2.011Loading LinearAlgebra : 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= print(); # input placeholder : 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 = print(); # input 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 og punktet E. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkjbW5HRiQ2JFEjMTJGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSShtZmVuY2VkR0YkNiQtRiM2JS1JI21pR0YkNiVRInhGJy8lJ2l0YWxpY0dRJXRydWVGJy9GMFEnaXRhbGljRictSSNtb0dGJDYtUSgmbWludXM7RidGLy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRi8lKXN0cmV0Y2h5R0ZGLyUqc3ltbWV0cmljR0ZGLyUobGFyZ2VvcEdGRi8lLm1vdmFibGVsaW1pdHNHRkYvJSdhY2NlbnRHRkYvJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0ZVLUYsNiRRIjFGJ0YvRi8tRkE2LVEiK0YnRi9GREZHRklGS0ZNRk9GUUZTRlYtRiw2JFEiMEYnRi8tRjM2JC1GIzYlLUY4NiVRInlGJ0Y7Rj5GZW4tRiw2JFEiNkYnRi9GL0ZALUYsNiRRIzYwRidGLy1GMzYkLUYjNiUtRjg2JVEiekYnRjtGPkZALUYsNiRRIjRGJ0YvRi8= = print(); # input placeholderJSFHVi har nu scenens tagplanens ligning \316\262: 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JSFHb)JSFHprint(); # input placeholderDen vandrette plan, xy-planen, har ligningen z = 0 og normalvektoren er vektoren 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 mellem to planer er vinklen mellem de to planers normalvektorer. L0klQ29zdkc2IiooLUkwZGVsYXlEb3RQcm9kdWN0RzYkJSpwcm90ZWN0ZWRHL0krbW9kdWxlbmFtZUdGJEksVHlwZXNldHRpbmdHNiRGKUkoX3N5c2xpYkdGJDYkST0jbW92ZXIobWkoImEiKSxtbygiJnJhcnI7IikpR0YkST0jbW92ZXIobWkoImIiKSxtbygiJnJhcnI7IikpR0YkIiIiLUkkYWJzR0YpNiNGMCEiIi1GNDYjRjFGNg==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 = print(); # input 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 = print(); # input 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 = print(); # input placeholderVinklen, som tagplanen danner med den vandrette plan er derfor: 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for linjen l:print(); # input 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= print(); # input placeholderLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Linjen l parameterfremstilling gennem centrum og 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til tangentplanen er vektoren 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= print(); # input placeholderJSFHLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Tangentplanen har dermed ligningen: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 \342\207\224 print(); # input 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=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 = print(); # input placeholderLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=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 = print(); # input placeholderLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Sk\303\246ringspunktet mellem l og tangentplanen til kuglen i punktet N er: 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planen z=6 tangerer kuglen i punktet (0,0,6) og kuglens centrum er (0,0,0), vil radius v\303\246re 6. 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= print(); # input placeholderDa 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er 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# input 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for denne ligning har ekstremumspunkter n\303\245r 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= print(); # input 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ser vi parabelen med grenene vendt nedad. Ligningen for kurven, som ekstremumspunkterne vil tilh\303\270rer, er 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Hvis grafen tangerer x-aksen i et punkt med positivt 1. koordinat, vil y'(x) = 0 og y(x) = 0. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEmc29sdmVGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2Li1JI21uR0YkNiRRIjBGJy9GM1Enbm9ybWFsRictSSNtb0dGJDYtUSI9RidGPi8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRi8lKXN0cmV0Y2h5R0ZGLyUqc3ltbWV0cmljR0ZGLyUobGFyZ2VvcEdGRi8lLm1vdmFibGVsaW1pdHNHRkYvJSdhY2NlbnRHRkYvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZVLUZBNi1RKiZ1bWludXMwO0YnRj5GREZHRklGS0ZNRk9GUS9GVFEsMC4yMjIyMjIyZW1GJy9GV0Zmbi1GOzYkUSIyRidGPi1GQTYtUSJ+RidGPkZERkdGSUZLRk1GT0ZRL0ZUUSYwLjBlbUYnL0ZXRl9vLUklbXN1cEdGJDYlLUYsNiVRInhGJ0YvRjItRiM2J0ZobkYvLyUrZXhlY3V0YWJsZUdGRi8lMGZvbnRfc3R5bGVfbmFtZUdRKDJEfk1hdGhGJ0YyLyUxc3VwZXJzY3JpcHRzaGlmdEdGPS1GQTYtUSIrRidGPkZERkdGSUZLRk1GT0ZRRmVuRmduRmhuRltvRmRvRmlvRj5GPkZpb0Y+ = print(); # input placeholderLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=JSFHJSFHI dette tilf\303\246lde vil x=1 bruges, fordi den er positiv.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 = print(); # input placeholderVi har nu fundet l\303\270sningen til differentialligningen: 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=JSFH