Решебник по алгебре 11 класс Никольский Параграф 4. Производная Задание 2

\[\boxed{\mathbf{2}\mathbf{.}}\]

\[s = t^{2};\ \ t_{1} = t;\ \ t_{2} = t + \mathrm{\Delta}t.\]

\[\textbf{а)}\ s\left( t_{1} \right) = s(t) = t^{2};\]

\[s\left( t_{2} \right) = s(t + \mathrm{\Delta}t) = (t + \mathrm{\Delta}t)^{2} =\]

\[= t^{2} + 2t \cdot \mathrm{\Delta}t + (\mathrm{\Delta}t)^{2};\]

\[\mathrm{\Delta}s = s\left( t_{2} \right) - s\left( t_{1} \right) = t^{2} +\]

\[+ 2t \cdot \mathrm{\Delta}t + (\mathrm{\Delta}t)^{2} - t^{2} =\]

\[= 2t \cdot \mathrm{\Delta}t + (\mathrm{\Delta}t)^{2}.\]

\[\textbf{б)}\ \mathrm{\Delta}t = t_{2} - t_{1} =\]

\[= (t + \mathrm{\Delta}t) - t = \mathrm{\Delta}t;\]

\[\mathrm{\Delta}s = 2t \cdot \mathrm{\Delta}t + (\mathrm{\Delta}t)^{2}\ (из\ п.\ а);\]

\[v_{ср} = \frac{\mathrm{\Delta}s}{\mathrm{\Delta}t} = \frac{2t \cdot \mathrm{\Delta}t + (\mathrm{\Delta}t)^{2}}{\mathrm{\Delta}t} =\]

\[= 2t + \mathrm{\Delta}t.\ \]

\[\textbf{в)}\ v = \lim_{\mathrm{\Delta}t \rightarrow 0}v_{ср} = \lim_{\mathrm{\Delta}t \rightarrow 0}(2t + \mathrm{\Delta}t) =\]

\[= \lim_{\mathrm{\Delta}t \rightarrow 0}{2t} + \lim_{\mathrm{\Delta}t \rightarrow 0}{\mathrm{\Delta}t} = 2t.\]

\[\textbf{г)}\ t = 1:\]

\[v(1) = 2 \cdot 1 = 2.\]