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МатематикаРешебник по алгебре и начала математического анализа 11 класс Колягин Задание 964
Задание 964
\[1)\ \left\{ \begin{matrix} x + 3y - 3 \geq 0\ \ \ \ \ \\ 2x + 3y - 12 \leq 0 \\ x \geq 0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ 0 \leq y \leq 2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \]
\[x + 3y - 3 \geq 0\]
\[3y \geq 3 - x\]
\[y \geq 1 - \frac{1}{3}x\]
\[y(0) = 1 - 0 = 1;\]
\[y(3) = 1 - 1 = 0.\]
\[2x + 3y - 12 \leq 0\]
\[3y \leq 12 - 2x\]
\[y \leq 4 - \frac{2}{3}x\]
\[y(0) = 4 - 0 = 4;\]
\[y(3) = 4 - 2 = 2.\]
\[S = 3 \bullet 1 + \frac{1}{2} \bullet 3 \bullet 1 + \frac{1}{2} \bullet 3 \bullet 2 =\]
\[= 3 + 1,5 + 3 = 7,5.\]
\[Ответ:\ \ 7,5.\]
\[2)\ \left\{ \begin{matrix} x - y + 1 \leq 0\ \ \ \ \ \ \ \\ 5x - 3y + 15 \geq 0 \\ x \leq 0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ 0 \leq y \leq 2,5\ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \]
\[x - y + 1 \leq 0\]
\[y \geq x + 1\]
\[y( - 1) = - 1 + 1 = 0;\]
\[y(0) = 0 + 1 = 1.\]
\[5x - 3y + 15 \geq 0\]
\[3y \leq 5x + 15\]
\[y \leq \frac{5}{3}x + 5\]
\[y( - 3) = - 5 + 5 = 0;\]
\[y(0) = 0 + 5 = 5.\]
\[S = 3 \bullet 2,5 - \frac{1}{2} \bullet 1,5 \bullet 2,5 - \frac{1}{2} \bullet 1 \bullet 1 =\]
\[= 7,5 - 1,875 - 0,5 = 5,125.\]
\[Ответ:\ \ 5,125.\]
\[3)\ \left\{ \begin{matrix} (x + y - 1)(x + y - 3) \leq 0 \\ x - y \leq 0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ x \geq 0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \]
\[(x + y - 1)(x + y - 3) \leq 0\]
\[x + y - 1 \geq 0\]
\[y \geq 1 - x;\]
\[\ x + y - 3 \leq 0\ \ \]
\[y \leq 3 - x;\]
\[1 - x \leq y \leq 3 - x.\]
\[y(0) = 1 - 0 = 1;\ \ \ \]
\[y(0) = 3 - 0 = 3;\]
\[y(1) = 1 - 1 = 0;\]
\[y(1) = 3 - 1 = 2.\]
\[x - y \leq 0\]
\[y \geq x\]
\[y(0) = 0;\]
\[y(1) = 1.\]
\[S =\]
\[= 1 \bullet 1 + \frac{1}{2} \bullet 1 \bullet 1 + \frac{1}{4} \bullet 1 \bullet 1 + \frac{1}{4} \bullet 1 \bullet 1 =\]
\[= 1 + 0,5 + 0,25 + 0,25 = 2.\]
\[Ответ:\ \ 2.\]
\[4)\ \left\{ \begin{matrix} (2x - y + 3)(4x + 3y - 9) \leq 0 \\ x + y \geq 0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ - 1 \leq y \leq 3\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \]
\[(2x - y + 3)(4x + 3y - 9) \leq 0\]
\[2x - y + 3 \geq 0\]
\[y \leq 2x + 3\]
\[y \leq 2x + 3.\ \ \]
\[4x + 3y - 9 \leq 0\]
\[3y \leq 9 - 4x\]
\[\ y \leq 3 - \frac{4}{3}\text{x.}\]
\[y(0) = 0 + 3 = 3;\]
\[y(0) = 3 - 0 = 3;\]
\[y(1) = 2 + 3 = 5;\]
\[y(3) = 3 - 4 = - 1.\]
\[x + y \geq 0\]
\[y \geq - x;\]
\[y(0) = 0;\ \ \ \]
\[y(1) = - 1.\]
\[S =\]
\[= 4 \bullet 4 - \frac{1}{2} \bullet 2 \bullet 1 - \frac{1}{2} \bullet 3 \bullet 4 - \frac{1}{2} \bullet 2 \bullet 2 =\]
\[= 16 - 1 - 6 - 2 = 10 - 3 = 7.\]
\[Ответ:\ \ 7.\]
