График функции
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Точки пересечения с осью координат X
График функции пересекает ось X при f = 0tan ( 3 x ) = 0 \tan{\left(3 x \right)} = 0 tan ( 3 x ) = 0 Решаем это уравнение Аналитическое решение x 1 = 0 x_{1} = 0 x 1 = 0 Численное решение x 1 = 59.6902604182061 x_{1} = 59.6902604182061 x 1 = 59.6902604182061 x 2 = 74.3510261349584 x_{2} = 74.3510261349584 x 2 = 74.3510261349584 x 3 = 21.9911485751286 x_{3} = 21.9911485751286 x 3 = 21.9911485751286 x 4 = − 55.5014702134197 x_{4} = -55.5014702134197 x 4 = − 55.5014702134197 x 5 = − 87.9645943005142 x_{5} = -87.9645943005142 x 5 = − 87.9645943005142 x 6 = 83.7758040957278 x_{6} = 83.7758040957278 x 6 = 83.7758040957278 x 7 = 56.5486677646163 x_{7} = 56.5486677646163 x 7 = 56.5486677646163 x 8 = − 75.398223686155 x_{8} = -75.398223686155 x 8 = − 75.398223686155 x 9 = − 99.4837673636768 x_{9} = -99.4837673636768 x 9 = − 99.4837673636768 x 10 = − 96.342174710087 x_{10} = -96.342174710087 x 10 = − 96.342174710087 x 11 = 26.1799387799149 x_{11} = 26.1799387799149 x 11 = 26.1799387799149 x 12 = − 33.5103216382911 x_{12} = -33.5103216382911 x 12 = − 33.5103216382911 x 13 = − 43.9822971502571 x_{13} = -43.9822971502571 x 13 = − 43.9822971502571 x 14 = − 85.870199198121 x_{14} = -85.870199198121 x 14 = − 85.870199198121 x 15 = 48.1710873550435 x_{15} = 48.1710873550435 x 15 = 48.1710873550435 x 16 = − 90.0589894029074 x_{16} = -90.0589894029074 x 16 = − 90.0589894029074 x 17 = 39.7935069454707 x_{17} = 39.7935069454707 x 17 = 39.7935069454707 x 18 = 6.28318530717959 x_{18} = 6.28318530717959 x 18 = 6.28318530717959 x 19 = 68.0678408277789 x_{19} = 68.0678408277789 x 19 = 68.0678408277789 x 20 = − 30.3687289847013 x_{20} = -30.3687289847013 x 20 = − 30.3687289847013 x 21 = 72.2566310325652 x_{21} = 72.2566310325652 x 21 = 72.2566310325652 x 22 = − 52.3598775598299 x_{22} = -52.3598775598299 x 22 = − 52.3598775598299 x 23 = 10.471975511966 x_{23} = 10.471975511966 x 23 = 10.471975511966 x 24 = 2.0943951023932 x_{24} = 2.0943951023932 x 24 = 2.0943951023932 x 25 = − 72.2566310325652 x_{25} = -72.2566310325652 x 25 = − 72.2566310325652 x 26 = − 53.4070751110265 x_{26} = -53.4070751110265 x 26 = − 53.4070751110265 x 27 = − 79.5870138909414 x_{27} = -79.5870138909414 x 27 = − 79.5870138909414 x 28 = − 26.1799387799149 x_{28} = -26.1799387799149 x 28 = − 26.1799387799149 x 29 = − 21.9911485751286 x_{29} = -21.9911485751286 x 29 = − 21.9911485751286 x 30 = 54.4542726622231 x_{30} = 54.4542726622231 x 30 = 54.4542726622231 x 31 = − 31.4159265358979 x_{31} = -31.4159265358979 x 31 = − 31.4159265358979 x 32 = − 48.1710873550435 x_{32} = -48.1710873550435 x 32 = − 48.1710873550435 x 33 = 98.4365698124802 x_{33} = 98.4365698124802 x 33 = 98.4365698124802 x 34 = − 92.1533845053006 x_{34} = -92.1533845053006 x 34 = − 92.1533845053006 x 35 = − 4.18879020478639 x_{35} = -4.18879020478639 x 35 = − 4.18879020478639 x 36 = − 39.7935069454707 x_{36} = -39.7935069454707 x 36 = − 39.7935069454707 x 37 = 34.5575191894877 x_{37} = 34.5575191894877 x 37 = 34.5575191894877 x 38 = − 74.3510261349584 x_{38} = -74.3510261349584 x 38 = − 74.3510261349584 x 39 = − 63.8790506229925 x_{39} = -63.8790506229925 x 39 = − 63.8790506229925 x 40 = − 94.2477796076938 x_{40} = -94.2477796076938 x 40 = − 94.2477796076938 x 41 = 12.5663706143592 x_{41} = 12.5663706143592 x 41 = 12.5663706143592 x 42 = − 24.0855436775217 x_{42} = -24.0855436775217 x 42 = − 24.0855436775217 x 43 = 80.634211442138 x_{43} = 80.634211442138 x 43 = 80.634211442138 x 44 = 90.0589894029074 x_{44} = 90.0589894029074 x 44 = 90.0589894029074 x 45 = − 50.2654824574367 x_{45} = -50.2654824574367 x 45 = − 50.2654824574367 x 46 = − 97.3893722612836 x_{46} = -97.3893722612836 x 46 = − 97.3893722612836 x 47 = 15.707963267949 x_{47} = 15.707963267949 x 47 = 15.707963267949 x 48 = − 35.6047167406843 x_{48} = -35.6047167406843 x 48 = − 35.6047167406843 x 49 = 96.342174710087 x_{49} = 96.342174710087 x 49 = 96.342174710087 x 50 = 63.8790506229925 x_{50} = 63.8790506229925 x 50 = 63.8790506229925 x 51 = 8.37758040957278 x_{51} = 8.37758040957278 x 51 = 8.37758040957278 x 52 = 24.0855436775217 x_{52} = 24.0855436775217 x 52 = 24.0855436775217 x 53 = − 7.33038285837618 x_{53} = -7.33038285837618 x 53 = − 7.33038285837618 x 54 = 0 x_{54} = 0 x 54 = 0 x 55 = − 13.6135681655558 x_{55} = -13.6135681655558 x 55 = − 13.6135681655558 x 56 = 19.8967534727354 x_{56} = 19.8967534727354 x 56 = 19.8967534727354 x 57 = − 37.6991118430775 x_{57} = -37.6991118430775 x 57 = − 37.6991118430775 x 58 = 61.7846555205993 x_{58} = 61.7846555205993 x 58 = 61.7846555205993 x 59 = 65.9734457253857 x_{59} = 65.9734457253857 x 59 = 65.9734457253857 x 60 = 85.870199198121 x_{60} = 85.870199198121 x 60 = 85.870199198121 x 61 = − 61.7846555205993 x_{61} = -61.7846555205993 x 61 = − 61.7846555205993 x 62 = 81.6814089933346 x_{62} = 81.6814089933346 x 62 = 81.6814089933346 x 63 = 17.8023583703422 x_{63} = 17.8023583703422 x 63 = 17.8023583703422 x 64 = − 19.8967534727354 x_{64} = -19.8967534727354 x 64 = − 19.8967534727354 x 65 = − 41.8879020478639 x_{65} = -41.8879020478639 x 65 = − 41.8879020478639 x 66 = 41.8879020478639 x_{66} = 41.8879020478639 x 66 = 41.8879020478639 x 67 = − 81.6814089933346 x_{67} = -81.6814089933346 x 67 = − 81.6814089933346 x 68 = 28.2743338823081 x_{68} = 28.2743338823081 x 68 = 28.2743338823081 x 69 = − 46.0766922526503 x_{69} = -46.0766922526503 x 69 = − 46.0766922526503 x 70 = 100.530964914873 x_{70} = 100.530964914873 x 70 = 100.530964914873 x 71 = 52.3598775598299 x_{71} = 52.3598775598299 x 71 = 52.3598775598299 x 72 = − 77.4926187885482 x_{72} = -77.4926187885482 x 72 = − 77.4926187885482 x 73 = 70.162235930172 x_{73} = 70.162235930172 x 73 = 70.162235930172 x 74 = − 2.0943951023932 x_{74} = -2.0943951023932 x 74 = − 2.0943951023932 x 75 = 4.18879020478639 x_{75} = 4.18879020478639 x 75 = 4.18879020478639 x 76 = 76.4454212373516 x_{76} = 76.4454212373516 x 76 = 76.4454212373516 x 77 = − 70.162235930172 x_{77} = -70.162235930172 x 77 = − 70.162235930172 x 78 = − 57.5958653158129 x_{78} = -57.5958653158129 x 78 = − 57.5958653158129 x 79 = 87.9645943005142 x_{79} = 87.9645943005142 x 79 = 87.9645943005142 x 80 = 58.6430628670095 x_{80} = 58.6430628670095 x 80 = 58.6430628670095 x 81 = − 11.5191730631626 x_{81} = -11.5191730631626 x 81 = − 11.5191730631626 x 82 = 32.4631240870945 x_{82} = 32.4631240870945 x 82 = 32.4631240870945 x 83 = 37.6991118430775 x_{83} = 37.6991118430775 x 83 = 37.6991118430775 x 84 = − 28.2743338823081 x_{84} = -28.2743338823081 x 84 = − 28.2743338823081 x 85 = − 6.28318530717959 x_{85} = -6.28318530717959 x 85 = − 6.28318530717959 x 86 = 43.9822971502571 x_{86} = 43.9822971502571 x 86 = 43.9822971502571 x 87 = − 17.8023583703422 x_{87} = -17.8023583703422 x 87 = − 17.8023583703422 x 88 = 94.2477796076938 x_{88} = 94.2477796076938 x 88 = 94.2477796076938 x 89 = − 68.0678408277789 x_{89} = -68.0678408277789 x 89 = − 68.0678408277789 x 90 = − 59.6902604182061 x_{90} = -59.6902604182061 x 90 = − 59.6902604182061 x 91 = − 9.42477796076938 x_{91} = -9.42477796076938 x 91 = − 9.42477796076938 x 92 = − 15.707963267949 x_{92} = -15.707963267949 x 92 = − 15.707963267949 x 93 = 30.3687289847013 x_{93} = 30.3687289847013 x 93 = 30.3687289847013 x 94 = 78.5398163397448 x_{94} = 78.5398163397448 x 94 = 78.5398163397448 x 95 = 92.1533845053006 x_{95} = 92.1533845053006 x 95 = 92.1533845053006 x 96 = − 65.9734457253857 x_{96} = -65.9734457253857 x 96 = − 65.9734457253857 x 97 = 36.6519142918809 x_{97} = 36.6519142918809 x 97 = 36.6519142918809 x 98 = 50.2654824574367 x_{98} = 50.2654824574367 x 98 = 50.2654824574367 x 99 = 46.0766922526503 x_{99} = 46.0766922526503 x 99 = 46.0766922526503 x 100 = − 83.7758040957278 x_{100} = -83.7758040957278 x 100 = − 83.7758040957278 x 101 = 14.6607657167524 x_{101} = 14.6607657167524 x 101 = 14.6607657167524
Точки пересечения с осью координат Y
График пересекает ось Y, когда x равняется 0:tan ( 3 ⋅ 0 ) \tan{\left(3 \cdot 0 \right)} tan ( 3 ⋅ 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 3 tan 2 ( 3 x ) + 3 = 0 3 \tan^{2}{\left(3 x \right)} + 3 = 0 3 tan 2 ( 3 x ) + 3 = 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 18 ( tan 2 ( 3 x ) + 1 ) tan ( 3 x ) = 0 18 \left(\tan^{2}{\left(3 x \right)} + 1\right) \tan{\left(3 x \right)} = 0 18 ( 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: [ 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 ( 3 x ) = ⟨ − ∞ , ∞ ⟩ \lim_{x \to -\infty} \tan{\left(3 x \right)} = \left\langle -\infty, \infty\right\rangle x → − ∞ lim tan ( 3 x ) = ⟨ − ∞ , ∞ ⟩ Возьмём предел y = ⟨ − ∞ , ∞ ⟩ y = \left\langle -\infty, \infty\right\rangle y = ⟨ − ∞ , ∞ ⟩ lim x → ∞ tan ( 3 x ) = ⟨ − ∞ , ∞ ⟩ \lim_{x \to \infty} \tan{\left(3 x \right)} = \left\langle -\infty, \infty\right\rangle x → ∞ lim tan ( 3 x ) = ⟨ − ∞ , ∞ ⟩ Возьмём предел y = ⟨ − ∞ , ∞ ⟩ y = \left\langle -\infty, \infty\right\rangle y = ⟨ − ∞ , ∞ ⟩
Наклонные асимптоты
Наклонную асимптоту можно найти, подсчитав предел функции tan(3*x), делённой на x при x->+oo и x ->-oolim x → − ∞ ( tan ( 3 x ) x ) = lim x → − ∞ ( tan ( 3 x ) x ) \lim_{x \to -\infty}\left(\frac{\tan{\left(3 x \right)}}{x}\right) = \lim_{x \to -\infty}\left(\frac{\tan{\left(3 x \right)}}{x}\right) x → − ∞ lim ( x tan ( 3 x ) ) = x → − ∞ lim ( x tan ( 3 x ) ) Возьмём предел y = x lim x → − ∞ ( tan ( 3 x ) x ) y = x \lim_{x \to -\infty}\left(\frac{\tan{\left(3 x \right)}}{x}\right) y = x x → − ∞ lim ( x tan ( 3 x ) ) lim x → ∞ ( tan ( 3 x ) x ) = lim x → ∞ ( tan ( 3 x ) x ) \lim_{x \to \infty}\left(\frac{\tan{\left(3 x \right)}}{x}\right) = \lim_{x \to \infty}\left(\frac{\tan{\left(3 x \right)}}{x}\right) x → ∞ lim ( x tan ( 3 x ) ) = x → ∞ lim ( x tan ( 3 x ) ) Возьмём предел y = x lim x → ∞ ( tan ( 3 x ) x ) y = x \lim_{x \to \infty}\left(\frac{\tan{\left(3 x \right)}}{x}\right) y = x x → ∞ lim ( x tan ( 3 x ) )
Чётность и нечётность функции
Проверим функци чётна или нечётна с помощью соотношений f = f(-x) и f = -f(-x).tan ( 3 x ) = − tan ( 3 x ) \tan{\left(3 x \right)} = - \tan{\left(3 x \right)} tan ( 3 x ) = − tan ( 3 x ) tan ( 3 x ) = tan ( 3 x ) \tan{\left(3 x \right)} = \tan{\left(3 x \right)} tan ( 3 x ) = tan ( 3 x ) 
