График функции
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Точки пересечения с осью координат X
График функции пересекает ось X при f = 0− cos ( 2 x ) + 1 = 0 - \cos{\left (2 x \right )} + 1 = 0 − cos ( 2 x ) + 1 = 0 Решаем это уравнение Аналитическое решение x 1 = 0 x_{1} = 0 x 1 = 0 x 2 = π x_{2} = \pi x 2 = π Численное решение x 1 = 53.4070756765 x_{1} = 53.4070756765 x 1 = 53.4070756765 x 2 = − 6371388.66252 x_{2} = -6371388.66252 x 2 = − 6371388.66252 x 3 = 37.6991120192 x_{3} = 37.6991120192 x 3 = 37.6991120192 x 4 = − 28.2743337166 x_{4} = -28.2743337166 x 4 = − 28.2743337166 x 5 = − 106.814150358 x_{5} = -106.814150358 x 5 = − 106.814150358 x 6 = 72.2566310277 x_{6} = 72.2566310277 x 6 = 72.2566310277 x 7 = − 15.7079632965 x_{7} = -15.7079632965 x 7 = − 15.7079632965 x 8 = 91.1061867314 x_{8} = 91.1061867314 x 8 = 91.1061867314 x 9 = − 47.1238901511 x_{9} = -47.1238901511 x 9 = − 47.1238901511 x 10 = 6.28318528425 x_{10} = 6.28318528425 x 10 = 6.28318528425 x 11 = 59.6902605977 x_{11} = 59.6902605977 x 11 = 59.6902605977 x 12 = − 84.8230014101 x_{12} = -84.8230014101 x 12 = − 84.8230014101 x 13 = 47.1238895894 x_{13} = 47.1238895894 x 13 = 47.1238895894 x 14 = 97.3893725149 x_{14} = 97.3893725149 x 14 = 97.3893725149 x 15 = 43.9822971694 x_{15} = 43.9822971694 x 15 = 43.9822971694 x 16 = − 50.2654822953 x_{16} = -50.2654822953 x 16 = − 50.2654822953 x 17 = 3.14159287686 x_{17} = 3.14159287686 x 17 = 3.14159287686 x 18 = − 62.8318528379 x_{18} = -62.8318528379 x 18 = − 62.8318528379 x 19 = − 81.681409038 x_{19} = -81.681409038 x 19 = − 81.681409038 x 20 = − 100.530964673 x_{20} = -100.530964673 x 20 = − 100.530964673 x 21 = 75.3982239389 x_{21} = 75.3982239389 x 21 = 75.3982239389 x 22 = − 87.9645943588 x_{22} = -87.9645943588 x 22 = − 87.9645943588 x 23 = 34.5575190305 x_{23} = 34.5575190305 x 23 = 34.5575190305 x 24 = − 40.840704266 x_{24} = -40.840704266 x 24 = − 40.840704266 x 25 = 31.4159267865 x_{25} = 31.4159267865 x 25 = 31.4159267865 x 26 = 91.1061871584 x_{26} = 91.1061871584 x 26 = 91.1061871584 x 27 = − 25.1327414731 x_{27} = -25.1327414731 x 27 = − 25.1327414731 x 28 = 97.3893727097 x_{28} = 97.3893727097 x 28 = 97.3893727097 x 29 = 69.1150385886 x_{29} = 69.1150385886 x 29 = 69.1150385886 x 30 = − 25.1327416321 x_{30} = -25.1327416321 x 30 = − 25.1327416321 x 31 = 87.9645943358 x_{31} = 87.9645943358 x 31 = 87.9645943358 x 32 = 69.1150381602 x_{32} = 69.1150381602 x 32 = 69.1150381602 x 33 = 9.42477821024 x_{33} = 9.42477821024 x 33 = 9.42477821024 x 34 = − 69.1150386737 x_{34} = -69.1150386737 x 34 = − 69.1150386737 x 35 = 12.5663704519 x_{35} = 12.5663704519 x 35 = 12.5663704519 x 36 = − 3.14159311568 x_{36} = -3.14159311568 x 36 = − 3.14159311568 x 37 = 18.8495554002 x_{37} = 18.8495554002 x 37 = 18.8495554002 x 38 = − 69.1150386253 x_{38} = -69.1150386253 x 38 = − 69.1150386253 x 39 = 3.14159244884 x_{39} = 3.14159244884 x 39 = 3.14159244884 x 40 = − 31.415926796 x_{40} = -31.415926796 x 40 = − 31.415926796 x 41 = − 72.2566308741 x_{41} = -72.2566308741 x 41 = − 72.2566308741 x 42 = 62.8318524523 x_{42} = 62.8318524523 x 42 = 62.8318524523 x 43 = − 94.247779453 x_{43} = -94.247779453 x 43 = − 94.247779453 x 44 = 0 x_{44} = 0 x 44 = 0 x 45 = − 40.8407046898 x_{45} = -40.8407046898 x 45 = − 40.8407046898 x 46 = 31.4159271479 x_{46} = 31.4159271479 x 46 = 31.4159271479 x 47 = − 47.1238900493 x_{47} = -47.1238900493 x 47 = − 47.1238900493 x 48 = − 1734.15914476 x_{48} = -1734.15914476 x 48 = − 1734.15914476 x 49 = 25.1327410189 x_{49} = 25.1327410189 x 49 = 25.1327410189 x 50 = − 18.8495561207 x_{50} = -18.8495561207 x 50 = − 18.8495561207 x 51 = 75.3982241945 x_{51} = 75.3982241945 x 51 = 75.3982241945 x 52 = 25.1327414478 x_{52} = 25.1327414478 x 52 = 25.1327414478 x 53 = 62.8318528327 x_{53} = 62.8318528327 x 53 = 62.8318528327 x 54 = − 12.5663703661 x_{54} = -12.5663703661 x 54 = − 12.5663703661 x 55 = − 31.4159267052 x_{55} = -31.4159267052 x 55 = − 31.4159267052 x 56 = 53.4070753627 x_{56} = 53.4070753627 x 56 = 53.4070753627 x 57 = 15.7079634407 x_{57} = 15.7079634407 x 57 = 15.7079634407 x 58 = − 34.5575189426 x_{58} = -34.5575189426 x 58 = − 34.5575189426 x 59 = − 21.9911485865 x_{59} = -21.9911485865 x 59 = − 21.9911485865 x 60 = 84.8230014093 x_{60} = 84.8230014093 x 60 = 84.8230014093 x 61 = 100.530964767 x_{61} = 100.530964767 x 61 = 100.530964767 x 62 = − 78.5398160958 x_{62} = -78.5398160958 x 62 = − 78.5398160958 x 63 = 65.9734457529 x_{63} = 65.9734457529 x 63 = 65.9734457529 x 64 = 56.5486676091 x_{64} = 56.5486676091 x 64 = 56.5486676091 x 65 = − 6.28318513794 x_{65} = -6.28318513794 x 65 = − 6.28318513794 x 66 = 40.8407039199 x_{66} = 40.8407039199 x 66 = 40.8407039199 x 67 = − 12.5663700417 x_{67} = -12.5663700417 x 67 = − 12.5663700417 x 68 = − 75.398223862 x_{68} = -75.398223862 x 68 = − 75.398223862 x 69 = 40.8407042561 x_{69} = 40.8407042561 x 69 = 40.8407042561 x 70 = 84.8230010167 x_{70} = 84.8230010167 x 70 = 84.8230010167 x 71 = 94.2477796094 x_{71} = 94.2477796094 x 71 = 94.2477796094 x 72 = 9.4247785908 x_{72} = 9.4247785908 x 72 = 9.4247785908 x 73 = − 43.9822971746 x_{73} = -43.9822971746 x 73 = − 43.9822971746 x 74 = 50.2654824463 x_{74} = 50.2654824463 x 74 = 50.2654824463 x 75 = − 65.973445765 x_{75} = -65.973445765 x 75 = − 65.973445765 x 76 = 18.8495556796 x_{76} = 18.8495556796 x 76 = 18.8495556796 x 77 = − 3.14159289677 x_{77} = -3.14159289677 x 77 = − 3.14159289677 x 78 = − 91.1061872003 x_{78} = -91.1061872003 x 78 = − 91.1061872003 x 79 = − 97.3893724404 x_{79} = -97.3893724404 x 79 = − 97.3893724404 x 80 = − 59.6902604576 x_{80} = -59.6902604576 x 80 = − 59.6902604576 x 81 = − 91.1061872013 x_{81} = -91.1061872013 x 81 = − 91.1061872013 x 82 = − 37.6991118772 x_{82} = -37.6991118772 x 82 = − 37.6991118772 x 83 = 78.5398161878 x_{83} = 78.5398161878 x 83 = 78.5398161878 x 84 = 47.1238900184 x_{84} = 47.1238900184 x 84 = 47.1238900184 x 85 = 21.9911485852 x_{85} = 21.9911485852 x 85 = 21.9911485852 x 86 = − 56.5486675192 x_{86} = -56.5486675192 x 86 = − 56.5486675192 x 87 = − 84.8230018263 x_{87} = -84.8230018263 x 87 = − 84.8230018263 x 88 = − 53.4070752836 x_{88} = -53.4070752836 x 88 = − 53.4070752836 x 89 = − 9.42477812668 x_{89} = -9.42477812668 x 89 = − 9.42477812668 x 90 = − 62.8318532584 x_{90} = -62.8318532584 x 90 = − 62.8318532584 x 91 = − 34.5575189701 x_{91} = -34.5575189701 x 91 = − 34.5575189701 x 92 = − 18.8495556944 x_{92} = -18.8495556944 x 92 = − 18.8495556944 x 93 = 28.2743338652 x_{93} = 28.2743338652 x 93 = 28.2743338652 x 94 = 81.6814091761 x_{94} = 81.6814091761 x 94 = 81.6814091761
Точки пересечения с осью координат Y
График пересекает ось Y, когда x равняется 0:− cos ( 0 ⋅ 2 ) + 1 - \cos{\left (0 \cdot 2 \right )} + 1 − cos ( 0 ⋅ 2 ) + 1 f ( 0 ) = 0 f{\left (0 \right )} = 0 f ( 0 ) = 0 (0, 0)
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 sin ( 2 x ) = 0 2 \sin{\left (2 x \right )} = 0 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, 0) pi
(--, 2)
2 Интервалы возрастания и убывания функции: x 2 = 0 x_{2} = 0 x 2 = 0 x 2 = π 2 x_{2} = \frac{\pi}{2} x 2 = 2 π [0, pi/2] (-oo, 0] U [pi/2, oo)
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 cos ( 2 x ) = 0 4 \cos{\left (2 x \right )} = 0 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: (-oo, pi/4] U [3*pi/4, oo) [pi/4, 3*pi/4]
Горизонтальные асимптоты
Горизонтальные асимптоты найдём с помощью пределов данной функции при x->+oo и x->-oolim x → − ∞ ( − cos ( 2 x ) + 1 ) = ⟨ 0 , 2 ⟩ \lim_{x \to -\infty}\left(- \cos{\left (2 x \right )} + 1\right) = \langle 0, 2\rangle x → − ∞ lim ( − cos ( 2 x ) + 1 ) = ⟨ 0 , 2 ⟩ Возьмём предел y = ⟨ 0 , 2 ⟩ y = \langle 0, 2\rangle y = ⟨ 0 , 2 ⟩ lim x → ∞ ( − cos ( 2 x ) + 1 ) = ⟨ 0 , 2 ⟩ \lim_{x \to \infty}\left(- \cos{\left (2 x \right )} + 1\right) = \langle 0, 2\rangle x → ∞ lim ( − cos ( 2 x ) + 1 ) = ⟨ 0 , 2 ⟩ Возьмём предел y = ⟨ 0 , 2 ⟩ y = \langle 0, 2\rangle y = ⟨ 0 , 2 ⟩
Наклонные асимптоты
Наклонную асимптоту можно найти, подсчитав предел функции 1 - cos(2*x), делённой на x при x->+oo и x ->-oolim x → − ∞ ( 1 x ( − cos ( 2 x ) + 1 ) ) = 0 \lim_{x \to -\infty}\left(\frac{1}{x} \left(- \cos{\left (2 x \right )} + 1\right)\right) = 0 x → − ∞ lim ( x 1 ( − cos ( 2 x ) + 1 ) ) = 0 Возьмём предел lim x → ∞ ( 1 x ( − cos ( 2 x ) + 1 ) ) = 0 \lim_{x \to \infty}\left(\frac{1}{x} \left(- \cos{\left (2 x \right )} + 1\right)\right) = 0 x → ∞ lim ( x 1 ( − cos ( 2 x ) + 1 ) ) = 0 Возьмём предел
Чётность и нечётность функции
Проверим функци чётна или нечётна с помощью соотношений f = f(-x) и f = -f(-x).− cos ( 2 x ) + 1 = − cos ( 2 x ) + 1 - \cos{\left (2 x \right )} + 1 = - \cos{\left (2 x \right )} + 1 − cos ( 2 x ) + 1 = − cos ( 2 x ) + 1 − cos ( 2 x ) + 1 = − − 1 cos ( 2 x ) − 1 - \cos{\left (2 x \right )} + 1 = - -1 \cos{\left (2 x \right )} - 1 − cos ( 2 x ) + 1 = − − 1 cos ( 2 x ) − 1 