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 = 0cot 2 ( x ) = 0 \cot^{2}{\left (x \right )} = 0 cot 2 ( x ) = 0 Решаем это уравнение Аналитическое решение x 1 = π 2 x_{1} = \frac{\pi}{2} x 1 = 2 π Численное решение x 1 = − 29.8451300929 x_{1} = -29.8451300929 x 1 = − 29.8451300929 x 2 = 67.544242417 x_{2} = 67.544242417 x 2 = 67.544242417 x 3 = − 32.9867236108 x_{3} = -32.9867236108 x 3 = − 32.9867236108 x 4 = 39.2699089742 x_{4} = 39.2699089742 x 4 = 39.2699089742 x 5 = − 73.8274272793 x_{5} = -73.8274272793 x 5 = − 73.8274272793 x 6 = 76.9690193554 x_{6} = 76.9690193554 x 6 = 76.9690193554 x 7 = − 17.2787599203 x_{7} = -17.2787599203 x 7 = − 17.2787599203 x 8 = − 4.71238821767 x_{8} = -4.71238821767 x 8 = − 4.71238821767 x 9 = − 36.1283154116 x_{9} = -36.1283154116 x 9 = − 36.1283154116 x 10 = 17.2787603779 x_{10} = 17.2787603779 x 10 = 17.2787603779 x 11 = 7.85398175057 x_{11} = 7.85398175057 x 11 = 7.85398175057 x 12 = − 64.4026490866 x_{12} = -64.4026490866 x 12 = − 64.4026490866 x 13 = − 61.2610570877 x_{13} = -61.2610570877 x 13 = − 61.2610570877 x 14 = − 86.39379767 x_{14} = -86.39379767 x 14 = − 86.39379767 x 15 = 48.6946858295 x_{15} = 48.6946858295 x 15 = 48.6946858295 x 16 = − 14.1371668305 x_{16} = -14.1371668305 x 16 = − 14.1371668305 x 17 = − 70.6858340007 x_{17} = -70.6858340007 x 17 = − 70.6858340007 x 18 = 10.9955735773 x_{18} = 10.9955735773 x 18 = 10.9955735773 x 19 = − 98.9601693978 x_{19} = -98.9601693978 x 19 = − 98.9601693978 x 20 = 80.1106131141 x_{20} = 80.1106131141 x 20 = 80.1106131141 x 21 = 92.6769829953 x_{21} = 92.6769829953 x 21 = 92.6769829953 x 22 = 14.1371671242 x_{22} = 14.1371671242 x 22 = 14.1371671242 x 23 = 64.4026493028 x_{23} = 64.4026493028 x 23 = 64.4026493028 x 24 = − 23.5619450188 x_{24} = -23.5619450188 x 24 = − 23.5619450188 x 25 = − 67.5442421799 x_{25} = -67.5442421799 x 25 = − 67.5442421799 x 26 = 29.8451303316 x_{26} = 29.8451303316 x 26 = 29.8451303316 x 27 = − 39.2699085039 x_{27} = -39.2699085039 x 27 = − 39.2699085039 x 28 = 54.9778707632 x_{28} = 54.9778707632 x 28 = 54.9778707632 x 29 = 95.8185760749 x_{29} = 95.8185760749 x 29 = 95.8185760749 x 30 = 89.5353910016 x_{30} = 89.5353910016 x 30 = 89.5353910016 x 31 = − 51.8362786881 x_{31} = -51.8362786881 x 31 = − 51.8362786881 x 32 = − 20.4203519195 x_{32} = -20.4203519195 x 32 = − 20.4203519195 x 33 = 4.71238866313 x_{33} = 4.71238866313 x 33 = 4.71238866313 x 34 = 32.9867221706 x_{34} = 32.9867221706 x 34 = 32.9867221706 x 35 = 86.3937978832 x_{35} = 86.3937978832 x 35 = 86.3937978832 x 36 = 45.5530938326 x_{36} = 45.5530938326 x 36 = 45.5530938326 x 37 = 36.128315673 x_{37} = 36.128315673 x 37 = 36.128315673 x 38 = 58.1194643519 x_{38} = 58.1194643519 x 38 = 58.1194643519 x 39 = − 1.57079643839 x_{39} = -1.57079643839 x 39 = − 1.57079643839 x 40 = − 80.1106125739 x_{40} = -80.1106125739 x 40 = − 80.1106125739 x 41 = − 26.7035368126 x_{41} = -26.7035368126 x 41 = − 26.7035368126 x 42 = − 45.5530935993 x_{42} = -45.5530935993 x 42 = − 45.5530935993 x 43 = 70.6858344125 x_{43} = 70.6858344125 x 43 = 70.6858344125 x 44 = 83.2522061689 x_{44} = 83.2522061689 x 44 = 83.2522061689 x 45 = 23.5619452484 x_{45} = 23.5619452484 x 45 = 23.5619452484 x 46 = 73.8274274938 x_{46} = 73.8274274938 x 46 = 73.8274274938 x 47 = − 58.1194639928 x_{47} = -58.1194639928 x 47 = − 58.1194639928 x 48 = 61.2610575711 x_{48} = 61.2610575711 x 48 = 61.2610575711 x 49 = 51.8362789127 x_{49} = 51.8362789127 x 49 = 51.8362789127 x 50 = − 10.9955750165 x_{50} = -10.9955750165 x 50 = − 10.9955750165 x 51 = − 76.9690208015 x_{51} = -76.9690208015 x 51 = − 76.9690208015 x 52 = 98.9601679471 x_{52} = 98.9601679471 x 52 = 98.9601679471 x 53 = − 95.8185758678 x_{53} = -95.8185758678 x 53 = − 95.8185758678 x 54 = − 42.4115005031 x_{54} = -42.4115005031 x 54 = − 42.4115005031 x 55 = 1.57079666439 x_{55} = 1.57079666439 x 55 = 1.57079666439 x 56 = 26.7035372464 x_{56} = 26.7035372464 x 56 = 26.7035372464 x 57 = − 92.6769825939 x_{57} = -92.6769825939 x 57 = − 92.6769825939 x 58 = − 83.2522056717 x_{58} = -83.2522056717 x 58 = − 83.2522056717 x 59 = − 48.6946854069 x_{59} = -48.6946854069 x 59 = − 48.6946854069 x 60 = 42.4115007224 x_{60} = 42.4115007224 x 60 = 42.4115007224 x 61 = − 7.85398149105 x_{61} = -7.85398149105 x 61 = − 7.85398149105 x 62 = − 89.5353907604 x_{62} = -89.5353907604 x 62 = − 89.5353907604 x 63 = 20.420352142 x_{63} = 20.420352142 x 63 = 20.420352142 x 64 = − 54.9778722058 x_{64} = -54.9778722058 x 64 = − 54.9778722058
Точки пересечения с осью координат Y
График пересекает ось Y, когда x равняется 0:cot 2 ( 0 ) \cot^{2}{\left (0 \right )} cot 2 ( 0 ) f ( 0 ) = ∞ ~ f{\left (0 \right )} = \tilde{\infty} f ( 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 cot 2 ( x ) − 2 ) cot ( x ) = 0 \left(- 2 \cot^{2}{\left (x \right )} - 2\right) \cot{\left (x \right )} = 0 ( − 2 cot 2 ( x ) − 2 ) cot ( x ) = 0 Решаем это уравнение x 1 = π 2 x_{1} = \frac{\pi}{2} x 1 = 2 π pi
(--, 0)
2 Интервалы возрастания и убывания функции: x 1 = π 2 x_{1} = \frac{\pi}{2} x 1 = 2 π [pi/2, oo) (-oo, pi/2]
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 ( cot 2 ( x ) + 1 ) ( 3 cot 2 ( x ) + 1 ) = 0 2 \left(\cot^{2}{\left (x \right )} + 1\right) \left(3 \cot^{2}{\left (x \right )} + 1\right) = 0 2 ( cot 2 ( x ) + 1 ) ( 3 cot 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 → − ∞ cot 2 ( x ) y = \lim_{x \to -\infty} \cot^{2}{\left (x \right )} y = x → − ∞ lim cot 2 ( x ) True Возьмём предел y = lim x → ∞ cot 2 ( x ) y = \lim_{x \to \infty} \cot^{2}{\left (x \right )} y = x → ∞ lim cot 2 ( x )
Наклонные асимптоты
Наклонную асимптоту можно найти, подсчитав предел функции cot(x)^2, делённой на x при x->+oo и x ->-ooTrue Возьмём предел y = x lim x → − ∞ ( 1 x cot 2 ( x ) ) y = x \lim_{x \to -\infty}\left(\frac{1}{x} \cot^{2}{\left (x \right )}\right) y = x x → − ∞ lim ( x 1 cot 2 ( x ) ) True Возьмём предел y = x lim x → ∞ ( 1 x cot 2 ( x ) ) y = x \lim_{x \to \infty}\left(\frac{1}{x} \cot^{2}{\left (x \right )}\right) y = x x → ∞ lim ( x 1 cot 2 ( x ) )
Чётность и нечётность функции
Проверим функци чётна или нечётна с помощью соотношений f = f(-x) и f = -f(-x).cot 2 ( x ) = cot 2 ( x ) \cot^{2}{\left (x \right )} = \cot^{2}{\left (x \right )} cot 2 ( x ) = cot 2 ( x ) cot 2 ( x ) = − cot 2 ( x ) \cot^{2}{\left (x \right )} = - \cot^{2}{\left (x \right )} cot 2 ( x ) = − cot 2 ( x ) 