График функции
-2.0 -1.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 -10000 10000
Точки пересечения с осью координат X
График функции пересекает ось X при f = 0x 3 cot ( x ) = 0 x^{3} \cot{\left (x \right )} = 0 x 3 cot ( x ) = 0 Решаем это уравнение Аналитическое решение x 1 = 0 x_{1} = 0 x 1 = 0 x 2 = π 2 x_{2} = \frac{\pi}{2} x 2 = 2 π Численное решение x 1 = − 54.9778714378 x_{1} = -54.9778714378 x 1 = − 54.9778714378 x 2 = 39.2699081699 x_{2} = 39.2699081699 x 2 = 39.2699081699 x 3 = 51.8362787842 x_{3} = 51.8362787842 x 3 = 51.8362787842 x 4 = 86.3937979737 x_{4} = 86.3937979737 x 4 = 86.3937979737 x 5 = − 17.2787595947 x_{5} = -17.2787595947 x 5 = − 17.2787595947 x 6 = 45.5530934771 x_{6} = 45.5530934771 x 6 = 45.5530934771 x 7 = 61.261056745 x_{7} = 61.261056745 x 7 = 61.261056745 x 8 = 83.2522053201 x_{8} = 83.2522053201 x 8 = 83.2522053201 x 9 = − 70.6858347058 x_{9} = -70.6858347058 x 9 = − 70.6858347058 x 10 = − 89.5353906273 x_{10} = -89.5353906273 x 10 = − 89.5353906273 x 11 = 92.6769832809 x_{11} = 92.6769832809 x 11 = 92.6769832809 x 12 = 76.9690200129 x_{12} = 76.9690200129 x 12 = 76.9690200129 x 13 = − 32.9867228627 x_{13} = -32.9867228627 x 13 = − 32.9867228627 x 14 = 17.2787595947 x_{14} = 17.2787595947 x 14 = 17.2787595947 x 15 = − 48.6946861306 x_{15} = -48.6946861306 x 15 = − 48.6946861306 x 16 = − 80.1106126665 x_{16} = -80.1106126665 x 16 = − 80.1106126665 x 17 = − 42.4115008235 x_{17} = -42.4115008235 x 17 = − 42.4115008235 x 18 = − 58.1194640914 x_{18} = -58.1194640914 x 18 = − 58.1194640914 x 19 = 1.57079632679 x_{19} = 1.57079632679 x 19 = 1.57079632679 x 20 = − 95.8185759345 x_{20} = -95.8185759345 x 20 = − 95.8185759345 x 21 = 95.8185759345 x_{21} = 95.8185759345 x 21 = 95.8185759345 x 22 = − 36.1283155163 x_{22} = -36.1283155163 x 22 = − 36.1283155163 x 23 = − 64.4026493986 x_{23} = -64.4026493986 x 23 = − 64.4026493986 x 24 = 36.1283155163 x_{24} = 36.1283155163 x 24 = 36.1283155163 x 25 = − 61.261056745 x_{25} = -61.261056745 x 25 = − 61.261056745 x 26 = − 92.6769832809 x_{26} = -92.6769832809 x 26 = − 92.6769832809 x 27 = 32.9867228627 x_{27} = 32.9867228627 x 27 = 32.9867228627 x 28 = − 14.1371669412 x_{28} = -14.1371669412 x 28 = − 14.1371669412 x 29 = 80.1106126665 x_{29} = 80.1106126665 x 29 = 80.1106126665 x 30 = 4.71238898038 x_{30} = 4.71238898038 x 30 = 4.71238898038 x 31 = 10.9955742876 x_{31} = 10.9955742876 x 31 = 10.9955742876 x 32 = 7.85398163397 x_{32} = 7.85398163397 x 32 = 7.85398163397 x 33 = 23.5619449019 x_{33} = 23.5619449019 x 33 = 23.5619449019 x 34 = − 39.2699081699 x_{34} = -39.2699081699 x 34 = − 39.2699081699 x 35 = 64.4026493986 x_{35} = 64.4026493986 x 35 = 64.4026493986 x 36 = − 73.8274273594 x_{36} = -73.8274273594 x 36 = − 73.8274273594 x 37 = 20.4203522483 x_{37} = 20.4203522483 x 37 = 20.4203522483 x 38 = − 26.7035375555 x_{38} = -26.7035375555 x 38 = − 26.7035375555 x 39 = − 83.2522053201 x_{39} = -83.2522053201 x 39 = − 83.2522053201 x 40 = − 98.9601685881 x_{40} = -98.9601685881 x 40 = − 98.9601685881 x 41 = 48.6946861306 x_{41} = 48.6946861306 x 41 = 48.6946861306 x 42 = 14.1371669412 x_{42} = 14.1371669412 x 42 = 14.1371669412 x 43 = 98.9601685881 x_{43} = 98.9601685881 x 43 = 98.9601685881 x 44 = − 45.5530934771 x_{44} = -45.5530934771 x 44 = − 45.5530934771 x 45 = − 51.8362787842 x_{45} = -51.8362787842 x 45 = − 51.8362787842 x 46 = − 67.5442420522 x_{46} = -67.5442420522 x 46 = − 67.5442420522 x 47 = 54.9778714378 x_{47} = 54.9778714378 x 47 = 54.9778714378 x 48 = 26.7035375555 x_{48} = 26.7035375555 x 48 = 26.7035375555 x 49 = − 86.3937979737 x_{49} = -86.3937979737 x 49 = − 86.3937979737 x 50 = − 20.4203522483 x_{50} = -20.4203522483 x 50 = − 20.4203522483 x 51 = − 7.85398163397 x_{51} = -7.85398163397 x 51 = − 7.85398163397 x 52 = − 4.71238898038 x_{52} = -4.71238898038 x 52 = − 4.71238898038 x 53 = − 76.9690200129 x_{53} = -76.9690200129 x 53 = − 76.9690200129 x 54 = 89.5353906273 x_{54} = 89.5353906273 x 54 = 89.5353906273 x 55 = − 10.9955742876 x_{55} = -10.9955742876 x 55 = − 10.9955742876 x 56 = − 1.57079632679 x_{56} = -1.57079632679 x 56 = − 1.57079632679 x 57 = − 23.5619449019 x_{57} = -23.5619449019 x 57 = − 23.5619449019 x 58 = 73.8274273594 x_{58} = 73.8274273594 x 58 = 73.8274273594 x 59 = 70.6858347058 x_{59} = 70.6858347058 x 59 = 70.6858347058 x 60 = 29.8451302091 x_{60} = 29.8451302091 x 60 = 29.8451302091 x 61 = 42.4115008235 x_{61} = 42.4115008235 x 61 = 42.4115008235 x 62 = 67.5442420522 x_{62} = 67.5442420522 x 62 = 67.5442420522 x 63 = 58.1194640914 x_{63} = 58.1194640914 x 63 = 58.1194640914 x 64 = − 29.8451302091 x_{64} = -29.8451302091 x 64 = − 29.8451302091
Точки пересечения с осью координат Y
График пересекает ось Y, когда x равняется 0:0 3 cot ( 0 ) 0^{3} \cot{\left (0 \right )} 0 3 cot ( 0 ) f ( 0 ) = N a N f{\left (0 \right )} = \mathrm{NaN} f ( 0 ) = NaN
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 x 3 ( − cot 2 ( x ) − 1 ) + 3 x 2 cot ( x ) = 0 x^{3} \left(- \cot^{2}{\left (x \right )} - 1\right) + 3 x^{2} \cot{\left (x \right )} = 0 x 3 ( − cot 2 ( x ) − 1 ) + 3 x 2 cot ( x ) = 0 Решаем это уравнение x 1 = 1.13943133004 x_{1} = 1.13943133004 x 1 = 1.13943133004 x 2 = − 1.13943133004 x_{2} = -1.13943133004 x 2 = − 1.13943133004 (1.13943133004, 0.680896441741587) (-1.13943133004, 0.680896441741587) Интервалы возрастания и убывания функции: x 2 = 1.13943133004 x_{2} = 1.13943133004 x 2 = 1.13943133004 x 2 = − 1.13943133004 x_{2} = -1.13943133004 x 2 = − 1.13943133004 (-oo, -1.13943133004] [1.13943133004, oo)
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 x ( x 2 ( cot 2 ( x ) + 1 ) cot ( x ) − 3 x ( cot 2 ( x ) + 1 ) + 3 cot ( x ) ) = 0 2 x \left(x^{2} \left(\cot^{2}{\left (x \right )} + 1\right) \cot{\left (x \right )} - 3 x \left(\cot^{2}{\left (x \right )} + 1\right) + 3 \cot{\left (x \right )}\right) = 0 2 x ( x 2 ( cot 2 ( x ) + 1 ) cot ( x ) − 3 x ( cot 2 ( x ) + 1 ) + 3 cot ( x ) ) = 0 Решаем это уравнение x 1 = 61.2121250521 x_{1} = 61.2121250521 x 1 = 61.2121250521 x 2 = 4.13955655465 x_{2} = 4.13955655465 x 2 = 4.13955655465 x 3 = − 26.5916600509 x_{3} = -26.5916600509 x 3 = − 26.5916600509 x 4 = 10.7290749998 x_{4} = 10.7290749998 x 4 = 10.7290749998 x 5 = − 73.7868143864 x_{5} = -73.7868143864 x 5 = − 73.7868143864 x 6 = − 86.3590872003 x_{6} = -86.3590872003 x 6 = − 86.3590872003 x 7 = − 4.13955655465 x_{7} = -4.13955655465 x 7 = − 4.13955655465 x 8 = − 36.0454677896 x_{8} = -36.0454677896 x 8 = − 36.0454677896 x 9 = 23.4352987676 x_{9} = 23.4352987676 x 9 = 23.4352987676 x 10 = − 92.6446240794 x_{10} = -92.6446240794 x 10 = − 92.6446240794 x 11 = 0.679249465841 x_{11} = 0.679249465841 x 11 = 0.679249465841 x 12 = 58.0678920019 x_{12} = 58.0678920019 x 12 = 58.0678920019 x 13 = − 42.340882721 x_{13} = -42.340882721 x 13 = − 42.340882721 x 14 = − 70.6434188276 x_{14} = -70.6434188276 x 14 = − 70.6434188276 x 15 = − 89.5018968492 x_{15} = -89.5018968492 x 15 = − 89.5018968492 x 16 = 86.3590872003 x_{16} = 86.3590872003 x 16 = 86.3590872003 x 17 = − 39.1936616084 x_{17} = -39.1936616084 x 17 = − 39.1936616084 x 18 = 73.7868143864 x_{18} = 73.7868143864 x 18 = 73.7868143864 x 19 = 95.7872769864 x_{19} = 95.7872769864 x 19 = 95.7872769864 x 20 = 54.9233580388 x_{20} = 54.9233580388 x 20 = 54.9233580388 x 21 = − 54.9233580388 x_{21} = -54.9233580388 x 21 = − 54.9233580388 x 22 = 64.3561010883 x_{22} = 64.3561010883 x 22 = 64.3561010883 x 23 = 89.5018968492 x_{23} = 89.5018968492 x 23 = 89.5018968492 x 24 = − 95.7872769864 x_{24} = -95.7872769864 x 24 = − 95.7872769864 x 25 = 20.2744761824 x_{25} = 20.2744761824 x 25 = 20.2744761824 x 26 = − 7.48842693978 x_{26} = -7.48842693978 x 26 = − 7.48842693978 x 27 = 98.9298626396 x_{27} = 98.9298626396 x 27 = 98.9298626396 x 28 = − 83.2161858173 x_{28} = -83.2161858173 x 28 = − 83.2161858173 x 29 = 67.4998558719 x_{29} = 67.4998558719 x 29 = 67.4998558719 x 30 = 36.0454677896 x_{30} = 36.0454677896 x 30 = 36.0454677896 x 31 = 7.48842693978 x_{31} = 7.48842693978 x 31 = 7.48842693978 x 32 = 80.0731819272 x_{32} = 80.0731819272 x 32 = 80.0731819272 x 33 = 26.5916600509 x_{33} = 26.5916600509 x 33 = 26.5916600509 x 34 = − 67.4998558719 x_{34} = -67.4998558719 x 34 = − 67.4998558719 x 35 = 48.6331554374 x_{35} = 48.6331554374 x 35 = 48.6331554374 x 36 = − 17.1068332728 x_{36} = -17.1068332728 x 36 = − 17.1068332728 x 37 = − 80.0731819272 x_{37} = -80.0731819272 x 37 = − 80.0731819272 x 38 = − 76.9300630019 x_{38} = -76.9300630019 x 38 = − 76.9300630019 x 39 = 70.6434188276 x_{39} = 70.6434188276 x 39 = 70.6434188276 x 40 = − 29.7449466925 x_{40} = -29.7449466925 x 40 = − 29.7449466925 x 41 = 76.9300630019 x_{41} = 76.9300630019 x 41 = 76.9300630019 x 42 = − 48.6331554374 x_{42} = -48.6331554374 x 42 = − 48.6331554374 x 43 = 29.7449466925 x_{43} = 29.7449466925 x 43 = 29.7449466925 x 44 = − 23.4352987676 x_{44} = -23.4352987676 x 44 = − 23.4352987676 x 45 = − 51.7784686744 x_{45} = -51.7784686744 x 45 = − 51.7784686744 x 46 = − 58.0678920019 x_{46} = -58.0678920019 x 46 = − 58.0678920019 x 47 = 32.896026005 x_{47} = 32.896026005 x 47 = 32.896026005 x 48 = 45.4873310872 x_{48} = 45.4873310872 x 48 = 45.4873310872 x 49 = 39.1936616084 x_{49} = 39.1936616084 x 49 = 39.1936616084 x 50 = 83.2161858173 x_{50} = 83.2161858173 x 50 = 83.2161858173 x 51 = − 32.896026005 x_{51} = -32.896026005 x 51 = − 32.896026005 x 52 = − 61.2121250521 x_{52} = -61.2121250521 x 52 = − 61.2121250521 x 53 = − 45.4873310872 x_{53} = -45.4873310872 x 53 = − 45.4873310872 x 54 = − 98.9298626396 x_{54} = -98.9298626396 x 54 = − 98.9298626396 x 55 = − 10.7290749998 x_{55} = -10.7290749998 x 55 = − 10.7290749998 x 56 = 92.6446240794 x_{56} = 92.6446240794 x 56 = 92.6446240794 x 57 = 42.340882721 x_{57} = 42.340882721 x 57 = 42.340882721 x 58 = 51.7784686744 x_{58} = 51.7784686744 x 58 = 51.7784686744 x 59 = − 0.679249465841 x_{59} = -0.679249465841 x 59 = − 0.679249465841 x 60 = − 13.928018615 x_{60} = -13.928018615 x 60 = − 13.928018615 x 61 = − 64.3561010883 x_{61} = -64.3561010883 x 61 = − 64.3561010883 x 62 = 17.1068332728 x_{62} = 17.1068332728 x 62 = 17.1068332728 x 63 = − 20.2744761824 x_{63} = -20.2744761824 x 63 = − 20.2744761824 x 64 = 13.928018615 x_{64} = 13.928018615 x 64 = 13.928018615 Интервалы выпуклости и вогнутости: [-0.679249465841, 0.679249465841] (-oo, -98.9298626396]
Горизонтальные асимптоты
Горизонтальные асимптоты найдём с помощью пределов данной функции при x->+oo и x->-ooTrue Возьмём предел y = lim x → − ∞ ( x 3 cot ( x ) ) y = \lim_{x \to -\infty}\left(x^{3} \cot{\left (x \right )}\right) y = x → − ∞ lim ( x 3 cot ( x ) ) True Возьмём предел y = lim x → ∞ ( x 3 cot ( x ) ) y = \lim_{x \to \infty}\left(x^{3} \cot{\left (x \right )}\right) y = x → ∞ lim ( x 3 cot ( x ) )
Наклонные асимптоты
Наклонную асимптоту можно найти, подсчитав предел функции x^3*cot(x), делённой на x при x->+oo и x ->-ooTrue Возьмём предел y = x lim x → − ∞ ( x 2 cot ( x ) ) y = x \lim_{x \to -\infty}\left(x^{2} \cot{\left (x \right )}\right) y = x x → − ∞ lim ( x 2 cot ( x ) ) True Возьмём предел y = x lim x → ∞ ( x 2 cot ( x ) ) y = x \lim_{x \to \infty}\left(x^{2} \cot{\left (x \right )}\right) y = x x → ∞ lim ( x 2 cot ( x ) )
Чётность и нечётность функции
Проверим функци чётна или нечётна с помощью соотношений f = f(-x) и f = -f(-x).x 3 cot ( x ) = x 3 cot ( x ) x^{3} \cot{\left (x \right )} = x^{3} \cot{\left (x \right )} x 3 cot ( x ) = x 3 cot ( x ) x 3 cot ( x ) = − x 3 cot ( x ) x^{3} \cot{\left (x \right )} = - x^{3} \cot{\left (x \right )} x 3 cot ( x ) = − x 3 cot ( x ) 