taylor e^{x/2}

Lösung

taylor

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

1 + \frac{1}{2} x + \frac{1}{8} x^{2} + \frac{1}{48} x^{3} + \frac{1}{384} x^{4} + \dots

Schritte zur Lösung

Apply the Taylor Formula

= 1 + \frac{\frac{d}{d x} ( e ^{\frac{x}{2}} ) ( 0 )}{1 !} x + \frac{\frac{d ^{2}}{d x ^{2}} ( e ^{\frac{x}{2}} ) ( 0 )}{2 !} x^{2} + \frac{\frac{d ^{3}}{d x ^{3}} ( e ^{\frac{x}{2}} ) ( 0 )}{3 !} x^{3} + \dots

Evaluate Derivatives

= 1 + \frac{\frac{1}{2}}{1 !} x + \frac{\frac{1}{4}}{2 !} x^{2} + \frac{\frac{1}{8}}{3 !} x^{3} + \frac{\frac{1}{16}}{4 !} x^{4} + \dots

Fasse zusammen

= 1 + \frac{1}{2} x + \frac{1}{8} x^{2} + \frac{1}{48} x^{3} + \frac{1}{384} x^{4} + \dots

im pl i c i t \frac{d y}{d x}, y^{3} = 4 x

\frac{\partial}{\partial x} (\frac{- 3 x}{x ^{2} + y ^{2}})

\int \frac{1}{x ^{2} + b} d x

\int e^{3} d x

x y^{^{'}} - y = - x^{2} + x - 1