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 = 0tan ( 3 x − 1 ) = 0 \tan{\left (3 x - 1 \right )} = 0 tan ( 3 x − 1 ) = 0 Решаем это уравнение Аналитическое решение x 1 = 1 3 x_{1} = \frac{1}{3} x 1 = 3 1 Численное решение x 1 = − 87.6312609672 x_{1} = -87.6312609672 x 1 = − 87.6312609672 x 2 = 99.817100697 x_{2} = 99.817100697 x 2 = 99.817100697 x 3 = 70.4955692635 x_{3} = 70.4955692635 x 3 = 70.4955692635 x 4 = 96.6755080434 x_{4} = 96.6755080434 x 4 = 96.6755080434 x 5 = − 67.7345074944 x_{5} = -67.7345074944 x 5 = − 67.7345074944 x 6 = − 21.6578152418 x_{6} = -21.6578152418 x 6 = − 21.6578152418 x 7 = 53.7404084444 x_{7} = 53.7404084444 x 7 = 53.7404084444 x 8 = − 76.112087904 x_{8} = -76.112087904 x 8 = − 76.112087904 x 9 = − 78.2064830064 x_{9} = -78.2064830064 x 9 = − 78.2064830064 x 10 = 86.2035325315 x_{10} = 86.2035325315 x 10 = 86.2035325315 x 11 = − 3.85545687145 x_{11} = -3.85545687145 x 11 = − 3.85545687145 x 12 = 44.3156304836 x_{12} = 44.3156304836 x 12 = 44.3156304836 x 13 = − 96.0088413768 x_{13} = -96.0088413768 x 13 = − 96.0088413768 x 14 = 2.42772843573 x_{14} = 2.42772843573 x 14 = 2.42772843573 x 15 = 42.2212353812 x_{15} = 42.2212353812 x 15 = 42.2212353812 x 16 = − 71.9232976992 x_{16} = -71.9232976992 x 16 = − 71.9232976992 x 17 = − 100.197631582 x_{17} = -100.197631582 x 17 = − 100.197631582 x 18 = − 69.8289025968 x_{18} = -69.8289025968 x 18 = − 69.8289025968 x 19 = − 59.3569270849 x_{19} = -59.3569270849 x 19 = − 59.3569270849 x 20 = − 23.7522103442 x_{20} = -23.7522103442 x 20 = − 23.7522103442 x 21 = − 36.3185809585 x_{21} = -36.3185809585 x 21 = − 36.3185809585 x 22 = − 89.7256560696 x_{22} = -89.7256560696 x 22 = − 89.7256560696 x 23 = − 17.469025037 x_{23} = -17.469025037 x 23 = − 17.469025037 x 24 = 68.4011741611 x_{24} = 68.4011741611 x 24 = 68.4011741611 x 25 = − 32.1297907538 x_{25} = -32.1297907538 x 25 = − 32.1297907538 x 26 = − 30.0353956514 x_{26} = -30.0353956514 x 26 = − 30.0353956514 x 27 = 13.9469014989 x_{27} = 13.9469014989 x 27 = 13.9469014989 x 28 = − 65.6401123921 x_{28} = -65.6401123921 x 28 = − 65.6401123921 x 29 = − 85.5368658648 x_{29} = -85.5368658648 x 29 = − 85.5368658648 x 30 = 0.333333333333 x_{30} = 0.333333333333 x 30 = 0.333333333333 x 31 = 92.4867178386 x_{31} = 92.4867178386 x 31 = 92.4867178386 x 32 = 79.9203472243 x_{32} = 79.9203472243 x 32 = 79.9203472243 x 33 = 6.61651864051 x_{33} = 6.61651864051 x 33 = 6.61651864051 x 34 = − 19.5634201394 x_{34} = -19.5634201394 x 34 = − 19.5634201394 x 35 = 31.7492598692 x_{35} = 31.7492598692 x 35 = 31.7492598692 x 36 = − 43.6489638169 x_{36} = -43.6489638169 x 36 = − 43.6489638169 x 37 = − 12.233037281 x_{37} = -12.233037281 x 37 = − 12.233037281 x 38 = 20.2300868061 x_{38} = 20.2300868061 x 38 = 20.2300868061 x 39 = − 45.7433589193 x_{39} = -45.7433589193 x 39 = − 45.7433589193 x 40 = 50.5988157908 x_{40} = 50.5988157908 x 40 = 50.5988157908 x 41 = 88.2979276338 x_{41} = 88.2979276338 x 41 = 88.2979276338 x 42 = 40.1268402788 x_{42} = 40.1268402788 x 42 = 40.1268402788 x 43 = 82.0147423267 x_{43} = 82.0147423267 x 43 = 82.0147423267 x 44 = − 74.0176928016 x_{44} = -74.0176928016 x 44 = − 74.0176928016 x 45 = 62.1179888539 x_{45} = 62.1179888539 x 45 = 62.1179888539 x 46 = 73.6371619171 x_{46} = 73.6371619171 x 46 = 73.6371619171 x 47 = − 39.4601736121 x_{47} = -39.4601736121 x 47 = − 39.4601736121 x 48 = − 56.2153344313 x_{48} = -56.2153344313 x 48 = − 56.2153344313 x 49 = 72.5899643659 x_{49} = 72.5899643659 x 49 = 72.5899643659 x 50 = − 93.9144462744 x_{50} = -93.9144462744 x 50 = − 93.9144462744 x 51 = − 80.3008781088 x_{51} = -80.3008781088 x 51 = − 80.3008781088 x 52 = − 63.5457172897 x_{52} = -63.5457172897 x 52 = − 63.5457172897 x 53 = 11.8525063965 x_{53} = 11.8525063965 x 53 = 11.8525063965 x 54 = − 15.3746299346 x_{54} = -15.3746299346 x 54 = − 15.3746299346 x 55 = 57.9291986491 x_{55} = 57.9291986491 x 55 = 57.9291986491 x 56 = 51.646013342 x_{56} = 51.646013342 x 56 = 51.646013342 x 57 = − 54.1209393289 x_{57} = -54.1209393289 x 57 = − 54.1209393289 x 58 = 4.52212353812 x_{58} = 4.52212353812 x 58 = 4.52212353812 x 59 = − 5.94985197385 x_{59} = -5.94985197385 x 59 = − 5.94985197385 x 60 = − 61.4513221873 x_{60} = -61.4513221873 x 60 = − 61.4513221873 x 61 = − 27.941000549 x_{61} = -27.941000549 x 61 = − 27.941000549 x 62 = − 98.1032364791 x_{62} = -98.1032364791 x 62 = − 98.1032364791 x 63 = 55.8348035468 x_{63} = 55.8348035468 x 63 = 55.8348035468 x 64 = − 1.76106176906 x_{64} = -1.76106176906 x 64 = − 1.76106176906 x 65 = − 10.1386421786 x_{65} = -10.1386421786 x 65 = − 10.1386421786 x 66 = 60.0235937515 x_{66} = 60.0235937515 x 66 = 60.0235937515 x 67 = − 34.2241858562 x_{67} = -34.2241858562 x 67 = − 34.2241858562 x 68 = − 41.5545687145 x_{68} = -41.5545687145 x 68 = − 41.5545687145 x 69 = − 81.34807566 x_{69} = -81.34807566 x 69 = − 81.34807566 x 70 = − 25.8466054466 x_{70} = -25.8466054466 x 70 = − 25.8466054466 x 71 = 9.7581112941 x_{71} = 9.7581112941 x 71 = 9.7581112941 x 72 = − 47.8377540217 x_{72} = -47.8377540217 x 72 = − 47.8377540217 x 73 = 90.3923227362 x_{73} = 90.3923227362 x 73 = 90.3923227362 x 74 = − 52.0265442265 x_{74} = -52.0265442265 x 74 = − 52.0265442265 x 75 = 46.410025586 x_{75} = 46.410025586 x 75 = 46.410025586 x 76 = 29.6548647668 x_{76} = 29.6548647668 x 76 = 29.6548647668 x 77 = 22.3244819085 x_{77} = 22.3244819085 x 77 = 22.3244819085 x 78 = 94.581112941 x_{78} = 94.581112941 x 78 = 94.581112941 x 79 = 33.8436549716 x_{79} = 33.8436549716 x 79 = 33.8436549716 x 80 = − 14.3274323834 x_{80} = -14.3274323834 x 80 = − 14.3274323834 x 81 = − 8.04424707624 x_{81} = -8.04424707624 x 81 = − 8.04424707624 x 82 = 24.4188770109 x_{82} = 24.4188770109 x 82 = 24.4188770109 x 83 = 16.0412966013 x_{83} = 16.0412966013 x 83 = 16.0412966013 x 84 = − 49.9321491241 x_{84} = -49.9321491241 x 84 = − 49.9321491241 x 85 = 38.0324451764 x_{85} = 38.0324451764 x 85 = 38.0324451764 x 86 = 64.2123839563 x_{86} = 64.2123839563 x 86 = 64.2123839563 x 87 = 66.3067790587 x_{87} = 66.3067790587 x 87 = 66.3067790587 x 88 = − 37.3657785097 x_{88} = -37.3657785097 x 88 = − 37.3657785097 x 89 = 84.1091374291 x_{89} = 84.1091374291 x 89 = 84.1091374291 x 90 = 28.6076672156 x_{90} = 28.6076672156 x 90 = 28.6076672156 x 91 = 18.1356917037 x_{91} = 18.1356917037 x 91 = 18.1356917037 x 92 = 48.5044206884 x_{92} = 48.5044206884 x 92 = 48.5044206884 x 93 = 75.7315570195 x_{93} = 75.7315570195 x 93 = 75.7315570195 x 94 = − 91.820051172 x_{94} = -91.820051172 x 94 = − 91.820051172 x 95 = 26.5132721132 x_{95} = 26.5132721132 x 95 = 26.5132721132 x 96 = 97.7227055946 x_{96} = 97.7227055946 x 96 = 97.7227055946 x 97 = 77.8259521219 x_{97} = 77.8259521219 x 97 = 77.8259521219 x 98 = − 58.3097295337 x_{98} = -58.3097295337 x 98 = − 58.3097295337 x 99 = 7.66371619171 x_{99} = 7.66371619171 x 99 = 7.66371619171 x 100 = − 83.4424707624 x_{100} = -83.4424707624 x 100 = − 83.4424707624 x 101 = 35.938050074 x_{101} = 35.938050074 x 101 = 35.938050074
Точки пересечения с осью координат Y
График пересекает ось Y, когда x равняется 0:tan ( − 1 + 0 ⋅ 3 ) \tan{\left (-1 + 0 \cdot 3 \right )} tan ( − 1 + 0 ⋅ 3 ) f ( 0 ) = − tan ( 1 ) f{\left (0 \right )} = - \tan{\left (1 \right )} f ( 0 ) = − tan ( 1 ) (0, -tan(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 3 tan 2 ( 3 x − 1 ) + 3 = 0 3 \tan^{2}{\left (3 x - 1 \right )} + 3 = 0 3 tan 2 ( 3 x − 1 ) + 3 = 0 Решаем это уравнение
Горизонтальные асимптоты
Горизонтальные асимптоты найдём с помощью пределов данной функции при x->+oo и x->-ooTrue Возьмём предел y = lim x → − ∞ tan ( 3 x − 1 ) y = \lim_{x \to -\infty} \tan{\left (3 x - 1 \right )} y = x → − ∞ lim tan ( 3 x − 1 ) True Возьмём предел y = lim x → ∞ tan ( 3 x − 1 ) y = \lim_{x \to \infty} \tan{\left (3 x - 1 \right )} y = x → ∞ lim tan ( 3 x − 1 )
Наклонные асимптоты
Наклонную асимптоту можно найти, подсчитав предел функции tan(3*x - 1), делённой на x при x->+oo и x ->-ooTrue Возьмём предел y = x lim x → − ∞ ( 1 x tan ( 3 x − 1 ) ) y = x \lim_{x \to -\infty}\left(\frac{1}{x} \tan{\left (3 x - 1 \right )}\right) y = x x → − ∞ lim ( x 1 tan ( 3 x − 1 ) ) True Возьмём предел y = x lim x → ∞ ( 1 x tan ( 3 x − 1 ) ) y = x \lim_{x \to \infty}\left(\frac{1}{x} \tan{\left (3 x - 1 \right )}\right) y = x x → ∞ lim ( x 1 tan ( 3 x − 1 ) )
Чётность и нечётность функции
Проверим функци чётна или нечётна с помощью соотношений f = f(-x) и f = -f(-x).tan ( 3 x − 1 ) = − tan ( 3 x + 1 ) \tan{\left (3 x - 1 \right )} = - \tan{\left (3 x + 1 \right )} tan ( 3 x − 1 ) = − tan ( 3 x + 1 ) tan ( 3 x − 1 ) = − − 1 tan ( 3 x + 1 ) \tan{\left (3 x - 1 \right )} = - -1 \tan{\left (3 x + 1 \right )} tan ( 3 x − 1 ) = − − 1 tan ( 3 x + 1 ) 