Решебник по алгебре 11 класс Никольский Параграф 5. Применение производной Задание 43

\[\boxed{\mathbf{43}\mathbf{.}}\]

\[f\left( x_{0} + \mathrm{\Delta}x \right) \approx f\left( x_{0} \right) + f^{'}\left( x_{0} \right)\mathrm{\Delta}x\]

\[\textbf{а)}\ tg\ 47{^\circ}\]

\[x_{0} = 45{^\circ};\]

\[f\left( x_{0} \right) = 1;\]

\[\mathrm{\Delta}x = 1;\]

\[\mathrm{\Delta}x = 45{^\circ} - 47{^\circ} = - 2{^\circ} \approx 0,035;\]

\[f^{'}(x) = \frac{1}{\cos^{2}x};\]

\[f^{'}(45{^\circ}) = 2.\]

\[tg\ 47{^\circ} = 1 + 2 \cdot 0,035 = 1,07.\]

\[\textbf{б)}\ tg\ 2{^\circ}\]

\[x_{0} = 0{^\circ}\]

\[f\left( x_{0} \right) = 0;\]

\[\mathrm{\Delta}x = 2{^\circ} - 0{^\circ} = 2{^\circ} \approx 0,035;\]

\[f^{'}(x) = \frac{1}{\cos^{2}x};\]

\[f^{'}(0{^\circ}) = 1;\]

\[tg\ 2{^\circ} = 0 + 1 \cdot 0,035 = 0,035.\]

\[\textbf{в)}\ ctg\ 46{^\circ}\ \]

\[x_{0} = 45{^\circ};\]

\[f\left( x_{0} \right) = 1;\]

\[\mathrm{\Delta}x = 46{^\circ} - 45{^\circ} = 1{^\circ} \approx 0,0175;\]

\[f^{'}(x) = - \frac{1}{\sin^{2}x};\]

\[f^{'}(45{^\circ}) = - 2;\]

\[ctg\ 46{^\circ}\ \approx 1 - 2 \cdot 0,0175 =\]

\[= 1 - 0,035 = 0,965.\]

\[\textbf{г)}\ ctg\ 88{^\circ}\ \]

\[x_{0} = 90{^\circ};\]

\[f\left( x_{0} \right) = 0;\]

\[\mathrm{\Delta}x = - 2{^\circ} \approx 0,035;\]

\[f^{'}(x) = - \frac{1}{\sin^{2}x};\]

\[f^{'}(90{^\circ}) = - 1;\]

\[ctg\ 88{^\circ} = 0,035.\]