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1.
Räumliche Grafische Darstellung von f(x) auf zwei Achsen.
f
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x
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−
9
25
x
2
+
81
{\displaystyle \mathrm {f} (x)={\sqrt {-{\frac {9}{25}}x^{2}+81}}}
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225
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9
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225
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225
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4241
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3
{\displaystyle {\begin{aligned}&\mathrm {Rotationsvolumen\ der\ Funktion\ um\ die\ y{-}Achse\ im\ Intervall\ I=[0;y_{0}]} \\y&={\sqrt {-{\frac {9}{25}}x^{2}+81}}\\y^{2}&=-{\frac {9}{25}}x^{2}+81\\y^{2}-81&=-{\frac {9}{25}}x^{2}\\x^{2}&=-{\frac {25}{9}}y^{2}+225\\x&={\sqrt {-{\frac {25}{9}}y^{2}+225}}\\\mathrm {f} (y)&={\sqrt {-{\frac {25}{9}}y^{2}+225}}\0円&={\sqrt {-{\frac {25}{9}}y^{2}+225}}\0円&=-{\frac {25}{9}}y^{2}+225\\y_{0,1}&=9\\y_{0,2}&=-9\ \mathrm {(entf{\ddot {a}}llt,da\ {-}9\notin [0,y_{0}])} \\V_{Y}&=\pi \int \limits _{0}^{9}\mathrm {f} (y),円\mathrm {d} y\\V_{Y}&=4241\ m^{3}\end{aligned}}}
b
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{\displaystyle {\begin{aligned}&\mathrm {Der\ Anstieg\ an\ der\ Stelle\ x_{0}\ ist\ die\ Ableitung\ von\ f(x)\ an\ x_{0}.} \\y&=mx+n\\m&=\mathrm {f} '(x_{0})\\&=-{\frac {9}{25}}x_{0}\cdot (-{\frac {1}{\sqrt {-{\frac {9}{25}}{x_{0}}^{2}+81}}})\\&={\frac {{\frac {9}{25}}x_{0}}{\sqrt {-{\frac {9}{25}}{x_{0}}^{2}+81}}}\\\ &\ \\y&=\left({\frac {{\frac {9}{25}}x_{0}}{\sqrt {-{\frac {9}{25}}{x_{0}}^{2}+81}}}\right)x+n\\n&=y-\left({\frac {{\frac {9}{25}}x_{0}}{\sqrt {-{\frac {9}{25}}{x_{0}}^{2}+81}}}\right)x\\n&={\sqrt {-{\frac {9}{25}}{x_{0}}^{2}+81}}-\left({\frac {{\frac {9}{25}}x_{0}}{\sqrt {-{\frac {9}{25}}{x_{0}}^{2}+81}}}\right)x\\n&={\frac {-{\frac {9}{25}}{x_{0}}^{2}+81+{\frac {9}{25}}{x_{0}}^{2}}{\sqrt {-{\frac {9}{25}}{x_{0}}^{2}+81}}}\\n&={\frac {81}{\sqrt {-{\frac {9}{25}}{x_{0}}^{2}+81}}}\\\ &\ \\\mathrm {t} _{x_{0}}(x)&=\left({\frac {{\frac {9}{25}}x_{0}}{\sqrt {-{\frac {9}{25}}{x_{0}}^{2}+81}}}\right)x+{\frac {81}{\sqrt {-{\frac {9}{25}}{x_{0}}^{2}+81}}}\end{aligned}}}
c
tan
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9
tan
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m
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90
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70
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arctan
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m
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0,369
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n
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y
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5,494
4
{\displaystyle {\begin{aligned}\tan \alpha _{1}&={\frac {2}{9}}\\\tan \alpha _{2}&=m_{t}=\mathrm {f} '(9)\\\ &\ \\\mathrm {g} (x)&=mx+n\\m&=\tan \alpha _{Anstieg}\\m&=\tan \left(90^{\circ }-\alpha -\alpha _{2}\right)\\m&=\tan \left(90^{\circ }-70{,}9^{\circ }-\arctan \mathrm {f} '(9)\right)\\\left[m\right]&=0{,}3693\\\ &\ \\n&=y_{0}=5{,}4944\end{aligned}}}
d
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90
∘
{\displaystyle \alpha =90^{\circ }}
