График функции
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Точки пересечения с осью координат X
График функции пересекает ось X при f = 0sin ( x ) + tan ( x ) = 0 \sin{\left (x \right )} + \tan{\left (x \right )} = 0 sin ( x ) + tan ( x ) = 0 Решаем это уравнение Аналитическое решение x 1 = 0 x_{1} = 0 x 1 = 0 x 2 = − π x_{2} = - \pi x 2 = − π x 3 = π x_{3} = \pi x 3 = π x 4 = 2 π x_{4} = 2 \pi x 4 = 2 π Численное решение x 1 = − 94.2477796077 x_{1} = -94.2477796077 x 1 = − 94.2477796077 x 2 = − 9.42485522091 x_{2} = -9.42485522091 x 2 = − 9.42485522091 x 3 = − 15.7079741661 x_{3} = -15.7079741661 x 3 = − 15.7079741661 x 4 = − 28.2742578152 x_{4} = -28.2742578152 x 4 = − 28.2742578152 x 5 = 81.6814089933 x_{5} = 81.6814089933 x 5 = 81.6814089933 x 6 = − 21.9911516408 x_{6} = -21.9911516408 x 6 = − 21.9911516408 x 7 = 15.7080436046 x_{7} = 15.7080436046 x 7 = 15.7080436046 x 8 = 59.6903450187 x_{8} = 59.6903450187 x 8 = 59.6903450187 x 9 = − 12.5663706144 x_{9} = -12.5663706144 x 9 = − 12.5663706144 x 10 = 78.5397471518 x_{10} = 78.5397471518 x 10 = 78.5397471518 x 11 = 72.2566292957 x_{11} = 72.2566292957 x 11 = 72.2566292957 x 12 = 100.530964915 x_{12} = 100.530964915 x 12 = 100.530964915 x 13 = 50.2654824574 x_{13} = 50.2654824574 x 13 = 50.2654824574 x 14 = 37.6991118431 x_{14} = 37.6991118431 x 14 = 37.6991118431 x 15 = 28.2743275333 x_{15} = 28.2743275333 x 15 = 28.2743275333 x 16 = − 25.1327412287 x_{16} = -25.1327412287 x 16 = − 25.1327412287 x 17 = − 43.9822971503 x_{17} = -43.9822971503 x 17 = − 43.9822971503 x 18 = 25.1327412287 x_{18} = 25.1327412287 x 18 = 25.1327412287 x 19 = 21.9911516422 x_{19} = 21.9911516422 x 19 = 21.9911516422 x 20 = − 81.6814089933 x_{20} = -81.6814089933 x 20 = − 81.6814089933 x 21 = − 59.6902757895 x_{21} = -59.6902757895 x 21 = − 59.6902757895 x 22 = 87.9645943005 x_{22} = 87.9645943005 x 22 = 87.9645943005 x 23 = 69.115038379 x_{23} = 69.115038379 x 23 = 69.115038379 x 24 = − 31.4159265359 x_{24} = -31.4159265359 x 24 = − 31.4159265359 x 25 = − 72.2565592477 x_{25} = -72.2565592477 x 25 = − 72.2565592477 x 26 = − 53.4071564235 x_{26} = -53.4071564235 x 26 = − 53.4071564235 x 27 = 56.5486677646 x_{27} = 56.5486677646 x 27 = 56.5486677646 x 28 = − 75.3982236862 x_{28} = -75.3982236862 x 28 = − 75.3982236862 x 29 = − 69.115038379 x_{29} = -69.115038379 x 29 = − 69.115038379 x 30 = − 6.28318530718 x_{30} = -6.28318530718 x 30 = − 6.28318530718 x 31 = 6.28318530718 x_{31} = 6.28318530718 x 31 = 6.28318530718 x 32 = 75.3982236862 x_{32} = 75.3982236862 x 32 = 75.3982236862 x 33 = 62.8318530718 x_{33} = 62.8318530718 x 33 = 62.8318530718 x 34 = − 87.9645943005 x_{34} = -87.9645943005 x 34 = − 87.9645943005 x 35 = 84.8228649671 x_{35} = 84.8228649671 x 35 = 84.8228649671 x 36 = 18.8495559215 x_{36} = 18.8495559215 x 36 = 18.8495559215 x 37 = − 97.3894576418 x_{37} = -97.3894576418 x 37 = − 97.3894576418 x 38 = − 50.2654824574 x_{38} = -50.2654824574 x 38 = − 50.2654824574 x 39 = − 56.5486677646 x_{39} = -56.5486677646 x 39 = − 56.5486677646 x 40 = 65.9734548227 x_{40} = 65.9734548227 x 40 = 65.9734548227 x 41 = 12.5663706144 x_{41} = 12.5663706144 x 41 = 12.5663706144 x 42 = − 62.8318530718 x_{42} = -62.8318530718 x 42 = − 62.8318530718 x 43 = − 65.9734547149 x_{43} = -65.9734547149 x 43 = − 65.9734547149 x 44 = − 18.8495559215 x_{44} = -18.8495559215 x 44 = − 18.8495559215 x 45 = − 37.6991118431 x_{45} = -37.6991118431 x 45 = − 37.6991118431 x 46 = 34.5574459697 x_{46} = 34.5574459697 x 46 = 34.5574459697 x 47 = 31.4159265359 x_{47} = 31.4159265359 x 47 = 31.4159265359 x 48 = − 100.530964915 x_{48} = -100.530964915 x 48 = − 100.530964915 x 49 = 0 x_{49} = 0 x 49 = 0 x 50 = 43.9822971503 x_{50} = 43.9822971503 x 50 = 43.9822971503 x 51 = 94.2477796077 x_{51} = 94.2477796077 x 51 = 94.2477796077
Точки пересечения с осью координат Y
График пересекает ось Y, когда x равняется 0:sin ( 0 ) + tan ( 0 ) \sin{\left (0 \right )} + \tan{\left (0 \right )} sin ( 0 ) + tan ( 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 cos ( x ) + tan 2 ( x ) + 1 = 0 \cos{\left (x \right )} + \tan^{2}{\left (x \right )} + 1 = 0 cos ( x ) + tan 2 ( x ) + 1 = 0 Решаем это уравнение x 1 = π x_{1} = \pi x 1 = π (pi, 0) Интервалы возрастания и убывания функции:
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 ( tan 2 ( x ) + 1 ) tan ( x ) − sin ( x ) = 0 2 \left(\tan^{2}{\left (x \right )} + 1\right) \tan{\left (x \right )} - \sin{\left (x \right )} = 0 2 ( tan 2 ( x ) + 1 ) tan ( x ) − sin ( x ) = 0 Решаем это уравнение x 1 = − 94.2477796077 x_{1} = -94.2477796077 x 1 = − 94.2477796077 x 2 = 31.4159265359 x_{2} = 31.4159265359 x 2 = 31.4159265359 x 3 = 81.6814089933 x_{3} = 81.6814089933 x 3 = 81.6814089933 x 4 = 84.8230016469 x_{4} = 84.8230016469 x 4 = 84.8230016469 x 5 = − 53.407075111 x_{5} = -53.407075111 x 5 = − 53.407075111 x 6 = 65.9734457254 x_{6} = 65.9734457254 x 6 = 65.9734457254 x 7 = 3.14159265359 x_{7} = 3.14159265359 x 7 = 3.14159265359 x 8 = 15.7079632679 x_{8} = 15.7079632679 x 8 = 15.7079632679 x 9 = 100.530964915 x_{9} = 100.530964915 x 9 = 100.530964915 x 10 = 50.2654824574 x_{10} = 50.2654824574 x 10 = 50.2654824574 x 11 = − 3.14159265359 x_{11} = -3.14159265359 x 11 = − 3.14159265359 x 12 = 40.8407044967 x_{12} = 40.8407044967 x 12 = 40.8407044967 x 13 = − 59.6902604182 x_{13} = -59.6902604182 x 13 = − 59.6902604182 x 14 = 97.3893722613 x_{14} = 97.3893722613 x 14 = 97.3893722613 x 15 = 78.5398163397 x_{15} = 78.5398163397 x 15 = 78.5398163397 x 16 = − 25.1327412287 x_{16} = -25.1327412287 x 16 = − 25.1327412287 x 17 = − 43.9822971503 x_{17} = -43.9822971503 x 17 = − 43.9822971503 x 18 = 25.1327412287 x_{18} = 25.1327412287 x 18 = 25.1327412287 x 19 = − 81.6814089933 x_{19} = -81.6814089933 x 19 = − 81.6814089933 x 20 = − 91.1061869541 x_{20} = -91.1061869541 x 20 = − 91.1061869541 x 21 = 87.9645943005 x_{21} = 87.9645943005 x 21 = 87.9645943005 x 22 = 69.115038379 x_{22} = 69.115038379 x 22 = 69.115038379 x 23 = − 34.5575191895 x_{23} = -34.5575191895 x 23 = − 34.5575191895 x 24 = 28.2743338823 x_{24} = 28.2743338823 x 24 = 28.2743338823 x 25 = − 31.4159265359 x_{25} = -31.4159265359 x 25 = − 31.4159265359 x 26 = − 100.530964915 x_{26} = -100.530964915 x 26 = − 100.530964915 x 27 = − 28.2743338823 x_{27} = -28.2743338823 x 27 = − 28.2743338823 x 28 = 72.2566310326 x_{28} = 72.2566310326 x 28 = 72.2566310326 x 29 = 56.5486677646 x_{29} = 56.5486677646 x 29 = 56.5486677646 x 30 = − 75.3982236862 x_{30} = -75.3982236862 x 30 = − 75.3982236862 x 31 = − 69.115038379 x_{31} = -69.115038379 x 31 = − 69.115038379 x 32 = − 6.28318530718 x_{32} = -6.28318530718 x 32 = − 6.28318530718 x 33 = − 9.42477796077 x_{33} = -9.42477796077 x 33 = − 9.42477796077 x 34 = 6.28318530718 x_{34} = 6.28318530718 x 34 = 6.28318530718 x 35 = 75.3982236862 x_{35} = 75.3982236862 x 35 = 75.3982236862 x 36 = − 65.9734457254 x_{36} = -65.9734457254 x 36 = − 65.9734457254 x 37 = − 87.9645943005 x_{37} = -87.9645943005 x 37 = − 87.9645943005 x 38 = − 72.2566310326 x_{38} = -72.2566310326 x 38 = − 72.2566310326 x 39 = 18.8495559215 x_{39} = 18.8495559215 x 39 = 18.8495559215 x 40 = − 84.8230016469 x_{40} = -84.8230016469 x 40 = − 84.8230016469 x 41 = 9.42477796077 x_{41} = 9.42477796077 x 41 = 9.42477796077 x 42 = − 50.2654824574 x_{42} = -50.2654824574 x 42 = − 50.2654824574 x 43 = − 56.5486677646 x_{43} = -56.5486677646 x 43 = − 56.5486677646 x 44 = 91.1061869541 x_{44} = 91.1061869541 x 44 = 91.1061869541 x 45 = 59.6902604182 x_{45} = 59.6902604182 x 45 = 59.6902604182 x 46 = − 47.1238898038 x_{46} = -47.1238898038 x 46 = − 47.1238898038 x 47 = 12.5663706144 x_{47} = 12.5663706144 x 47 = 12.5663706144 x 48 = − 62.8318530718 x_{48} = -62.8318530718 x 48 = − 62.8318530718 x 49 = 62.8318530718 x_{49} = 62.8318530718 x 49 = 62.8318530718 x 50 = − 18.8495559215 x_{50} = -18.8495559215 x 50 = − 18.8495559215 x 51 = − 12.5663706144 x_{51} = -12.5663706144 x 51 = − 12.5663706144 x 52 = − 37.6991118431 x_{52} = -37.6991118431 x 52 = − 37.6991118431 x 53 = − 97.3893722613 x_{53} = -97.3893722613 x 53 = − 97.3893722613 x 54 = 94.2477796077 x_{54} = 94.2477796077 x 54 = 94.2477796077 x 55 = 34.5575191895 x_{55} = 34.5575191895 x 55 = 34.5575191895 x 56 = − 21.9911485751 x_{56} = -21.9911485751 x 56 = − 21.9911485751 x 57 = 21.9911485751 x_{57} = 21.9911485751 x 57 = 21.9911485751 x 58 = 37.6991118431 x_{58} = 37.6991118431 x 58 = 37.6991118431 x 59 = 53.407075111 x_{59} = 53.407075111 x 59 = 53.407075111 x 60 = − 78.5398163397 x_{60} = -78.5398163397 x 60 = − 78.5398163397 x 61 = 0 x_{61} = 0 x 61 = 0 x 62 = 43.9822971503 x_{62} = 43.9822971503 x 62 = 43.9822971503 x 63 = − 40.8407044967 x_{63} = -40.8407044967 x 63 = − 40.8407044967 x 64 = − 15.7079632679 x_{64} = -15.7079632679 x 64 = − 15.7079632679 x 65 = 47.1238898038 x_{65} = 47.1238898038 x 65 = 47.1238898038 Интервалы выпуклости и вогнутости: [100.530964915, oo) (-oo, -100.530964915]
Горизонтальные асимптоты
Горизонтальные асимптоты найдём с помощью пределов данной функции при x->+oo и x->-ooTrue Возьмём предел y = lim x → − ∞ ( sin ( x ) + tan ( x ) ) y = \lim_{x \to -\infty}\left(\sin{\left (x \right )} + \tan{\left (x \right )}\right) y = x → − ∞ lim ( sin ( x ) + tan ( x ) ) True Возьмём предел y = lim x → ∞ ( sin ( x ) + tan ( x ) ) y = \lim_{x \to \infty}\left(\sin{\left (x \right )} + \tan{\left (x \right )}\right) y = x → ∞ lim ( sin ( x ) + tan ( x ) )
Наклонные асимптоты
Наклонную асимптоту можно найти, подсчитав предел функции sin(x) + tan(x), делённой на x при x->+oo и x ->-ooTrue Возьмём предел y = x lim x → − ∞ ( 1 x ( sin ( x ) + tan ( x ) ) ) y = x \lim_{x \to -\infty}\left(\frac{1}{x} \left(\sin{\left (x \right )} + \tan{\left (x \right )}\right)\right) y = x x → − ∞ lim ( x 1 ( sin ( x ) + tan ( x ) ) ) True Возьмём предел y = x lim x → ∞ ( 1 x ( sin ( x ) + tan ( x ) ) ) y = x \lim_{x \to \infty}\left(\frac{1}{x} \left(\sin{\left (x \right )} + \tan{\left (x \right )}\right)\right) y = x x → ∞ lim ( x 1 ( sin ( x ) + tan ( x ) ) )
Чётность и нечётность функции
Проверим функци чётна или нечётна с помощью соотношений f = f(-x) и f = -f(-x).sin ( x ) + tan ( x ) = − sin ( x ) − tan ( x ) \sin{\left (x \right )} + \tan{\left (x \right )} = - \sin{\left (x \right )} - \tan{\left (x \right )} sin ( x ) + tan ( x ) = − sin ( x ) − tan ( x ) sin ( x ) + tan ( x ) = − − 1 sin ( x ) − − tan ( x ) \sin{\left (x \right )} + \tan{\left (x \right )} = - -1 \sin{\left (x \right )} - - \tan{\left (x \right )} sin ( x ) + tan ( x ) = − − 1 sin ( x ) − − tan ( x ) 