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. Aufgabe (Differentiation)(a) Bestimmen Sie die Taylorreihe der Funktionim Entwicklungspunkt sowie den Konvergenzradius von .
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Q. . Aufgabe (Differentiation)(a) Bestimmen Sie die Taylorreihe der Funktionim Entwicklungspunkt sowie den Konvergenzradius von .
- Write Function and Point: Write down the function and the development point.,
- Find Taylor Series Expansion: Find the Taylor series expansion of around . The Taylor series expansion of at is given by:
- Calculate Derivatives at : Calculate the derivatives of at ....We will calculate these derivatives and evaluate them at .
- Calculate : Calculate .
- Calculate : Calculate .f'(x) = \frac{d}{dx}\left[\frac{-27x^3}{1+3x}\right] = -27\left(\frac{3x^2(1+3x) - x^3(3)}{(1+3x)^2}\right)\(\newline\$f'(x_0) = f'(0) = -27\left(\frac{3\cdot 0^2(1+3\cdot 0) - 0^3(3)}{(1+3\cdot 0)^2}\right) = 0\)
- Calculate \(f''(x_0)\): Calculate \(f''(x_0)\).\(\newline\)\(f''(x) = \frac{d^2}{dx^2}\left[\frac{-27x^3}{1+3x}\right]\)\(\newline\)To calculate this, we need to apply the quotient rule again.
- Apply Quotient Rule: Calculate \(f''(x_0)\) using the quotient rule.\(\newline\)\(f''(x) = \frac{d}{dx}\left[-27\left(3x^2(1+3x) - x^3(3)\right)/(1+3x)^2\right]\)\(\newline\)\(f''(x_0) = f''(0) = \frac{d}{dx}\left[-27\left(3x^2(1+3x) - x^3(3)\right)/(1+3x)^2\right]\) at \(x=0\)\(\newline\)We need to simplify this derivative before evaluating at \(x=0\).
- Simplify and Evaluate \(f''(x_0)\): Simplify \(f''(x)\) and evaluate at \(x_0\).
\(f''(x) = \frac{d}{dx}[-81x^2 + 243x^3]/(1+3x)^2\)
\(f''(x_0) = f''(0) = \frac{d}{dx}[-81x^2 + 243x^3]/(1+3x)^2\) at \(x=0\)
\(f''(x_0) = -162x + 729x^2\) at \(x=0\)
\(f''(x_0) = -162\times 0 + 729\times 0^2 = 0\)
