График функции
0 2 4 6 8 -8 -6 -4 -2 -10 10 -100 100
Точки пересечения с осью координат X
График функции пересекает ось X при f = 0− tan ( 2 x ) + 1 = 0 - \tan{\left (2 x \right )} + 1 = 0 − tan ( 2 x ) + 1 = 0 Решаем это уравнение Аналитическое решение x 1 = π 8 x_{1} = \frac{\pi}{8} x 1 = 8 π Численное решение x 1 = − 100.138265833 x_{1} = -100.138265833 x 1 = − 100.138265833 x 2 = 1.96349540849 x_{2} = 1.96349540849 x 2 = 1.96349540849 x 3 = − 31.0232274542 x_{3} = -31.0232274542 x 3 = − 31.0232274542 x 4 = 86.7864970554 x_{4} = 86.7864970554 x 4 = 86.7864970554 x 5 = 93.0696823626 x_{5} = 93.0696823626 x 5 = 93.0696823626 x 6 = − 82.8595062384 x_{6} = -82.8595062384 x 6 = − 82.8595062384 x 7 = − 73.4347282777 x_{7} = -73.4347282777 x 7 = − 73.4347282777 x 8 = − 7.46128255228 x_{8} = -7.46128255228 x 8 = − 7.46128255228 x 9 = 71.0785337875 x_{9} = 71.0785337875 x 9 = 71.0785337875 x 10 = − 46.7311907221 x_{10} = -46.7311907221 x 10 = − 46.7311907221 x 11 = 88.3572933822 x_{11} = 88.3572933822 x 11 = 88.3572933822 x 12 = − 13.7444678595 x_{12} = -13.7444678595 x 12 = − 13.7444678595 x 13 = − 45.1603943954 x_{13} = -45.1603943954 x 13 = − 45.1603943954 x 14 = 56.9413668463 x_{14} = 56.9413668463 x 14 = 56.9413668463 x 15 = − 1.1780972451 x_{15} = -1.1780972451 x 15 = − 1.1780972451 x 16 = − 38.8772090882 x_{16} = -38.8772090882 x 16 = − 38.8772090882 x 17 = − 24.740042147 x_{17} = -24.740042147 x 17 = − 24.740042147 x 18 = − 27.8816348006 x_{18} = -27.8816348006 x 18 = − 27.8816348006 x 19 = 60.0829594999 x_{19} = 60.0829594999 x 19 = 60.0829594999 x 20 = 53.7997741927 x_{20} = 53.7997741927 x 20 = 53.7997741927 x 21 = 9.81747704247 x_{21} = 9.81747704247 x 21 = 9.81747704247 x 22 = 74.2201264411 x_{22} = 74.2201264411 x 22 = 74.2201264411 x 23 = 12.9590696961 x_{23} = 12.9590696961 x 23 = 12.9590696961 x 24 = 8.24668071567 x_{24} = 8.24668071567 x 24 = 8.24668071567 x 25 = − 59.2975613365 x_{25} = -59.2975613365 x 25 = − 59.2975613365 x 26 = 42.8041999052 x_{26} = 42.8041999052 x 26 = 42.8041999052 x 27 = 96.2112750162 x_{27} = 96.2112750162 x 27 = 96.2112750162 x 28 = − 79.7179135848 x_{28} = -79.7179135848 x 28 = − 79.7179135848 x 29 = 82.074108075 x_{29} = 82.074108075 x 29 = 82.074108075 x 30 = 14.5298660229 x_{30} = 14.5298660229 x 30 = 14.5298660229 x 31 = 5.10508806208 x_{31} = 5.10508806208 x 31 = 5.10508806208 x 32 = − 71.8639319509 x_{32} = -71.8639319509 x 32 = − 71.8639319509 x 33 = − 67.1515429705 x_{33} = -67.1515429705 x 33 = − 67.1515429705 x 34 = 20.81305133 x_{34} = 20.81305133 x 34 = 20.81305133 x 35 = − 29.4524311274 x_{35} = -29.4524311274 x 35 = − 29.4524311274 x 36 = − 2.74889357189 x_{36} = -2.74889357189 x 36 = − 2.74889357189 x 37 = − 90.7134878724 x_{37} = -90.7134878724 x 37 = − 90.7134878724 x 38 = 52.2289778659 x_{38} = 52.2289778659 x 38 = 52.2289778659 x 39 = − 95.4258768528 x_{39} = -95.4258768528 x 39 = − 95.4258768528 x 40 = 78.9325154214 x_{40} = 78.9325154214 x 40 = 78.9325154214 x 41 = − 65.5807466437 x_{41} = -65.5807466437 x 41 = − 65.5807466437 x 42 = − 89.1426915456 x_{42} = -89.1426915456 x 42 = − 89.1426915456 x 43 = 67.9369411339 x_{43} = 67.9369411339 x 43 = 67.9369411339 x 44 = − 86.001098892 x_{44} = -86.001098892 x 44 = − 86.001098892 x 45 = 97.782071343 x_{45} = 97.782071343 x 45 = 97.782071343 x 46 = − 9.03207887907 x_{46} = -9.03207887907 x 46 = − 9.03207887907 x 47 = 31.8086256176 x_{47} = 31.8086256176 x 47 = 31.8086256176 x 48 = 80.5033117482 x_{48} = 80.5033117482 x 48 = 80.5033117482 x 49 = − 75.0055246045 x_{49} = -75.0055246045 x 49 = − 75.0055246045 x 50 = − 96.9966731796 x_{50} = -96.9966731796 x 50 = − 96.9966731796 x 51 = 64.7953484803 x_{51} = 64.7953484803 x 51 = 64.7953484803 x 52 = − 68.7223392973 x_{52} = -68.7223392973 x 52 = − 68.7223392973 x 53 = 22.3838476568 x_{53} = 22.3838476568 x 53 = 22.3838476568 x 54 = − 56.1559686829 x_{54} = -56.1559686829 x 54 = − 56.1559686829 x 55 = 0.392699081699 x_{55} = 0.392699081699 x 55 = 0.392699081699 x 56 = − 37.3064127614 x_{56} = -37.3064127614 x 56 = − 37.3064127614 x 57 = 41.2334035784 x_{57} = 41.2334035784 x 57 = 41.2334035784 x 58 = 100.923663997 x_{58} = 100.923663997 x 58 = 100.923663997 x 59 = − 5.89048622548 x_{59} = -5.89048622548 x 59 = − 5.89048622548 x 60 = − 57.7267650097 x_{60} = -57.7267650097 x 60 = − 57.7267650097 x 61 = 49.0873852123 x_{61} = 49.0873852123 x 61 = 49.0873852123 x 62 = − 49.8727833757 x_{62} = -49.8727833757 x 62 = − 49.8727833757 x 63 = − 23.1692458202 x_{63} = -23.1692458202 x 63 = − 23.1692458202 x 64 = 19.2422550032 x_{64} = 19.2422550032 x 64 = 19.2422550032 x 65 = 58.5121631731 x_{65} = 58.5121631731 x 65 = 58.5121631731 x 66 = 66.3661448071 x_{66} = 66.3661448071 x 66 = 66.3661448071 x 67 = − 35.7356164346 x_{67} = -35.7356164346 x 67 = − 35.7356164346 x 68 = − 21.5984494934 x_{68} = -21.5984494934 x 68 = − 21.5984494934 x 69 = − 20.0276531666 x_{69} = -20.0276531666 x 69 = − 20.0276531666 x 70 = − 43.5895980686 x_{70} = -43.5895980686 x 70 = − 43.5895980686 x 71 = 50.6581815391 x_{71} = 50.6581815391 x 71 = 50.6581815391 x 72 = 94.6404786894 x_{72} = 94.6404786894 x 72 = 94.6404786894 x 73 = − 64.0099503169 x_{73} = -64.0099503169 x 73 = − 64.0099503169 x 74 = 72.6493301143 x_{74} = 72.6493301143 x 74 = 72.6493301143 x 75 = − 42.0188017418 x_{75} = -42.0188017418 x 75 = − 42.0188017418 x 76 = 36.521014598 x_{76} = 36.521014598 x 76 = 36.521014598 x 77 = − 34.1648201078 x_{77} = -34.1648201078 x 77 = − 34.1648201078 x 78 = 6.67588438888 x_{78} = 6.67588438888 x 78 = 6.67588438888 x 79 = − 93.855080526 x_{79} = -93.855080526 x 79 = − 93.855080526 x 80 = − 51.4435797025 x_{80} = -51.4435797025 x 80 = − 51.4435797025 x 81 = 34.9502182712 x_{81} = 34.9502182712 x 81 = 34.9502182712 x 82 = 28.667032964 x_{82} = 28.667032964 x 82 = 28.667032964 x 83 = − 87.5718952188 x_{83} = -87.5718952188 x 83 = − 87.5718952188 x 84 = − 12.1736715327 x_{84} = -12.1736715327 x 84 = − 12.1736715327 x 85 = 27.0962366372 x_{85} = 27.0962366372 x 85 = 27.0962366372 x 86 = − 60.8683576633 x_{86} = -60.8683576633 x 86 = − 60.8683576633 x 87 = 30.2378292908 x_{87} = 30.2378292908 x 87 = 30.2378292908 x 88 = − 16.886060513 x_{88} = -16.886060513 x 88 = − 16.886060513 x 89 = 75.7909227679 x_{89} = 75.7909227679 x 89 = 75.7909227679 x 90 = 38.0918109248 x_{90} = 38.0918109248 x 90 = 38.0918109248 x 91 = − 15.3152641863 x_{91} = -15.3152641863 x 91 = − 15.3152641863 x 92 = 23.9546439836 x_{92} = 23.9546439836 x 92 = 23.9546439836 x 93 = 85.2157007286 x_{93} = 85.2157007286 x 93 = 85.2157007286 x 94 = 16.1006623496 x_{94} = 16.1006623496 x 94 = 16.1006623496 x 95 = − 78.147117258 x_{95} = -78.147117258 x 95 = − 78.147117258 x 96 = 44.374996232 x_{96} = 44.374996232 x 96 = 44.374996232 x 97 = 45.9457925588 x_{97} = 45.9457925588 x 97 = 45.9457925588 x 98 = − 53.0143760293 x_{98} = -53.0143760293 x 98 = − 53.0143760293 x 99 = − 81.2887099116 x_{99} = -81.2887099116 x 99 = − 81.2887099116 x 100 = 89.928089709 x_{100} = 89.928089709 x 100 = 89.928089709 x 101 = 63.2245521535 x_{101} = 63.2245521535 x 101 = 63.2245521535
Точки пересечения с осью координат Y
График пересекает ось Y, когда x равняется 0:− tan ( 0 ⋅ 2 ) + 1 - \tan{\left (0 \cdot 2 \right )} + 1 − tan ( 0 ⋅ 2 ) + 1 f ( 0 ) = 1 f{\left (0 \right )} = 1 f ( 0 ) = 1 (0, 1)
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: (-oo, 0] [0, oo)
Горизонтальные асимптоты
Горизонтальные асимптоты найдём с помощью пределов данной функции при x->+oo и x->-ooTrue Возьмём предел y = lim x → − ∞ ( − tan ( 2 x ) + 1 ) y = \lim_{x \to -\infty}\left(- \tan{\left (2 x \right )} + 1\right) y = x → − ∞ lim ( − tan ( 2 x ) + 1 ) True Возьмём предел y = lim x → ∞ ( − tan ( 2 x ) + 1 ) y = \lim_{x \to \infty}\left(- \tan{\left (2 x \right )} + 1\right) y = x → ∞ lim ( − tan ( 2 x ) + 1 )
Наклонные асимптоты
Наклонную асимптоту можно найти, подсчитав предел функции 1 - tan(2*x), делённой на x при x->+oo и x ->-ooTrue Возьмём предел y = x lim x → − ∞ ( 1 x ( − tan ( 2 x ) + 1 ) ) y = x \lim_{x \to -\infty}\left(\frac{1}{x} \left(- \tan{\left (2 x \right )} + 1\right)\right) y = x x → − ∞ lim ( x 1 ( − tan ( 2 x ) + 1 ) ) True Возьмём предел y = x lim x → ∞ ( 1 x ( − tan ( 2 x ) + 1 ) ) y = x \lim_{x \to \infty}\left(\frac{1}{x} \left(- \tan{\left (2 x \right )} + 1\right)\right) y = x x → ∞ lim ( x 1 ( − tan ( 2 x ) + 1 ) )
Чётность и нечётность функции
Проверим функци чётна или нечётна с помощью соотношений f = f(-x) и f = -f(-x).− tan ( 2 x ) + 1 = tan ( 2 x ) + 1 - \tan{\left (2 x \right )} + 1 = \tan{\left (2 x \right )} + 1 − tan ( 2 x ) + 1 = tan ( 2 x ) + 1 − tan ( 2 x ) + 1 = − tan ( 2 x ) − 1 - \tan{\left (2 x \right )} + 1 = - \tan{\left (2 x \right )} - 1 − tan ( 2 x ) + 1 = − tan ( 2 x ) − 1 