График функции
0 -10000 -8000 -6000 -4000 -2000 2000 4000 6000 8000 10000 0.5 -0.5
Точки пересечения с осью координат X
График функции пересекает ось X при f = 01 3 cos ( 2 x ) = 0 \frac{1}{3} \cos{\left (2 x \right )} = 0 3 1 cos ( 2 x ) = 0 Решаем это уравнение Аналитическое решение x 1 = π 4 x_{1} = \frac{\pi}{4} x 1 = 4 π x 2 = 3 π 4 x_{2} = \frac{3 \pi}{4} x 2 = 4 3 π Численное решение x 1 = − 77.7544181763 x_{1} = -77.7544181763 x 1 = − 77.7544181763 x 2 = 90.3207887907 x_{2} = 90.3207887907 x 2 = 90.3207887907 x 3 = 22.7765467385 x_{3} = 22.7765467385 x 3 = 22.7765467385 x 4 = − 93.4623814443 x_{4} = -93.4623814443 x 4 = − 93.4623814443 x 5 = 77.7544181763 x_{5} = 77.7544181763 x 5 = 77.7544181763 x 6 = − 13.3517687778 x_{6} = -13.3517687778 x 6 = − 13.3517687778 x 7 = − 71.4712328692 x_{7} = -71.4712328692 x 7 = − 71.4712328692 x 8 = 33.7721210261 x_{8} = 33.7721210261 x 8 = 33.7721210261 x 9 = − 47.9092879672 x_{9} = -47.9092879672 x 9 = − 47.9092879672 x 10 = 66.7588438888 x_{10} = 66.7588438888 x 10 = 66.7588438888 x 11 = 162.577419823 x_{11} = 162.577419823 x 11 = 162.577419823 x 12 = 69.9004365424 x_{12} = 69.9004365424 x 12 = 69.9004365424 x 13 = − 25.9181393921 x_{13} = -25.9181393921 x 13 = − 25.9181393921 x 14 = 63.6172512352 x_{14} = 63.6172512352 x 14 = 63.6172512352 x 15 = 30.6305283725 x_{15} = 30.6305283725 x 15 = 30.6305283725 x 16 = − 49.480084294 x_{16} = -49.480084294 x 16 = − 49.480084294 x 17 = 84.0376034835 x_{17} = 84.0376034835 x 17 = 84.0376034835 x 18 = 54.1924732744 x_{18} = 54.1924732744 x 18 = 54.1924732744 x 19 = 2.35619449019 x_{19} = 2.35619449019 x 19 = 2.35619449019 x 20 = − 1147.46671672 x_{20} = -1147.46671672 x 20 = − 1147.46671672 x 21 = − 33.7721210261 x_{21} = -33.7721210261 x 21 = − 33.7721210261 x 22 = 10.2101761242 x_{22} = 10.2101761242 x 22 = 10.2101761242 x 23 = 87.1791961371 x_{23} = 87.1791961371 x 23 = 87.1791961371 x 24 = 76.1836218496 x_{24} = 76.1836218496 x 24 = 76.1836218496 x 25 = 49.480084294 x_{25} = 49.480084294 x 25 = 49.480084294 x 26 = − 2.35619449019 x_{26} = -2.35619449019 x 26 = − 2.35619449019 x 27 = − 5.49778714378 x_{27} = -5.49778714378 x 27 = − 5.49778714378 x 28 = − 55.7632696012 x_{28} = -55.7632696012 x 28 = − 55.7632696012 x 29 = 60.4756585816 x_{29} = 60.4756585816 x 29 = 60.4756585816 x 30 = − 54.1924732744 x_{30} = -54.1924732744 x 30 = − 54.1924732744 x 31 = − 38.4845100065 x_{31} = -38.4845100065 x 31 = − 38.4845100065 x 32 = − 46.3384916404 x_{32} = -46.3384916404 x 32 = − 46.3384916404 x 33 = 40.0553063333 x_{33} = 40.0553063333 x 33 = 40.0553063333 x 34 = 41.6261026601 x_{34} = 41.6261026601 x 34 = 41.6261026601 x 35 = − 32.2013246993 x_{35} = -32.2013246993 x 35 = − 32.2013246993 x 36 = − 79.3252145031 x_{36} = -79.3252145031 x 36 = − 79.3252145031 x 37 = − 18.0641577581 x_{37} = -18.0641577581 x 37 = − 18.0641577581 x 38 = − 62.0464549084 x_{38} = -62.0464549084 x 38 = − 62.0464549084 x 39 = 44.7676953137 x_{39} = 44.7676953137 x 39 = 44.7676953137 x 40 = 46.3384916404 x_{40} = 46.3384916404 x 40 = 46.3384916404 x 41 = − 11.780972451 x_{41} = -11.780972451 x 41 = − 11.780972451 x 42 = 27.4889357189 x_{42} = 27.4889357189 x 42 = 27.4889357189 x 43 = 1106.62601223 x_{43} = 1106.62601223 x 43 = 1106.62601223 x 44 = 85.6083998103 x_{44} = 85.6083998103 x 44 = 85.6083998103 x 45 = 32.2013246993 x_{45} = 32.2013246993 x 45 = 32.2013246993 x 46 = 74.6128255228 x_{46} = 74.6128255228 x 46 = 74.6128255228 x 47 = − 63.6172512352 x_{47} = -63.6172512352 x 47 = − 63.6172512352 x 48 = − 76.1836218496 x_{48} = -76.1836218496 x 48 = − 76.1836218496 x 49 = 18.0641577581 x_{49} = 18.0641577581 x 49 = 18.0641577581 x 50 = − 99.7455667515 x_{50} = -99.7455667515 x 50 = − 99.7455667515 x 51 = − 60.4756585816 x_{51} = -60.4756585816 x 51 = − 60.4756585816 x 52 = − 90.3207887907 x_{52} = -90.3207887907 x 52 = − 90.3207887907 x 53 = − 16.4933614313 x_{53} = -16.4933614313 x 53 = − 16.4933614313 x 54 = − 69.9004365424 x_{54} = -69.9004365424 x 54 = − 69.9004365424 x 55 = 88.7499924639 x_{55} = 88.7499924639 x 55 = 88.7499924639 x 56 = 3.92699081699 x_{56} = 3.92699081699 x 56 = 3.92699081699 x 57 = 11.780972451 x_{57} = 11.780972451 x 57 = 11.780972451 x 58 = 98.1747704247 x_{58} = 98.1747704247 x 58 = 98.1747704247 x 59 = − 19.6349540849 x_{59} = -19.6349540849 x 59 = − 19.6349540849 x 60 = 38.4845100065 x_{60} = 38.4845100065 x 60 = 38.4845100065 x 61 = 24.3473430653 x_{61} = 24.3473430653 x 61 = 24.3473430653 x 62 = 62.0464549084 x_{62} = 62.0464549084 x 62 = 62.0464549084 x 63 = − 84.0376034835 x_{63} = -84.0376034835 x 63 = − 84.0376034835 x 64 = − 35.3429173529 x_{64} = -35.3429173529 x 64 = − 35.3429173529 x 65 = − 41.6261026601 x_{65} = -41.6261026601 x 65 = − 41.6261026601 x 66 = − 91.8915851175 x_{66} = -91.8915851175 x 66 = − 91.8915851175 x 67 = 82.4668071567 x_{67} = 82.4668071567 x 67 = 82.4668071567 x 68 = 96.6039740979 x_{68} = 96.6039740979 x 68 = 96.6039740979 x 69 = 25.9181393921 x_{69} = 25.9181393921 x 69 = 25.9181393921 x 70 = − 27.4889357189 x_{70} = -27.4889357189 x 70 = − 27.4889357189 x 71 = 384.059701901 x_{71} = 384.059701901 x 71 = 384.059701901 x 72 = − 82.4668071567 x_{72} = -82.4668071567 x 72 = − 82.4668071567 x 73 = − 10.2101761242 x_{73} = -10.2101761242 x 73 = − 10.2101761242 x 74 = − 40.0553063333 x_{74} = -40.0553063333 x 74 = − 40.0553063333 x 75 = − 85.6083998103 x_{75} = -85.6083998103 x 75 = − 85.6083998103 x 76 = − 57.334065928 x_{76} = -57.334065928 x 76 = − 57.334065928 x 77 = − 98.1747704247 x_{77} = -98.1747704247 x 77 = − 98.1747704247 x 78 = 47.9092879672 x_{78} = 47.9092879672 x 78 = 47.9092879672 x 79 = 16.4933614313 x_{79} = 16.4933614313 x 79 = 16.4933614313 x 80 = − 3.92699081699 x_{80} = -3.92699081699 x 80 = − 3.92699081699 x 81 = 68.3296402156 x_{81} = 68.3296402156 x 81 = 68.3296402156 x 82 = 19.6349540849 x_{82} = 19.6349540849 x 82 = 19.6349540849 x 83 = 5.49778714378 x_{83} = 5.49778714378 x 83 = 5.49778714378 x 84 = 99.7455667515 x_{84} = 99.7455667515 x 84 = 99.7455667515 x 85 = 52.6216769476 x_{85} = 52.6216769476 x 85 = 52.6216769476 x 86 = − 24.3473430653 x_{86} = -24.3473430653 x 86 = − 24.3473430653 x 87 = − 68.3296402156 x_{87} = -68.3296402156 x 87 = − 68.3296402156 x 88 = 55.7632696012 x_{88} = 55.7632696012 x 88 = 55.7632696012 x 89 = 91.8915851175 x_{89} = 91.8915851175 x 89 = 91.8915851175 x 90 = 8.63937979737 x_{90} = 8.63937979737 x 90 = 8.63937979737
Точки пересечения с осью координат Y
График пересекает ось Y, когда x равняется 0:1 3 cos ( 0 ⋅ 2 ) \frac{1}{3} \cos{\left (0 \cdot 2 \right )} 3 1 cos ( 0 ⋅ 2 ) f ( 0 ) = 1 3 f{\left (0 \right )} = \frac{1}{3} f ( 0 ) = 3 1 (0, 1/3)
Экстремумы функции
Для того, чтобы найти экстремумы, нужно решить уравнениеd d x f ( x ) = 0 \frac{d}{d x} f{\left (x \right )} = 0 d x d f ( x ) = 0 d d x f ( x ) = \frac{d}{d x} f{\left (x \right )} = d x d f ( x ) = Первая производная − 2 3 sin ( 2 x ) = 0 - \frac{2}{3} \sin{\left (2 x \right )} = 0 − 3 2 sin ( 2 x ) = 0 Решаем это уравнение x 1 = 0 x_{1} = 0 x 1 = 0 x 2 = π 2 x_{2} = \frac{\pi}{2} x 2 = 2 π (0, 1/3) pi
(--, -1/3)
2 Интервалы возрастания и убывания функции: x 2 = π 2 x_{2} = \frac{\pi}{2} x 2 = 2 π x 2 = 0 x_{2} = 0 x 2 = 0 (-oo, 0] U [pi/2, oo) [0, pi/2]
Точки перегибов
Найдем точки перегибов, для этого надо решить уравнениеd 2 d x 2 f ( x ) = 0 \frac{d^{2}}{d x^{2}} f{\left (x \right )} = 0 d x 2 d 2 f ( x ) = 0 d 2 d x 2 f ( x ) = \frac{d^{2}}{d x^{2}} f{\left (x \right )} = d x 2 d 2 f ( x ) = Вторая производная − 4 3 cos ( 2 x ) = 0 - \frac{4}{3} \cos{\left (2 x \right )} = 0 − 3 4 cos ( 2 x ) = 0 Решаем это уравнение x 1 = π 4 x_{1} = \frac{\pi}{4} x 1 = 4 π x 2 = 3 π 4 x_{2} = \frac{3 \pi}{4} x 2 = 4 3 π Интервалы выпуклости и вогнутости: [pi/4, 3*pi/4] (-oo, pi/4] U [3*pi/4, oo)
Горизонтальные асимптоты
Горизонтальные асимптоты найдём с помощью пределов данной функции при x->+oo и x->-oolim x → − ∞ ( 1 3 cos ( 2 x ) ) = ⟨ − 1 3 , 1 3 ⟩ \lim_{x \to -\infty}\left(\frac{1}{3} \cos{\left (2 x \right )}\right) = \langle - \frac{1}{3}, \frac{1}{3}\rangle x → − ∞ lim ( 3 1 cos ( 2 x ) ) = ⟨ − 3 1 , 3 1 ⟩ Возьмём предел y = ⟨ − 1 3 , 1 3 ⟩ y = \langle - \frac{1}{3}, \frac{1}{3}\rangle y = ⟨ − 3 1 , 3 1 ⟩ lim x → ∞ ( 1 3 cos ( 2 x ) ) = ⟨ − 1 3 , 1 3 ⟩ \lim_{x \to \infty}\left(\frac{1}{3} \cos{\left (2 x \right )}\right) = \langle - \frac{1}{3}, \frac{1}{3}\rangle x → ∞ lim ( 3 1 cos ( 2 x ) ) = ⟨ − 3 1 , 3 1 ⟩ Возьмём предел y = ⟨ − 1 3 , 1 3 ⟩ y = \langle - \frac{1}{3}, \frac{1}{3}\rangle y = ⟨ − 3 1 , 3 1 ⟩
Наклонные асимптоты
Наклонную асимптоту можно найти, подсчитав предел функции cos(2*x)/3, делённой на x при x->+oo и x ->-oolim x → − ∞ ( 1 3 x cos ( 2 x ) ) = 0 \lim_{x \to -\infty}\left(\frac{1}{3 x} \cos{\left (2 x \right )}\right) = 0 x → − ∞ lim ( 3 x 1 cos ( 2 x ) ) = 0 Возьмём предел lim x → ∞ ( 1 3 x cos ( 2 x ) ) = 0 \lim_{x \to \infty}\left(\frac{1}{3 x} \cos{\left (2 x \right )}\right) = 0 x → ∞ lim ( 3 x 1 cos ( 2 x ) ) = 0 Возьмём предел
Чётность и нечётность функции
Проверим функци чётна или нечётна с помощью соотношений f = f(-x) и f = -f(-x).1 3 cos ( 2 x ) = 1 3 cos ( 2 x ) \frac{1}{3} \cos{\left (2 x \right )} = \frac{1}{3} \cos{\left (2 x \right )} 3 1 cos ( 2 x ) = 3 1 cos ( 2 x ) 1 3 cos ( 2 x ) = − 1 3 cos ( 2 x ) \frac{1}{3} \cos{\left (2 x \right )} = - \frac{1}{3} \cos{\left (2 x \right )} 3 1 cos ( 2 x ) = − 3 1 cos ( 2 x ) 