taylor sin(x/2),pi

Lösung

taylor

sin (\frac{x}{2}), π

1 - \frac{1}{8} (x - π)^{2} + \frac{1}{384} (x - π)^{4} - \frac{1}{46080} (x - π)^{6} + \frac{1}{10321920} (x - π)^{8} + \dots

Schritte zur Lösung

Apply the Taylor Formula

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

Evaluate Derivatives

= 1 + \frac{0}{1 !} (x - π) + \frac{- \frac{1}{4}}{2 !} (x - π)^{2} + \frac{0}{3 !} (x - π)^{3} + \frac{\frac{1}{16}}{4 !} (x - π)^{4} + \frac{0}{5 !} (x - π)^{5} + \frac{- \frac{1}{64}}{6 !} (x - π)^{6} + \frac{0}{7 !} (x - π)^{7} + \frac{\frac{1}{256}}{8 !} (x - π)^{8} + \dots

Fasse zusammen

= 1 - \frac{1}{8} (x - π)^{2} + \frac{1}{384} (x - π)^{4} - \frac{1}{46080} (x - π)^{6} + \frac{1}{10321920} (x - π)^{8} + \dots

d er i v a t i v e \frac{1}{x ^{2} - 1}

\int e^{1} d x

d er i v a t i v e arctan (\frac{1 + x}{1 - x})

y^{^{''}} - 3 y^{^{'}} + 2 y = 2 e^{t}

\int cos (a) x d x