miniaufgabe.js
==== 12. Februar 2024 bis 16. Februar 2024 ====
=== Dienstag 13. Februar 2024 ===
Leiten Sie die Funktion $f(x)=x^n$ mit $n=2$ oder $n=3$ mit Hilfe des Grenzwertes des Differenzenquotienten ab.
Für $f(x)=x^2$:
\[
f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h} = \lim_{h \to 0} \frac{(x+h)^2-x^2}{h} = \\
\lim_{h \to 0} \frac{x^2+2hx+h^2 -x^2}{h} = \lim_{h \to 0} \frac{2hx+h^2}{h} = \\
\lim_{h \to 0} \frac{h(2x+h)}{h} = \lim_{h \to 0} (2x+h) = 2x
\]
Analog für $f(x)=x^3$:
\[
f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h} = \lim_{h \to 0} \frac{(x+h)^3-x^3}{h} = \\
\lim_{h \to 0} \frac{x^2+3hx^2+3h^2x+h^3 -x^3}{h} = \lim_{h \to 0} \frac{3hx^2+3h^2x+h^3}{h} = \\
\lim_{h \to 0} \frac{h(3x^2+3hx+h^2)}{h} = \lim_{h \to 0} (3x^2+3hx+h^2) = 3x^2
\]
=== Mittwoch 14. Februar 2024 ===
Leiten Sie von Hand und ohne Unterlagen ab:
miniAufgabe("#exonurpolynome","#solnurpolynome",
[["a) $f(x)=\\frac{4}{3}x^{11}-\\frac{1}{4}x^{6}-\\frac{4}{3}x\\qquad$ b) $f(x)=-\\frac{1}{6}x^{3}+\\frac{3}{7}x^{2}+\\frac{1}{7}x\\qquad$ ", "a) $f'(x)=\\frac{44}{3}x^{10}-\\frac{3}{2}x^{5}-\\frac{4}{3}\\qquad$ b) $f'(x)=-\\frac{1}{2}x^{2}+\\frac{6}{7}x+\\frac{1}{7}\\qquad$ "], ["a) $f(x)=-\\frac{1}{8}x^{11}-\\frac{3}{7}x^{2}-\\frac{4}{3}x\\qquad$ b) $f(x)=\\frac{1}{3}x^{12}+\\frac{1}{9}x^{5}+\\frac{1}{6}x\\qquad$ ", "a) $f'(x)=-\\frac{11}{8}x^{10}-\\frac{6}{7}x-\\frac{4}{3}\\qquad$ b) $f'(x)=4x^{11}+\\frac{5}{9}x^{4}+\\frac{1}{6}\\qquad$ "], ["a) $f(x)=\\frac{3}{4}x^{12}-\\frac{1}{4}x^{4}-\\frac{1}{9}x\\qquad$ b) $f(x)=-\\frac{4}{7}x^{9}+\\frac{1}{5}x^{5}+\\frac{1}{4}x\\qquad$ ", "a) $f'(x)=9x^{11}-x^{3}-\\frac{1}{9}\\qquad$ b) $f'(x)=-\\frac{36}{7}x^{8}+x^{4}+\\frac{1}{4}\\qquad$ "], ["a) $f(x)=-\\frac{1}{2}x^{8}-\\frac{1}{6}x^{2}-\\frac{1}{5}\\qquad$ b) $f(x)=\\frac{1}{6}x^{11}-\\frac{1}{6}x^{3}+\\frac{1}{8}\\qquad$ ", "a) $f'(x)=-4x^{7}-\\frac{1}{3}x\\qquad$ b) $f'(x)=\\frac{11}{6}x^{10}-\\frac{1}{2}x^{2}\\qquad$ "], ["a) $f(x)=\\frac{1}{4}x^{9}+\\frac{2}{3}x^{3}+\\frac{1}{8}\\qquad$ b) $f(x)=\\frac{2}{5}x^{10}+\\frac{2}{7}x^{8}+\\frac{1}{6}\\qquad$ ", "a) $f'(x)=\\frac{9}{4}x^{8}+2x^{2}\\qquad$ b) $f'(x)=4x^{9}+\\frac{16}{7}x^{7}\\qquad$ "], ["a) $f(x)=\\frac{1}{2}x^{9}-\\frac{1}{9}x^{3}-\\frac{3}{7}x\\qquad$ b) $f(x)=\\frac{3}{2}x^{9}+\\frac{1}{2}x^{8}+\\frac{4}{9}x\\qquad$ ", "a) $f'(x)=\\frac{9}{2}x^{8}-\\frac{1}{3}x^{2}-\\frac{3}{7}\\qquad$ b) $f'(x)=\\frac{27}{2}x^{8}+4x^{7}+\\frac{4}{9}\\qquad$ "], ["a) $f(x)=-\\frac{2}{5}x^{4}+\\frac{2}{5}x^{2}-\\frac{1}{2}x\\qquad$ b) $f(x)=-\\frac{3}{4}x^{11}+\\frac{2}{9}x^{5}-\\frac{1}{5}x\\qquad$ ", "a) $f'(x)=-\\frac{8}{5}x^{3}+\\frac{4}{5}x-\\frac{1}{2}\\qquad$ b) $f'(x)=-\\frac{33}{4}x^{10}+\\frac{10}{9}x^{4}-\\frac{1}{5}\\qquad$ "], ["a) $f(x)=-\\frac{3}{8}x^{4}+\\frac{1}{4}x^{2}+\\frac{1}{8}\\qquad$ b) $f(x)=\\frac{1}{4}x^{10}-\\frac{2}{5}x^{9}-\\frac{1}{2}\\qquad$ ", "a) $f'(x)=-\\frac{3}{2}x^{3}+\\frac{1}{2}x\\qquad$ b) $f'(x)=\\frac{5}{2}x^{9}-\\frac{18}{5}x^{8}\\qquad$ "], ["a) $f(x)=-\\frac{2}{7}x^{11}-\\frac{4}{7}x^{5}-\\frac{1}{4}\\qquad$ b) $f(x)=-\\frac{1}{2}x^{6}+\\frac{2}{9}x^{4}-\\frac{1}{2}\\qquad$ ", "a) $f'(x)=-\\frac{22}{7}x^{10}-\\frac{20}{7}x^{4}\\qquad$ b) $f'(x)=-3x^{5}+\\frac{8}{9}x^{3}\\qquad$ "], ["a) $f(x)=-\\frac{1}{7}x^{10}-\\frac{4}{7}x^{6}+\\frac{4}{3}\\qquad$ b) $f(x)=-\\frac{1}{4}x^{6}-\\frac{1}{2}x^{4}-\\frac{2}{9}x\\qquad$ ", "a) $f'(x)=-\\frac{10}{7}x^{9}-\\frac{24}{7}x^{5}\\qquad$ b) $f'(x)=-\\frac{3}{2}x^{5}-2x^{3}-\\frac{2}{9}\\qquad$ "]],
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ruby ableiten-von-hand.rb 4