График функции
0 -3000 -2500 -2000 -1500 -1000 -500 500 1000 1500 2000 2500 3000 0.5 1.5
Точки пересечения с осью координат X
График функции пересекает ось X при f = 0cos ( sin ( 3 x ) ) = 0 \cos{\left (\sin{\left (3 x \right )} \right )} = 0 cos ( sin ( 3 x ) ) = 0 Решаем это уравнение
Точки пересечения с осью координат Y
График пересекает ось Y, когда x равняется 0:cos ( sin ( 0 ⋅ 3 ) ) \cos{\left (\sin{\left (0 \cdot 3 \right )} \right )} cos ( sin ( 0 ⋅ 3 ) ) f ( 0 ) = 1 f{\left (0 \right )} = 1 f ( 0 ) = 1 (0, 1)
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 − 3 sin ( sin ( 3 x ) ) cos ( 3 x ) = 0 - 3 \sin{\left (\sin{\left (3 x \right )} \right )} \cos{\left (3 x \right )} = 0 − 3 sin ( sin ( 3 x ) ) cos ( 3 x ) = 0 Решаем это уравнение x 1 = 0 x_{1} = 0 x 1 = 0 x 2 = π 6 x_{2} = \frac{\pi}{6} x 2 = 6 π x 3 = π 3 x_{3} = \frac{\pi}{3} x 3 = 3 π x 4 = π 2 x_{4} = \frac{\pi}{2} x 4 = 2 π (0, 1) pi
(--, cos(1))
6 pi
(--, 1)
3 pi
(--, cos(1))
2 Интервалы возрастания и убывания функции: x 4 = π 6 x_{4} = \frac{\pi}{6} x 4 = 6 π x 4 = π 2 x_{4} = \frac{\pi}{2} x 4 = 2 π x 4 = 0 x_{4} = 0 x 4 = 0 x 4 = π 3 x_{4} = \frac{\pi}{3} x 4 = 3 π [pi/2, oo) (-oo, pi/6] U [pi/3, 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 9 ( sin ( 3 x ) sin ( sin ( 3 x ) ) − cos 2 ( 3 x ) cos ( sin ( 3 x ) ) ) = 0 9 \left(\sin{\left (3 x \right )} \sin{\left (\sin{\left (3 x \right )} \right )} - \cos^{2}{\left (3 x \right )} \cos{\left (\sin{\left (3 x \right )} \right )}\right) = 0 9 ( sin ( 3 x ) sin ( sin ( 3 x ) ) − cos 2 ( 3 x ) cos ( sin ( 3 x ) ) ) = 0 Решаем это уравнение x 1 = − 96.0946198833 x_{1} = -96.0946198833 x 1 = − 96.0946198833 x 2 = − 21.7435937483 x_{2} = -21.7435937483 x 2 = − 21.7435937483 x 3 = − 50.0179276306 x_{3} = -50.0179276306 x 3 = − 50.0179276306 x 4 = 25.9323839531 x_{4} = 25.9323839531 x 4 = 25.9323839531 x 5 = − 74.1034713081 x_{5} = -74.1034713081 x 5 = − 74.1034713081 x 6 = 66.2210005522 x_{6} = 66.2210005522 x 6 = 66.2210005522 x 7 = − 77.7401736154 x_{7} = -77.7401736154 x 7 = − 77.7401736154 x 8 = − 43.7347423234 x_{8} = -43.7347423234 x 8 = − 43.7347423234 x 9 = − 1.84684027558 x_{9} = -1.84684027558 x 9 = − 1.84684027558 x 10 = 86.1177540249 x_{10} = 86.1177540249 x 10 = 86.1177540249 x 11 = − 23.8379888507 x_{11} = -23.8379888507 x 11 = − 23.8379888507 x 12 = 30.1211741579 x_{12} = 30.1211741579 x 12 = 30.1211741579 x 13 = − 65.7258908986 x_{13} = -65.7258908986 x 13 = − 65.7258908986 x 14 = 44.2298519771 x_{14} = 44.2298519771 x 14 = 44.2298519771 x 15 = 47.9235325282 x_{15} = 47.9235325282 x 15 = 47.9235325282 x 16 = − 84.0233589225 x_{16} = -84.0233589225 x 16 = − 84.0233589225 x 17 = − 91.9058296785 x_{17} = -91.9058296785 x 17 = − 91.9058296785 x 18 = − 41.0882593235 x_{18} = -41.0882593235 x 18 = − 41.0882593235 x 19 = − 35.8522715675 x_{19} = -35.8522715675 x 19 = − 35.8522715675 x 20 = 40.0410617723 x_{20} = 40.0410617723 x 20 = 40.0410617723 x 21 = − 79.8345687178 x_{21} = -79.8345687178 x 21 = − 79.8345687178 x 22 = 64.1266054498 x_{22} = 64.1266054498 x 22 = 64.1266054498 x 23 = − 11.76672789 x_{23} = -11.76672789 x 23 = − 11.76672789 x 24 = − 81.9289638201 x_{24} = -81.9289638201 x 24 = − 81.9289638201 x 25 = 54.2067178354 x_{25} = 54.2067178354 x 25 = 54.2067178354 x 26 = 62.0322103474 x_{26} = 62.0322103474 x 26 = 62.0322103474 x 27 = − 53.6546299378 x_{27} = -53.6546299378 x 27 = − 53.6546299378 x 28 = 69.9146811034 x_{28} = 69.9146811034 x 28 = 69.9146811034 x 29 = − 25.9323839531 x_{29} = -25.9323839531 x 29 = − 25.9323839531 x 30 = − 69.9146811034 x_{30} = -69.9146811034 x 30 = − 69.9146811034 x 31 = − 47.9235325282 x_{31} = -47.9235325282 x 31 = − 47.9235325282 x 32 = 96.0946198833 x_{32} = 96.0946198833 x 32 = 96.0946198833 x 33 = − 52.6074323866 x_{33} = -52.6074323866 x 33 = − 52.6074323866 x 34 = 22.2387034019 x_{34} = 22.2387034019 x 34 = 22.2387034019 x 35 = 18.0499131972 x_{35} = 18.0499131972 x 35 = 18.0499131972 x 36 = 94.0002247809 x_{36} = 94.0002247809 x 36 = 94.0002247809 x 37 = − 99.7313221905 x_{37} = -99.7313221905 x 37 = − 99.7313221905 x 38 = − 3.94123537797 x_{38} = -3.94123537797 x 38 = − 3.94123537797 x 39 = − 97.1418174345 x_{39} = -97.1418174345 x 39 = − 97.1418174345 x 40 = − 72.0090762058 x_{40} = -72.0090762058 x 40 = − 72.0090762058 x 41 = 32.2155692603 x_{41} = 32.2155692603 x 41 = 32.2155692603 x 42 = − 9.67233278758 x_{42} = -9.67233278758 x 42 = − 9.67233278758 x 43 = − 31.6634813627 x_{43} = -31.6634813627 x 43 = − 31.6634813627 x 44 = − 37.9466666699 x_{44} = -37.9466666699 x 44 = − 37.9466666699 x 45 = 12.3188157875 x_{45} = 12.3188157875 x 45 = 12.3188157875 x 46 = − 74.5985809618 x_{46} = -74.5985809618 x 46 = − 74.5985809618 x 47 = − 22.7907912995 x_{47} = -22.7907912995 x 47 = − 22.7907912995 x 48 = 72.0090762058 x_{48} = 72.0090762058 x 48 = 72.0090762058 x 49 = 50.0179276306 x_{49} = 50.0179276306 x 49 = 50.0179276306 x 50 = − 158.374385058 x_{50} = -158.374385058 x 50 = − 158.374385058 x 51 = − 8.62513523639 x_{51} = -8.62513523639 x 51 = − 8.62513523639 x 52 = 68.3153956546 x_{52} = 68.3153956546 x 52 = 68.3153956546 x 53 = 11.2716182363 x_{53} = 11.2716182363 x 53 = 11.2716182363 x 54 = 37.9466666699 x_{54} = 37.9466666699 x 54 = 37.9466666699 x 55 = − 28.0267790555 x_{55} = -28.0267790555 x 55 = − 28.0267790555 x 56 = 0.247554826815 x_{56} = 0.247554826815 x 56 = 0.247554826815 x 57 = − 6.03563048036 x_{57} = -6.03563048036 x 57 = − 6.03563048036 x 58 = 74.1034713081 x_{58} = 74.1034713081 x 58 = 74.1034713081 x 59 = − 40.0410617723 x_{59} = -40.0410617723 x 59 = − 40.0410617723 x 60 = 78.2922615129 x_{60} = 78.2922615129 x 60 = 78.2922615129 x 61 = 6.03563048036 x_{61} = 6.03563048036 x 61 = 6.03563048036 x 62 = 81.9289638201 x_{62} = 81.9289638201 x 62 = 81.9289638201 x 63 = 98.1890149857 x_{63} = 98.1890149857 x 63 = 98.1890149857 x 64 = − 62.0322103474 x_{64} = -62.0322103474 x 64 = − 62.0322103474 x 65 = 88.2121491273 x_{65} = 88.2121491273 x 65 = 88.2121491273 x 66 = 52.112322733 x_{66} = 52.112322733 x 66 = 52.112322733 x 67 = − 1291.99422335 x_{67} = -1291.99422335 x 67 = − 1291.99422335 x 68 = 20.1443082996 x_{68} = 20.1443082996 x 68 = 20.1443082996 x 69 = 56.3011129378 x_{69} = 56.3011129378 x 69 = 56.3011129378 x 70 = 2.34194992921 x_{70} = 2.34194992921 x 70 = 2.34194992921 x 71 = − 45.8291374258 x_{71} = -45.8291374258 x 71 = − 45.8291374258 x 72 = 42.1354568747 x_{72} = 42.1354568747 x 72 = 42.1354568747 x 73 = 8.13002558276 x_{73} = 8.13002558276 x 73 = 8.13002558276 x 74 = − 13.8611229924 x_{74} = -13.8611229924 x 74 = − 13.8611229924 x 75 = − 18.0499131972 x_{75} = -18.0499131972 x 75 = − 18.0499131972 x 76 = 91.3537417809 x_{76} = 91.3537417809 x 76 = 91.3537417809 x 77 = 34.3099643627 x_{77} = 34.3099643627 x 77 = 34.3099643627 x 78 = 3.94123537797 x_{78} = 3.94123537797 x 78 = 3.94123537797 x 79 = 100.283410088 x_{79} = 100.283410088 x 79 = 100.283410088 x 80 = − 59.937815245 x_{80} = -59.937815245 x 80 = − 59.937815245 x 81 = − 57.8434201426 x_{81} = -57.8434201426 x 81 = − 57.8434201426 x 82 = 76.1978664105 x_{82} = 76.1978664105 x 82 = 76.1978664105 x 83 = − 94.0002247809 x_{83} = -94.0002247809 x 83 = − 94.0002247809 x 84 = 46.3242470795 x_{84} = 46.3242470795 x 84 = 46.3242470795 x 85 = 84.0233589225 x_{85} = 84.0233589225 x 85 = 84.0233589225 x 86 = − 55.7490250402 x_{86} = -55.7490250402 x 86 = − 55.7490250402 x 87 = 15.9555180948 x_{87} = 15.9555180948 x 87 = 15.9555180948 x 88 = − 89.8114345761 x_{88} = -89.8114345761 x 88 = − 89.8114345761 x 89 = − 33.7578764651 x_{89} = -33.7578764651 x 89 = − 33.7578764651 x 90 = − 67.820286001 x_{90} = -67.820286001 x 90 = − 67.820286001 x 91 = 8.62513523639 x_{91} = 8.62513523639 x 91 = 8.62513523639 x 92 = − 793.52818898 x_{92} = -793.52818898 x 92 = − 793.52818898 x 93 = − 18.6020010947 x_{93} = -18.6020010947 x 93 = − 18.6020010947 x 94 = 90.3065442297 x_{94} = 90.3065442297 x 94 = 90.3065442297 x 95 = 24.3330985043 x_{95} = 24.3330985043 x 95 = 24.3330985043 x 96 = − 15.9555180948 x_{96} = -15.9555180948 x 96 = − 15.9555180948 x 97 = − 87.7170394737 x_{97} = -87.7170394737 x 97 = − 87.7170394737 x 98 = 59.937815245 x_{98} = 59.937815245 x 98 = 59.937815245 x 99 = 28.0267790555 x_{99} = 28.0267790555 x 99 = 28.0267790555 x 100 = 10.2244206852 x_{100} = 10.2244206852 x 100 = 10.2244206852 Интервалы выпуклости и вогнутости: [91.3537417809, oo) (-oo, -1291.99422335]
Горизонтальные асимптоты
Горизонтальные асимптоты найдём с помощью пределов данной функции при x->+oo и x->-oolim x → − ∞ cos ( sin ( 3 x ) ) = cos ( sin ( ∞ ~ ) ) \lim_{x \to -\infty} \cos{\left (\sin{\left (3 x \right )} \right )} = \cos{\left (\sin{\left (\tilde{\infty} \right )} \right )} x → − ∞ lim cos ( sin ( 3 x ) ) = cos ( sin ( ∞ ~ ) ) Возьмём предел y = cos ( sin ( ∞ ~ ) ) y = \cos{\left (\sin{\left (\tilde{\infty} \right )} \right )} y = cos ( sin ( ∞ ~ ) ) lim x → ∞ cos ( sin ( 3 x ) ) = cos ( sin ( ∞ ~ ) ) \lim_{x \to \infty} \cos{\left (\sin{\left (3 x \right )} \right )} = \cos{\left (\sin{\left (\tilde{\infty} \right )} \right )} x → ∞ lim cos ( sin ( 3 x ) ) = cos ( sin ( ∞ ~ ) ) Возьмём предел y = cos ( sin ( ∞ ~ ) ) y = \cos{\left (\sin{\left (\tilde{\infty} \right )} \right )} y = cos ( sin ( ∞ ~ ) )
Наклонные асимптоты
Наклонную асимптоту можно найти, подсчитав предел функции cos(sin(3*x)), делённой на x при x->+oo и x ->-oolim x → − ∞ ( 1 x cos ( sin ( 3 x ) ) ) = ∞ ~ sin ( sin ( ∞ ~ ) ) cos ( ∞ ~ ) \lim_{x \to -\infty}\left(\frac{1}{x} \cos{\left (\sin{\left (3 x \right )} \right )}\right) = \tilde{\infty} \sin{\left (\sin{\left (\tilde{\infty} \right )} \right )} \cos{\left (\tilde{\infty} \right )} x → − ∞ lim ( x 1 cos ( sin ( 3 x ) ) ) = ∞ ~ sin ( sin ( ∞ ~ ) ) cos ( ∞ ~ ) Возьмём предел y = ∞ ~ x sin ( sin ( ∞ ~ ) ) cos ( ∞ ~ ) y = \tilde{\infty} x \sin{\left (\sin{\left (\tilde{\infty} \right )} \right )} \cos{\left (\tilde{\infty} \right )} y = ∞ ~ x sin ( sin ( ∞ ~ ) ) cos ( ∞ ~ ) lim x → ∞ ( 1 x cos ( sin ( 3 x ) ) ) = ∞ ~ sin ( sin ( ∞ ~ ) ) cos ( ∞ ~ ) \lim_{x \to \infty}\left(\frac{1}{x} \cos{\left (\sin{\left (3 x \right )} \right )}\right) = \tilde{\infty} \sin{\left (\sin{\left (\tilde{\infty} \right )} \right )} \cos{\left (\tilde{\infty} \right )} x → ∞ lim ( x 1 cos ( sin ( 3 x ) ) ) = ∞ ~ sin ( sin ( ∞ ~ ) ) cos ( ∞ ~ ) Возьмём предел y = ∞ ~ x sin ( sin ( ∞ ~ ) ) cos ( ∞ ~ ) y = \tilde{\infty} x \sin{\left (\sin{\left (\tilde{\infty} \right )} \right )} \cos{\left (\tilde{\infty} \right )} y = ∞ ~ x sin ( sin ( ∞ ~ ) ) cos ( ∞ ~ )
Чётность и нечётность функции
Проверим функци чётна или нечётна с помощью соотношений f = f(-x) и f = -f(-x).cos ( sin ( 3 x ) ) = cos ( sin ( 3 x ) ) \cos{\left (\sin{\left (3 x \right )} \right )} = \cos{\left (\sin{\left (3 x \right )} \right )} cos ( sin ( 3 x ) ) = cos ( sin ( 3 x ) ) cos ( sin ( 3 x ) ) = − cos ( sin ( 3 x ) ) \cos{\left (\sin{\left (3 x \right )} \right )} = - \cos{\left (\sin{\left (3 x \right )} \right )} cos ( sin ( 3 x ) ) = − cos ( sin ( 3 x ) ) 