Решебник по алгебре 11 класс Никольский Параграф 5. Применение производной Задание 123

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\[\mathbf{Формула\ Тейлора:}\]

\[\textbf{а)}\ y = \sin x;\ \ n = 7:\]

\[\sin x =\]

\[= x - \frac{x^{3}}{3!} + \frac{x^{5}}{5!} - \frac{x^{7}}{7!} + R_{9}(x);\]

\[R_{9}(x) = \frac{\cos c}{9!}x^{9}.\]

\[\textbf{б)}\ y = \cos x;\ \ n = 7:\]

\[\cos x =\]

\[= 1 - \frac{x^{2}}{2!} + \frac{x^{4}}{4!} - \frac{x^{6}}{6!} + R_{8}(x);\]

\[R_{8}(x) = \frac{\cos c}{8!}x^{8}.\]

\[\textbf{в)}\ y = tg\ x;\ \ n = 5:\]

\[tg\ x = x + \frac{x^{3}}{3} + \frac{7x^{5}}{60} + R_{6}(x).\]

\[\textbf{г)}\ y = e^{x};\ \ n = 8:\]

\[R_{9}(x) = \frac{e^{c}}{9!}x^{9}.\]

\[\textbf{д)}\ y = \ln(1 + x);\ \ n = 5:\]

\[R_{6}(x) = - \frac{120}{(1 + c)^{6}}.\]

\[\textbf{е)}\ y = \frac{1}{1 + x};\ \ n = 5:\]

\[R_{6}(x) = \frac{720}{(1 + c)^{7}}.\]