Решебник по алгебре 11 класс Никольский Параграф 4. Производная Задание 7

\[\boxed{\mathbf{7}\mathbf{.}}\]

\[\textbf{а)}\ f(x) = x^{2};\ \ x \in R\]

\[\mathrm{\Delta}f = (x + \mathrm{\Delta}x)^{2} - x^{2} =\]

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

\[\frac{\mathrm{\Delta}f}{\mathrm{\Delta}x} = \frac{(2x + \mathrm{\Delta}x) \cdot \mathrm{\Delta}x}{\mathrm{\Delta}x} = 2x + \mathrm{\Delta}x.\]

\[При\ x \rightarrow 0:\]

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

\[\textbf{б)}\ f^{'}(x) = 2x;\]

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

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

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

\[x = 2 \rightarrow f^{'}(2) = 4;\]

\[x = - 2 \rightarrow f^{'}( - 2) = - 4;\]

\[x = 3 \rightarrow f^{'}(3) = 9;\]

\[x = - 3 \rightarrow f^{'}( - 3) = - 9.\]

\[\textbf{в)}\ f^{'}(x) = 2x;\]

\[2x = 0\]

\[x = 0.\]

\[2x = 1\]

\[x = \frac{1}{2} = 0,5.\]

\[2x = 3\]

\[x = \frac{3}{2} = 1,5.\]