ГДЗ по алгебре и начала математического анализа 11 класс Колягин Задание 161

\[1)\ s(t) = \frac{3}{2}t^{2}:\]

\[s^{'}(t) = \lim_{h \rightarrow 0}\frac{\frac{3}{2}(t + h)^{2} - \frac{3}{2}t^{2}}{h} =\]

\[= \lim_{h \rightarrow 0}\frac{\frac{3}{2}t^{2} + 3th + \frac{3}{2}h^{2} - \frac{3}{2}t^{2}}{h} =\]

\[= \lim_{h \rightarrow 0}\frac{3th + \frac{3}{2}h^{2}}{h} =\]

\[= \lim_{h \rightarrow 0}\left( 3t + \frac{3}{2}h \right) = 3t.\]

\[Ответ:\ \ 3t.\]

\[2)\ s(t) = 5t^{2}:\]

\[s^{'}(t) = \lim_{h \rightarrow 0}\frac{5(t + h)^{2} - 5t^{2}}{h} =\]

\[= \lim_{h \rightarrow 0}\frac{5t^{2} + 10th + 5h^{2} - 5t^{2}}{h} =\]

\[= \lim_{h \rightarrow 0}\frac{10th + 5h^{2}}{h} =\]

\[= \lim_{h \rightarrow 0}(10t + 5h) = 10t.\]

\[Ответ:\ \ 10t.\]