График функции
0 -2000 -1500 -1000 -500 500 1000 1500 2000 0 20000
Точки пересечения с осью координат X
График функции пересекает ось X при f = 0tan 2 ( x ) = 0 \tan^{2}{\left (x \right )} = 0 tan 2 ( x ) = 0 Решаем это уравнение Аналитическое решение x 1 = 0 x_{1} = 0 x 1 = 0 Численное решение x 1 = 53.4070756505 x_{1} = 53.4070756505 x 1 = 53.4070756505 x 2 = 72.2566310277 x_{2} = 72.2566310277 x 2 = 72.2566310277 x 3 = 47.1238887522 x_{3} = 47.1238887522 x 3 = 47.1238887522 x 4 = − 47.1238903089 x_{4} = -47.1238903089 x 4 = − 47.1238903089 x 5 = − 56.5486672889 x_{5} = -56.5486672889 x 5 = − 56.5486672889 x 6 = 100.530964739 x_{6} = 100.530964739 x 6 = 100.530964739 x 7 = 94.2477796094 x_{7} = 94.2477796094 x 7 = 94.2477796094 x 8 = 97.3893728286 x_{8} = 97.3893728286 x 8 = 97.3893728286 x 9 = 18.8495554527 x_{9} = 18.8495554527 x 9 = 18.8495554527 x 10 = − 97.3893724933 x_{10} = -97.3893724933 x 10 = − 97.3893724933 x 11 = 25.1327401464 x_{11} = 25.1327401464 x 11 = 25.1327401464 x 12 = − 100.530964462 x_{12} = -100.530964462 x 12 = − 100.530964462 x 13 = − 75.3982239115 x_{13} = -75.3982239115 x 13 = − 75.3982239115 x 14 = 56.5486675771 x_{14} = 56.5486675771 x 14 = 56.5486675771 x 15 = − 31.4159267483 x_{15} = -31.4159267483 x 15 = − 31.4159267483 x 16 = 50.2654824463 x_{16} = 50.2654824463 x 16 = 50.2654824463 x 17 = 59.6902606508 x_{17} = 59.6902606508 x 17 = 59.6902606508 x 18 = − 62.8318519641 x_{18} = -62.8318519641 x 18 = − 62.8318519641 x 19 = 12.5663704146 x_{19} = 12.5663704146 x 19 = 12.5663704146 x 20 = − 94.2477794214 x_{20} = -94.2477794214 x 20 = − 94.2477794214 x 21 = − 81.6814090389 x_{21} = -81.6814090389 x 21 = − 81.6814090389 x 22 = − 65.9734457647 x_{22} = -65.9734457647 x 22 = − 65.9734457647 x 23 = 40.8407040394 x_{23} = 40.8407040394 x 23 = 40.8407040394 x 24 = 84.8230012118 x_{24} = 84.8230012118 x 24 = 84.8230012118 x 25 = − 6.28318509494 x_{25} = -6.28318509494 x 25 = − 6.28318509494 x 26 = − 43.9822971744 x_{26} = -43.9822971744 x 26 = − 43.9822971744 x 27 = − 21.9911485864 x_{27} = -21.9911485864 x 27 = − 21.9911485864 x 28 = − 69.1150388968 x_{28} = -69.1150388968 x 28 = − 69.1150388968 x 29 = 37.6991120688 x_{29} = 37.6991120688 x 29 = 37.6991120688 x 30 = 81.6814092329 x_{30} = 81.6814092329 x 30 = 81.6814092329 x 31 = 65.9734457532 x_{31} = 65.9734457532 x 31 = 65.9734457532 x 32 = 6.28318528408 x_{32} = 6.28318528408 x 32 = 6.28318528408 x 33 = 34.5575189959 x_{33} = 34.5575189959 x 33 = 34.5575189959 x 34 = − 53.4070753298 x_{34} = -53.4070753298 x 34 = − 53.4070753298 x 35 = 91.1061859604 x_{35} = 91.1061859604 x 35 = 91.1061859604 x 36 = − 37.6991118776 x_{36} = -37.6991118776 x 36 = − 37.6991118776 x 37 = − 84.8230005709 x_{37} = -84.8230005709 x 37 = − 84.8230005709 x 38 = 62.8318526257 x_{38} = 62.8318526257 x 38 = 62.8318526257 x 39 = 87.9645943363 x_{39} = 87.9645943363 x 39 = 87.9645943363 x 40 = 59.6902602145 x_{40} = 59.6902602145 x 40 = 59.6902602145 x 41 = − 15.7079632968 x_{41} = -15.7079632968 x 41 = − 15.7079632968 x 42 = 9.42477847374 x_{42} = 9.42477847374 x 42 = 9.42477847374 x 43 = − 3.14159313419 x_{43} = -3.14159313419 x 43 = − 3.14159313419 x 44 = 75.3982242393 x_{44} = 75.3982242393 x 44 = 75.3982242393 x 45 = 21.9911485852 x_{45} = 21.9911485852 x 45 = 21.9911485852 x 46 = 15.7079634869 x_{46} = 15.7079634869 x 46 = 15.7079634869 x 47 = 78.5398161582 x_{47} = 78.5398161582 x 47 = 78.5398161582 x 48 = 28.2743338651 x_{48} = 28.2743338651 x 48 = 28.2743338651 x 49 = − 87.9645943582 x_{49} = -87.9645943582 x 49 = − 87.9645943582 x 50 = − 34.5575187016 x_{50} = -34.5575187016 x 50 = − 34.5575187016 x 51 = − 72.2566308399 x_{51} = -72.2566308399 x 51 = − 72.2566308399 x 52 = − 28.2743336767 x_{52} = -28.2743336767 x 52 = − 28.2743336767 x 53 = 31.4159270619 x_{53} = 31.4159270619 x 53 = 31.4159270619 x 54 = − 12.5663701141 x_{54} = -12.5663701141 x 54 = − 12.5663701141 x 55 = − 78.5398158758 x_{55} = -78.5398158758 x 55 = − 78.5398158758 x 56 = − 59.6902604583 x_{56} = -59.6902604583 x 56 = − 59.6902604583 x 57 = − 50.2654822583 x_{57} = -50.2654822583 x 57 = − 50.2654822583 x 58 = − 91.106187485 x_{58} = -91.106187485 x 58 = − 91.106187485 x 59 = 69.1150373568 x_{59} = 69.1150373568 x 59 = 69.1150373568 x 60 = − 9.4247781668 x_{60} = -9.4247781668 x 60 = − 9.4247781668 x 61 = 0 x_{61} = 0 x 61 = 0 x 62 = 3.14159153946 x_{62} = 3.14159153946 x 62 = 3.14159153946 x 63 = 43.9822971696 x_{63} = 43.9822971696 x 63 = 43.9822971696 x 64 = − 25.1327417214 x_{64} = -25.1327417214 x 64 = − 25.1327417214
Точки пересечения с осью координат Y
График пересекает ось Y, когда x равняется 0:tan 2 ( 0 ) \tan^{2}{\left (0 \right )} tan 2 ( 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 tan 2 ( x ) + 2 ) tan ( x ) = 0 \left(2 \tan^{2}{\left (x \right )} + 2\right) \tan{\left (x \right )} = 0 ( 2 tan 2 ( x ) + 2 ) tan ( x ) = 0 Решаем это уравнение x 1 = 0 x_{1} = 0 x 1 = 0 (0, 0) Интервалы возрастания и убывания функции: x 1 = 0 x_{1} = 0 x 1 = 0 [0, oo) (-oo, 0]
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 ) ( 3 tan 2 ( x ) + 1 ) = 0 2 \left(\tan^{2}{\left (x \right )} + 1\right) \left(3 \tan^{2}{\left (x \right )} + 1\right) = 0 2 ( tan 2 ( x ) + 1 ) ( 3 tan 2 ( x ) + 1 ) = 0 Решаем это уравнение
Горизонтальные асимптоты
Горизонтальные асимптоты найдём с помощью пределов данной функции при x->+oo и x->-ooTrue Возьмём предел y = lim x → − ∞ tan 2 ( x ) y = \lim_{x \to -\infty} \tan^{2}{\left (x \right )} y = x → − ∞ lim tan 2 ( x ) True Возьмём предел y = lim x → ∞ tan 2 ( x ) y = \lim_{x \to \infty} \tan^{2}{\left (x \right )} y = x → ∞ lim tan 2 ( x )
Наклонные асимптоты
Наклонную асимптоту можно найти, подсчитав предел функции tan(x)^2, делённой на x при x->+oo и x ->-ooTrue Возьмём предел y = x lim x → − ∞ ( 1 x tan 2 ( x ) ) y = x \lim_{x \to -\infty}\left(\frac{1}{x} \tan^{2}{\left (x \right )}\right) y = x x → − ∞ lim ( x 1 tan 2 ( x ) ) True Возьмём предел y = x lim x → ∞ ( 1 x tan 2 ( x ) ) y = x \lim_{x \to \infty}\left(\frac{1}{x} \tan^{2}{\left (x \right )}\right) y = x x → ∞ lim ( x 1 tan 2 ( x ) )
Чётность и нечётность функции
Проверим функци чётна или нечётна с помощью соотношений f = f(-x) и f = -f(-x).tan 2 ( x ) = tan 2 ( x ) \tan^{2}{\left (x \right )} = \tan^{2}{\left (x \right )} tan 2 ( x ) = tan 2 ( x ) tan 2 ( x ) = − tan 2 ( x ) \tan^{2}{\left (x \right )} = - \tan^{2}{\left (x \right )} tan 2 ( x ) = − tan 2 ( x ) 