tangent (x^2+1)/(x+e^x+2)

Lösung

tangente von

\frac{x ^{2} + 1}{x + e ^{x} + 2}

y = \frac{a _{0}^{2} - e ^{a_{0}} a _{0}^{2} + 2 e ^{a_{0}} a _{0} + 4 a _{0} - e ^{a_{0}} - 1}{( a _{0} + e ^{a_{0}} + 2 ) ^{2}} x + \frac{a _{0}^{3} e ^{a_{0}} - a _{0}^{2} e ^{a_{0}} - 2 a _{0}^{2} + 2 a _{0} + a _{0} e ^{a_{0}} + e ^{a_{0}} + 2}{( a _{0} + e ^{a_{0}} + 2 ) ^{2}}

Schritte zur Lösung

Berechne den Graphen der Tangente für den allgemeinen Punkt

x = a_{0}

Finde den Tangentenpunkt:

(a_{0}, \frac{a _{0}^{2} + 1}{a _{0} + e ^{a_{0}} + 2})

Finde die Steigung von

y = \frac{x ^{2} + 1}{x + e ^{x} + 2} : \frac{d y}{d x} = \frac{x ^{2} - e ^{x} x ^{2} + 2 e ^{x} x + 4 x - e ^{x} - 1}{( x + e ^{x} + 2 ) ^{2}}

EN : T i tl e G e n er a l Eq u a t i o n Sl o p e A t P o in t 2 Eq : m = \frac{a _{0}^{2} - e ^{a_{0}} a _{0}^{2} + 2 e ^{a_{0}} a _{0} + 4 a _{0} - e ^{a_{0}} - 1}{( a _{0} + e ^{a_{0}} + 2 ) ^{2}}

Finde den Graphen mit Steigung m=

\frac{a _{0}^{2} - e ^{a_{0}} a _{0}^{2} + 2 e ^{a_{0}} a _{0} + 4 a _{0} - e ^{a_{0}} - 1}{( a _{0} + e ^{a_{0}} + 2 ) ^{2}}

, der durch den Punkt

(a_{0}, \frac{a _{0}^{2} + 1}{a _{0} + e ^{a_{0}} + 2})

verläuft:

y = \frac{a _{0}^{2} - e ^{a_{0}} a _{0}^{2} + 2 e ^{a_{0}} a _{0} + 4 a _{0} - e ^{a_{0}} - 1}{( a _{0} + e ^{a_{0}} + 2 ) ^{2}} x + \frac{a _{0}^{3} e ^{a_{0}} - a _{0}^{2} e ^{a_{0}} - 2 a _{0}^{2} + 2 a _{0} + a _{0} e ^{a_{0}} + e ^{a_{0}} + 2}{( a _{0} + e ^{a_{0}} + 2 ) ^{2}}

y = \frac{a _{0}^{2} - e ^{a_{0}} a _{0}^{2} + 2 e ^{a_{0}} a _{0} + 4 a _{0} - e ^{a_{0}} - 1}{( a _{0} + e ^{a_{0}} + 2 ) ^{2}} x + \frac{a _{0}^{3} e ^{a_{0}} - a _{0}^{2} e ^{a_{0}} - 2 a _{0}^{2} + 2 a _{0} + a _{0} e ^{a_{0}} + e ^{a_{0}} + 2}{( a _{0} + e ^{a_{0}} + 2 ) ^{2}}

\int (- x^{2} - 3 x) d x

d er i v a t i v e W (A) = \frac{A}{6}

d er i v a t i v e f (x) = \frac{4 x ^{3}}{( 3 - 2 x ) ^{5}}

x \to 1 lim (x^{2} - 5)

y^{^{'}} = y + x