График функции
0 -40 -30 -20 -10 10 20 30 40 50 0 4
Точки пересечения с осью координат X
График функции пересекает ось X при f = 01 − cos ( x ) = 0 1 - \cos{\left(x \right)} = 0 1 − 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 = − 12.5663703112531 x_{1} = -12.5663703112531 x 1 = − 12.5663703112531 x 2 = − 43.9822976246252 x_{2} = -43.9822976246252 x 2 = − 43.9822976246252 x 3 = 56.5486668532011 x_{3} = 56.5486668532011 x 3 = 56.5486668532011 x 4 = 56.5486682809363 x_{4} = 56.5486682809363 x 4 = 56.5486682809363 x 5 = 87.964594335905 x_{5} = 87.964594335905 x 5 = 87.964594335905 x 6 = 6.28318579821791 x_{6} = 6.28318579821791 x 6 = 6.28318579821791 x 7 = 12.5663704426592 x_{7} = 12.5663704426592 x 7 = 12.5663704426592 x 8 = 12.5663710110881 x_{8} = 12.5663710110881 x 8 = 12.5663710110881 x 9 = − 18.8495563230046 x_{9} = -18.8495563230046 x 9 = − 18.8495563230046 x 10 = 87.9645938121814 x_{10} = 87.9645938121814 x 10 = 87.9645938121814 x 11 = 50.2654821322586 x_{11} = 50.2654821322586 x 11 = 50.2654821322586 x 12 = 37.6991113348642 x_{12} = 37.6991113348642 x 12 = 37.6991113348642 x 13 = − 62.831852673202 x_{13} = -62.831852673202 x 13 = − 62.831852673202 x 14 = 62.8318527849002 x_{14} = 62.8318527849002 x 14 = 62.8318527849002 x 15 = − 87.964593928489 x_{15} = -87.964593928489 x 15 = − 87.964593928489 x 16 = − 43.9822971745392 x_{16} = -43.9822971745392 x 16 = − 43.9822971745392 x 17 = 6.28318626747926 x_{17} = 6.28318626747926 x 17 = 6.28318626747926 x 18 = − 31.4159267157965 x_{18} = -31.4159267157965 x 18 = − 31.4159267157965 x 19 = − 25.1327415297174 x_{19} = -25.1327415297174 x 19 = − 25.1327415297174 x 20 = 37.6991120311338 x_{20} = 37.6991120311338 x 20 = 37.6991120311338 x 21 = − 50.265482641087 x_{21} = -50.265482641087 x 21 = − 50.265482641087 x 22 = − 75.3982238741744 x_{22} = -75.3982238741744 x 22 = − 75.3982238741744 x 23 = − 6.28318555849548 x_{23} = -6.28318555849548 x 23 = − 6.28318555849548 x 24 = 100.530964759815 x_{24} = 100.530964759815 x 24 = 100.530964759815 x 25 = 94.2477800892631 x_{25} = 94.2477800892631 x 25 = 94.2477800892631 x 26 = 81.6814084860076 x_{26} = 81.6814084860076 x 26 = 81.6814084860076 x 27 = − 31.4159260208155 x_{27} = -31.4159260208155 x 27 = − 31.4159260208155 x 28 = 81.6814085860518 x_{28} = 81.6814085860518 x 28 = 81.6814085860518 x 29 = − 43.9822967932182 x_{29} = -43.9822967932182 x 29 = − 43.9822967932182 x 30 = 50.2654829439723 x_{30} = 50.2654829439723 x 30 = 50.2654829439723 x 31 = 100.530965157364 x_{31} = 100.530965157364 x 31 = 100.530965157364 x 32 = − 6.2831858160515 x_{32} = -6.2831858160515 x 32 = − 6.2831858160515 x 33 = − 81.6814075578313 x_{33} = -81.6814075578313 x 33 = − 81.6814075578313 x 34 = − 56.5486674685864 x_{34} = -56.5486674685864 x 34 = − 56.5486674685864 x 35 = − 94.2477794452815 x_{35} = -94.2477794452815 x 35 = − 94.2477794452815 x 36 = 94.2477796093523 x_{36} = 94.2477796093523 x 36 = 94.2477796093523 x 37 = − 69.1150379045123 x_{37} = -69.1150379045123 x 37 = − 69.1150379045123 x 38 = 37.6991115173992 x_{38} = 37.6991115173992 x 38 = 37.6991115173992 x 39 = 75.3982232188727 x_{39} = 75.3982232188727 x 39 = 75.3982232188727 x 40 = 50.2654824463392 x_{40} = 50.2654824463392 x 40 = 50.2654824463392 x 41 = 6.28318500093652 x_{41} = 6.28318500093652 x 41 = 6.28318500093652 x 42 = − 75.3982231720141 x_{42} = -75.3982231720141 x 42 = − 75.3982231720141 x 43 = − 94.2477801171671 x_{43} = -94.2477801171671 x 43 = − 94.2477801171671 x 44 = − 37.6991121287155 x_{44} = -37.6991121287155 x 44 = − 37.6991121287155 x 45 = 12.5663711301703 x_{45} = 12.5663711301703 x 45 = 12.5663711301703 x 46 = 25.1327408328211 x_{46} = 25.1327408328211 x 46 = 25.1327408328211 x 47 = − 94.2477797298079 x_{47} = -94.2477797298079 x 47 = − 94.2477797298079 x 48 = − 50.2654822863493 x_{48} = -50.2654822863493 x 48 = − 50.2654822863493 x 49 = − 62.8318534787248 x_{49} = -62.8318534787248 x 49 = − 62.8318534787248 x 50 = − 18.8495552124105 x_{50} = -18.8495552124105 x 50 = − 18.8495552124105 x 51 = 0 x_{51} = 0 x 51 = 0 x 52 = 18.8495564031971 x_{52} = 18.8495564031971 x 52 = 18.8495564031971 x 53 = − 18.8495555173448 x_{53} = -18.8495555173448 x 53 = − 18.8495555173448 x 54 = − 100.530964626003 x_{54} = -100.530964626003 x 54 = − 100.530964626003 x 55 = 43.9822966661001 x_{55} = 43.9822966661001 x 55 = 43.9822966661001 x 56 = − 56.5486682426592 x_{56} = -56.5486682426592 x 56 = − 56.5486682426592 x 57 = 62.8318535568358 x_{57} = 62.8318535568358 x 57 = 62.8318535568358 x 58 = 87.9645946044253 x_{58} = 87.9645946044253 x 58 = 87.9645946044253 x 59 = − 31.4159260507536 x_{59} = -31.4159260507536 x 59 = − 31.4159260507536 x 60 = 56.5486680806249 x_{60} = 56.5486680806249 x 60 = 56.5486680806249 x 61 = − 81.6814092565354 x_{61} = -81.6814092565354 x 61 = − 81.6814092565354 x 62 = − 81.6814084945807 x_{62} = -81.6814084945807 x 62 = − 81.6814084945807 x 63 = 43.9822974733639 x_{63} = 43.9822974733639 x 63 = 43.9822974733639 x 64 = − 87.9645947692094 x_{64} = -87.9645947692094 x 64 = − 87.9645947692094 x 65 = − 37.6991113479743 x_{65} = -37.6991113479743 x 65 = − 37.6991113479743 x 66 = 31.4159260648825 x_{66} = 31.4159260648825 x 66 = 31.4159260648825 x 67 = − 50.2654829667315 x_{67} = -50.2654829667315 x 67 = − 50.2654829667315 x 68 = 43.9822971694647 x_{68} = 43.9822971694647 x 68 = 43.9822971694647 x 69 = − 25.1327407505866 x_{69} = -25.1327407505866 x 69 = − 25.1327407505866 x 70 = − 75.3982231045728 x_{70} = -75.3982231045728 x 70 = − 75.3982231045728 x 71 = − 6.2831851275477 x_{71} = -6.2831851275477 x 71 = − 6.2831851275477 x 72 = − 69.1150386869085 x_{72} = -69.1150386869085 x 72 = − 69.1150386869085 x 73 = 69.115038794053 x_{73} = 69.115038794053 x 73 = 69.115038794053 x 74 = 81.6814091897036 x_{74} = 81.6814091897036 x 74 = 81.6814091897036 x 75 = 6.28318528420851 x_{75} = 6.28318528420851 x 75 = 6.28318528420851 x 76 = − 87.9645943586158 x_{76} = -87.9645943586158 x 76 = − 87.9645943586158 x 77 = 69.1150379887504 x_{77} = 69.1150379887504 x 77 = 69.1150379887504 x 78 = − 12.5663710889626 x_{78} = -12.5663710889626 x 78 = − 12.5663710889626 x 79 = − 81.6814090382277 x_{79} = -81.6814090382277 x 79 = − 81.6814090382277 x 80 = 31.4159268459961 x_{80} = 31.4159268459961 x 80 = 31.4159268459961 x 81 = − 37.6991118772631 x_{81} = -37.6991118772631 x 81 = − 37.6991118772631 x 82 = 25.1327416384075 x_{82} = 25.1327416384075 x 82 = 25.1327416384075 x 83 = 18.8495556275525 x_{83} = 18.8495556275525 x 83 = 18.8495556275525 x 84 = 75.3982240031607 x_{84} = 75.3982240031607 x 84 = 75.3982240031607 x 85 = 56.5486676011951 x_{85} = 56.5486676011951 x 85 = 56.5486676011951 x 86 = 94.2477792651059 x_{86} = 94.2477792651059 x 86 = 94.2477792651059
Точки пересечения с осью координат Y
График пересекает ось Y, когда x равняется 0:1 − cos ( 0 ) 1 - \cos{\left(0 \right)} 1 − 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 sin ( x ) = 0 \sin{\left(x \right)} = 0 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, 2) 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 cos ( x ) = 0 \cos{\left(x \right)} = 0 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 → − ∞ ( 1 − cos ( x ) ) = ⟨ 0 , 2 ⟩ \lim_{x \to -\infty}\left(1 - \cos{\left(x \right)}\right) = \left\langle 0, 2\right\rangle x → − ∞ lim ( 1 − cos ( x ) ) = ⟨ 0 , 2 ⟩ Возьмём предел y = ⟨ 0 , 2 ⟩ y = \left\langle 0, 2\right\rangle y = ⟨ 0 , 2 ⟩ lim x → ∞ ( 1 − cos ( x ) ) = ⟨ 0 , 2 ⟩ \lim_{x \to \infty}\left(1 - \cos{\left(x \right)}\right) = \left\langle 0, 2\right\rangle x → ∞ lim ( 1 − cos ( x ) ) = ⟨ 0 , 2 ⟩ Возьмём предел y = ⟨ 0 , 2 ⟩ y = \left\langle 0, 2\right\rangle y = ⟨ 0 , 2 ⟩
Наклонные асимптоты
Наклонную асимптоту можно найти, подсчитав предел функции 1 - cos(x), делённой на x при x->+oo и x ->-oolim x → − ∞ ( 1 − cos ( x ) x ) = 0 \lim_{x \to -\infty}\left(\frac{1 - \cos{\left(x \right)}}{x}\right) = 0 x → − ∞ lim ( x 1 − cos ( x ) ) = 0 Возьмём предел lim x → ∞ ( 1 − cos ( x ) x ) = 0 \lim_{x \to \infty}\left(\frac{1 - \cos{\left(x \right)}}{x}\right) = 0 x → ∞ lim ( x 1 − cos ( x ) ) = 0 Возьмём предел
Чётность и нечётность функции
Проверим функци чётна или нечётна с помощью соотношений f = f(-x) и f = -f(-x).1 − cos ( x ) = 1 − cos ( x ) 1 - \cos{\left(x \right)} = 1 - \cos{\left(x \right)} 1 − cos ( x ) = 1 − cos ( x ) 1 − cos ( x ) = cos ( x ) − 1 1 - \cos{\left(x \right)} = \cos{\left(x \right)} - 1 1 − cos ( x ) = cos ( x ) − 1 
