRzR0ZFLyUnYWNjZW50R0ZFLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjE2JlEieUYnRjRGN0Y6RkFGNEZBRkE= seleccionado en la Figura 3, y el origen. Mueva las barras deslizantes en ambas figuras y trate de comprender la informaci\303\263n que est\303\241 proporcionando cada gr\303\241fica. 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Figura 3. Gr\303\241fica de 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. La barra deslizante de abajo permite seleccionar diferentes puntos sobre la par\303\241bola. use DocumentTools in # Enter Maple commands to be executed when the specified # action is carried out on the component. # Use: # Do( %component_name ); # and # Do( %component_name = value ); # to set and get properties of the component. # You can also use arbitrary expressions # involving components, e.g.: # Do( %target = %input1 + 2*%input2 ); # Note the %-prefix to each component name. # See ?CustomizingComponents for more information. Do(b=%Slider0); texto:=plots[textplot]([b+0.5,b^2-1,sprintf("(%2.4f,%2.4f)",b,b^2-1)]): texto1:=plots[textplot]([b+0.5,sqrt(b^4-b^2+1+0.5),sprintf("(%2.4f,%2.4f)",b,sqrt(b^4-b^2+1))]): f:=plot(x^2-1,x=-2..2,color=red); l:=plottools[line]([0,0],[b,b^2-1],color=blue); Do(%Plot0=plots[display]({f,l,texto},labelfont=["TIMES NEW ROMAN",13],labels=["x","f(x)"]) ); d:=plot(sqrt(x^4-x^2+1),x=-2..2); lv:=plottools[line]([b,0],[b,sqrt(b^4-b^2+1)],color=black,linestyle=dash); lh:=plottools[line]([b,sqrt(b^4-b^2+1)],[0,sqrt(b^4-b^2+1)],color=black,linestyle=dash); Do(%Plot1=plots[display]({d,lv,lh,texto1})); Do(%Area=evalf(b,4)); Do(%Area1=evalf(sqrt(b^4-b^2+1),4)); Do(%Area2=evalf(b,4)); Do(%Area3=evalf(b^2-1,4)); end use; 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Figura 4. Gr\303\241fica de 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. La coordenada vertical proporciona la distancia entre elorigen y el punto seleccionado en la figura de la izquierda. Observe que tanto en la Figura 3 como en la Figura 4 se muestran las coordenadas del punto correspondiente en cada gr\303\241fica. Note que, en ambas gr\303\241ficas, la coordenada LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JlEieEYnLyUlc2l6ZUdRIzE2RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0Y2USdub3JtYWxGJw== de cada punto es la misma, mientras que la segunda coordenada del par es diferente en cada una de ellas. \302\277Qu\303\251 represente la segunda coordenada en cada uno de los puntos de las Figuras 3 y 4?

En la Figura 3 la segunda coordenada del punto proporciona el valor de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiZkYnLyUlc2l6ZUdRIzE2RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNiQtRiw2JlEieEYnRi9GMkY1L0Y2USdub3JtYWxGJ0YvRkBGQA==, mientras que en la Figura 4, la segunda coordenada del punto indica la distancia LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiZEYnLyUlc2l6ZUdRIzE2RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNiQtRiw2JlEieEYnRi9GMkY1L0Y2USdub3JtYWxGJ0YvRkBGQA== entre el origen y el punto LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYlLUYjNictSSNtaUdGJDYmUSJ4RicvJSVzaXplR1EjMTZGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYuUSIsRidGNC9GO1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjkvJSlzdHJldGNoeUdGRS8lKnN5bW1ldHJpY0dGRS8lKGxhcmdlb3BHRkUvJS5tb3ZhYmxlbGltaXRzR0ZFLyUnYWNjZW50R0ZFLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjE2JlEiZkYnRjRGN0Y6LUYsNiUtRiM2JEYwRkFGNEZBRkFGNEZBRkE= de la Figura 3. \302\241Aseg\303\272rese de comprender esto bien!

Al mover la barra deslizante que est\303\241 abajo de la Figura 3 cambia el valor de la coordenada LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JlEieEYnLyUlc2l6ZUdRIzE2RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0Y2USdub3JtYWxGJw== en ambas gr\303\241ficas. Por ejemplo, mueva la barra y observe que si LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEieEYnLyUlc2l6ZUdRIzE2RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiPUYnRi8vRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZALyUpc3RyZXRjaHlHRkAvJSpzeW1tZXRyaWNHRkAvJShsYXJnZW9wR0ZALyUubW92YWJsZWxpbWl0c0dGQC8lJ2FjY2VudEdGQC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRk8tSSNtbkdGJDYlUScwLjc1MDBGJ0YvRjxGPA==, en la gr\303\241fica de la izquierda el segmento de l\303\255nea azul va desde el punto LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYlLUYjNiYtSSNtbkdGJDYlUSIwRicvJSVzaXplR1EjMTZGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYuUSIsRidGNEY3LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR1EldHJ1ZUYnLyUpc3RyZXRjaHlHRkAvJSpzeW1tZXRyaWNHRkAvJShsYXJnZW9wR0ZALyUubW92YWJsZWxpbWl0c0dGQC8lJ2FjY2VudEdGQC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnRjBGN0Y0RjdGNw== hasta el punto LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYlLUYjNictSSNtbkdGJDYlUScwLjc1MDBGJy8lJXNpemVHUSMxNkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi5RIixGJ0Y0RjcvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHUSV0cnVlRicvJSlzdHJldGNoeUdGQC8lKnN5bW1ldHJpY0dGQC8lKGxhcmdlb3BHRkAvJS5tb3ZhYmxlbGltaXRzR0ZALyUnYWNjZW50R0ZALyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjs2LlEqJnVtaW51czA7RidGNEY3Rj4vRkJGQEZERkZGSEZKRkwvRk9RLDAuMjIyMjIyMmVtRicvRlJGWS1GMTYlUScwLjQzNzVGJ0Y0RjdGN0Y0RjdGNw==, mientras que en la figura de la derecha, las coordenadas LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYlLUYjNiYtSSNtbkdGJDYlUScwLjc1MDBGJy8lJXNpemVHUSMxNkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi5RIixGJ0Y0RjcvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHUSV0cnVlRicvJSlzdHJldGNoeUdGQC8lKnN5bW1ldHJpY0dGQC8lKGxhcmdlb3BHRkAvJS5tb3ZhYmxlbGltaXRzR0ZALyUnYWNjZW50R0ZALyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjE2JVEnMC44NjgzRidGNEY3RjdGNEY3Rjc= indican que la distancia entre los puntos de la figura de la izquierda es LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEiZEYnLyUlc2l6ZUdRIzE2RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNiQtSSNtbkdGJDYlUScwLjc1MDBGJ0YvL0Y2USdub3JtYWxGJ0ZBRi9GQS1JI21vR0YkNi5RIj1GJ0YvRkEvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkkvJSlzdHJldGNoeUdGSS8lKnN5bW1ldHJpY0dGSS8lKGxhcmdlb3BHRkkvJS5tb3ZhYmxlbGltaXRzR0ZJLyUnYWNjZW50R0ZJLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGWC1GPjYlUScwLjg2ODNGJ0YvRkFGQQ== u. Mueva la barra deslizante hasta encontrar la m\303\255nima distancia entre el punto LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYlLUYjNiYtSSNtbkdGJDYlUSIwRicvJSVzaXplR1EjMTZGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYuUSIsRidGNEY3LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR1EldHJ1ZUYnLyUpc3RyZXRjaHlHRkAvJSpzeW1tZXRyaWNHRkAvJShsYXJnZW9wR0ZALyUubW92YWJsZWxpbWl0c0dGQC8lJ2FjY2VudEdGQC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnRjBGN0Y0RjdGNw== y un punto LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYlLUYjNiYtSSNtaUdGJDYmUSJ4RicvJSVzaXplR1EjMTZGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYuUSIsRidGNC9GO1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjkvJSlzdHJldGNoeUdGRS8lKnN5bW1ldHJpY0dGRS8lKGxhcmdlb3BHRkUvJS5tb3ZhYmxlbGltaXRzR0ZFLyUnYWNjZW50R0ZFLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjE2JlEieUYnRjRGN0Y6RkFGNEZBRkE= de la gr\303\241fica de 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 -recuerde que la gr\303\241fica de la derecha le servir\303\241 para encontrar la distancia m\303\255nima-. \302\277Es \303\272nico ese punto? \302\277Por qu\303\251?

En este caso, dada la simetr\303\255a que se observa en la gr\303\241fica de la izquierda, hay dos puntos cuya distancia es m\303\255nima; las coordenadas de los puntos son (0.7045,-0.5036) y (-0.7045,-0.5036). La distancia m\303\255nima es de 0.8660 u.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Observe que la distancia m\303\255nima entre los puntos de la Figura 3 corresponde a los puntos m\303\255nimos en la Figura 4. \302\277Comprende por qu\303\251 es as\303\255 esto? Note tambi\303\251n que en la Figura 4, en cada uno de los puntos donde la funci\303\263n toma sus valores m\303\255nimos, se puede trazar rectas tangentes horizontales; en cambio, en el punto LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYlLUYjNiYtSSNtaUdGJDYmUSJ4RicvJSVzaXplR1EjMTZGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYuUSIsRidGNC9GO1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjkvJSlzdHJldGNoeUdGRS8lKnN5bW1ldHJpY0dGRS8lKGxhcmdlb3BHRkUvJS5tb3ZhYmxlbGltaXRzR0ZFLyUnYWNjZW50R0ZFLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjE2JlEieUYnRjRGN0Y6RkFGNEZBRkE= de la Figura 3 donde la distancia es m\303\255nima al origen, aunque puede trazarse una recta tangente, \303\251sta no es horizontal. Observe en las figuras de abajo los segmentos de recta color verde: 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Comentarios: Este ejemplo, de mayor dificultad al del problema de juguete, requiri\303\263 nuevamente de seguir los pasos b\303\241sicos de an\303\241lisis que hay que realizar en los problemas de optimizaci\303\263n, y que fueron resaltados en color verde en la secci\303\263n Problema aplicado 1: Un problema de juguete sobre optimizaci\303\263n. Trate de identificar cada uno de esos pasos b\303\241sicos en el problema anterior de distancia. Nuevamente, en este problema se ha recurrido al an\303\241lisis gr\303\241fico para resolver el problema de optimizaci\303\263n. Esto se ha hecho as\303\255 con fines de que usted adquiera una comprensi\303\263n gr\303\241fica del proceso, lo cual ser\303\241 de mucha ayuda cuando aprenda c\303\263mo resolver problemas de optimizacion utilizando m\303\251todos anal\303\255ticos. Observe que en este caso la funci\303\263n que se est\303\241 minimizando, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2J1EiZEYnLyUlc2l6ZUdRIzE2RicvJSdpdGFsaWNHUSV0cnVlRicvJSp1bmRlcmxpbmVHRjQvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYmLUYjNiUtRiw2J1EieEYnRi9GMkY1RjdGNS9GOFEnbm9ybWFsRidGL0Y1RkJGNUZC, tiene la caracter\303\255stica de que en los puntos donde se alcanzan valores m\303\255nimos pueden trazarse rectas horizontales. El valor de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JlEieEYnLyUlc2l6ZUdRIzE2RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0Y2USdub3JtYWxGJw== que minimiza la distancia no se encuentra en los extremos, tal como sucedi\303\263 en el problema de juguete. Note tambi\303\251n lo relacionado con la simetr\303\255a del problema y la existencia de dos valores m\303\255nimos para este problema.

Ahora estudiar\303\241 un problema cl\303\241sico del c\303\241lculo diferencial: Encontrar el volumen m\303\241ximo de una caja, sin tapadera, que se forma al cortar cuadrados de las esquinas de una l\303\241mina y luego doblar los lados. Abajo se muestran tres figuras: En la figura del extremo izquierdo puede ver la forma que adquiere la l\303\241mina cuando se var\303\255a el tama\303\261o del cuadrado que se corta en cada una de sus esquinas; con la barra deslizante puede variar el tama\303\261o de los cuadrados. En la figura del centro se muestra la misma l\303\241mina, pero ahora se han coloreado, para su comodidad, cada uno de los lados de la caja que se forma al doblar la l\303\241mina. En la figura del extremo derecho puede observar la forma final de la caja. Note que puede rotar la caja del extremo derecho para observarla desde diferentes \303\241ngulos. 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 Mueva la barra deslizante para variar el tama\303\261o de los cuadrados que se recortan en cada una de las esquinas de la l\303\241mina. use DocumentTools in # Enter Maple commands to be executed when the specified # action is carried out on the component. # Use: # Do( %component_name ); # and # Do( %component_name = value ); # to set and get properties of the component. # You can also use arbitrary expressions # involving components, e.g.: # Do( %target = %input1 + 2*%input2 ); # Note the %-prefix to each component name. # See ?CustomizingComponents for more information. Do(f=%Slider3); hojac:=plottools[polygon]([[-f,0],[f,0],[f,5-f],[5,5-f],[5,15+f],[f,15+f],[f,20],[-f,20], [-f,15+f],[-5,15+f], [-5,5-f],[-f,5-f]], color="Aqua"); Do(%Plot4=plots[display](hojac, axes=none,scaling=constrained)); cara1:=plottools[polygon]([[-f,0], [f,0], [f,5-f],[-f,5-f]], color="Green"); cara2:=plottools[polygon]([[f, 5-f],[5,5-f],[5,15+f],[f,15+f]], color="Red"); cara3:=plottools[polygon]([[-f,15+f], [f,15+f], [f,20],[-f,20]], color="Green"); cara4:=plottools[polygon]([[-5, 5-f],[-f,5-f],[-f,15+f],[-5,15+f]], color="Red"); cara5:=plottools[polygon]([[-f, 5-f],[f,5-f],[f,15+f],[-f,15+f]], color="Yellow"); Do(%Plot5=plots[display]({cara1,cara2,cara3,cara4,cara5},axes=none,scaling=constrained)); Tcara1:=plottools[polygon]([[-f,0,0], [f,0,0], [f,0,5-f],[-f,0,5-f]], color="Green"); Tcara2:=plottools[polygon]([[f,0, 0],[f,10+2*f,0],[f,10+2*f,5-f],[f,0,5-f]], color="Red"); Tcara3:=plottools[polygon]([[-f,10+2*f,0], [f,10+2*f,0], [f,10+2*f,5-f],[-f,10+2*f,5-f]], color="Green"); Tcara4:=plottools[polygon]([[-f, 0,0],[-f,10+2*f,0],[-f,10+2*f,5-f],[-f,0,5-f]], color="Red"); Tcara5:=plottools[polygon]([[-f,0, 0],[f,0,0],[f,10+2*f,0],[-f,10+2*f,0]], color="Yellow"); Do(%Plot6=plots[display]({Tcara1,Tcara5,Tcara3,Tcara2,Tcara4},axes=none,scaling=constrained)); end use; LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= 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 Coloque el rat\303\263n sobre la figura de abajo, y mientras mantiene presionado el bot\303\263n izquierdo, puede rotar la caja. Si la figura no rota, presione el bot\303\263n 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LSUlUk9PVEc2Jy0lKUJPVU5EU19YRzYjJCIiISEiIi0lKUJPVU5EU19ZR0YnLSUtQk9VTkRTX1dJRFRIRzYjJCIkIUdGKi0lLkJPVU5EU19IRUlHSFRHNiMkIiQrJEYqLSUpQ0hJTERSRU5HNiI= que aparece en el men\303\272 y luego rote la figura. 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 Suponga que la caja la va a construir a partir de una l\303\241mina que mide 20 u de largo y 10 u de ancho. A continuaci\303\263n se indican los tres pasos b\303\241sicos sugeridos en esta gu\303\255a para resolver gr\303\241ficamente el problema:

Denote como LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JlEieEYnLyUlc2l6ZUdRIzE2RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0Y2USdub3JtYWxGJw== el lado del cuadrado que cortar\303\241 en cada esquina, tal como se indica en la figura de abajo. MFNWtKUb<ob<R=MDLCdNFZdk:lW>T:xy_<x?c?CVLCTJcDXoXuux=SEaiYYyXvYXkovDqgw?CW=tb?beQt?CGuMhLQucKcS[daAs;KUR_VL;TeqFfGHjSuCURtowDKhkWEO;XTSuS_IFUwDOTSEvSOcwSeHKBQKwYSDXaRtSR;CIk=stWTFCW@cstOFMAhU=hdeSU=gWwvbcSVWuw[ILMFcke>ES^swBmiX=BkMvkas]mYa_ioiviqeumgNmgBLMPYqfDZ[pZ:?o;^jTRbG:ZJNGCf@:>j^Wl_>fs`[qwlLqijFggYmBAcIWhgY]Ho]BqsPpuFpcyOtTpqCHlVxw`i`ha__PvHqsr`riYZ<gs@F_nQjboebqp]glbplmNtvV]gQd@yjDIwiWeTypR`ajX`gVvYpcs`uyVyGWtWOgIww?oiY>ee^gw?hsQutYdVptv`aNih[PcaXqL^xPqq;XpMxgmwc@NyTWhjhquPxOi^OpaAX`JPxvP]qhnyOniP`Mwv\\nqI>[o^_lF[iYc_VtSYtkviyOnsXfIgvTYjSyhZOhcvyd^naGpwik`qkfI_Dgmk`mR`pPA[o^eDonm?yJAgLQqaf]EVaUaqgVadv\\]XvMvlkiyUXpg@y`fliapgHixiyX>uSOccXirowy@]SIeT?hW>[o^c^_inptQQelhqRpjAqc^QtAHvCamRVoV_hoongoeJXeLYeVyw;iiSp]RyvwFlCVg=x^MFe_Hjtab@qnAnugGos?f@Awro^`xf]YcsqyAXhb_peVb]VhpqoUXilNdhIamafaIn=ilCNyJAixfehvrmikR@o[glGvy@wgp_[oy_X>vpxeGwe_OfYfwWylF^xE```NgR_tfqanQnY_mSq\\cguXplcIwrGpgIkTyrCw\\AXwgYuxFlKPm?nmSyqXferxiGn^d?m]G[uAtaYuPFosNcqwvX`[Vxy?n`?iqQwwVI\\[>mbHvvV^oOapap`xpqGlw?wShyeqoHImK@ipwgsQbDHsu?pmV^wy\\DYik_woqmngnqvjHvchnirw]R@o[pxkQtMxjIffnPlKqsEHuvn_X@^vxnRQhhWaaNsWYaI^sUPuAXqAVpgxn]Oi\\xqvXB=HKUhH]T]=iRYcLMcQEFXKSEuwwCIJIHmwuVEChUSieFMYR]SSyMdimsEmtUMSwwvvEirstBaRqEGgCEDEim=H=gewIexEhsEhIeYj=H=idImBEUbcaGuMVKCf`UfR=y^yu;qxNOWV_ecGia]F_iWO?tp_dLeSfuT@oG\\AXsGwmaYnqeiAGbIeQmIHwsvaE?]gamSCKtPIuOEihAVTYSc_ua?SmoXNOs?IST;Bb=UpAgqKfcGGe=VWCU=]Ww]W^Sb=Gs]uc_shVQdvSxFgxr_UIqIU_CF[FI_IL_uuugoARbGU@?RYsIXeyv?CoSsqcTYyXYeXweBUCf`Qh=ccZUFNaHJ]HpuwhgEbQvQmx\\YbOgCpAxjMXMiSPaB[oVKcSGEsEsGNUVk?UJWccusXcx_YYnItdQEi[tSIUWOg?ME[ov<GhhKVV]B]oTjGUXUh;oxV\\jTDrP<W>PmoqMfyTYepXTs[qpUQl]ur>LM[pV?LjW]YjEyhQqEQMjPOvPPK<RbHJJLg<g[:@d=:ugpWTa_I<?bJuHnohp_v^ISIOFZ[tGCgrcbaExDSiwYywKc:QipoXYEerSYPQhaseFmxhiSpoWoAWHIxOwgveFY[yZyBg[RW_irQvEGWiuWq?Df=vJuDMaxfIiPwSYogHetygxUmEGIIDwxfkYaqxUYdy[y=UckOutSI]iI]GDdibaABpcTikgNMgvKrHuvLudoub_]hpqxbwvHUtmoVaOW:Cfh=rcMYhaTgACJ;VWKUUuQhlNciOUMlWUqdxmQtRw]LaQJXaqsPysYSiDYMET:=oeQoBPxKXNauQyHv@`vsyvWxUS<UQaq>`M:=o;ZG::\\jteOclUK@UVxv=<RbXMNEXp`Qayx:\\jT<J[p:f`:JJLgBf`:::Z:@d=>>:bJ<Bjh]NUx`HwmpA_<NB[ggWUMWdacID?bJUBZ>d=q^E?hkouxWygauvWxGi[yYhXp_pqwiOqCYZPw^YvyryrmgtOwdo^yIIinAwKaqaiaiIx:H[oQyLo_QX`oylyGmFhkonyIVevFuDQwuOwc`yYPsy>wGvwpXulqsuvwfpswgmtflfGu\\yw@@diPnqgqeFq[qZeAr\\quuquoHxmoekgrdgytwlVIiaOxAwi^?ir`t^Ae<>f?gasafl_yrgxLFn@N]FG`GwwYyewYhkGwow[<g[n>vnfauQeAPvLadg^t:WjHWtm`cOQbG^`ba`wVsNg_Nu:YtnGgtQds[II;UxuGy_UwErOawEogDwsqewTwyoYihQhHMRG[tGED[Ii[kGOOd]sRfWs_KhoIBZ[t:;;U::>[oQEsF[sxL]T:=gU;UkqSTSCIMy<;FNABB>;:?\\J;ZBf`::::>>\\Z::eIgmuV_YaYIXcR:=gUktPigCIsMoE?=W^eryyREwyiOh;URG[HxWtaWgygy\\[t:K:<BZZB::::;^Z::?L:w_UPVas\\icmCB>Oi;eDiERY]iOcQ[j;:B<::ZjHtOvv]I_HYs=@d=f[d@a=ifkonE_jT>bbHZJNG>BjJ?BZhxq`pn^OgDYnB>[o:J:BB>;;:::;^Z::?L:;jQ@ocHOSPmlAO<LRG<XCqSWPC?UACf@[bT=g;;<b;B[ojJ[PYkGoLKELAXqMSgyBaaUKQexGcK_xnstrgy^IbfaWcisbQThUiVuxc[t:qRoCYZuVTmrycelEiymhZGwhchawh__dkInnWef>p]glbpLJGXuhttwmo\\TUY<R]hkLIQVHjNxvOHpgdyMdYOHypDQRquWEq?ayUyXQmnZqtxHjmXliUXImJtQUQDqgQXWynJUJULUUMYPTue=WhAsXPnTHMEIkCIqM@xx`TldmwdMGYKhmqaPXg;mxuaDd=b?eHaUuHOvluRpgecMw^mCYKmnpwCur<qsptWEEXmItkIk_mqIQwwqMRIUDEou=YhyqRDuSMUjemJUJUTs^mSFZ`aw@fr>orAWgWqq_VuEHgr_w>o^QXxxvn=`uLpu\\qiInwEax:qaiIkFivaIs>?fG`htAhHfkpgfb_fHqalavC`kGw[vXuLWx:akKIxF_cIqrEahYhswF]cvxVHa<_vJIvDUYKEIyfEivf_tmic`KWZeSRkyGoufAEdySUUfVyVUyg`wglChIoxhYS@kTpyvTEiCOiXsr:MBeYXRoIjWHuqhI]n<QMHqw=uOtyUGaWE`@LWguSNqWGesuaQuuvOiyeIt?XyQYW^\\toAQ@LLRuo]`XqmwCAR@mLW`u`qw=upglqwHXxhWKXStxsstxXeqmIw]LRTqTjlte<LdIu]ILaqyNit]ILAQOyQmOtr]lXPxvxQmjTYF\\v>Qj]=NIdn_amUQJU\\WNpsHIrHaknyPQamdUQpap`xpqElw=wShyeqoHqpmdUSpuUmqfYyVXuthQcQSimKWlmV]yjqq_Mp<TYNixrJIXlnEwXPVuyxVxU;YtwXx:Im]DQdep\\MxjInFpTlLuSqpvyxSeqcAVphxn]SI]xQxXB<VWppxLWwYsqmwxmsfIPJhYotk;`QrlY;xveUmTPlcHodiNjpY\\dQmaMQUScxxVxOQtq;xmcPrfyWZiqqav:urFqYmevyxWcio^MYb]VcyqKxO[qQaLqEEWCATDim=HMfawIax`qwn=o@TPVEy^=pn<q\\Tq:ToRLwmaM=Dut@ugyljxx>\\o=iuHhmMIlghWKxu:YY@MTXqTCdWEEOOarpeVbQNHtlUMxwmkQDThuSc@koasM=XgemWeleqK`hqNPswmoFqTgQw?PqXtr:MJemsn]XumO[yQfUwYXh@h\\txsxQhHIqEXsrxu_PeoXvbXwkKZeQRelJusXtkRyU]AMBirg<Q@LL:=oQMWNpUdIPJhPVMsk<l:LRw`vKMqt]Q\\ussyk;`mWyjXxyYqOrlJLrEyICugtIHJ?BeAtDaEoITIWVZ<KoQtNYvR<c<giKNwk_[^JmmQdOBVZGXwNIo^WsPVwZ^t:WbAPyWywpp[<g[ZJ>>:<<>J<<SEb[tGeSCaLI`NZ\\t:<;C>jJ;::B[::[Z>D::oU\\of`_VdMs]CCBcP>>pXfygyu`Vep@eAFn@N]PIoMObg^t:NB;>NBB<<:::JJB<<:Z:teUtIgyRMHocvK_B>O]T:JZB>:::=OVMoMunLhOU]?LRg\\t:<;C>jJ?B\\WZ@TqdP_tYxyvXDS:=o;<<j;J[>J::ZB::>>\\Z:JRuQpDU<<WJuuuqULytQ`qpQUlevOHs:ARoQy\\YxIeYdDyjyjMpuXUw^MnXUJTPmLH?Twj]kd`sKuKw\\lvayIyNIiQQ`qnQTlQyeIsMySatwuuUlewoptBaRG\\P>xkv]NsHx@avExUFmoypQsMvCTW\\Mn@LMHyJx<lvtyvMxCySTUK[pJTLthhUCYQiaNYyyxQuuIuIemDmpfExRLKo\\omuRIpTitUseRbhjT<WrdrMxjI\\QJ`UqeUlinidTiXyPypfXkJUJoePsDUwLr]isqToUmMGisCiMq]N]TNCYUieNY=y<hPLLWZpoXtYvDYyYyMxswmOcQwdpUquxWIsUtW^tMwAWoMr`Urj=YX\\P\\pyxav;yJMhMulsXutXqUGdxpiwPyr>eKfpjeqqsisvyrYmy_alEenmuqR]qLDXh]naeOkLt?yxvySy@yEHWlQVjTLo<XcEjx=xqtYmduOHosioemoSEpuIyayqqYYsarayQYxVhYqitky\\yjmnDxTkQNNARgUUImmFYrIxQXxkydynmyYyKxAVhPypypaxQPMsumydyo`xOyIuY\\y;myOYXIivytYfYVSyqquoZUKo<TeMjx=vCxNlyYihqper=dqJyKwisYaSYmyExuwiumTLpELfI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\302\277Cu\303\241l es la funci\303\263n LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiVkYnLyUlc2l6ZUdRIzE2RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNiQtRiw2JlEieEYnRi9GMkY1L0Y2USdub3JtYWxGJ0YvRkBGQA== que permite calcular el volumen de la caja que se construye a partir de la l\303\241mina recortada?

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. Vea la figura para auxiliarse en la construcci\303\263n de la funci\303\263n: 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Nota: Observe que en este problema no hemos necesitado introducir ning\303\272n sistema de coordenadas para resolverlo.

\302\277Cu\303\241les son los valores LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JlEieEYnLyUlc2l6ZUdRIzE2RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0Y2USdub3JtYWxGJw== que tienen sentido en este problema para la funci\303\263n LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiVkYnLyUlc2l6ZUdRIzE2RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNiQtRiw2JlEieEYnRi9GMkY1L0Y2USdub3JtYWxGJ0YvRkBGQA==?

Al ir variando la barra deslizante de la figura de abajo es claro que se obtiene una caja si 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. \302\277Qu\303\251 sucede si LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEieEYnLyUlc2l6ZUdRIzE2RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiPUYnRi8vRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZALyUpc3RyZXRjaHlHRkAvJSpzeW1tZXRyaWNHRkAvJShsYXJnZW9wR0ZALyUubW92YWJsZWxpbWl0c0dGQC8lJ2FjY2VudEdGQC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRk8tSSNtbkdGJDYlUSIwRidGL0Y8Rjw= o LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEieEYnLyUlc2l6ZUdRIzE2RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiPUYnRi8vRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZALyUpc3RyZXRjaHlHRkAvJSpzeW1tZXRyaWNHRkAvJShsYXJnZW9wR0ZALyUubW92YWJsZWxpbWl0c0dGQC8lJ2FjY2VudEdGQC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRk8tSSNtbkdGJDYlUSI1RidGL0Y8Rjw=? Mueva la barra deslizante y vea lo que sucede cuando LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEieEYnLyUlc2l6ZUdRIzE2RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiPUYnRi8vRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZALyUpc3RyZXRjaHlHRkAvJSpzeW1tZXRyaWNHRkAvJShsYXJnZW9wR0ZALyUubW92YWJsZWxpbWl0c0dGQC8lJ2FjY2VudEdGQC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRk8tSSNtbkdGJDYlUSIwRidGL0Y8Rjw= o LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEieEYnLyUlc2l6ZUdRIzE2RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiPUYnRi8vRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZALyUpc3RyZXRjaHlHRkAvJSpzeW1tZXRyaWNHRkAvJShsYXJnZW9wR0ZALyUubW92YWJsZWxpbWl0c0dGQC8lJ2FjY2VudEdGQC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRk8tSSNtbkdGJDYlUSI1RidGL0Y8Rjw=. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=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 use DocumentTools in # Enter Maple commands to be executed when the specified # action is carried out on the component. # Use: # Do( %component_name ); # and # Do( %component_name = value ); # to set and get properties of the component. # You can also use arbitrary expressions # involving components, e.g.: # Do( %target = %input1 + 2*%input2 ); # Note the %-prefix to each component name. # See ?CustomizingComponents for more information. Do(g=%Slider4); hojac:=plottools[polygon]([[-g,0],[g,0],[g,5-g],[5,5-g],[5,15+g],[g,15+g],[g,20],[-g,20], [-g,15+g],[-5,15+g], [-5,5-g],[-g,5-g]], color="Aqua"); Do(%Plot7=plots[display](hojac, axes=none,scaling=constrained)); cara1:=plottools[polygon]([[-g,0], [g,0], [g,5-g],[-g,5-g]], color="Green"); cara2:=plottools[polygon]([[g, 5-g],[5,5-g],[5,15+g],[g,15+g]], color="Red"); cara3:=plottools[polygon]([[-g,15+g], [g,15+g], [g,20],[-g,20]], color="Green"); cara4:=plottools[polygon]([[-5, 5-g],[-g,5-g],[-g,15+g],[-5,15+g]], color="Red"); cara5:=plottools[polygon]([[-g, 5-g],[g,5-g],[g,15+g],[-g,15+g]], color="Yellow"); Do(%Plot8=plots[display]({cara1,cara2,cara3,cara4,cara5},axes=none,scaling=constrained)); Tcara1:=plottools[polygon]([[-g,0,0], [g,0,0], [g,0,5-g],[-g,0,5-g]], color="Green"); Tcara2:=plottools[polygon]([[g,0, 0],[g,10+2*g,0],[g,10+2*g,5-g],[g,0,5-g]], color="Red"); Tcara3:=plottools[polygon]([[-g,10+2*g,0], [g,10+2*g,0], [g,10+2*g,5-g],[-g,10+2*g,5-g]], color="Green"); Tcara4:=plottools[polygon]([[-g, 0,0],[-g,10+2*g,0],[-g,10+2*g,5-g],[-g,0,5-g]], color="Red"); Tcara5:=plottools[polygon]([[-g,0, 0],[g,0,0],[g,10+2*g,0],[-g,10+2*g,0]], color="Yellow"); Do(%Plot9=plots[display]({Tcara1,Tcara5,Tcara3,Tcara2,Tcara4},axes=none,scaling=constrained)); end use; Recuerde: 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=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

En este problema se desea encontrar el valor m\303\241ximo de la funci\303\263n 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. Abajo se muestra la gr\303\241fica de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiVkYnLyUlc2l6ZUdRIzE2RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNiQtRiw2JlEieEYnRi9GMkY1L0Y2USdub3JtYWxGJ0YvRkBGQA==. \302\277Cu\303\241l es el volumen m\303\241ximo que puede obtenerse para la caja? Mueva la barra deslizante para encontrar el valor buscado. Note que la gr\303\241fica se ha construido en el intervalo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbW5HRiQ2JVEiMEYnLyUlc2l6ZUdRIzE2RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LlEmJmxlcTtGJ0YvRjIvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjsvJSlzdHJldGNoeUdGOy8lKnN5bW1ldHJpY0dGOy8lKGxhcmdlb3BHRjsvJS5tb3ZhYmxlbGltaXRzR0Y7LyUnYWNjZW50R0Y7LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSi1JI21pR0YkNiZRInhGJ0YvLyUnaXRhbGljR1EldHJ1ZUYnL0YzUSdpdGFsaWNGJ0Y1LUYsNiVRIzUuRidGL0YyRjI= 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use DocumentTools in # Enter Maple commands to be executed when the specified # action is carried out on the component. # Use: # Do( %component_name ); # and # Do( %component_name = value ); # to set and get properties of the component. # You can also use arbitrary expressions # involving components, e.g.: # Do( %target = %input1 + 2*%input2 ); # Note the %-prefix to each component name. # See ?CustomizingComponents for more information. Do(z=%Slider5); vdex:=plot((10-2*x)*(20-2*x)*x,x=0..5,y=0..200, color=red, thickness=2); lv2:=plottools[line]([z,0],[z,z*(10-2*z)*(20-2*z)],color=black,linestyle=dash); lh2:=plottools[line]([z,z*(10-2*z)*(20-2*z)],[0,z*(10-2*z)*(20-2*z)],color=black,linestyle=dash); textof:=plots[textplot]([z+0.5,z*(10-2*z)*(20-2*z),sprintf("(%2.4f,%2.4f)",z,z*(10-2*z)*(20-2*z))]): Do(%Plot10=plots[display]({vdex,lv2,lh2,textof},labels=["x","V(x)"],axes=normal)); end use;

Al ir moviendo la barra deslizante es claro que el valor m\303\241ximo de la funci\303\263n LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiVkYnLyUlc2l6ZUdRIzE2RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNiQtRiw2JlEieEYnRi9GMkY1L0Y2USdub3JtYWxGJ0YvRkBGQA== ocurre en LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEieEYnLyUlc2l6ZUdRIzE2RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiPUYnRi8vRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZALyUpc3RyZXRjaHlHRkAvJSpzeW1tZXRyaWNHRkAvJShsYXJnZW9wR0ZALyUubW92YWJsZWxpbWl0c0dGQC8lJ2FjY2VudEdGQC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRk8tSSNtbkdGJDYlUScyLjExMTFGJ0YvRjxGPA==, por lo que 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.

Con la informaci\303\263n de la gr\303\241fica se puede determinar las dimensiones de la caja con el volumen m\303\241ximo. \302\277Cu\303\241les son las dimensiones de la caja con volumen m\303\241ximo?

Aproximando a dos cifras decimales, las dimensiones de la caja con volumen m\303\241ximo son las que se muestran en la figura de abajo: 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JSFH Mueva la barra deslizante de la figura de abajo y observe que en la gr\303\241fica de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiVkYnLyUlc2l6ZUdRIzE2RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNiQtRiw2JlEieEYnRi9GMkY1L0Y2USdub3JtYWxGJ0YvRkBGQA== se puede dibujar una recta tangente horizontal en el m\303\241ximo valor LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEiVkYnLyUlc2l6ZUdRIzE2RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtRi82JlEkbWF4RidGMi9GNlEmZmFsc2VGJy9GOVEnbm9ybWFsRidGMkY1RjgvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0ZC. Note los signos de la pendiente de la recta tangente antes y despu\303\251s de alcanzar LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiVkYnLyUlc2l6ZUdRIzE2RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNiQtRiw2JlEieEYnRi9GMkY1L0Y2USdub3JtYWxGJ0YvRkBGQA== su m\303\241ximo valor LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEiVkYnLyUlc2l6ZUdRIzE2RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtRi82JlEkbWF4RidGMi9GNlEmZmFsc2VGJy9GOVEnbm9ybWFsRidGMkY1RjgvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0ZC. Observe el signo de la pendiente de la recta tangente, justo antes y justo despu\303\251s de que la funci\303\263n alcanza el valor m\303\241ximo. 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use DocumentTools in # Enter Maple commands to be executed when the specified # action is carried out on the component. # Use: # Do( %component_name ); # and # Do( %component_name = value ); # to set and get properties of the component. # You can also use arbitrary expressions # involving components, e.g.: # Do( %target = %input1 + 2*%input2 ); # Note the %-prefix to each component name. # See ?CustomizingComponents for more information. Do(z2=%Slider8); vdex:=plot((10-2*x)*(20-2*x)*x,x=0..5, y=0..220,color=red, thickness=2); lv22:=plottools[line]([z2,0],[z2,z2*(10-2*z2)*(20-2*z2)],color=black,linestyle=dash); lh22:=plottools[line]([z2,z2*(10-2*z2)*(20-2*z2)],[0,z2*(10-2*z2)*(20-2*z2)],color=black,linestyle=dash); textof2:=plots[textplot]([z2+0.75,z2*(10-2*z2)*(20-2*z2)-15,sprintf("(%2.4f,%2.4f)",z2,z2*(10-2*z2)*(20-2*z2))]): m:=12*z2^2-120*z2+200; yo:=4*z2^3-60*z2^2+200*z2; tangente:=plot(m*(x-z2)+yo, x=0..5,y=0..220,color=blue); Do(%Plot13=plots[display]({vdex,lv22,lh22,textof2,tangente},axes=normal,labels=["x","V(x)"] )); end use;

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Si en lugar de buscar el valor m\303\241ximo de una funci\303\263n arbitraria, cuya derivada existe en todo punto, se busca el valor m\303\255nimo de ella, \302\277cu\303\241les ser\303\255an los signos de la pendiente de la recta tangente poco antes de alcanzar el valor m\303\255nimo, en el valor m\303\255nimo, y poco despu\303\251s de alcanzar el valor m\303\255nimo? Utilice la figura de abajo para realizar el an\303\241lisis.

Antes de alcanzar el valor m\303\255nimo, la pendiente de la recta tangente es negativa. En el valor m\303\255nimo, la recta tangente tiene pendiente nula. Poco despu\303\251s de alcanzar el valor m\303\255nimo, la pendiente de la recta tangente es positiva.

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)BOUNDS_XG6#$"$q&!""-%)BOUNDS_YG6#$"$?"!""-%-BOUNDS_WIDTHG6#$"%+L!""-%.BOUNDS_HEIGHTG6#$"%!y$!""-%)CHILDRENG6" use DocumentTools in # Enter Maple commands to be executed when the specified # action is carried out on the component. # Use: # Do( %component_name ); # and # Do( %component_name = value ); # to set and get properties of the component. # You can also use arbitrary expressions # involving components, e.g.: # Do( %target = %input1 + 2*%input2 ); # Note the %-prefix to each component name. # See ?CustomizingComponents for more information. Do(z3=%Slider9); vdex2:=plot(-(10-2*x)*(20-2*x)*x,x=0..5, y=-220..0,color=red, thickness=2); m2:=-12*z3^2+120*z3-200; yo2:=-4*z3^3+60*z3^2-200*z3; tangente2:=plot(m2*(x-z3)+yo2, x=0..5,y=-220..0,color=blue); Do(%Plot14=plots[display]({vdex2,tangente2},axes=normal)); end use;

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