График функции
0 -90 -80 -70 -60 -50 -40 -30 -20 -10 10 20 30 40 50 -5000 5000
Точки пересечения с осью координат X
График функции пересекает ось X при f = 0x 2 tan ( x ) = 0 x^{2} \tan{\left(x \right)} = 0 x 2 tan ( x ) = 0 Решаем это уравнение Аналитическое решение x 1 = 0 x_{1} = 0 x 1 = 0 Численное решение x 1 = 59.6902604182061 x_{1} = 59.6902604182061 x 1 = 59.6902604182061 x 2 = − 97.3893722612836 x_{2} = -97.3893722612836 x 2 = − 97.3893722612836 x 3 = − 75.398223686155 x_{3} = -75.398223686155 x 3 = − 75.398223686155 x 4 = − 31.4159265358979 x_{4} = -31.4159265358979 x 4 = − 31.4159265358979 x 5 = 97.3893722612836 x_{5} = 97.3893722612836 x 5 = 97.3893722612836 x 6 = − 28.2743338823081 x_{6} = -28.2743338823081 x 6 = − 28.2743338823081 x 7 = 72.2566310325652 x_{7} = 72.2566310325652 x 7 = 72.2566310325652 x 8 = 47.1238898038469 x_{8} = 47.1238898038469 x 8 = 47.1238898038469 x 9 = 12.5663706143592 x_{9} = 12.5663706143592 x 9 = 12.5663706143592 x 10 = − 84.8230016469244 x_{10} = -84.8230016469244 x 10 = − 84.8230016469244 x 11 = 56.5486677646163 x_{11} = 56.5486677646163 x 11 = 56.5486677646163 x 12 = 0 x_{12} = 0 x 12 = 0 x 13 = 69.1150383789755 x_{13} = 69.1150383789755 x 13 = 69.1150383789755 x 14 = 81.6814089933346 x_{14} = 81.6814089933346 x 14 = 81.6814089933346 x 15 = 75.398223686155 x_{15} = 75.398223686155 x 15 = 75.398223686155 x 16 = − 40.8407044966673 x_{16} = -40.8407044966673 x 16 = − 40.8407044966673 x 17 = − 81.6814089933346 x_{17} = -81.6814089933346 x 17 = − 81.6814089933346 x 18 = 91.106186954104 x_{18} = 91.106186954104 x 18 = 91.106186954104 x 19 = − 94.2477796076938 x_{19} = -94.2477796076938 x 19 = − 94.2477796076938 x 20 = − 53.4070751110265 x_{20} = -53.4070751110265 x 20 = − 53.4070751110265 x 21 = 34.5575191894877 x_{21} = 34.5575191894877 x 21 = 34.5575191894877 x 22 = 40.8407044966673 x_{22} = 40.8407044966673 x 22 = 40.8407044966673 x 23 = − 62.8318530717959 x_{23} = -62.8318530717959 x 23 = − 62.8318530717959 x 24 = 21.9911485751286 x_{24} = 21.9911485751286 x 24 = 21.9911485751286 x 25 = 15.707963267949 x_{25} = 15.707963267949 x 25 = 15.707963267949 x 26 = − 15.707963267949 x_{26} = -15.707963267949 x 26 = − 15.707963267949 x 27 = − 21.9911485751286 x_{27} = -21.9911485751286 x 27 = − 21.9911485751286 x 28 = 18.8495559215388 x_{28} = 18.8495559215388 x 28 = 18.8495559215388 x 29 = − 3.14159265358979 x_{29} = -3.14159265358979 x 29 = − 3.14159265358979 x 30 = 3.14159265358979 x_{30} = 3.14159265358979 x 30 = 3.14159265358979 x 31 = 94.2477796076938 x_{31} = 94.2477796076938 x 31 = 94.2477796076938 x 32 = − 50.2654824574367 x_{32} = -50.2654824574367 x 32 = − 50.2654824574367 x 33 = 100.530964914873 x_{33} = 100.530964914873 x 33 = 100.530964914873 x 34 = − 78.5398163397448 x_{34} = -78.5398163397448 x 34 = − 78.5398163397448 x 35 = 31.4159265358979 x_{35} = 31.4159265358979 x 35 = 31.4159265358979 x 36 = − 43.9822971502571 x_{36} = -43.9822971502571 x 36 = − 43.9822971502571 x 37 = − 37.6991118430775 x_{37} = -37.6991118430775 x 37 = − 37.6991118430775 x 38 = − 59.6902604182061 x_{38} = -59.6902604182061 x 38 = − 59.6902604182061 x 39 = 37.6991118430775 x_{39} = 37.6991118430775 x 39 = 37.6991118430775 x 40 = − 100.530964914873 x_{40} = -100.530964914873 x 40 = − 100.530964914873 x 41 = 84.8230016469244 x_{41} = 84.8230016469244 x 41 = 84.8230016469244 x 42 = 87.9645943005142 x_{42} = 87.9645943005142 x 42 = 87.9645943005142 x 43 = − 12.5663706143592 x_{43} = -12.5663706143592 x 43 = − 12.5663706143592 x 44 = 62.8318530717959 x_{44} = 62.8318530717959 x 44 = 62.8318530717959 x 45 = − 69.1150383789755 x_{45} = -69.1150383789755 x 45 = − 69.1150383789755 x 46 = 9.42477796076938 x_{46} = 9.42477796076938 x 46 = 9.42477796076938 x 47 = − 18.8495559215388 x_{47} = -18.8495559215388 x 47 = − 18.8495559215388 x 48 = 50.2654824574367 x_{48} = 50.2654824574367 x 48 = 50.2654824574367 x 49 = − 6.28318530717959 x_{49} = -6.28318530717959 x 49 = − 6.28318530717959 x 50 = 43.9822971502571 x_{50} = 43.9822971502571 x 50 = 43.9822971502571 x 51 = 6.28318530717959 x_{51} = 6.28318530717959 x 51 = 6.28318530717959 x 52 = 53.4070751110265 x_{52} = 53.4070751110265 x 52 = 53.4070751110265 x 53 = − 56.5486677646163 x_{53} = -56.5486677646163 x 53 = − 56.5486677646163 x 54 = − 9.42477796076938 x_{54} = -9.42477796076938 x 54 = − 9.42477796076938 x 55 = − 87.9645943005142 x_{55} = -87.9645943005142 x 55 = − 87.9645943005142 x 56 = 25.1327412287183 x_{56} = 25.1327412287183 x 56 = 25.1327412287183 x 57 = − 25.1327412287183 x_{57} = -25.1327412287183 x 57 = − 25.1327412287183 x 58 = 65.9734457253857 x_{58} = 65.9734457253857 x 58 = 65.9734457253857 x 59 = 28.2743338823081 x_{59} = 28.2743338823081 x 59 = 28.2743338823081 x 60 = − 91.106186954104 x_{60} = -91.106186954104 x 60 = − 91.106186954104 x 61 = − 47.1238898038469 x_{61} = -47.1238898038469 x 61 = − 47.1238898038469 x 62 = − 72.2566310325652 x_{62} = -72.2566310325652 x 62 = − 72.2566310325652 x 63 = 78.5398163397448 x_{63} = 78.5398163397448 x 63 = 78.5398163397448 x 64 = − 65.9734457253857 x_{64} = -65.9734457253857 x 64 = − 65.9734457253857 x 65 = − 34.5575191894877 x_{65} = -34.5575191894877 x 65 = − 34.5575191894877
Точки пересечения с осью координат Y
График пересекает ось Y, когда x равняется 0:0 2 tan ( 0 ) 0^{2} \tan{\left(0 \right)} 0 2 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 x 2 ( tan 2 ( x ) + 1 ) + 2 x tan ( x ) = 0 x^{2} \left(\tan^{2}{\left(x \right)} + 1\right) + 2 x \tan{\left(x \right)} = 0 x 2 ( tan 2 ( x ) + 1 ) + 2 x tan ( x ) = 0 Решаем это уравнение x 1 = 0 x_{1} = 0 x 1 = 0 (0, 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 ( x 2 ( tan 2 ( x ) + 1 ) tan ( x ) + 2 x ( tan 2 ( x ) + 1 ) + tan ( x ) ) = 0 2 \left(x^{2} \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + 2 x \left(\tan^{2}{\left(x \right)} + 1\right) + \tan{\left(x \right)}\right) = 0 2 ( x 2 ( tan 2 ( x ) + 1 ) tan ( x ) + 2 x ( tan 2 ( x ) + 1 ) + tan ( x ) ) = 0 Решаем это уравнение x 1 = 91.0842354292587 x_{1} = 91.0842354292587 x 1 = 91.0842354292587 x 2 = − 97.3688368609732 x_{2} = -97.3688368609732 x 2 = − 97.3688368609732 x 3 = − 69.086103133906 x_{3} = -69.086103133906 x 3 = − 69.086103133906 x 4 = 75.3716994163885 x_{4} = 75.3716994163885 x 4 = 75.3716994163885 x 5 = 5.96726435810305 x_{5} = 5.96726435810305 x 5 = 5.96726435810305 x 6 = 69.086103133906 x_{6} = 69.086103133906 x 6 = 69.086103133906 x 7 = 47.0814548431779 x_{7} = 47.0814548431779 x 7 = 47.0814548431779 x 8 = 43.93683212641 x_{8} = 43.93683212641 x 8 = 43.93683212641 x 9 = − 84.7994242285037 x_{9} = -84.7994242285037 x 9 = − 84.7994242285037 x 10 = − 15.5808081405648 x_{10} = -15.5808081405648 x 10 = − 15.5808081405648 x 11 = 59.6567572450692 x_{11} = 59.6567572450692 x 11 = 59.6567572450692 x 12 = 84.7994242285037 x_{12} = 84.7994242285037 x 12 = 84.7994242285037 x 13 = − 18.7435508863884 x_{13} = -18.7435508863884 x 13 = − 18.7435508863884 x 14 = − 43.93683212641 x_{14} = -43.93683212641 x 14 = − 43.93683212641 x 15 = 0 x_{15} = 0 x 15 = 0 x 16 = − 81.6569248399483 x_{16} = -81.6569248399483 x 16 = − 81.6569248399483 x 17 = − 56.513303680917 x_{17} = -56.513303680917 x 17 = − 56.513303680917 x 18 = − 2.51787226577809 x_{18} = -2.51787226577809 x 18 = − 2.51787226577809 x 19 = 94.2265597445368 x_{19} = 94.2265597445368 x 19 = 94.2265597445368 x 20 = 9.21332735720748 x_{20} = 9.21332735720748 x 20 = 9.21332735720748 x 21 = 50.2256989613876 x_{21} = 50.2256989613876 x 21 = 50.2256989613876 x 22 = 2.51787226577809 x_{22} = 2.51787226577809 x 22 = 2.51787226577809 x 23 = − 28.2036276766186 x_{23} = -28.2036276766186 x 23 = − 28.2036276766186 x 24 = − 100.511071202847 x_{24} = -100.511071202847 x 24 = − 100.511071202847 x 25 = − 31.3522859596756 x_{25} = -31.3522859596756 x 25 = − 31.3522859596756 x 26 = 31.3522859596756 x_{26} = 31.3522859596756 x 26 = 31.3522859596756 x 27 = 56.513303680917 x_{27} = 56.513303680917 x 27 = 56.513303680917 x 28 = 78.5143529238898 x_{28} = 78.5143529238898 x 28 = 78.5143529238898 x 29 = 28.2036276766186 x_{29} = 28.2036276766186 x 29 = 28.2036276766186 x 30 = − 87.9418588589466 x_{30} = -87.9418588589466 x 30 = − 87.9418588589466 x 31 = 25.0532054465023 x_{31} = 25.0532054465023 x 31 = 25.0532054465023 x 32 = − 59.6567572450692 x_{32} = -59.6567572450692 x 32 = − 59.6567572450692 x 33 = − 37.6460725978858 x_{33} = -37.6460725978858 x 33 = − 37.6460725978858 x 34 = − 78.5143529238898 x_{34} = -78.5143529238898 x 34 = − 78.5143529238898 x 35 = 34.4996607566446 x_{35} = 34.4996607566446 x 35 = 34.4996607566446 x 36 = − 53.3696312584227 x_{36} = -53.3696312584227 x 36 = − 53.3696312584227 x 37 = − 62.8000247676753 x_{37} = -62.8000247676753 x 37 = − 62.8000247676753 x 38 = − 75.3716994163885 x_{38} = -75.3716994163885 x 38 = − 75.3716994163885 x 39 = 65.9431328173515 x_{39} = 65.9431328173515 x 39 = 65.9431328173515 x 40 = − 34.4996607566446 x_{40} = -34.4996607566446 x 40 = − 34.4996607566446 x 41 = − 12.4075419598293 x_{41} = -12.4075419598293 x 41 = − 12.4075419598293 x 42 = 12.4075419598293 x_{42} = 12.4075419598293 x 42 = 12.4075419598293 x 43 = − 40.7917435045268 x_{43} = -40.7917435045268 x 43 = − 40.7917435045268 x 44 = 87.9418588589466 x_{44} = 87.9418588589466 x 44 = 87.9418588589466 x 45 = 37.6460725978858 x_{45} = 37.6460725978858 x 45 = 37.6460725978858 x 46 = 53.3696312584227 x_{46} = 53.3696312584227 x 46 = 53.3696312584227 x 47 = 15.5808081405648 x_{47} = 15.5808081405648 x 47 = 15.5808081405648 x 48 = − 94.2265597445368 x_{48} = -94.2265597445368 x 48 = − 94.2265597445368 x 49 = 21.9002649847656 x_{49} = 21.9002649847656 x 49 = 21.9002649847656 x 50 = − 91.0842354292587 x_{50} = -91.0842354292587 x 50 = − 91.0842354292587 x 51 = − 47.0814548431779 x_{51} = -47.0814548431779 x 51 = − 47.0814548431779 x 52 = − 5.96726435810305 x_{52} = -5.96726435810305 x 52 = − 5.96726435810305 x 53 = − 9.21332735720748 x_{53} = -9.21332735720748 x 53 = − 9.21332735720748 x 54 = − 25.0532054465023 x_{54} = -25.0532054465023 x 54 = − 25.0532054465023 x 55 = − 65.9431328173515 x_{55} = -65.9431328173515 x 55 = − 65.9431328173515 x 56 = 97.3688368609732 x_{56} = 97.3688368609732 x 56 = 97.3688368609732 x 57 = − 72.2289536776301 x_{57} = -72.2289536776301 x 57 = − 72.2289536776301 x 58 = 100.511071202847 x_{58} = 100.511071202847 x 58 = 100.511071202847 x 59 = 40.7917435045268 x_{59} = 40.7917435045268 x 59 = 40.7917435045268 x 60 = 72.2289536776301 x_{60} = 72.2289536776301 x 60 = 72.2289536776301 x 61 = 62.8000247676753 x_{61} = 62.8000247676753 x 61 = 62.8000247676753 x 62 = − 50.2256989613876 x_{62} = -50.2256989613876 x 62 = − 50.2256989613876 x 63 = 18.7435508863884 x_{63} = 18.7435508863884 x 63 = 18.7435508863884 x 64 = − 21.9002649847656 x_{64} = -21.9002649847656 x 64 = − 21.9002649847656 x 65 = 81.6569248399483 x_{65} = 81.6569248399483 x 65 = 81.6569248399483 Интервалы выпуклости и вогнутости: [ 100.511071202847 , ∞ ) \left[100.511071202847, \infty\right) [ 100.511071202847 , ∞ ) ( − ∞ , − 100.511071202847 ] \left(-\infty, -100.511071202847\right] ( − ∞ , − 100.511071202847 ]
Горизонтальные асимптоты
Горизонтальные асимптоты найдём с помощью пределов данной функции при x->+oo и x->-oolim x → − ∞ ( x 2 tan ( x ) ) = lim x → − ∞ ( x 2 tan ( x ) ) \lim_{x \to -\infty}\left(x^{2} \tan{\left(x \right)}\right) = \lim_{x \to -\infty}\left(x^{2} \tan{\left(x \right)}\right) x → − ∞ lim ( x 2 tan ( x ) ) = x → − ∞ lim ( x 2 tan ( x ) ) Возьмём предел y = lim x → − ∞ ( x 2 tan ( x ) ) y = \lim_{x \to -\infty}\left(x^{2} \tan{\left(x \right)}\right) y = x → − ∞ lim ( x 2 tan ( x ) ) lim x → ∞ ( x 2 tan ( x ) ) = lim x → ∞ ( x 2 tan ( x ) ) \lim_{x \to \infty}\left(x^{2} \tan{\left(x \right)}\right) = \lim_{x \to \infty}\left(x^{2} \tan{\left(x \right)}\right) x → ∞ lim ( x 2 tan ( x ) ) = x → ∞ lim ( x 2 tan ( x ) ) Возьмём предел y = lim x → ∞ ( x 2 tan ( x ) ) y = \lim_{x \to \infty}\left(x^{2} \tan{\left(x \right)}\right) y = x → ∞ lim ( x 2 tan ( x ) )
Наклонные асимптоты
Наклонную асимптоту можно найти, подсчитав предел функции x^2*tan(x), делённой на x при x->+oo и x ->-oolim x → − ∞ ( x tan ( x ) ) = lim x → − ∞ ( x tan ( x ) ) \lim_{x \to -\infty}\left(x \tan{\left(x \right)}\right) = \lim_{x \to -\infty}\left(x \tan{\left(x \right)}\right) x → − ∞ lim ( x tan ( x ) ) = x → − ∞ lim ( x tan ( x ) ) Возьмём предел y = x lim x → − ∞ ( x tan ( x ) ) y = x \lim_{x \to -\infty}\left(x \tan{\left(x \right)}\right) y = x x → − ∞ lim ( x tan ( x ) ) lim x → ∞ ( x tan ( x ) ) = lim x → ∞ ( x tan ( x ) ) \lim_{x \to \infty}\left(x \tan{\left(x \right)}\right) = \lim_{x \to \infty}\left(x \tan{\left(x \right)}\right) x → ∞ lim ( x tan ( x ) ) = x → ∞ lim ( x tan ( x ) ) Возьмём предел y = x lim x → ∞ ( x tan ( x ) ) y = x \lim_{x \to \infty}\left(x \tan{\left(x \right)}\right) y = x x → ∞ lim ( x tan ( x ) )
Чётность и нечётность функции
Проверим функци чётна или нечётна с помощью соотношений f = f(-x) и f = -f(-x).x 2 tan ( x ) = − x 2 tan ( x ) x^{2} \tan{\left(x \right)} = - x^{2} \tan{\left(x \right)} x 2 tan ( x ) = − x 2 tan ( x ) x 2 tan ( x ) = x 2 tan ( x ) x^{2} \tan{\left(x \right)} = x^{2} \tan{\left(x \right)} x 2 tan ( x ) = x 2 tan ( x ) 
