График функции
0 2 4 6 8 -8 -6 -4 -2 -10 10 -100 100
Область определения функции
Точки, в которых функция точно неопределена:x 1 = 0 x_{1} = 0 x 1 = 0
Точки пересечения с осью координат X
График функции пересекает ось X при f = 0tan ( x ) − 1 x = 0 \tan{\left (x \right )} - \frac{1}{x} = 0 tan ( x ) − x 1 = 0 Решаем это уравнение Численное решение x 1 = 6.43729817917 x_{1} = 6.43729817917 x 1 = 6.43729817917 x 2 = − 91.1171613945 x_{2} = -91.1171613945 x 2 = − 91.1171613945 x 3 = 65.9885986985 x_{3} = 65.9885986985 x 3 = 65.9885986985 x 4 = − 44.0050179208 x_{4} = -44.0050179208 x 4 = − 44.0050179208 x 5 = 56.5663442798 x_{5} = 56.5663442798 x 5 = 56.5663442798 x 6 = − 72.2704670603 x_{6} = -72.2704670603 x 6 = − 72.2704670603 x 7 = 50.2853663378 x_{7} = 50.2853663378 x 7 = 50.2853663378 x 8 = 47.1450977368 x_{8} = 47.1450977368 x 8 = 47.1450977368 x 9 = 91.1171613945 x_{9} = 91.1171613945 x 9 = 91.1171613945 x 10 = − 94.258388345 x_{10} = -94.258388345 x 10 = − 94.258388345 x 11 = 97.3996388791 x_{11} = 97.3996388791 x 11 = 97.3996388791 x 12 = − 6.43729817917 x_{12} = -6.43729817917 x 12 = − 6.43729817917 x 13 = − 12.6452872239 x_{13} = -12.6452872239 x 13 = − 12.6452872239 x 14 = 37.7256128278 x_{14} = 37.7256128278 x 14 = 37.7256128278 x 15 = 81.6936492356 x_{15} = 81.6936492356 x 15 = 81.6936492356 x 16 = 94.258388345 x_{16} = 94.258388345 x 16 = 94.258388345 x 17 = − 75.4114834888 x_{17} = -75.4114834888 x 17 = − 75.4114834888 x 18 = − 65.9885986985 x_{18} = -65.9885986985 x 18 = − 65.9885986985 x 19 = − 3.42561845948 x_{19} = -3.42561845948 x 19 = − 3.42561845948 x 20 = 25.1724463266 x_{20} = 25.1724463266 x 20 = 25.1724463266 x 21 = 69.1295029739 x_{21} = 69.1295029739 x 21 = 69.1295029739 x 22 = − 34.5864242153 x_{22} = -34.5864242153 x 22 = − 34.5864242153 x 23 = − 59.7070073053 x_{23} = -59.7070073053 x 23 = − 59.7070073053 x 24 = − 18.9024099569 x_{24} = -18.9024099569 x 24 = − 18.9024099569 x 25 = − 15.7712848748 x_{25} = -15.7712848748 x 25 = − 15.7712848748 x 26 = 34.5864242153 x_{26} = 34.5864242153 x 26 = 34.5864242153 x 27 = 100.540910787 x_{27} = 100.540910787 x 27 = 100.540910787 x 28 = 53.4257904774 x_{28} = 53.4257904774 x 28 = 53.4257904774 x 29 = − 22.0364967279 x_{29} = -22.0364967279 x 29 = − 22.0364967279 x 30 = 62.8477631945 x_{30} = 62.8477631945 x 30 = 62.8477631945 x 31 = 3.42561845948 x_{31} = 3.42561845948 x 31 = 3.42561845948 x 32 = 28.3096428545 x_{32} = 28.3096428545 x 32 = 28.3096428545 x 33 = − 81.6936492356 x_{33} = -81.6936492356 x 33 = − 81.6936492356 x 34 = − 47.1450977368 x_{34} = -47.1450977368 x 34 = − 47.1450977368 x 35 = 40.8651703305 x_{35} = 40.8651703305 x 35 = 40.8651703305 x 36 = − 9.52933440536 x_{36} = -9.52933440536 x 36 = − 9.52933440536 x 37 = − 62.8477631945 x_{37} = -62.8477631945 x 37 = − 62.8477631945 x 38 = 72.2704670603 x_{38} = 72.2704670603 x 38 = 72.2704670603 x 39 = − 25.1724463266 x_{39} = -25.1724463266 x 39 = − 25.1724463266 x 40 = 44.0050179208 x_{40} = 44.0050179208 x 40 = 44.0050179208 x 41 = 12.6452872239 x_{41} = 12.6452872239 x 41 = 12.6452872239 x 42 = − 100.540910787 x_{42} = -100.540910787 x 42 = − 100.540910787 x 43 = 9.52933440536 x_{43} = 9.52933440536 x 43 = 9.52933440536 x 44 = − 31.4477146375 x_{44} = -31.4477146375 x 44 = − 31.4477146375 x 45 = − 97.3996388791 x_{45} = -97.3996388791 x 45 = − 97.3996388791 x 46 = − 84.834788718 x_{46} = -84.834788718 x 46 = − 84.834788718 x 47 = 22.0364967279 x_{47} = 22.0364967279 x 47 = 22.0364967279 x 48 = 84.834788718 x_{48} = 84.834788718 x 48 = 84.834788718 x 49 = 59.7070073053 x_{49} = 59.7070073053 x 49 = 59.7070073053 x 50 = 15.7712848748 x_{50} = 15.7712848748 x 50 = 15.7712848748 x 51 = 18.9024099569 x_{51} = 18.9024099569 x 51 = 18.9024099569 x 52 = − 40.8651703305 x_{52} = -40.8651703305 x 52 = − 40.8651703305 x 53 = − 50.2853663378 x_{53} = -50.2853663378 x 53 = − 50.2853663378 x 54 = 31.4477146375 x_{54} = 31.4477146375 x 54 = 31.4477146375 x 55 = − 53.4257904774 x_{55} = -53.4257904774 x 55 = − 53.4257904774 x 56 = 0.860333589019 x_{56} = 0.860333589019 x 56 = 0.860333589019 x 57 = − 37.7256128278 x_{57} = -37.7256128278 x 57 = − 37.7256128278 x 58 = 75.4114834888 x_{58} = 75.4114834888 x 58 = 75.4114834888 x 59 = − 69.1295029739 x_{59} = -69.1295029739 x 59 = − 69.1295029739 x 60 = − 56.5663442798 x_{60} = -56.5663442798 x 60 = − 56.5663442798 x 61 = 78.5525459842 x_{61} = 78.5525459842 x 61 = 78.5525459842 x 62 = − 78.5525459842 x_{62} = -78.5525459842 x 62 = − 78.5525459842 x 63 = − 87.9759605525 x_{63} = -87.9759605525 x 63 = − 87.9759605525 x 64 = 87.9759605525 x_{64} = 87.9759605525 x 64 = 87.9759605525 x 65 = − 28.3096428545 x_{65} = -28.3096428545 x 65 = − 28.3096428545
Точки пересечения с осью координат Y
График пересекает ось Y, когда x равняется 0:tan ( 0 ) − ∞ ~ \tan{\left (0 \right )} - \tilde{\infty} tan ( 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 tan 2 ( x ) + 1 + 1 x 2 = 0 \tan^{2}{\left (x \right )} + 1 + \frac{1}{x^{2}} = 0 tan 2 ( x ) + 1 + x 2 1 = 0 Решаем это уравнение
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 ) − 1 x 3 ) = 0 2 \left(\left(\tan^{2}{\left (x \right )} + 1\right) \tan{\left (x \right )} - \frac{1}{x^{3}}\right) = 0 2 ( ( tan 2 ( x ) + 1 ) tan ( x ) − x 3 1 ) = 0 Решаем это уравнение x 1 = − 3.17285930451 x_{1} = -3.17285930451 x 1 = − 3.17285930451 x 2 = − 12.5668744838 x_{2} = -12.5668744838 x 2 = − 12.5668744838 x 3 = − 50.2654903313 x_{3} = -50.2654903313 x 3 = − 50.2654903313 x 4 = − 87.9645957697 x_{4} = -87.9645957697 x 4 = − 87.9645957697 x 5 = − 75.3982260192 x_{5} = -75.3982260192 x 5 = − 75.3982260192 x 6 = 75.3982260192 x_{6} = 75.3982260192 x 6 = 75.3982260192 x 7 = − 34.5575434205 x_{7} = -34.5575434205 x 7 = − 34.5575434205 x 8 = 34.5575434205 x_{8} = 34.5575434205 x 8 = 34.5575434205 x 9 = 25.1328042195 x_{9} = 25.1328042195 x 9 = 25.1328042195 x 10 = 21.9912426017 x_{10} = 21.9912426017 x 10 = 21.9912426017 x 11 = − 69.1150414079 x_{11} = -69.1150414079 x 11 = − 69.1150414079 x 12 = − 21.9912426017 x_{12} = -21.9912426017 x 12 = − 21.9912426017 x 13 = − 100.530965899 x_{13} = -100.530965899 x 13 = − 100.530965899 x 14 = 78.5398184038 x_{14} = 78.5398184038 x 14 = 78.5398184038 x 15 = 31.4159587873 x_{15} = 31.4159587873 x 15 = 31.4159587873 x 16 = 56.5486732947 x_{16} = 56.5486732947 x 16 = 56.5486732947 x 17 = 6.28720892709 x_{17} = 6.28720892709 x 17 = 6.28720892709 x 18 = 43.9823089037 x_{18} = 43.9823089037 x 18 = 43.9823089037 x 19 = 3.17285930451 x_{19} = 3.17285930451 x 19 = 3.17285930451 x 20 = − 43.9823089037 x_{20} = -43.9823089037 x 20 = − 43.9823089037 x 21 = − 31.4159587873 x_{21} = -31.4159587873 x 21 = − 31.4159587873 x 22 = − 72.2566336833 x_{22} = -72.2566336833 x 22 = − 72.2566336833 x 23 = − 65.9734492079 x_{23} = -65.9734492079 x 23 = − 65.9734492079 x 24 = − 78.5398184038 x_{24} = -78.5398184038 x 24 = − 78.5398184038 x 25 = − 56.5486732947 x_{25} = -56.5486732947 x 25 = − 56.5486732947 x 26 = 37.6991305071 x_{26} = 37.6991305071 x 26 = 37.6991305071 x 27 = 91.1061882765 x_{27} = 91.1061882765 x 27 = 91.1061882765 x 28 = 100.530965899 x_{28} = 100.530965899 x 28 = 100.530965899 x 29 = 97.3893733439 x_{29} = 97.3893733439 x 29 = 97.3893733439 x 30 = 28.2743781229 x_{30} = 28.2743781229 x 30 = 28.2743781229 x 31 = 18.8497052306 x_{31} = 18.8497052306 x 31 = 18.8497052306 x 32 = 9.42597200589 x_{32} = 9.42597200589 x 32 = 9.42597200589 x 33 = 94.2477808022 x_{33} = 94.2477808022 x 33 = 94.2477808022 x 34 = − 47.1238993599 x_{34} = -47.1238993599 x 34 = − 47.1238993599 x 35 = 12.5668744838 x_{35} = 12.5668744838 x 35 = 12.5668744838 x 36 = 87.9645957697 x_{36} = 87.9645957697 x 36 = 87.9645957697 x 37 = 69.1150414079 x_{37} = 69.1150414079 x 37 = 69.1150414079 x 38 = 40.8407191765 x_{38} = 40.8407191765 x 38 = 40.8407191765 x 39 = 50.2654903313 x_{39} = 50.2654903313 x 39 = 50.2654903313 x 40 = 59.6902651203 x_{40} = 59.6902651203 x 40 = 59.6902651203 x 41 = 47.1238993599 x_{41} = 47.1238993599 x 41 = 47.1238993599 x 42 = − 6.28720892709 x_{42} = -6.28720892709 x 42 = − 6.28720892709 x 43 = − 37.6991305071 x_{43} = -37.6991305071 x 43 = − 37.6991305071 x 44 = 72.2566336833 x_{44} = 72.2566336833 x 44 = 72.2566336833 x 45 = − 91.1061882765 x_{45} = -91.1061882765 x 45 = − 91.1061882765 x 46 = − 18.8497052306 x_{46} = -18.8497052306 x 46 = − 18.8497052306 x 47 = − 84.8230032855 x_{47} = -84.8230032855 x 47 = − 84.8230032855 x 48 = 81.6814108283 x_{48} = 81.6814108283 x 48 = 81.6814108283 x 49 = 62.8318571032 x_{49} = 62.8318571032 x 49 = 62.8318571032 x 50 = − 53.4070816756 x_{50} = -53.4070816756 x 50 = − 53.4070816756 x 51 = − 81.6814108283 x_{51} = -81.6814108283 x 51 = − 81.6814108283 x 52 = 15.7082212675 x_{52} = 15.7082212675 x 52 = 15.7082212675 x 53 = − 59.6902651203 x_{53} = -59.6902651203 x 53 = − 59.6902651203 x 54 = 84.8230032855 x_{54} = 84.8230032855 x 54 = 84.8230032855 x 55 = − 97.3893733439 x_{55} = -97.3893733439 x 55 = − 97.3893733439 x 56 = − 15.7082212675 x_{56} = -15.7082212675 x 56 = − 15.7082212675 x 57 = − 94.2477808022 x_{57} = -94.2477808022 x 57 = − 94.2477808022 x 58 = − 62.8318571032 x_{58} = -62.8318571032 x 58 = − 62.8318571032 x 59 = − 28.2743781229 x_{59} = -28.2743781229 x 59 = − 28.2743781229 x 60 = − 9.42597200589 x_{60} = -9.42597200589 x 60 = − 9.42597200589 x 61 = 53.4070816756 x_{61} = 53.4070816756 x 61 = 53.4070816756 x 62 = 65.9734492079 x_{62} = 65.9734492079 x 62 = 65.9734492079 x 63 = − 40.8407191765 x_{63} = -40.8407191765 x 63 = − 40.8407191765 x 64 = − 25.1328042195 x_{64} = -25.1328042195 x 64 = − 25.1328042195 x 1 = 0 x_{1} = 0 x 1 = 0 lim x → 0 − ( 2 ( ( tan 2 ( x ) + 1 ) tan ( x ) − 1 x 3 ) ) = ∞ \lim_{x \to 0^-}\left(2 \left(\left(\tan^{2}{\left (x \right )} + 1\right) \tan{\left (x \right )} - \frac{1}{x^{3}}\right)\right) = \infty x → 0 − lim ( 2 ( ( tan 2 ( x ) + 1 ) tan ( x ) − x 3 1 ) ) = ∞ lim x → 0 + ( 2 ( ( tan 2 ( x ) + 1 ) tan ( x ) − 1 x 3 ) ) = − ∞ \lim_{x \to 0^+}\left(2 \left(\left(\tan^{2}{\left (x \right )} + 1\right) \tan{\left (x \right )} - \frac{1}{x^{3}}\right)\right) = -\infty x → 0 + lim ( 2 ( ( tan 2 ( x ) + 1 ) tan ( x ) − x 3 1 ) ) = − ∞ x 1 = 0 x_{1} = 0 x 1 = 0 Интервалы выпуклости и вогнутости: [100.530965899, oo) (-oo, -100.530965899]
Горизонтальные асимптоты
Горизонтальные асимптоты найдём с помощью пределов данной функции при x->+oo и x->-ooTrue Возьмём предел y = lim x → − ∞ ( tan ( x ) − 1 x ) y = \lim_{x \to -\infty}\left(\tan{\left (x \right )} - \frac{1}{x}\right) y = x → − ∞ lim ( tan ( x ) − x 1 ) True Возьмём предел y = lim x → ∞ ( tan ( x ) − 1 x ) y = \lim_{x \to \infty}\left(\tan{\left (x \right )} - \frac{1}{x}\right) y = x → ∞ lim ( tan ( x ) − x 1 )
Наклонные асимптоты
Наклонную асимптоту можно найти, подсчитав предел функции tan(x) - 1/x, делённой на x при x->+oo и x ->-ooTrue Возьмём предел y = x lim x → − ∞ ( 1 x ( tan ( x ) − 1 x ) ) y = x \lim_{x \to -\infty}\left(\frac{1}{x} \left(\tan{\left (x \right )} - \frac{1}{x}\right)\right) y = x x → − ∞ lim ( x 1 ( tan ( x ) − x 1 ) ) True Возьмём предел y = x lim x → ∞ ( 1 x ( tan ( x ) − 1 x ) ) y = x \lim_{x \to \infty}\left(\frac{1}{x} \left(\tan{\left (x \right )} - \frac{1}{x}\right)\right) y = x x → ∞ lim ( x 1 ( tan ( x ) − x 1 ) )
Чётность и нечётность функции
Проверим функци чётна или нечётна с помощью соотношений f = f(-x) и f = -f(-x).tan ( x ) − 1 x = − tan ( x ) + 1 x \tan{\left (x \right )} - \frac{1}{x} = - \tan{\left (x \right )} + \frac{1}{x} tan ( x ) − x 1 = − tan ( x ) + x 1 tan ( x ) − 1 x = − − 1 tan ( x ) − 1 x \tan{\left (x \right )} - \frac{1}{x} = - -1 \tan{\left (x \right )} - \frac{1}{x} tan ( x ) − x 1 = − − 1 tan ( x ) − x 1 