taylor sqrt(x),\at 4

Lösung

taylor

x, at 4

\frac{\frac{1}{2 at 4}}{1 !} (x - at 4) + \frac{- \frac{1}{4 ( at 4 ) ^{\frac{3}{2}}}}{2 !} (x - at 4)^{2} + \frac{\frac{3}{8 ( at 4 ) ^{\frac{5}{2}}}}{3 !} (x - at 4)^{3} + \frac{- \frac{15}{16 ( at 4 ) ^{\frac{7}{2}}}}{4 !} (x - at 4)^{4} + \dots

Schritte zur Lösung

Apply the Taylor Formula

= at 4 + \frac{\frac{d}{d x} ( x ) ( at 4 )}{1 !} (x - at 4) + \frac{\frac{d ^{2}}{d x ^{2}} ( x ) ( at 4 )}{2 !} (x - at 4)^{2} + \frac{\frac{d ^{3}}{d x ^{3}} ( x ) ( at 4 )}{3 !} (x - at 4)^{3} + \dots

Evaluate Derivatives

= at 4 + \frac{\frac{1}{2 at 4}}{1 !} (x - at 4) + \frac{- \frac{1}{4 ( at 4 ) ^{\frac{3}{2}}}}{2 !} (x - at 4)^{2} + \frac{\frac{3}{8 ( at 4 ) ^{\frac{5}{2}}}}{3 !} (x - at 4)^{3} + \frac{- \frac{15}{16 ( at 4 ) ^{\frac{7}{2}}}}{4 !} (x - at 4)^{4} + \dots

Fasse zusammen

= \frac{\frac{1}{2 at 4}}{1 !} (x - at 4) + \frac{- \frac{1}{4 ( at 4 ) ^{\frac{3}{2}}}}{2 !} (x - at 4)^{2} + \frac{\frac{3}{8 ( at 4 ) ^{\frac{5}{2}}}}{3 !} (x - at 4)^{3} + \frac{- \frac{15}{16 ( at 4 ) ^{\frac{7}{2}}}}{4 !} (x - at 4)^{4} + \dots

x \to 0 lim ((1 - 5 x)^{\frac{3}{x}})

d er i v a t i v e x^{3} csc (x)

\frac{d}{d x} (ln ((ln (x))^{2}))

\int \frac{1 - sin ^{3} ( x )}{sin ^{2} ( x )} d x

\frac{d}{d x} (\frac{2 x ^{2} - 3 x + 1}{x})