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Rechner für Taylor- und Maclaurin-Reihen (Potenzreihen)
Bestimme die Taylor-/Maclaurin-Reihe Schritt für Schritt
Der Rechner ermittelt die Entwicklung in eine Taylorreihe (oder Potenzreihe) der gegebenen Funktion um den angegebenen Punkt und zeigt die Schritte an. Sie können die Ordnung des Taylor-Polynoms angeben. Wenn Sie das Maclaurin-Polynom möchten, setzen Sie den Entwicklungspunkt einfach auf $$$0$$$.
Solution
Your input: calculate the Taylor (Maclaurin) series of $$$x^{3} - 3 x^{2}$$$ up to $$$n=5$$$
A Maclaurin series is given by $$$f\left(x\right)=\sum\limits_{k=0}^{\infty}\frac{f^{(k)}\left(a\right)}{k!}x^k$$$
In our case, $$$f\left(x\right) \approx P\left(x\right) = \sum\limits_{k=0}^{n}\frac{f^{(k)}\left(a\right)}{k!}x^k=\sum\limits_{k=0}^{5}\frac{f^{(k)}\left(a\right)}{k!}x^k$$$
So, what we need to do to get the desired polynomial is to calculate the derivatives, evaluate them at the given point, and plug the results into the given formula.
$$$f^{(0)}\left(x\right)=f\left(x\right)=x^{3} - 3 x^{2}$$$
Evaluate the function at the point: $$$f\left(0\right)=0$$$
Find the 1st derivative: $$$f^{(1)}\left(x\right)=\left(f^{(0)}\left(x\right)\right)^{\prime}=\left(x^{3} - 3 x^{2}\right)^{\prime}=3 x \left(x - 2\right)$$$ (steps can be seen here).
Evaluate the 1st derivative at the given point: $$$\left(f\left(0\right)\right)^{\prime }=0$$$
Find the 2nd derivative: $$$f^{(2)}\left(x\right)=\left(f^{(1)}\left(x\right)\right)^{\prime}=\left(3 x \left(x - 2\right)\right)^{\prime}=6 x - 6$$$ (steps can be seen here).
Evaluate the 2nd derivative at the given point: $$$\left(f\left(0\right)\right)^{\prime \prime }=-6$$$
Find the 3rd derivative: $$$f^{(3)}\left(x\right)=\left(f^{(2)}\left(x\right)\right)^{\prime}=\left(6 x - 6\right)^{\prime}=6$$$ (steps can be seen here).
Evaluate the 3rd derivative at the given point: $$$\left(f\left(0\right)\right)^{\prime \prime \prime }=6$$$
Find the 4th derivative: $$$f^{(4)}\left(x\right)=\left(f^{(3)}\left(x\right)\right)^{\prime}=\left(6\right)^{\prime}=0$$$ (steps can be seen here).
Evaluate the 4th derivative at the given point: $$$\left(f\left(0\right)\right)^{\prime \prime \prime \prime }=0$$$
Find the 5th derivative: $$$f^{(5)}\left(x\right)=\left(f^{(4)}\left(x\right)\right)^{\prime}=\left(0\right)^{\prime}=0$$$ (steps can be seen here).
Evaluate the 5th derivative at the given point: $$$\left(f\left(0\right)\right)^{\left(5\right)}=0$$$
Now, use the calculated values to get a polynomial:
$$$f\left(x\right)\approx\frac{0}{0!}x^{0}+\frac{0}{1!}x^{1}+\frac{-6}{2!}x^{2}+\frac{6}{3!}x^{3}+\frac{0}{4!}x^{4}+\frac{0}{5!}x^{5}$$$
Finally, after simplifying we get the final answer:
$$$f\left(x\right)\approx P\left(x\right) = -3x^{2}+x^{3}$$$
Answer: the Taylor (Maclaurin) series of $$$x^{3} - 3 x^{2}$$$ up to $$$n=5$$$ is $$$x^{3} - 3 x^{2}\approx P\left(x\right)=-3x^{2}+x^{3}$$$