miniaufgabe.js
==== 17. Juni 2024 bis 21. Juni 2024 ====
=== Dienstag 18. Juni 2024 ===
Gegeben ist der Graph einer Funktion $f(x)$. Sei $F(x)$ die Funktion, die die vorzeichenbehaftete Fläche zwischen dem Graphen von $f(x)$ und der $x$-Achse im Bereich zwischen 0 und $x$ angibt.
//Vorzeichenbehaftet heisst, dass Flächen im positiven $x$-Bereich unter der $x$-Achse negativ sind.//
{{lehrkraefte:blc:grafisch-integrieren.pdf}}
//Sie werden die Graphen ausgedruckt erhalten.//
=== Mittwoch 19. Juni 2024 ===
Berechnen Sie:miniAufgabe("#exopolynome_integrieren","#solpolynome_integrieren",
[["$\\displaystyle \\int_{2}^{3}\\left(-4x^{2}+2x-2\\right)\\mathrm{d}x$", "$\\displaystyle \\int_{2}^{3}\\left(-4x^{2}+2x-2\\right)\\mathrm{d}x = -\\frac{4}{3}x^{3}+x^{2}-2x \\,\\,\\Bigr\\rvert_{2}^{3} = $\\begin{multline*}-\\frac{4}{3} \\cdot \\left(3\\right)^{3}+1 \\cdot \\left(3\\right)^{2}-2 \\cdot 3-\\left(-\\frac{4}{3} \\cdot \\left(2\\right)^{3}+1 \\cdot \\left(2\\right)^{2}-2 \\cdot 2\\right) = \\\\\n-\\frac{4}{3} \\cdot 27+1 \\cdot 9-6+\\frac{4}{3} \\cdot \\left(2\\right)^{3}-1 \\cdot \\left(2\\right)^{2}+2 \\cdot 2 = \\\\\n-36+9-6+\\frac{4}{3} \\cdot 8-1 \\cdot 4+4 = \\\\\n-36+9-6+\\frac{32}{3}-4+4 = \\\\\n-\\frac{67}{3}\\\\\n\n\\end{multline*}"], ["$\\displaystyle \\int_{-2}^{2}\\left(-4x^{2}-3x-3\\right)\\mathrm{d}x$", "$\\displaystyle \\int_{-2}^{2}\\left(-4x^{2}-3x-3\\right)\\mathrm{d}x = -\\frac{4}{3}x^{3}-\\frac{3}{2}x^{2}-3x \\,\\,\\Bigr\\rvert_{-2}^{2} = $\\begin{multline*}-\\frac{4}{3} \\cdot \\left(2\\right)^{3}-\\frac{3}{2} \\cdot \\left(2\\right)^{2}-3 \\cdot 2-\\left(-\\frac{4}{3} \\cdot \\left(-2\\right)^{3}-\\frac{3}{2} \\cdot \\left(-2\\right)^{2}-3\\left(-2\\right)\\right) = \\\\\n-\\frac{4}{3} \\cdot 8-\\frac{3}{2} \\cdot 4-6+\\frac{4}{3} \\cdot \\left(-2\\right)^{3}+\\frac{3}{2} \\cdot \\left(-2\\right)^{2}+3\\left(-2\\right) = \\\\\n-\\frac{32}{3}-6-6+\\frac{4}{3}\\left(-8\\right)+\\frac{3}{2} \\cdot 4-6 = \\\\\n-\\frac{32}{3}-6-6-\\frac{32}{3}+6-6 = \\\\\n-\\frac{100}{3}\\\\\n\n\\end{multline*}"], ["$\\displaystyle \\int_{-3}^{-2}\\left(4x^{2}-2x+3\\right)\\mathrm{d}x$", "$\\displaystyle \\int_{-3}^{-2}\\left(4x^{2}-2x+3\\right)\\mathrm{d}x = \\frac{4}{3}x^{3}-x^{2}+3x \\,\\,\\Bigr\\rvert_{-3}^{-2} = $\\begin{multline*}\\frac{4}{3} \\cdot \\left(-2\\right)^{3}-1 \\cdot \\left(-2\\right)^{2}+3\\left(-2\\right)-\\left(\\frac{4}{3} \\cdot \\left(-3\\right)^{3}-1 \\cdot \\left(-3\\right)^{2}+3\\left(-3\\right)\\right) = \\\\\n\\frac{4}{3}\\left(-8\\right)-1 \\cdot 4-6-\\frac{4}{3} \\cdot \\left(-3\\right)^{3}+1 \\cdot \\left(-3\\right)^{2}-3\\left(-3\\right) = \\\\\n-\\frac{32}{3}-4-6-\\frac{4}{3}\\left(-27\\right)+1 \\cdot 9+9 = \\\\\n-\\frac{32}{3}-4-6+36+9+9 = \\\\\n\\frac{100}{3}\\\\\n\n\\end{multline*}"], ["$\\displaystyle \\int_{2}^{3}\\left(4x^{2}-4x-2\\right)\\mathrm{d}x$", "$\\displaystyle \\int_{2}^{3}\\left(4x^{2}-4x-2\\right)\\mathrm{d}x = \\frac{4}{3}x^{3}-2x^{2}-2x \\,\\,\\Bigr\\rvert_{2}^{3} = $\\begin{multline*}\\frac{4}{3} \\cdot \\left(3\\right)^{3}-2 \\cdot \\left(3\\right)^{2}-2 \\cdot 3-\\left(\\frac{4}{3} \\cdot \\left(2\\right)^{3}-2 \\cdot \\left(2\\right)^{2}-2 \\cdot 2\\right) = \\\\\n\\frac{4}{3} \\cdot 27-2 \\cdot 9-6-\\frac{4}{3} \\cdot \\left(2\\right)^{3}+2 \\cdot \\left(2\\right)^{2}+2 \\cdot 2 = \\\\\n36-18-6-\\frac{4}{3} \\cdot 8+2 \\cdot 4+4 = \\\\\n36-18-6-\\frac{32}{3}+8+4 = \\\\\n\\frac{40}{3}\\\\\n\n\\end{multline*}"], ["$\\displaystyle \\int_{2}^{3}\\left(-2x^{2}+2x+3\\right)\\mathrm{d}x$", "$\\displaystyle \\int_{2}^{3}\\left(-2x^{2}+2x+3\\right)\\mathrm{d}x = -\\frac{2}{3}x^{3}+x^{2}+3x \\,\\,\\Bigr\\rvert_{2}^{3} = $\\begin{multline*}-\\frac{2}{3} \\cdot \\left(3\\right)^{3}+1 \\cdot \\left(3\\right)^{2}+3 \\cdot 3-\\left(-\\frac{2}{3} \\cdot \\left(2\\right)^{3}+1 \\cdot \\left(2\\right)^{2}+3 \\cdot 2\\right) = \\\\\n-\\frac{2}{3} \\cdot 27+1 \\cdot 9+9+\\frac{2}{3} \\cdot \\left(2\\right)^{3}-1 \\cdot \\left(2\\right)^{2}-3 \\cdot 2 = \\\\\n-18+9+9+\\frac{2}{3} \\cdot 8-1 \\cdot 4-6 = \\\\\n-18+9+9+\\frac{16}{3}-4-6 = \\\\\n-\\frac{14}{3}\\\\\n\n\\end{multline*}"], ["$\\displaystyle \\int_{2}^{3}\\left(2x^{2}-3x-3\\right)\\mathrm{d}x$", "$\\displaystyle \\int_{2}^{3}\\left(2x^{2}-3x-3\\right)\\mathrm{d}x = \\frac{2}{3}x^{3}-\\frac{3}{2}x^{2}-3x \\,\\,\\Bigr\\rvert_{2}^{3} = $\\begin{multline*}\\frac{2}{3} \\cdot \\left(3\\right)^{3}-\\frac{3}{2} \\cdot \\left(3\\right)^{2}-3 \\cdot 3-\\left(\\frac{2}{3} \\cdot \\left(2\\right)^{3}-\\frac{3}{2} \\cdot \\left(2\\right)^{2}-3 \\cdot 2\\right) = \\\\\n\\frac{2}{3} \\cdot 27-\\frac{3}{2} \\cdot 9-9-\\frac{2}{3} \\cdot \\left(2\\right)^{3}+\\frac{3}{2} \\cdot \\left(2\\right)^{2}+3 \\cdot 2 = \\\\\n18-\\frac{27}{2}-9-\\frac{2}{3} \\cdot 8+\\frac{3}{2} \\cdot 4+6 = \\\\\n18-\\frac{27}{2}-9-\\frac{16}{3}+6+6 = \\\\\n\\frac{13}{6}\\\\\n\n\\end{multline*}"], ["$\\displaystyle \\int_{2}^{3}\\left(-4x^{2}-4x+3\\right)\\mathrm{d}x$", "$\\displaystyle \\int_{2}^{3}\\left(-4x^{2}-4x+3\\right)\\mathrm{d}x = -\\frac{4}{3}x^{3}-2x^{2}+3x \\,\\,\\Bigr\\rvert_{2}^{3} = $\\begin{multline*}-\\frac{4}{3} \\cdot \\left(3\\right)^{3}-2 \\cdot \\left(3\\right)^{2}+3 \\cdot 3-\\left(-\\frac{4}{3} \\cdot \\left(2\\right)^{3}-2 \\cdot \\left(2\\right)^{2}+3 \\cdot 2\\right) = \\\\\n-\\frac{4}{3} \\cdot 27-2 \\cdot 9+9+\\frac{4}{3} \\cdot \\left(2\\right)^{3}+2 \\cdot \\left(2\\right)^{2}-3 \\cdot 2 = \\\\\n-36-18+9+\\frac{4}{3} \\cdot 8+2 \\cdot 4-6 = \\\\\n-36-18+9+\\frac{32}{3}+8-6 = \\\\\n-\\frac{97}{3}\\\\\n\n\\end{multline*}"], ["$\\displaystyle \\int_{-3}^{2}\\left(-2x^{2}+4x+2\\right)\\mathrm{d}x$", "$\\displaystyle \\int_{-3}^{2}\\left(-2x^{2}+4x+2\\right)\\mathrm{d}x = -\\frac{2}{3}x^{3}+2x^{2}+2x \\,\\,\\Bigr\\rvert_{-3}^{2} = $\\begin{multline*}-\\frac{2}{3} \\cdot \\left(2\\right)^{3}+2 \\cdot \\left(2\\right)^{2}+2 \\cdot 2-\\left(-\\frac{2}{3} \\cdot \\left(-3\\right)^{3}+2 \\cdot \\left(-3\\right)^{2}+2\\left(-3\\right)\\right) = \\\\\n-\\frac{2}{3} \\cdot 8+2 \\cdot 4+4+\\frac{2}{3} \\cdot \\left(-3\\right)^{3}-2 \\cdot \\left(-3\\right)^{2}-2\\left(-3\\right) = \\\\\n-\\frac{16}{3}+8+4+\\frac{2}{3}\\left(-27\\right)-2 \\cdot 9+6 = \\\\\n-\\frac{16}{3}+8+4-18-18+6 = \\\\\n-\\frac{70}{3}\\\\\n\n\\end{multline*}"], ["$\\displaystyle \\int_{2}^{3}\\left(2x^{2}-4x+4\\right)\\mathrm{d}x$", "$\\displaystyle \\int_{2}^{3}\\left(2x^{2}-4x+4\\right)\\mathrm{d}x = \\frac{2}{3}x^{3}-2x^{2}+4x \\,\\,\\Bigr\\rvert_{2}^{3} = $\\begin{multline*}\\frac{2}{3} \\cdot \\left(3\\right)^{3}-2 \\cdot \\left(3\\right)^{2}+4 \\cdot 3-\\left(\\frac{2}{3} \\cdot \\left(2\\right)^{3}-2 \\cdot \\left(2\\right)^{2}+4 \\cdot 2\\right) = \\\\\n\\frac{2}{3} \\cdot 27-2 \\cdot 9+12-\\frac{2}{3} \\cdot \\left(2\\right)^{3}+2 \\cdot \\left(2\\right)^{2}-4 \\cdot 2 = \\\\\n18-18+12-\\frac{2}{3} \\cdot 8+2 \\cdot 4-8 = \\\\\n18-18+12-\\frac{16}{3}+8-8 = \\\\\n\\frac{20}{3}\\\\\n\n\\end{multline*}"], ["$\\displaystyle \\int_{-4}^{-3}\\left(-2x^{2}-4x-3\\right)\\mathrm{d}x$", "$\\displaystyle \\int_{-4}^{-3}\\left(-2x^{2}-4x-3\\right)\\mathrm{d}x = -\\frac{2}{3}x^{3}-2x^{2}-3x \\,\\,\\Bigr\\rvert_{-4}^{-3} = $\\begin{multline*}-\\frac{2}{3} \\cdot \\left(-3\\right)^{3}-2 \\cdot \\left(-3\\right)^{2}-3\\left(-3\\right)-\\left(-\\frac{2}{3} \\cdot \\left(-4\\right)^{3}-2 \\cdot \\left(-4\\right)^{2}-3\\left(-4\\right)\\right) = \\\\\n-\\frac{2}{3}\\left(-27\\right)-2 \\cdot 9+9+\\frac{2}{3} \\cdot \\left(-4\\right)^{3}+2 \\cdot \\left(-4\\right)^{2}+3\\left(-4\\right) = \\\\\n18-18+9+\\frac{2}{3}\\left(-64\\right)+2 \\cdot 16-12 = \\\\\n18-18+9-\\frac{128}{3}+32-12 = \\\\\n-\\frac{41}{3}\\\\\n\n\\end{multline*}"]],
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ruby polynome-integrieren.rb 1