miniaufgabe.js
=== Montag 10. Dezember 2018 ===
Die Grundmenge seien die natürlichen Zahlen bis 12 ($G=\{1,2,3,4,\ldots,10,11,12\}$). Bilden Sie für die gegebenen Mengen $A$ und $B$ folgende Mengen:
$A\cap B$, $A\cup B$, $A\setminus B$ und $\overline{A} \cap B$.miniAufgabe("#exomengenoperationen","#solmengenoperationen",
[["$A=\\{1, 2, 3, 5, 6, 12\\}$, $B=\\{2, 4, 5, 8\\}$", "$A\\cap B = \\{2, 5\\}$, $A\\cup B=\\{1, 2, 3, 4, 5, 6, 8, 12\\}$, $A \\setminus B = \\{1, 3, 6, 12\\}$, $\\overline{A} \\cap B = \\{4, 8\\}$"], ["$A=\\{4, 8, 9, 10\\}$, $B=\\{3, 6, 7, 9, 10\\}$", "$A\\cap B = \\{9, 10\\}$, $A\\cup B=\\{3, 4, 6, 7, 8, 9, 10\\}$, $A \\setminus B = \\{4, 8\\}$, $\\overline{A} \\cap B = \\{3, 6, 7\\}$"], ["$A=\\{2, 4, 5, 8, 9\\}$, $B=\\{2, 7, 8, 12\\}$", "$A\\cap B = \\{2, 8\\}$, $A\\cup B=\\{2, 4, 5, 7, 8, 9, 12\\}$, $A \\setminus B = \\{4, 5, 9\\}$, $\\overline{A} \\cap B = \\{7, 12\\}$"], ["$A=\\{6, 7, 10, 11\\}$, $B=\\{4, 6, 9, 10\\}$", "$A\\cap B = \\{6, 10\\}$, $A\\cup B=\\{4, 6, 7, 9, 10, 11\\}$, $A \\setminus B = \\{7, 11\\}$, $\\overline{A} \\cap B = \\{4, 9\\}$"], ["$A=\\{3, 4, 6, 7, 9, 11\\}$, $B=\\{2, 4, 8, 9, 10, 11\\}$", "$A\\cap B = \\{4, 9, 11\\}$, $A\\cup B=\\{2, 3, 4, 6, 7, 8, 9, 10, 11\\}$, $A \\setminus B = \\{3, 6, 7\\}$, $\\overline{A} \\cap B = \\{2, 8, 10\\}$"], ["$A=\\{1, 2, 4, 5, 7, 9\\}$, $B=\\{1, 5, 6, 7, 8\\}$", "$A\\cap B = \\{1, 5, 7\\}$, $A\\cup B=\\{1, 2, 4, 5, 6, 7, 8, 9\\}$, $A \\setminus B = \\{2, 4, 9\\}$, $\\overline{A} \\cap B = \\{6, 8\\}$"], ["$A=\\{1, 3, 4, 6, 9, 10\\}$, $B=\\{6, 8, 9, 10, 11\\}$", "$A\\cap B = \\{6, 9, 10\\}$, $A\\cup B=\\{1, 3, 4, 6, 8, 9, 10, 11\\}$, $A \\setminus B = \\{1, 3, 4\\}$, $\\overline{A} \\cap B = \\{8, 11\\}$"], ["$A=\\{3, 5, 9, 11\\}$, $B=\\{1, 2, 4, 6, 9, 11\\}$", "$A\\cap B = \\{9, 11\\}$, $A\\cup B=\\{1, 2, 3, 4, 5, 6, 9, 11\\}$, $A \\setminus B = \\{3, 5\\}$, $\\overline{A} \\cap B = \\{1, 2, 4, 6\\}$"], ["$A=\\{1, 2, 7, 8, 11, 12\\}$, $B=\\{2, 5, 6, 7, 12\\}$", "$A\\cap B = \\{2, 7, 12\\}$, $A\\cup B=\\{1, 2, 5, 6, 7, 8, 11, 12\\}$, $A \\setminus B = \\{1, 8, 11\\}$, $\\overline{A} \\cap B = \\{5, 6\\}$"], ["$A=\\{2, 3, 5, 7, 12\\}$, $B=\\{2, 3, 4, 6, 10, 11\\}$", "$A\\cap B = \\{2, 3\\}$, $A\\cup B=\\{2, 3, 4, 5, 6, 7, 10, 11, 12\\}$, $A \\setminus B = \\{5, 7, 12\\}$, $\\overline{A} \\cap B = \\{4, 6, 10, 11\\}$"]],
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=== Donnerstag 13. Dezember 2018 ===
Zusammenfassen, ausklammern, kürzen, Resultat als Vielfaches einer Potenz von $x$:
miniAufgabe("#exovereinfachenAusklammern","#solvereinfachenAusklammern",
[["$\\displaystyle \\frac{\\frac{1}{3} \\cdot x^{5} \\cdot \\frac{1}{x^{-2}} +\\frac{1}{3} \\cdot x^{8} \\cdot \\frac{1}{x^{-5}}}{\\frac{1}{5} \\cdot x^{5} \\cdot \\frac{1}{x^{-4}} +\\frac{1}{5} \\cdot x^{3} \\cdot \\frac{1}{x^{-12}}}$", "$\\displaystyle \\frac{\\frac{1}{3}\\cdot x^{7} +\\frac{1}{3}\\cdot x^{13}}{\\frac{1}{5}\\cdot x^{9} +\\frac{1}{5}\\cdot x^{15}} = \\frac{\\frac{1}{3}\\cdot x^{7} \\cdot \\left(1 +1\\cdot x^{6}\\right)}{\\frac{1}{5}\\cdot x^{9} \\cdot \\left(1 +1\\cdot x^{6}\\right)} = \\frac{1}{3} \\cdot 5 \\cdot x^{-2} = \\frac{5}{3}\\cdot x^{-2}$"], ["$\\displaystyle \\frac{\\frac{5}{6} \\cdot x^{7} \\cdot \\frac{1}{x^{3}} +\\frac{7}{6} \\cdot x^{4} \\cdot \\frac{1}{x^{-7}}}{\\frac{1}{6} \\cdot x^{3} \\cdot \\frac{1}{x^{-9}} +\\frac{7}{30} \\cdot x^{6} \\cdot \\frac{1}{x^{-13}}}$", "$\\displaystyle \\frac{\\frac{5}{6}\\cdot x^{4} +\\frac{7}{6}\\cdot x^{11}}{\\frac{1}{6}\\cdot x^{12} +\\frac{7}{30}\\cdot x^{19}} = \\frac{\\frac{1}{6}\\cdot x^{4} \\cdot \\left(5 +7\\cdot x^{7}\\right)}{\\frac{1}{30}\\cdot x^{12} \\cdot \\left(5 +7\\cdot x^{7}\\right)} = \\frac{1}{6} \\cdot 30 \\cdot x^{-8} = 5\\cdot x^{-8}$"], ["$\\displaystyle \\frac{\\frac{3}{4} \\cdot x^{6} \\cdot \\frac{1}{x^{-3}} +\\frac{3}{4} \\cdot x^{6} \\cdot \\frac{1}{x^{-7}}}{\\frac{3}{7} \\cdot x^{2} \\cdot \\frac{1}{x^{-4}} +\\frac{3}{7} \\cdot x^{7} \\cdot \\frac{1}{x^{-3}}}$", "$\\displaystyle \\frac{\\frac{3}{4}\\cdot x^{9} +\\frac{3}{4}\\cdot x^{13}}{\\frac{3}{7}\\cdot x^{6} +\\frac{3}{7}\\cdot x^{10}} = \\frac{\\frac{3}{4}\\cdot x^{9} \\cdot \\left(1 +1\\cdot x^{4}\\right)}{\\frac{3}{7}\\cdot x^{6} \\cdot \\left(1 +1\\cdot x^{4}\\right)} = \\frac{3}{4} \\cdot \\frac{7}{3} \\cdot x^{3} = \\frac{7}{4}\\cdot x^{3}$"], ["$\\displaystyle \\frac{\\frac{2}{5} \\cdot x^{7} \\cdot \\frac{1}{x^{3}} +\\frac{1}{4} \\cdot x^{5} \\cdot \\frac{1}{x^{-7}}}{\\frac{6}{7} \\cdot x^{8} \\cdot \\frac{1}{x^{-5}} +\\frac{15}{28} \\cdot x^{5} \\cdot \\frac{1}{x^{-16}}}$", "$\\displaystyle \\frac{\\frac{2}{5}\\cdot x^{4} +\\frac{1}{4}\\cdot x^{12}}{\\frac{6}{7}\\cdot x^{13} +\\frac{15}{28}\\cdot x^{21}} = \\frac{\\frac{1}{20}\\cdot x^{4} \\cdot \\left(8 +5\\cdot x^{8}\\right)}{\\frac{3}{28}\\cdot x^{13} \\cdot \\left(8 +5\\cdot x^{8}\\right)} = \\frac{1}{20} \\cdot \\frac{28}{3} \\cdot x^{-9} = \\frac{7}{15}\\cdot x^{-9}$"], ["$\\displaystyle \\frac{\\frac{4}{3} \\cdot x^{-2} \\cdot \\frac{1}{x^{-8}} +\\frac{2}{3} \\cdot x^{2} \\cdot \\frac{1}{x^{-2}}}{\\frac{8}{3} \\cdot x^{8} \\cdot \\frac{1}{x^{-9}} +\\frac{4}{3} \\cdot x^{3} \\cdot \\frac{1}{x^{-12}}}$", "$\\displaystyle \\frac{\\frac{4}{3}\\cdot x^{6} +\\frac{2}{3}\\cdot x^{4}}{\\frac{8}{3}\\cdot x^{17} +\\frac{4}{3}\\cdot x^{15}} = \\frac{\\frac{2}{3}\\cdot x^{4} \\cdot \\left(2\\cdot x^{2} +1\\right)}{\\frac{4}{3}\\cdot x^{15} \\cdot \\left(2\\cdot x^{2} +1\\right)} = \\frac{2}{3} \\cdot \\frac{3}{4} \\cdot x^{-11} = \\frac{1}{2}\\cdot x^{-11}$"], ["$\\displaystyle \\frac{\\frac{1}{3} \\cdot x^{9} \\cdot \\frac{1}{x^{-9}} +\\frac{8}{7} \\cdot x^{8} \\cdot \\frac{1}{x^{-3}}}{\\frac{7}{8} \\cdot x^{3} \\cdot \\frac{1}{x^{-8}} +3 \\cdot x^{4} \\cdot \\frac{1}{x^{0}}}$", "$\\displaystyle \\frac{\\frac{1}{3}\\cdot x^{18} +\\frac{8}{7}\\cdot x^{11}}{\\frac{7}{8}\\cdot x^{11} +3\\cdot x^{4}} = \\frac{\\frac{1}{21}\\cdot x^{11} \\cdot \\left(7\\cdot x^{7} +24\\right)}{\\frac{1}{8}\\cdot x^{4} \\cdot \\left(7\\cdot x^{7} +24\\right)} = \\frac{1}{21} \\cdot 8 \\cdot x^{7} = \\frac{8}{21}\\cdot x^{7}$"], ["$\\displaystyle \\frac{\\frac{1}{5} \\cdot x^{7} \\cdot \\frac{1}{x^{2}} +\\frac{8}{5} \\cdot x^{9} \\cdot \\frac{1}{x^{-5}}}{\\frac{5}{8} \\cdot x^{7} \\cdot \\frac{1}{x^{-4}} +5 \\cdot x^{9} \\cdot \\frac{1}{x^{-11}}}$", "$\\displaystyle \\frac{\\frac{1}{5}\\cdot x^{5} +\\frac{8}{5}\\cdot x^{14}}{\\frac{5}{8}\\cdot x^{11} +5\\cdot x^{20}} = \\frac{\\frac{1}{5}\\cdot x^{5} \\cdot \\left(1 +8\\cdot x^{9}\\right)}{\\frac{5}{8}\\cdot x^{11} \\cdot \\left(1 +8\\cdot x^{9}\\right)} = \\frac{1}{5} \\cdot \\frac{8}{5} \\cdot x^{-6} = \\frac{8}{25}\\cdot x^{-6}$"], ["$\\displaystyle \\frac{\\frac{3}{4} \\cdot x^{-3} \\cdot \\frac{1}{x^{-9}} +\\frac{5}{6} \\cdot x^{8} \\cdot \\frac{1}{x^{-8}}}{\\frac{6}{5} \\cdot x^{2} \\cdot \\frac{1}{x^{-7}} +\\frac{4}{3} \\cdot x^{7} \\cdot \\frac{1}{x^{-12}}}$", "$\\displaystyle \\frac{\\frac{3}{4}\\cdot x^{6} +\\frac{5}{6}\\cdot x^{16}}{\\frac{6}{5}\\cdot x^{9} +\\frac{4}{3}\\cdot x^{19}} = \\frac{\\frac{1}{12}\\cdot x^{6} \\cdot \\left(9 +10\\cdot x^{10}\\right)}{\\frac{2}{15}\\cdot x^{9} \\cdot \\left(9 +10\\cdot x^{10}\\right)} = \\frac{1}{12} \\cdot \\frac{15}{2} \\cdot x^{-3} = \\frac{5}{8}\\cdot x^{-3}$"], ["$\\displaystyle \\frac{\\frac{2}{3} \\cdot x^{2} \\cdot \\frac{1}{x^{-8}} +\\frac{4}{3} \\cdot x^{6} \\cdot \\frac{1}{x^{-2}}}{\\frac{8}{3} \\cdot x^{-4} \\cdot \\frac{1}{x^{-9}} +\\frac{16}{3} \\cdot x^{9} \\cdot \\frac{1}{x^{6}}}$", "$\\displaystyle \\frac{\\frac{2}{3}\\cdot x^{10} +\\frac{4}{3}\\cdot x^{8}}{\\frac{8}{3}\\cdot x^{5} +\\frac{16}{3}\\cdot x^{3}} = \\frac{\\frac{2}{3}\\cdot x^{8} \\cdot \\left(1\\cdot x^{2} +2\\right)}{\\frac{8}{3}\\cdot x^{3} \\cdot \\left(1\\cdot x^{2} +2\\right)} = \\frac{2}{3} \\cdot \\frac{3}{8} \\cdot x^{5} = \\frac{1}{4}\\cdot x^{5}$"], ["$\\displaystyle \\frac{\\frac{6}{5} \\cdot x^{4} \\cdot \\frac{1}{x^{-4}} +\\frac{5}{4} \\cdot x^{6} \\cdot \\frac{1}{x^{-6}}}{\\frac{9}{2} \\cdot x^{9} \\cdot \\frac{1}{x^{-5}} +\\frac{75}{16} \\cdot x^{3} \\cdot \\frac{1}{x^{-15}}}$", "$\\displaystyle \\frac{\\frac{6}{5}\\cdot x^{8} +\\frac{5}{4}\\cdot x^{12}}{\\frac{9}{2}\\cdot x^{14} +\\frac{75}{16}\\cdot x^{18}} = \\frac{\\frac{1}{20}\\cdot x^{8} \\cdot \\left(24 +25\\cdot x^{4}\\right)}{\\frac{3}{16}\\cdot x^{14} \\cdot \\left(24 +25\\cdot x^{4}\\right)} = \\frac{1}{20} \\cdot \\frac{16}{3} \\cdot x^{-6} = \\frac{4}{15}\\cdot x^{-6}$"]],
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=== Freitag 14. Dezember 2018 ===
Bestimmen Sie eine Formel, die aus der Nummer $n$ des Elements den Wert berechnet:
miniAufgabe("#exofolgenafgf","#solfolgenafgf",
[["$\\frac{20}{9},\\,-\\frac{10}{9},\\,\\frac{5}{9},\\,-\\frac{5}{18},\\,\\frac{5}{36}\\ldots$", "Konstanter Quotient von $-\\frac{1}{2}$, und damit ist $x_{n}=\\frac{20}{9}\\cdot\\left(-\\frac{1}{2}\\right)^{(n-1)}$."], ["$-\\frac{245}{27},\\,\\frac{35}{9},\\,-\\frac{5}{3},\\,\\frac{5}{7},\\,-\\frac{15}{49}\\ldots$", "Konstanter Quotient von $-\\frac{3}{7}$, und damit ist $x_{n}=-\\frac{245}{27}\\cdot\\left(-\\frac{3}{7}\\right)^{(n-1)}$."], ["$-\\frac{5}{8},\\,-\\frac{55}{24},\\,-\\frac{95}{24},\\,-\\frac{45}{8},\\,-\\frac{175}{24}\\ldots$", "Konstante Differenz von $-\\frac{5}{3}$, und damit ist $x_{n}=-\\frac{5}{8}-\\frac{5}{3}\\cdot(n-1)$."], ["$-\\frac{10}{9},\\,\\frac{5}{6},\\,-\\frac{5}{8},\\,\\frac{15}{32},\\,-\\frac{45}{128}\\ldots$", "Konstanter Quotient von $-\\frac{3}{4}$, und damit ist $x_{n}=-\\frac{10}{9}\\cdot\\left(-\\frac{3}{4}\\right)^{(n-1)}$."], ["$\\frac{175}{216},\\,-\\frac{35}{36},\\,\\frac{7}{6},\\,-\\frac{7}{5},\\,\\frac{42}{25}\\ldots$", "Konstanter Quotient von $-\\frac{6}{5}$, und damit ist $x_{n}=\\frac{175}{216}\\cdot\\left(-\\frac{6}{5}\\right)^{(n-1)}$."], ["$-\\frac{245}{12},\\,\\frac{35}{6},\\,-\\frac{5}{3},\\,\\frac{10}{21},\\,-\\frac{20}{147}\\ldots$", "Konstanter Quotient von $-\\frac{2}{7}$, und damit ist $x_{n}=-\\frac{245}{12}\\cdot\\left(-\\frac{2}{7}\\right)^{(n-1)}$."], ["$-\\frac{35}{36},\\,-\\frac{7}{6},\\,-\\frac{7}{5},\\,-\\frac{42}{25},\\,-\\frac{252}{125}\\ldots$", "Konstanter Quotient von $\\frac{6}{5}$, und damit ist $x_{n}=-\\frac{35}{36}\\cdot\\left(\\frac{6}{5}\\right)^{(n-1)}$."], ["$-\\frac{3}{5},\\,-\\frac{9}{5},\\,-3,\\,-\\frac{21}{5},\\,-\\frac{27}{5}\\ldots$", "Konstante Differenz von $-\\frac{6}{5}$, und damit ist $x_{n}=-\\frac{3}{5}-\\frac{6}{5}\\cdot(n-1)$."], ["$\\frac{4}{7},\\,\\frac{48}{35},\\,\\frac{76}{35},\\,\\frac{104}{35},\\,\\frac{132}{35}\\ldots$", "Konstante Differenz von $\\frac{4}{5}$, und damit ist $x_{n}=\\frac{4}{7}+\\frac{4}{5}\\cdot(n-1)$."], ["$-\\frac{7}{8},\\,-\\frac{19}{40},\\,-\\frac{3}{40},\\,\\frac{13}{40},\\,\\frac{29}{40}\\ldots$", "Konstante Differenz von $\\frac{2}{5}$, und damit ist $x_{n}=-\\frac{7}{8}+\\frac{2}{5}\\cdot(n-1)$."]],
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