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МатематикаРешебник по алгебре 11 класс Никольский Параграф 5. Применение производной Задание 42
Задание 42
\[\boxed{\mathbf{42}\mathbf{.}}\]
\[f\left( x_{0} + \mathrm{\Delta}x \right) \approx f\left( x_{0} \right) + f^{'}\left( x_{0} \right)\mathrm{\Delta}x\]
\[\textbf{а)}\sin{1{^\circ}};\]
\[x_{0} = 0;\]
\[f\left( x_{0} \right) = 0;\]
\[\mathrm{\Delta}x = 1{^\circ} \approx 0,0175;\]
\[f^{'}(x) = \cos x;\]
\[f^{'}\left( x_{0} \right) = 1.\]
\[\sin{1{^\circ}} \approx 0 + 1 \cdot 0,0175 = 0,0175.\]
\[\textbf{б)}\sin{2{^\circ}}\]
\[x_{0} = 0;\]
\[f\left( x_{0} \right) = 0;\]
\[\mathrm{\Delta}x = 2{^\circ} \approx 0,035;\]
\[f^{'}(x) = \cos x;\]
\[f^{'}\left( x_{0} \right) = 1.\]
\[\sin{1{^\circ}} \approx 0 + 1 \cdot 0,035 = 0,035.\]
\[\textbf{в)}\sin{31{^\circ}}\]
\[x_{0} = 30{^\circ};\]
\[f\left( x_{0} \right) = 0,5;\]
\[\mathrm{\Delta}x = 1{^\circ} \approx 0,0175;\]
\[f^{'}(x) = \cos x;\]
\[f^{'}\left( x_{0} \right) = \frac{\sqrt{3}}{2} \approx 0,866.\]
\[\sin{31{^\circ}} \approx 0,5 + 0,866 \cdot 0,0175 =\]
\[= 0,5 + 0,015 = 0,515.\]
\[\textbf{г)}\sin{29{^\circ}}\]
\[x_{0} = 30{^\circ};\]
\[f\left( x_{0} \right) = 0,5;\]
\[\mathrm{\Delta}x = - 1{^\circ} \approx - 0,0175;\]
\[f^{'}(x) = \cos x;\]
\[f^{'}\left( x_{0} \right) = \frac{\sqrt{3}}{2} \approx 0,866.\]
\[\sin{29{^\circ}} \approx 0,5 - 0,866 \cdot 0,0175 \approx\]
\[\approx 0,5 - 0,0152 = 0,4848.\]
\[\textbf{д)}\cos{91{^\circ}\ }\]
\[x_{0} = 90{^\circ};\]
\[f\left( x_{0} \right) = 0;\]
\[\mathrm{\Delta}x = 91{^\circ} - 90{^\circ} = 1{^\circ} \approx 0,0175;\]
\[f^{'}(x) = - \sin x;\]
\[f^{'}\left( x_{0} \right) = - 1.\]
\[\cos{91{^\circ}} \approx 0 - 1 \cdot 0,0175 =\]
\[= - 0,0175.\]
\[\textbf{е)}\cos{61{^\circ}}\]
\[x_{0} = 60{^\circ};\]
\[f\left( x_{0} \right) = 0,5;\]
\[\mathrm{\Delta}x = 1{^\circ} \approx 0,0175;\]
\[f^{'}(x) = - \sin x;\]
\[f^{'}\left( x_{0} \right) = - \frac{\sqrt{3}}{2} \approx - 0,866.\]
\[\cos{61{^\circ}} \approx 0,5 - 0,866 \cdot 0,0175 =\]
\[= 0,4848.\]
\[\textbf{ж)}\cos{59{^\circ}\ }\]
\[x_{0} = 60{^\circ};\]
\[f\left( x_{0} \right) = 0,5;\]
\[\mathrm{\Delta}x = - 1{^\circ} \approx - 0,0175;\]
\[f^{'}(x) = - \sin x;\]
\[f^{'}\left( x_{0} \right) = - \frac{\sqrt{3}}{2} \approx - 0,866.\]
\[\cos{59{^\circ}} \approx 0,5 + 0,866 \cdot 0,0175 =\]
\[= 0,515.\]
\[\textbf{з)}\cos{89{^\circ}}\]
\[x_{0} = 90{^\circ};\]
\[f\left( x_{0} \right) = 0;\]
\[\mathrm{\Delta}x = - 1{^\circ} \approx - 0,0175;\]
\[f^{'}(x) = - \sin x;\]
\[f^{'}\left( x_{0} \right) = - 1.\]
\[\cos{89{^\circ}} \approx 0 + 0,0175 \cdot 1 =\]
\[= 0,0175.\]
