Решебник по алгебре и начала математического анализа 11 класс Алимов Задание 858

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\[1)\ f(x) = x^{3}\text{\ \ }и\ \ x_{0} = 1:\]

\[f^{'}(x) = \left( x^{3} \right)^{'} = 3x^{2}\]

\[k = f^{'}(1) = 3 \bullet 1^{2} = 3.\]

\[Ответ:\ \ k = 3.\]

\[2)\ f(x) = \sin x\ и\ \ x_{0} = \frac{\pi}{4}:\]

\[f^{'}(x) = \left( \sin x \right)^{'} = \cos x\]

\[k = f^{'}\left( \frac{\pi}{4} \right) = \cos\frac{\pi}{4} = \frac{\sqrt{2}}{2}.\]

\[Ответ:\ \ k = \frac{\sqrt{2}}{2}.\]

\[3)\ f(x) = \ln x\ и\ \ x_{0} = 1:\]

\[f^{'}(x) = \left( \ln x \right) = \frac{1}{x}\]

\[k = f^{'}(1) = \frac{1}{1} = 1.\]

\[Ответ:\ \ k = 1.\]

\[4)\ f(x) = e^{x}\text{\ \ }и\ \ x_{0} = \ln 3:\]

\[f^{'}(x) = \left( e^{x} \right)^{'} = e^{x}\]

\[k = f^{'}\left( \ln 3 \right) = e^{\ln 3} = 3.\]

\[Ответ:\ \ k = 3.\]