График функции
0 -10000 -7500 -5000 -2500 2500 5000 7500 10000 12500 0.5 1.5
Точки пересечения с осью координат X
График функции пересекает ось X при f = 0cos ( sin ( x ) ) = 0 \cos{\left (\sin{\left (x \right )} \right )} = 0 cos ( sin ( x ) ) = 0 Решаем это уравнение
Точки пересечения с осью координат Y
График пересекает ось Y, когда x равняется 0:cos ( sin ( 0 ) ) \cos{\left (\sin{\left (0 \right )} \right )} cos ( sin ( 0 ) ) 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 − sin ( sin ( x ) ) cos ( x ) = 0 - \sin{\left (\sin{\left (x \right )} \right )} \cos{\left (x \right )} = 0 − sin ( sin ( x ) ) cos ( 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 x 3 = π x_{3} = \pi x 3 = π x 4 = 3 π 2 x_{4} = \frac{3 \pi}{2} x 4 = 2 3 π (0, 1) pi
(--, cos(1))
2 (pi, 1) 3*pi
(----, cos(1))
2 Интервалы возрастания и убывания функции: x 4 = π 2 x_{4} = \frac{\pi}{2} x 4 = 2 π x 4 = 3 π 2 x_{4} = \frac{3 \pi}{2} x 4 = 2 3 π x 4 = 0 x_{4} = 0 x 4 = 0 x 4 = π x_{4} = \pi x 4 = π [3*pi/2, oo) (-oo, pi/2] 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 sin ( x ) sin ( sin ( x ) ) − cos 2 ( x ) cos ( sin ( x ) ) = 0 \sin{\left (x \right )} \sin{\left (\sin{\left (x \right )} \right )} - \cos^{2}{\left (x \right )} \cos{\left (\sin{\left (x \right )} \right )} = 0 sin ( x ) sin ( sin ( x ) ) − cos 2 ( x ) cos ( sin ( x ) ) = 0 Решаем это уравнение x 1 = − 16.4506277484 x_{1} = -16.4506277484 x 1 = − 16.4506277484 x 2 = 62.0891885914 x_{2} = 62.0891885914 x 2 = 62.0891885914 x 3 = 11.8237061339 x_{3} = 11.8237061339 x 3 = 11.8237061339 x 4 = − 24.3900767483 x_{4} = -24.3900767483 x 4 = − 24.3900767483 x 5 = − 21.2484840947 x_{5} = -21.2484840947 x 5 = − 21.2484840947 x 6 = 74.6555592057 x_{6} = 74.6555592057 x 6 = 74.6555592057 x 7 = 2.39892817314 x_{7} = 2.39892817314 x 7 = 2.39892817314 x 8 = − 13.3090350948 x_{8} = -13.3090350948 x 8 = − 13.3090350948 x 9 = 93.5051151272 x_{9} = 93.5051151272 x 9 = 93.5051151272 x 10 = 32.1585910163 x_{10} = 32.1585910163 x 10 = 32.1585910163 x 11 = − 41.5833689771 x_{11} = -41.5833689771 x 11 = − 41.5833689771 x 12 = 82.4240734738 x_{12} = 82.4240734738 x 12 = 82.4240734738 x 13 = − 57.2913322451 x_{13} = -57.2913322451 x 13 = − 57.2913322451 x 14 = − 18.1068914411 x_{14} = -18.1068914411 x 14 = − 18.1068914411 x 15 = 19.592220402 x_{15} = 19.592220402 x 15 = 19.592220402 x 16 = − 19.592220402 x_{16} = -19.592220402 x 16 = − 19.592220402 x 17 = − 5.54052082673 x_{17} = -5.54052082673 x 17 = − 5.54052082673 x 18 = 49.522817977 x_{18} = 49.522817977 x 18 = 49.522817977 x 19 = − 35.3001836699 x_{19} = -35.3001836699 x 19 = − 35.3001836699 x 20 = 3.88425713404 x_{20} = 3.88425713404 x 20 = 3.88425713404 x 21 = 85.5656661274 x_{21} = 85.5656661274 x 21 = 85.5656661274 x 22 = − 55.8060032842 x_{22} = -55.8060032842 x 22 = − 55.8060032842 x 23 = − 91.8488514345 x_{23} = -91.8488514345 x 23 = − 91.8488514345 x 24 = − 68.3723738985 x_{24} = -68.3723738985 x 24 = − 68.3723738985 x 25 = 27.5316694019 x_{25} = 27.5316694019 x 25 = 27.5316694019 x 26 = 0.742664480446 x_{26} = 0.742664480446 x 26 = 0.742664480446 x 27 = − 33.814854709 x_{27} = -33.814854709 x 27 = − 33.814854709 x 28 = 8.68211348032 x_{28} = 8.68211348032 x 28 = 8.68211348032 x 29 = − 60.4329248987 x_{29} = -60.4329248987 x 29 = − 60.4329248987 x 30 = 47.8665542843 x_{30} = 47.8665542843 x 30 = 47.8665542843 x 31 = 46.3812253234 x_{31} = 46.3812253234 x 31 = 46.3812253234 x 32 = 99.7883004344 x_{32} = 99.7883004344 x 32 = 99.7883004344 x 33 = 55.8060032842 x_{33} = 55.8060032842 x 33 = 55.8060032842 x 34 = − 11.8237061339 x_{34} = -11.8237061339 x 34 = − 11.8237061339 x 35 = − 77.7971518593 x_{35} = -77.7971518593 x 35 = − 77.7971518593 x 36 = 68.3723738985 x_{36} = 68.3723738985 x 36 = 68.3723738985 x 37 = − 2.39892817314 x_{37} = -2.39892817314 x 37 = − 2.39892817314 x 38 = 25.8754057092 x_{38} = 25.8754057092 x 38 = 25.8754057092 x 39 = 98.1320367417 x_{39} = 98.1320367417 x 39 = 98.1320367417 x 40 = − 3.88425713404 x_{40} = -3.88425713404 x 40 = − 3.88425713404 x 41 = − 46.3812253234 x_{41} = -46.3812253234 x 41 = − 46.3812253234 x 42 = − 90.3635224737 x_{42} = -90.3635224737 x 42 = − 90.3635224737 x 43 = − 82.4240734738 x_{43} = -82.4240734738 x 43 = − 82.4240734738 x 44 = 5.54052082673 x_{44} = 5.54052082673 x 44 = 5.54052082673 x 45 = − 98.1320367417 x_{45} = -98.1320367417 x 45 = − 98.1320367417 x 46 = − 54.1497395915 x_{46} = -54.1497395915 x 46 = − 54.1497395915 x 47 = 69.8577028594 x_{47} = 69.8577028594 x 47 = 69.8577028594 x 48 = 54.1497395915 x_{48} = 54.1497395915 x 48 = 54.1497395915 x 49 = − 47.8665542843 x_{49} = -47.8665542843 x 49 = − 47.8665542843 x 50 = 30.6732620555 x_{50} = 30.6732620555 x 50 = 30.6732620555 x 51 = − 76.1408881666 x_{51} = -76.1408881666 x 51 = − 76.1408881666 x 52 = 90.3635224737 x_{52} = 90.3635224737 x 52 = 90.3635224737 x 53 = − 138.972741238 x_{53} = -138.972741238 x 53 = − 138.972741238 x 54 = − 99.7883004344 x_{54} = -99.7883004344 x 54 = − 99.7883004344 x 55 = 91.8488514345 x_{55} = 91.8488514345 x 55 = 91.8488514345 x 56 = 41.5833689771 x_{56} = 41.5833689771 x 56 = 41.5833689771 x 57 = 22.7338130556 x_{57} = 22.7338130556 x 57 = 22.7338130556 x 58 = 38.4417763235 x_{58} = 38.4417763235 x 58 = 38.4417763235 x 59 = 24.3900767483 x_{59} = 24.3900767483 x 59 = 24.3900767483 x 60 = − 10.1674424412 x_{60} = -10.1674424412 x 60 = − 10.1674424412 x 61 = − 25.8754057092 x_{61} = -25.8754057092 x 61 = − 25.8754057092 x 62 = 10.1674424412 x_{62} = 10.1674424412 x 62 = 10.1674424412 x 63 = 52.6644106306 x_{63} = 52.6644106306 x 63 = 52.6644106306 x 64 = − 79.2824808202 x_{64} = -79.2824808202 x 64 = − 79.2824808202 x 65 = − 69.8577028594 x_{65} = -69.8577028594 x 65 = − 69.8577028594 x 66 = − 71.5139665521 x_{66} = -71.5139665521 x 66 = − 71.5139665521 x 67 = 33.814854709 x_{67} = 33.814854709 x 67 = 33.814854709 x 68 = − 49.522817977 x_{68} = -49.522817977 x 68 = − 49.522817977 x 69 = 226.937335539 x_{69} = 226.937335539 x 69 = 226.937335539 x 70 = − 84.0803371665 x_{70} = -84.0803371665 x 70 = − 84.0803371665 x 71 = − 62.0891885914 x_{71} = -62.0891885914 x 71 = − 62.0891885914 x 72 = 88.707258781 x_{72} = 88.707258781 x 72 = 88.707258781 x 73 = 18.1068914411 x_{73} = 18.1068914411 x 73 = 18.1068914411 x 74 = − 27.5316694019 x_{74} = -27.5316694019 x 74 = − 27.5316694019 x 75 = 40.0980400162 x_{75} = 40.0980400162 x 75 = 40.0980400162 x 76 = − 85.5656661274 x_{76} = -85.5656661274 x 76 = − 85.5656661274 x 77 = − 63.5745175522 x_{77} = -63.5745175522 x 77 = − 63.5745175522 x 78 = − 38.4417763235 x_{78} = -38.4417763235 x 78 = − 38.4417763235 x 79 = 16.4506277484 x_{79} = 16.4506277484 x 79 = 16.4506277484 x 80 = − 93.5051151272 x_{80} = -93.5051151272 x 80 = − 93.5051151272 x 81 = 76.1408881666 x_{81} = 76.1408881666 x 81 = 76.1408881666 x 82 = 63.5745175522 x_{82} = 63.5745175522 x 82 = 63.5745175522 x 83 = 96.6467077808 x_{83} = 96.6467077808 x 83 = 96.6467077808 x 84 = 44.7249616307 x_{84} = 44.7249616307 x 84 = 44.7249616307 x 85 = − 32.1585910163 x_{85} = -32.1585910163 x 85 = − 32.1585910163 x 86 = − 40.0980400162 x_{86} = -40.0980400162 x 86 = − 40.0980400162 x 87 = 60.4329248987 x_{87} = 60.4329248987 x 87 = 60.4329248987 x 88 = 77.7971518593 x_{88} = 77.7971518593 x 88 = 77.7971518593 x 89 = 66.7161102058 x_{89} = 66.7161102058 x 89 = 66.7161102058 x 90 = 84.0803371665 x_{90} = 84.0803371665 x 90 = 84.0803371665 x 91 = 71.5139665521 x_{91} = 71.5139665521 x 91 = 71.5139665521 Интервалы выпуклости и вогнутости: [226.937335539, oo) (-oo, -99.7883004344]
Горизонтальные асимптоты
Горизонтальные асимптоты найдём с помощью пределов данной функции при x->+oo и x->-oolim x → − ∞ cos ( sin ( x ) ) = cos ( sin ( ∞ ~ ) ) \lim_{x \to -\infty} \cos{\left (\sin{\left (x \right )} \right )} = \cos{\left (\sin{\left (\tilde{\infty} \right )} \right )} x → − ∞ lim cos ( sin ( x ) ) = cos ( sin ( ∞ ~ ) ) Возьмём предел y = cos ( sin ( ∞ ~ ) ) y = \cos{\left (\sin{\left (\tilde{\infty} \right )} \right )} y = cos ( sin ( ∞ ~ ) ) lim x → ∞ cos ( sin ( x ) ) = cos ( sin ( ∞ ~ ) ) \lim_{x \to \infty} \cos{\left (\sin{\left (x \right )} \right )} = \cos{\left (\sin{\left (\tilde{\infty} \right )} \right )} x → ∞ lim cos ( sin ( 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(x)), делённой на x при x->+oo и x ->-oolim x → − ∞ ( 1 x cos ( sin ( x ) ) ) = ∞ ~ sin ( sin ( ∞ ~ ) ) cos ( ∞ ~ ) \lim_{x \to -\infty}\left(\frac{1}{x} \cos{\left (\sin{\left (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 ( 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 ( x ) ) ) = ∞ ~ sin ( sin ( ∞ ~ ) ) cos ( ∞ ~ ) \lim_{x \to \infty}\left(\frac{1}{x} \cos{\left (\sin{\left (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 ( 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 ( x ) ) = cos ( sin ( x ) ) \cos{\left (\sin{\left (x \right )} \right )} = \cos{\left (\sin{\left (x \right )} \right )} cos ( sin ( x ) ) = cos ( sin ( x ) ) cos ( sin ( x ) ) = − cos ( sin ( x ) ) \cos{\left (\sin{\left (x \right )} \right )} = - \cos{\left (\sin{\left (x \right )} \right )} cos ( sin ( x ) ) = − cos ( sin ( x ) ) 