Решебник по алгебре и начала математического анализа 11 класс Колягин Задание 355

\[1)\ f(x) = x^{2};\text{\ \ \ M}(1;\ 2):\]

\[F(x) = \frac{x^{2 + 1}}{2 + 1} + C = \frac{x^{3}}{3} + C;\]

\[2 = \frac{1^{3}}{3} + C\]

\[C = \frac{5}{3}.\]

\[Ответ:\ \ F(x) = \frac{x^{3}}{3} + \frac{5}{3}.\]

\[2)\ f(x) = x;\text{\ \ \ M}( - 1;\ 3):\]

\[F(x) = \frac{x^{1 + 1}}{1 + 1} + C = \frac{x^{2}}{2} + C;\]

\[3 = \frac{( - 1)^{2}}{2} + C\]

\[C = \frac{5}{2}.\]

\[Ответ:\ \ F(x) = \frac{x^{2}}{2} + \frac{5}{2}.\]

\[3)\ f(x) = \frac{1}{x};M(1;\ - 1):\]

\[F(x) = \ln|x| + C;\]

\[- 1 = \ln 1 + C\]

\[C = - 1.\]

\[Ответ:\ \ F(x) = \ln|x| - 1.\]

\[4)\ f(x) = \sqrt{x};\text{\ \ \ M}(9;\ 10):\]

\[F(x) = \frac{x^{\frac{1}{2} + 1}}{\frac{1}{2} + 1} + C = \frac{2}{3}x^{\frac{3}{2}} + C;\]

\[10 = \frac{2}{3}9^{\frac{3}{2}} + C\]

\[10 = \frac{2}{3} \bullet 3^{3} + C\]

\[10 = 18 + C\]

\[C = - 8.\]

\[Ответ:\ \ F(x) = \frac{2}{3}x^{\frac{3}{2}} - 8.\]