График функции
0 -80 -70 -60 -50 -40 -30 -20 -10 10 20 30 40 50 0 5
Точки пересечения с осью координат X
График функции пересекает ось X при f = 02 − 2 cos ( x ) = 0 2 - 2 \cos{\left(x \right)} = 0 2 − 2 cos ( x ) = 0 Решаем это уравнение Аналитическое решение x 1 = 0 x_{1} = 0 x 1 = 0 x 2 = 2 π x_{2} = 2 \pi x 2 = 2 π Численное решение x 1 = − 87.9645943586158 x_{1} = -87.9645943586158 x 1 = − 87.9645943586158 x 2 = 56.5486682809363 x_{2} = 56.5486682809363 x 2 = 56.5486682809363 x 3 = − 18.8495555173448 x_{3} = -18.8495555173448 x 3 = − 18.8495555173448 x 4 = 62.8318527849002 x_{4} = 62.8318527849002 x 4 = 62.8318527849002 x 5 = 75.3982240031607 x_{5} = 75.3982240031607 x 5 = 75.3982240031607 x 6 = 81.6814091897036 x_{6} = 81.6814091897036 x 6 = 81.6814091897036 x 7 = − 50.265482641087 x_{7} = -50.265482641087 x 7 = − 50.265482641087 x 8 = 25.1327416384075 x_{8} = 25.1327416384075 x 8 = 25.1327416384075 x 9 = 87.9645946044253 x_{9} = 87.9645946044253 x 9 = 87.9645946044253 x 10 = 75.3982232188727 x_{10} = 75.3982232188727 x 10 = 75.3982232188727 x 11 = − 87.964593928489 x_{11} = -87.964593928489 x 11 = − 87.964593928489 x 12 = − 31.4159260208155 x_{12} = -31.4159260208155 x 12 = − 31.4159260208155 x 13 = 6.28318579821791 x_{13} = 6.28318579821791 x 13 = 6.28318579821791 x 14 = 12.5663704426592 x_{14} = 12.5663704426592 x 14 = 12.5663704426592 x 15 = 69.1150379887504 x_{15} = 69.1150379887504 x 15 = 69.1150379887504 x 16 = 37.6991115173992 x_{16} = 37.6991115173992 x 16 = 37.6991115173992 x 17 = − 50.2654829667315 x_{17} = -50.2654829667315 x 17 = − 50.2654829667315 x 18 = 43.9822971694647 x_{18} = 43.9822971694647 x 18 = 43.9822971694647 x 19 = − 43.9822976246252 x_{19} = -43.9822976246252 x 19 = − 43.9822976246252 x 20 = 50.2654824463392 x_{20} = 50.2654824463392 x 20 = 50.2654824463392 x 21 = 37.6991113348642 x_{21} = 37.6991113348642 x 21 = 37.6991113348642 x 22 = − 25.1327415297174 x_{22} = -25.1327415297174 x 22 = − 25.1327415297174 x 23 = 43.9822966661001 x_{23} = 43.9822966661001 x 23 = 43.9822966661001 x 24 = − 94.2477794452815 x_{24} = -94.2477794452815 x 24 = − 94.2477794452815 x 25 = − 87.9645947692094 x_{25} = -87.9645947692094 x 25 = − 87.9645947692094 x 26 = − 37.6991118772631 x_{26} = -37.6991118772631 x 26 = − 37.6991118772631 x 27 = 25.1327408328211 x_{27} = 25.1327408328211 x 27 = 25.1327408328211 x 28 = 0 x_{28} = 0 x 28 = 0 x 29 = 50.2654829439723 x_{29} = 50.2654829439723 x 29 = 50.2654829439723 x 30 = 94.2477796093523 x_{30} = 94.2477796093523 x 30 = 94.2477796093523 x 31 = 62.8318535568358 x_{31} = 62.8318535568358 x 31 = 62.8318535568358 x 32 = − 18.8495552124105 x_{32} = -18.8495552124105 x 32 = − 18.8495552124105 x 33 = 43.9822974733639 x_{33} = 43.9822974733639 x 33 = 43.9822974733639 x 34 = 100.530964759815 x_{34} = 100.530964759815 x 34 = 100.530964759815 x 35 = 81.6814085860518 x_{35} = 81.6814085860518 x 35 = 81.6814085860518 x 36 = − 31.4159260507536 x_{36} = -31.4159260507536 x 36 = − 31.4159260507536 x 37 = − 37.6991121287155 x_{37} = -37.6991121287155 x 37 = − 37.6991121287155 x 38 = 6.28318528420851 x_{38} = 6.28318528420851 x 38 = 6.28318528420851 x 39 = 87.9645938121814 x_{39} = 87.9645938121814 x 39 = 87.9645938121814 x 40 = − 12.5663703112531 x_{40} = -12.5663703112531 x 40 = − 12.5663703112531 x 41 = 94.2477792651059 x_{41} = 94.2477792651059 x 41 = 94.2477792651059 x 42 = 18.8495564031971 x_{42} = 18.8495564031971 x 42 = 18.8495564031971 x 43 = − 12.5663710889626 x_{43} = -12.5663710889626 x 43 = − 12.5663710889626 x 44 = − 37.6991113479743 x_{44} = -37.6991113479743 x 44 = − 37.6991113479743 x 45 = 69.115038794053 x_{45} = 69.115038794053 x 45 = 69.115038794053 x 46 = − 31.4159267157965 x_{46} = -31.4159267157965 x 46 = − 31.4159267157965 x 47 = 56.5486680806249 x_{47} = 56.5486680806249 x 47 = 56.5486680806249 x 48 = − 25.1327407505866 x_{48} = -25.1327407505866 x 48 = − 25.1327407505866 x 49 = − 81.6814090382277 x_{49} = -81.6814090382277 x 49 = − 81.6814090382277 x 50 = − 6.2831851275477 x_{50} = -6.2831851275477 x 50 = − 6.2831851275477 x 51 = − 100.530964626003 x_{51} = -100.530964626003 x 51 = − 100.530964626003 x 52 = − 62.8318534787248 x_{52} = -62.8318534787248 x 52 = − 62.8318534787248 x 53 = − 6.2831858160515 x_{53} = -6.2831858160515 x 53 = − 6.2831858160515 x 54 = − 62.831852673202 x_{54} = -62.831852673202 x 54 = − 62.831852673202 x 55 = − 6.28318555849548 x_{55} = -6.28318555849548 x 55 = − 6.28318555849548 x 56 = − 75.3982231720141 x_{56} = -75.3982231720141 x 56 = − 75.3982231720141 x 57 = − 94.2477801171671 x_{57} = -94.2477801171671 x 57 = − 94.2477801171671 x 58 = − 43.9822971745392 x_{58} = -43.9822971745392 x 58 = − 43.9822971745392 x 59 = − 56.5486682426592 x_{59} = -56.5486682426592 x 59 = − 56.5486682426592 x 60 = 87.964594335905 x_{60} = 87.964594335905 x 60 = 87.964594335905 x 61 = 6.28318626747926 x_{61} = 6.28318626747926 x 61 = 6.28318626747926 x 62 = − 94.2477797298079 x_{62} = -94.2477797298079 x 62 = − 94.2477797298079 x 63 = 50.2654821322586 x_{63} = 50.2654821322586 x 63 = 50.2654821322586 x 64 = − 69.1150379045123 x_{64} = -69.1150379045123 x 64 = − 69.1150379045123 x 65 = − 56.5486674685864 x_{65} = -56.5486674685864 x 65 = − 56.5486674685864 x 66 = − 69.1150386869085 x_{66} = -69.1150386869085 x 66 = − 69.1150386869085 x 67 = 94.2477800892631 x_{67} = 94.2477800892631 x 67 = 94.2477800892631 x 68 = − 75.3982231045728 x_{68} = -75.3982231045728 x 68 = − 75.3982231045728 x 69 = 81.6814084860076 x_{69} = 81.6814084860076 x 69 = 81.6814084860076 x 70 = − 18.8495563230046 x_{70} = -18.8495563230046 x 70 = − 18.8495563230046 x 71 = 31.4159268459961 x_{71} = 31.4159268459961 x 71 = 31.4159268459961 x 72 = 37.6991120311338 x_{72} = 37.6991120311338 x 72 = 37.6991120311338 x 73 = 12.5663710110881 x_{73} = 12.5663710110881 x 73 = 12.5663710110881 x 74 = − 43.9822967932182 x_{74} = -43.9822967932182 x 74 = − 43.9822967932182 x 75 = 56.5486676011951 x_{75} = 56.5486676011951 x 75 = 56.5486676011951 x 76 = − 81.6814092565354 x_{76} = -81.6814092565354 x 76 = − 81.6814092565354 x 77 = 6.28318500093652 x_{77} = 6.28318500093652 x 77 = 6.28318500093652 x 78 = − 50.2654822863493 x_{78} = -50.2654822863493 x 78 = − 50.2654822863493 x 79 = − 75.3982238741744 x_{79} = -75.3982238741744 x 79 = − 75.3982238741744 x 80 = 100.530965157364 x_{80} = 100.530965157364 x 80 = 100.530965157364 x 81 = 31.4159260648825 x_{81} = 31.4159260648825 x 81 = 31.4159260648825 x 82 = − 81.6814084945807 x_{82} = -81.6814084945807 x 82 = − 81.6814084945807 x 83 = 18.8495556275525 x_{83} = 18.8495556275525 x 83 = 18.8495556275525 x 84 = 12.5663711301703 x_{84} = 12.5663711301703 x 84 = 12.5663711301703 x 85 = 56.5486668532011 x_{85} = 56.5486668532011 x 85 = 56.5486668532011
Точки пересечения с осью координат Y
График пересекает ось Y, когда x равняется 0:2 − 2 cos ( 0 ) 2 - 2 \cos{\left(0 \right)} 2 − 2 cos ( 0 ) 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 ( x ) = 0 2 \sin{\left(x \right)} = 0 2 sin ( x ) = 0 Решаем это уравнение x 1 = 0 x_{1} = 0 x 1 = 0 x 2 = π x_{2} = \pi x 2 = π (0, 0) (pi, 4) Интервалы возрастания и убывания функции: x 1 = 0 x_{1} = 0 x 1 = 0 x 1 = π x_{1} = \pi x 1 = π [ 0 , π ] \left[0, \pi\right] [ 0 , π ] ( − ∞ , 0 ] ∪ [ π , ∞ ) \left(-\infty, 0\right] \cup \left[\pi, \infty\right) ( − ∞ , 0 ] ∪ [ π , ∞ )
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 2 cos ( x ) = 0 2 \cos{\left(x \right)} = 0 2 cos ( x ) = 0 Решаем это уравнение x 1 = π 2 x_{1} = \frac{\pi}{2} x 1 = 2 π x 2 = 3 π 2 x_{2} = \frac{3 \pi}{2} x 2 = 2 3 π Интервалы выпуклости и вогнутости: ( − ∞ , π 2 ] ∪ [ 3 π 2 , ∞ ) \left(-\infty, \frac{\pi}{2}\right] \cup \left[\frac{3 \pi}{2}, \infty\right) ( − ∞ , 2 π ] ∪ [ 2 3 π , ∞ ) [ π 2 , 3 π 2 ] \left[\frac{\pi}{2}, \frac{3 \pi}{2}\right] [ 2 π , 2 3 π ]
Горизонтальные асимптоты
Горизонтальные асимптоты найдём с помощью пределов данной функции при x->+oo и x->-oolim x → − ∞ ( 2 − 2 cos ( x ) ) = ⟨ 0 , 4 ⟩ \lim_{x \to -\infty}\left(2 - 2 \cos{\left(x \right)}\right) = \left\langle 0, 4\right\rangle x → − ∞ lim ( 2 − 2 cos ( x ) ) = ⟨ 0 , 4 ⟩ Возьмём предел y = ⟨ 0 , 4 ⟩ y = \left\langle 0, 4\right\rangle y = ⟨ 0 , 4 ⟩ lim x → ∞ ( 2 − 2 cos ( x ) ) = ⟨ 0 , 4 ⟩ \lim_{x \to \infty}\left(2 - 2 \cos{\left(x \right)}\right) = \left\langle 0, 4\right\rangle x → ∞ lim ( 2 − 2 cos ( x ) ) = ⟨ 0 , 4 ⟩ Возьмём предел y = ⟨ 0 , 4 ⟩ y = \left\langle 0, 4\right\rangle y = ⟨ 0 , 4 ⟩
Наклонные асимптоты
Наклонную асимптоту можно найти, подсчитав предел функции -2*cos(x) + 2, делённой на x при x->+oo и x ->-oolim x → − ∞ ( 2 − 2 cos ( x ) x ) = 0 \lim_{x \to -\infty}\left(\frac{2 - 2 \cos{\left(x \right)}}{x}\right) = 0 x → − ∞ lim ( x 2 − 2 cos ( x ) ) = 0 Возьмём предел lim x → ∞ ( 2 − 2 cos ( x ) x ) = 0 \lim_{x \to \infty}\left(\frac{2 - 2 \cos{\left(x \right)}}{x}\right) = 0 x → ∞ lim ( x 2 − 2 cos ( x ) ) = 0 Возьмём предел
Чётность и нечётность функции
Проверим функци чётна или нечётна с помощью соотношений f = f(-x) и f = -f(-x).2 − 2 cos ( x ) = 2 − 2 cos ( x ) 2 - 2 \cos{\left(x \right)} = 2 - 2 \cos{\left(x \right)} 2 − 2 cos ( x ) = 2 − 2 cos ( x ) 2 − 2 cos ( x ) = 2 cos ( x ) − 2 2 - 2 \cos{\left(x \right)} = 2 \cos{\left(x \right)} - 2 2 − 2 cos ( x ) = 2 cos ( x ) − 2 
