График функции
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Точки пересечения с осью координат X
График функции пересекает ось X при f = 0tan ( 2 x ) = 0 \tan{\left(2 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 = − 45.553093477052 x_{1} = -45.553093477052 x 1 = − 45.553093477052 x 2 = − 86.3937979737193 x_{2} = -86.3937979737193 x 2 = − 86.3937979737193 x 3 = − 97.3893722612836 x_{3} = -97.3893722612836 x 3 = − 97.3893722612836 x 4 = − 72.2566310325652 x_{4} = -72.2566310325652 x 4 = − 72.2566310325652 x 5 = 31.4159265358979 x_{5} = 31.4159265358979 x 5 = 31.4159265358979 x 6 = 86.3937979737193 x_{6} = 86.3937979737193 x 6 = 86.3937979737193 x 7 = − 34.5575191894877 x_{7} = -34.5575191894877 x 7 = − 34.5575191894877 x 8 = 50.2654824574367 x_{8} = 50.2654824574367 x 8 = 50.2654824574367 x 9 = 62.8318530717959 x_{9} = 62.8318530717959 x 9 = 62.8318530717959 x 10 = − 69.1150383789755 x_{10} = -69.1150383789755 x 10 = − 69.1150383789755 x 11 = 1.5707963267949 x_{11} = 1.5707963267949 x 11 = 1.5707963267949 x 12 = 36.1283155162826 x_{12} = 36.1283155162826 x 12 = 36.1283155162826 x 13 = − 28.2743338823081 x_{13} = -28.2743338823081 x 13 = − 28.2743338823081 x 14 = 4.71238898038469 x_{14} = 4.71238898038469 x 14 = 4.71238898038469 x 15 = 95.8185759344887 x_{15} = 95.8185759344887 x 15 = 95.8185759344887 x 16 = − 7.85398163397448 x_{16} = -7.85398163397448 x 16 = − 7.85398163397448 x 17 = − 73.8274273593601 x_{17} = -73.8274273593601 x 17 = − 73.8274273593601 x 18 = − 23.5619449019235 x_{18} = -23.5619449019235 x 18 = − 23.5619449019235 x 19 = − 65.9734457253857 x_{19} = -65.9734457253857 x 19 = − 65.9734457253857 x 20 = − 100.530964914873 x_{20} = -100.530964914873 x 20 = − 100.530964914873 x 21 = 87.9645943005142 x_{21} = 87.9645943005142 x 21 = 87.9645943005142 x 22 = − 12.5663706143592 x_{22} = -12.5663706143592 x 22 = − 12.5663706143592 x 23 = − 91.106186954104 x_{23} = -91.106186954104 x 23 = − 91.106186954104 x 24 = − 37.6991118430775 x_{24} = -37.6991118430775 x 24 = − 37.6991118430775 x 25 = − 21.9911485751286 x_{25} = -21.9911485751286 x 25 = − 21.9911485751286 x 26 = − 42.4115008234622 x_{26} = -42.4115008234622 x 26 = − 42.4115008234622 x 27 = 100.530964914873 x_{27} = 100.530964914873 x 27 = 100.530964914873 x 28 = 59.6902604182061 x_{28} = 59.6902604182061 x 28 = 59.6902604182061 x 29 = − 36.1283155162826 x_{29} = -36.1283155162826 x 29 = − 36.1283155162826 x 30 = − 58.1194640914112 x_{30} = -58.1194640914112 x 30 = − 58.1194640914112 x 31 = 42.4115008234622 x_{31} = 42.4115008234622 x 31 = 42.4115008234622 x 32 = − 6.28318530717959 x_{32} = -6.28318530717959 x 32 = − 6.28318530717959 x 33 = 7.85398163397448 x_{33} = 7.85398163397448 x 33 = 7.85398163397448 x 34 = 28.2743338823081 x_{34} = 28.2743338823081 x 34 = 28.2743338823081 x 35 = − 14.1371669411541 x_{35} = -14.1371669411541 x 35 = − 14.1371669411541 x 36 = − 61.261056745001 x_{36} = -61.261056745001 x 36 = − 61.261056745001 x 37 = − 15.707963267949 x_{37} = -15.707963267949 x 37 = − 15.707963267949 x 38 = 20.4203522483337 x_{38} = 20.4203522483337 x 38 = 20.4203522483337 x 39 = 26.7035375555132 x_{39} = 26.7035375555132 x 39 = 26.7035375555132 x 40 = 78.5398163397448 x_{40} = 78.5398163397448 x 40 = 78.5398163397448 x 41 = − 51.8362787842316 x_{41} = -51.8362787842316 x 41 = − 51.8362787842316 x 42 = 65.9734457253857 x_{42} = 65.9734457253857 x 42 = 65.9734457253857 x 43 = − 94.2477796076938 x_{43} = -94.2477796076938 x 43 = − 94.2477796076938 x 44 = − 56.5486677646163 x_{44} = -56.5486677646163 x 44 = − 56.5486677646163 x 45 = − 50.2654824574367 x_{45} = -50.2654824574367 x 45 = − 50.2654824574367 x 46 = 14.1371669411541 x_{46} = 14.1371669411541 x 46 = 14.1371669411541 x 47 = − 31.4159265358979 x_{47} = -31.4159265358979 x 47 = − 31.4159265358979 x 48 = − 20.4203522483337 x_{48} = -20.4203522483337 x 48 = − 20.4203522483337 x 49 = − 78.5398163397448 x_{49} = -78.5398163397448 x 49 = − 78.5398163397448 x 50 = − 83.2522053201295 x_{50} = -83.2522053201295 x 50 = − 83.2522053201295 x 51 = − 67.5442420521806 x_{51} = -67.5442420521806 x 51 = − 67.5442420521806 x 52 = 72.2566310325652 x_{52} = 72.2566310325652 x 52 = 72.2566310325652 x 53 = − 80.1106126665397 x_{53} = -80.1106126665397 x 53 = − 80.1106126665397 x 54 = 48.6946861306418 x_{54} = 48.6946861306418 x 54 = 48.6946861306418 x 55 = − 1.5707963267949 x_{55} = -1.5707963267949 x 55 = − 1.5707963267949 x 56 = 0 x_{56} = 0 x 56 = 0 x 57 = 23.5619449019235 x_{57} = 23.5619449019235 x 57 = 23.5619449019235 x 58 = 43.9822971502571 x_{58} = 43.9822971502571 x 58 = 43.9822971502571 x 59 = − 75.398223686155 x_{59} = -75.398223686155 x 59 = − 75.398223686155 x 60 = − 17.2787595947439 x_{60} = -17.2787595947439 x 60 = − 17.2787595947439 x 61 = 53.4070751110265 x_{61} = 53.4070751110265 x 61 = 53.4070751110265 x 62 = 80.1106126665397 x_{62} = 80.1106126665397 x 62 = 80.1106126665397 x 63 = 56.5486677646163 x_{63} = 56.5486677646163 x 63 = 56.5486677646163 x 64 = 21.9911485751286 x_{64} = 21.9911485751286 x 64 = 21.9911485751286 x 65 = 64.4026493985908 x_{65} = 64.4026493985908 x 65 = 64.4026493985908 x 66 = 89.5353906273091 x_{66} = 89.5353906273091 x 66 = 89.5353906273091 x 67 = − 43.9822971502571 x_{67} = -43.9822971502571 x 67 = − 43.9822971502571 x 68 = 58.1194640914112 x_{68} = 58.1194640914112 x 68 = 58.1194640914112 x 69 = 40.8407044966673 x_{69} = 40.8407044966673 x 69 = 40.8407044966673 x 70 = 12.5663706143592 x_{70} = 12.5663706143592 x 70 = 12.5663706143592 x 71 = − 64.4026493985908 x_{71} = -64.4026493985908 x 71 = − 64.4026493985908 x 72 = − 3.14159265358979 x_{72} = -3.14159265358979 x 72 = − 3.14159265358979 x 73 = 75.398223686155 x_{73} = 75.398223686155 x 73 = 75.398223686155 x 74 = 84.8230016469244 x_{74} = 84.8230016469244 x 74 = 84.8230016469244 x 75 = 18.8495559215388 x_{75} = 18.8495559215388 x 75 = 18.8495559215388 x 76 = 67.5442420521806 x_{76} = 67.5442420521806 x 76 = 67.5442420521806 x 77 = 92.6769832808989 x_{77} = 92.6769832808989 x 77 = 92.6769832808989 x 78 = 6.28318530717959 x_{78} = 6.28318530717959 x 78 = 6.28318530717959 x 79 = 34.5575191894877 x_{79} = 34.5575191894877 x 79 = 34.5575191894877 x 80 = − 47.1238898038469 x_{80} = -47.1238898038469 x 80 = − 47.1238898038469 x 81 = − 25.1327412287183 x_{81} = -25.1327412287183 x 81 = − 25.1327412287183 x 82 = − 9.42477796076938 x_{82} = -9.42477796076938 x 82 = − 9.42477796076938 x 83 = 81.6814089933346 x_{83} = 81.6814089933346 x 83 = 81.6814089933346 x 84 = 73.8274273593601 x_{84} = 73.8274273593601 x 84 = 73.8274273593601 x 85 = − 95.8185759344887 x_{85} = -95.8185759344887 x 85 = − 95.8185759344887 x 86 = − 89.5353906273091 x_{86} = -89.5353906273091 x 86 = − 89.5353906273091 x 87 = − 59.6902604182061 x_{87} = -59.6902604182061 x 87 = − 59.6902604182061 x 88 = 37.6991118430775 x_{88} = 37.6991118430775 x 88 = 37.6991118430775 x 89 = 9.42477796076938 x_{89} = 9.42477796076938 x 89 = 9.42477796076938 x 90 = 29.845130209103 x_{90} = 29.845130209103 x 90 = 29.845130209103 x 91 = 45.553093477052 x_{91} = 45.553093477052 x 91 = 45.553093477052 x 92 = − 53.4070751110265 x_{92} = -53.4070751110265 x 92 = − 53.4070751110265 x 93 = − 87.9645943005142 x_{93} = -87.9645943005142 x 93 = − 87.9645943005142 x 94 = 97.3893722612836 x_{94} = 97.3893722612836 x 94 = 97.3893722612836 x 95 = 15.707963267949 x_{95} = 15.707963267949 x 95 = 15.707963267949 x 96 = − 29.845130209103 x_{96} = -29.845130209103 x 96 = − 29.845130209103 x 97 = − 39.2699081698724 x_{97} = -39.2699081698724 x 97 = − 39.2699081698724 x 98 = 70.6858347057703 x_{98} = 70.6858347057703 x 98 = 70.6858347057703 x 99 = 94.2477796076938 x_{99} = 94.2477796076938 x 99 = 94.2477796076938 x 100 = − 81.6814089933346 x_{100} = -81.6814089933346 x 100 = − 81.6814089933346 x 101 = 51.8362787842316 x_{101} = 51.8362787842316 x 101 = 51.8362787842316
Точки пересечения с осью координат Y
График пересекает ось Y, когда x равняется 0:tan ( 2 ⋅ 0 ) \tan{\left(2 \cdot 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 ( 2 x ) + 2 = 0 2 \tan^{2}{\left(2 x \right)} + 2 = 0 2 tan 2 ( 2 x ) + 2 = 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 8 ( tan 2 ( 2 x ) + 1 ) tan ( 2 x ) = 0 8 \left(\tan^{2}{\left(2 x \right)} + 1\right) \tan{\left(2 x \right)} = 0 8 ( tan 2 ( 2 x ) + 1 ) tan ( 2 x ) = 0 Решаем это уравнение x 1 = 0 x_{1} = 0 x 1 = 0 Интервалы выпуклости и вогнутости: [ 0 , ∞ ) \left[0, \infty\right) [ 0 , ∞ ) ( − ∞ , 0 ] \left(-\infty, 0\right] ( − ∞ , 0 ]
Горизонтальные асимптоты
Горизонтальные асимптоты найдём с помощью пределов данной функции при x->+oo и x->-oolim x → − ∞ tan ( 2 x ) = ⟨ − ∞ , ∞ ⟩ \lim_{x \to -\infty} \tan{\left(2 x \right)} = \left\langle -\infty, \infty\right\rangle x → − ∞ lim tan ( 2 x ) = ⟨ − ∞ , ∞ ⟩ Возьмём предел y = ⟨ − ∞ , ∞ ⟩ y = \left\langle -\infty, \infty\right\rangle y = ⟨ − ∞ , ∞ ⟩ lim x → ∞ tan ( 2 x ) = ⟨ − ∞ , ∞ ⟩ \lim_{x \to \infty} \tan{\left(2 x \right)} = \left\langle -\infty, \infty\right\rangle x → ∞ lim tan ( 2 x ) = ⟨ − ∞ , ∞ ⟩ Возьмём предел y = ⟨ − ∞ , ∞ ⟩ y = \left\langle -\infty, \infty\right\rangle y = ⟨ − ∞ , ∞ ⟩
Наклонные асимптоты
Наклонную асимптоту можно найти, подсчитав предел функции tan(2*x), делённой на x при x->+oo и x ->-oolim x → − ∞ ( tan ( 2 x ) x ) = lim x → − ∞ ( tan ( 2 x ) x ) \lim_{x \to -\infty}\left(\frac{\tan{\left(2 x \right)}}{x}\right) = \lim_{x \to -\infty}\left(\frac{\tan{\left(2 x \right)}}{x}\right) x → − ∞ lim ( x tan ( 2 x ) ) = x → − ∞ lim ( x tan ( 2 x ) ) Возьмём предел y = x lim x → − ∞ ( tan ( 2 x ) x ) y = x \lim_{x \to -\infty}\left(\frac{\tan{\left(2 x \right)}}{x}\right) y = x x → − ∞ lim ( x tan ( 2 x ) ) lim x → ∞ ( tan ( 2 x ) x ) = lim x → ∞ ( tan ( 2 x ) x ) \lim_{x \to \infty}\left(\frac{\tan{\left(2 x \right)}}{x}\right) = \lim_{x \to \infty}\left(\frac{\tan{\left(2 x \right)}}{x}\right) x → ∞ lim ( x tan ( 2 x ) ) = x → ∞ lim ( x tan ( 2 x ) ) Возьмём предел y = x lim x → ∞ ( tan ( 2 x ) x ) y = x \lim_{x \to \infty}\left(\frac{\tan{\left(2 x \right)}}{x}\right) y = x x → ∞ lim ( x 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{\left(2 x \right)} = - \tan{\left(2 x \right)} tan ( 2 x ) = − tan ( 2 x ) tan ( 2 x ) = tan ( 2 x ) \tan{\left(2 x \right)} = \tan{\left(2 x \right)} tan ( 2 x ) = tan ( 2 x ) 
