Benutzer:Hourssales/HA7

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1.

f ( x ) = − 9 25 x 2 + 81 {\displaystyle \mathrm {f} (x)={\sqrt {-{\frac {9}{25}}x^{2}+81}}} {\displaystyle \mathrm {f} (x)={\sqrt {-{\frac {9}{25}}x^{2}+81}}}

a

R o t a t i o n s v o l u m e n   d e r   F u n k t i o n   u m   d i e   y − A c h s e   i m   I n t e r v a l l   I = [ 0 ; y 0 ] y = − 9 25 x 2 + 81 y 2 = − 9 25 x 2 + 81 y 2 − 81 = − 9 25 x 2 x 2 = − 25 9 y 2 + 225 x = − 25 9 y 2 + 225 f ( y ) = − 25 9 y 2 + 225 0 = − 25 9 y 2 + 225 0 = − 25 9 y 2 + 225 y 0 , 1 = 9 y 0 , 2 = − 9   ( e n t f a ¨ l l t , d a   − 9 ∉ [ 0 , y 0 ] ) V Y = π ∫ 0 9 f ( y ) d y V Y = 4241   m 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}}} {\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

D e r   A n s t i e g   a n   d e r   S t e l l e   x 0   i s t   d i e   A b l e i t u n g   v o n   f ( x )   a n   x 0 . y = m x + n m = f ′ ( x 0 ) = − 9 25 x 0 ∗ ( − 1 − 9 25 x 0 2 + 81 ) = 9 25 x 0 − 9 25 x 0 2 + 81 y = ( 9 25 x 0 − 9 25 x 0 2 + 81 ) x + n n = y − ( 9 25 x 0 − 9 25 x 0 2 + 81 ) x n = − 9 25 x 0 2 + 81 − ( 9 25 x 0 − 9 25 x 0 2 + 81 ) x n = − 9 25 x 0 2 + 81 + 9 25 x 0 2 − 9 25 x 0 2 + 81 n = 8 1 − 9 25 x 0 2 + 81 t x 0 ( x ) = ( 9 25 x 0 − 9 25 x 0 2 + 81 ) x + 8 1 − 9 25 x 0 2 + 81 {\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}*(-{\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 {8}{1}}{\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 {8}{1}}{\sqrt {-{\frac {9}{25}}{x_{0}}^{2}+81}}\end{aligned}}} {\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}*(-{\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 {8}{1}}{\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 {8}{1}}{\sqrt {-{\frac {9}{25}}{x_{0}}^{2}+81}}\end{aligned}}}

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