График функции
0 2 4 6 8 -8 -6 -4 -2 -10 10 -500 500
Точки пересечения с осью координат X
График функции пересекает ось X при f = 0− 2 tan ( 3 x ) = 0 - 2 \tan{\left (3 x \right )} = 0 − 2 tan ( 3 x ) = 0 Решаем это уравнение Аналитическое решение x 1 = 0 x_{1} = 0 x 1 = 0 Численное решение x 1 = 87.9645943005 x_{1} = 87.9645943005 x 1 = 87.9645943005 x 2 = 48.171087355 x_{2} = 48.171087355 x 2 = 48.171087355 x 3 = 58.643062867 x_{3} = 58.643062867 x 3 = 58.643062867 x 4 = 52.3598775598 x_{4} = 52.3598775598 x 4 = 52.3598775598 x 5 = 78.5398163397 x_{5} = 78.5398163397 x 5 = 78.5398163397 x 6 = − 21.9911485751 x_{6} = -21.9911485751 x 6 = − 21.9911485751 x 7 = 14.6607657168 x_{7} = 14.6607657168 x 7 = 14.6607657168 x 8 = 92.1533845053 x_{8} = 92.1533845053 x 8 = 92.1533845053 x 9 = 63.879050623 x_{9} = 63.879050623 x 9 = 63.879050623 x 10 = 56.5486677646 x_{10} = 56.5486677646 x 10 = 56.5486677646 x 11 = 36.6519142919 x_{11} = 36.6519142919 x 11 = 36.6519142919 x 12 = 26.1799387799 x_{12} = 26.1799387799 x 12 = 26.1799387799 x 13 = − 2.09439510239 x_{13} = -2.09439510239 x 13 = − 2.09439510239 x 14 = − 33.5103216383 x_{14} = -33.5103216383 x 14 = − 33.5103216383 x 15 = 39.7935069455 x_{15} = 39.7935069455 x 15 = 39.7935069455 x 16 = 32.4631240871 x_{16} = 32.4631240871 x 16 = 32.4631240871 x 17 = 68.0678408278 x_{17} = 68.0678408278 x 17 = 68.0678408278 x 18 = − 92.1533845053 x_{18} = -92.1533845053 x 18 = − 92.1533845053 x 19 = 4.18879020479 x_{19} = 4.18879020479 x 19 = 4.18879020479 x 20 = 12.5663706144 x_{20} = 12.5663706144 x 20 = 12.5663706144 x 21 = − 19.8967534727 x_{21} = -19.8967534727 x 21 = − 19.8967534727 x 22 = − 97.3893722613 x_{22} = -97.3893722613 x 22 = − 97.3893722613 x 23 = − 90.0589894029 x_{23} = -90.0589894029 x 23 = − 90.0589894029 x 24 = 17.8023583703 x_{24} = 17.8023583703 x 24 = 17.8023583703 x 25 = − 4.18879020479 x_{25} = -4.18879020479 x 25 = − 4.18879020479 x 26 = − 55.5014702134 x_{26} = -55.5014702134 x 26 = − 55.5014702134 x 27 = 30.3687289847 x_{27} = 30.3687289847 x 27 = 30.3687289847 x 28 = 41.8879020479 x_{28} = 41.8879020479 x 28 = 41.8879020479 x 29 = − 13.6135681656 x_{29} = -13.6135681656 x 29 = − 13.6135681656 x 30 = 61.7846555206 x_{30} = 61.7846555206 x 30 = 61.7846555206 x 31 = − 61.7846555206 x_{31} = -61.7846555206 x 31 = − 61.7846555206 x 32 = − 77.4926187885 x_{32} = -77.4926187885 x 32 = − 77.4926187885 x 33 = 8.37758040957 x_{33} = 8.37758040957 x 33 = 8.37758040957 x 34 = 54.4542726622 x_{34} = 54.4542726622 x 34 = 54.4542726622 x 35 = − 52.3598775598 x_{35} = -52.3598775598 x 35 = − 52.3598775598 x 36 = − 46.0766922527 x_{36} = -46.0766922527 x 36 = − 46.0766922527 x 37 = 43.9822971503 x_{37} = 43.9822971503 x 37 = 43.9822971503 x 38 = 59.6902604182 x_{38} = 59.6902604182 x 38 = 59.6902604182 x 39 = 83.7758040957 x_{39} = 83.7758040957 x 39 = 83.7758040957 x 40 = − 37.6991118431 x_{40} = -37.6991118431 x 40 = − 37.6991118431 x 41 = 21.9911485751 x_{41} = 21.9911485751 x 41 = 21.9911485751 x 42 = 70.1622359302 x_{42} = 70.1622359302 x 42 = 70.1622359302 x 43 = 19.8967534727 x_{43} = 19.8967534727 x 43 = 19.8967534727 x 44 = 0 x_{44} = 0 x 44 = 0 x 45 = − 94.2477796077 x_{45} = -94.2477796077 x 45 = − 94.2477796077 x 46 = − 17.8023583703 x_{46} = -17.8023583703 x 46 = − 17.8023583703 x 47 = − 68.0678408278 x_{47} = -68.0678408278 x 47 = − 68.0678408278 x 48 = 15.7079632679 x_{48} = 15.7079632679 x 48 = 15.7079632679 x 49 = − 57.5958653158 x_{49} = -57.5958653158 x 49 = − 57.5958653158 x 50 = 50.2654824574 x_{50} = 50.2654824574 x 50 = 50.2654824574 x 51 = − 53.407075111 x_{51} = -53.407075111 x 51 = − 53.407075111 x 52 = − 63.879050623 x_{52} = -63.879050623 x 52 = − 63.879050623 x 53 = − 59.6902604182 x_{53} = -59.6902604182 x 53 = − 59.6902604182 x 54 = 28.2743338823 x_{54} = 28.2743338823 x 54 = 28.2743338823 x 55 = − 43.9822971503 x_{55} = -43.9822971503 x 55 = − 43.9822971503 x 56 = − 41.8879020479 x_{56} = -41.8879020479 x 56 = − 41.8879020479 x 57 = − 81.6814089933 x_{57} = -81.6814089933 x 57 = − 81.6814089933 x 58 = 72.2566310326 x_{58} = 72.2566310326 x 58 = 72.2566310326 x 59 = − 6.28318530718 x_{59} = -6.28318530718 x 59 = − 6.28318530718 x 60 = 46.0766922527 x_{60} = 46.0766922527 x 60 = 46.0766922527 x 61 = − 65.9734457254 x_{61} = -65.9734457254 x 61 = − 65.9734457254 x 62 = − 72.2566310326 x_{62} = -72.2566310326 x 62 = − 72.2566310326 x 63 = − 11.5191730632 x_{63} = -11.5191730632 x 63 = − 11.5191730632 x 64 = 85.8701991981 x_{64} = 85.8701991981 x 64 = 85.8701991981 x 65 = 96.3421747101 x_{65} = 96.3421747101 x 65 = 96.3421747101 x 66 = − 24.0855436775 x_{66} = -24.0855436775 x 66 = − 24.0855436775 x 67 = 94.2477796077 x_{67} = 94.2477796077 x 67 = 94.2477796077 x 68 = − 15.7079632679 x_{68} = -15.7079632679 x 68 = − 15.7079632679 x 69 = 100.530964915 x_{69} = 100.530964915 x 69 = 100.530964915 x 70 = − 70.1622359302 x_{70} = -70.1622359302 x 70 = − 70.1622359302 x 71 = 81.6814089933 x_{71} = 81.6814089933 x 71 = 81.6814089933 x 72 = 98.4365698125 x_{72} = 98.4365698125 x 72 = 98.4365698125 x 73 = 65.9734457254 x_{73} = 65.9734457254 x 73 = 65.9734457254 x 74 = − 30.3687289847 x_{74} = -30.3687289847 x 74 = − 30.3687289847 x 75 = 10.471975512 x_{75} = 10.471975512 x 75 = 10.471975512 x 76 = 74.351026135 x_{76} = 74.351026135 x 76 = 74.351026135 x 77 = − 79.5870138909 x_{77} = -79.5870138909 x 77 = − 79.5870138909 x 78 = 90.0589894029 x_{78} = 90.0589894029 x 78 = 90.0589894029 x 79 = − 83.7758040957 x_{79} = -83.7758040957 x 79 = − 83.7758040957 x 80 = − 39.7935069455 x_{80} = -39.7935069455 x 80 = − 39.7935069455 x 81 = − 31.4159265359 x_{81} = -31.4159265359 x 81 = − 31.4159265359 x 82 = 2.09439510239 x_{82} = 2.09439510239 x 82 = 2.09439510239 x 83 = − 85.8701991981 x_{83} = -85.8701991981 x 83 = − 85.8701991981 x 84 = − 75.3982236862 x_{84} = -75.3982236862 x 84 = − 75.3982236862 x 85 = − 9.42477796077 x_{85} = -9.42477796077 x 85 = − 9.42477796077 x 86 = 6.28318530718 x_{86} = 6.28318530718 x 86 = 6.28318530718 x 87 = − 87.9645943005 x_{87} = -87.9645943005 x 87 = − 87.9645943005 x 88 = 76.4454212374 x_{88} = 76.4454212374 x 88 = 76.4454212374 x 89 = − 99.4837673637 x_{89} = -99.4837673637 x 89 = − 99.4837673637 x 90 = − 96.3421747101 x_{90} = -96.3421747101 x 90 = − 96.3421747101 x 91 = − 50.2654824574 x_{91} = -50.2654824574 x 91 = − 50.2654824574 x 92 = − 48.171087355 x_{92} = -48.171087355 x 92 = − 48.171087355 x 93 = − 26.1799387799 x_{93} = -26.1799387799 x 93 = − 26.1799387799 x 94 = − 35.6047167407 x_{94} = -35.6047167407 x 94 = − 35.6047167407 x 95 = 24.0855436775 x_{95} = 24.0855436775 x 95 = 24.0855436775 x 96 = 80.6342114421 x_{96} = 80.6342114421 x 96 = 80.6342114421 x 97 = 34.5575191895 x_{97} = 34.5575191895 x 97 = 34.5575191895 x 98 = − 7.33038285838 x_{98} = -7.33038285838 x 98 = − 7.33038285838 x 99 = 37.6991118431 x_{99} = 37.6991118431 x 99 = 37.6991118431 x 100 = − 74.351026135 x_{100} = -74.351026135 x 100 = − 74.351026135 x 101 = − 28.2743338823 x_{101} = -28.2743338823 x 101 = − 28.2743338823
Точки пересечения с осью координат Y
График пересекает ось Y, когда x равняется 0:− 2 tan ( 0 ⋅ 3 ) - 2 \tan{\left (0 \cdot 3 \right )} − 2 tan ( 0 ⋅ 3 ) 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 − 6 tan 2 ( 3 x ) − 6 = 0 - 6 \tan^{2}{\left (3 x \right )} - 6 = 0 − 6 tan 2 ( 3 x ) − 6 = 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 − 36 ( tan 2 ( 3 x ) + 1 ) tan ( 3 x ) = 0 - 36 \left(\tan^{2}{\left (3 x \right )} + 1\right) \tan{\left (3 x \right )} = 0 − 36 ( tan 2 ( 3 x ) + 1 ) tan ( 3 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 → − ∞ ( − 2 tan ( 3 x ) ) y = \lim_{x \to -\infty}\left(- 2 \tan{\left (3 x \right )}\right) y = x → − ∞ lim ( − 2 tan ( 3 x ) ) True Возьмём предел y = lim x → ∞ ( − 2 tan ( 3 x ) ) y = \lim_{x \to \infty}\left(- 2 \tan{\left (3 x \right )}\right) y = x → ∞ lim ( − 2 tan ( 3 x ) )
Наклонные асимптоты
Наклонную асимптоту можно найти, подсчитав предел функции -2*tan(3*x), делённой на x при x->+oo и x ->-ooTrue Возьмём предел y = x lim x → − ∞ ( − 2 x tan ( 3 x ) ) y = x \lim_{x \to -\infty}\left(- \frac{2}{x} \tan{\left (3 x \right )}\right) y = x x → − ∞ lim ( − x 2 tan ( 3 x ) ) True Возьмём предел y = x lim x → ∞ ( − 2 x tan ( 3 x ) ) y = x \lim_{x \to \infty}\left(- \frac{2}{x} \tan{\left (3 x \right )}\right) y = x x → ∞ lim ( − x 2 tan ( 3 x ) )
Чётность и нечётность функции
Проверим функци чётна или нечётна с помощью соотношений f = f(-x) и f = -f(-x).− 2 tan ( 3 x ) = 2 tan ( 3 x ) - 2 \tan{\left (3 x \right )} = 2 \tan{\left (3 x \right )} − 2 tan ( 3 x ) = 2 tan ( 3 x ) − 2 tan ( 3 x ) = − 2 tan ( 3 x ) - 2 \tan{\left (3 x \right )} = - 2 \tan{\left (3 x \right )} − 2 tan ( 3 x ) = − 2 tan ( 3 x ) 