График функции
0 -90 -80 -70 -60 -50 -40 -30 -20 -10 10 5 -5
Точки пересечения с осью координат X
График функции пересекает ось X при f = 02 sin ( 3 x ) = 0 2 \sin{\left(3 x \right)} = 0 2 sin ( 3 x ) = 0 Решаем это уравнение Аналитическое решение x 1 = 0 x_{1} = 0 x 1 = 0 x 2 = π 3 x_{2} = \frac{\pi}{3} x 2 = 3 π Численное решение x 1 = − 10.471975511966 x_{1} = -10.471975511966 x 1 = − 10.471975511966 x 2 = − 92.1533845053006 x_{2} = -92.1533845053006 x 2 = − 92.1533845053006 x 3 = − 99.4837673636768 x_{3} = -99.4837673636768 x 3 = − 99.4837673636768 x 4 = − 72.2566310325652 x_{4} = -72.2566310325652 x 4 = − 72.2566310325652 x 5 = 48.1710873550435 x_{5} = 48.1710873550435 x 5 = 48.1710873550435 x 6 = 50.2654824574367 x_{6} = 50.2654824574367 x 6 = 50.2654824574367 x 7 = − 29.3215314335047 x_{7} = -29.3215314335047 x 7 = − 29.3215314335047 x 8 = 90.0589894029074 x_{8} = 90.0589894029074 x 8 = 90.0589894029074 x 9 = 26.1799387799149 x_{9} = 26.1799387799149 x 9 = 26.1799387799149 x 10 = − 28.2743338823081 x_{10} = -28.2743338823081 x 10 = − 28.2743338823081 x 11 = 569.675467850949 x_{11} = 569.675467850949 x 11 = 569.675467850949 x 12 = − 65.9734457253857 x_{12} = -65.9734457253857 x 12 = − 65.9734457253857 x 13 = 87.9645943005142 x_{13} = 87.9645943005142 x 13 = 87.9645943005142 x 14 = 41.8879020478639 x_{14} = 41.8879020478639 x 14 = 41.8879020478639 x 15 = 80.634211442138 x_{15} = 80.634211442138 x 15 = 80.634211442138 x 16 = − 37.6991118430775 x_{16} = -37.6991118430775 x 16 = − 37.6991118430775 x 17 = − 21.9911485751286 x_{17} = -21.9911485751286 x 17 = − 21.9911485751286 x 18 = 100.530964914873 x_{18} = 100.530964914873 x 18 = 100.530964914873 x 19 = − 83.7758040957278 x_{19} = -83.7758040957278 x 19 = − 83.7758040957278 x 20 = 92.1533845053006 x_{20} = 92.1533845053006 x 20 = 92.1533845053006 x 21 = − 77.4926187885482 x_{21} = -77.4926187885482 x 21 = − 77.4926187885482 x 22 = − 46.0766922526503 x_{22} = -46.0766922526503 x 22 = − 46.0766922526503 x 23 = 59.6902604182061 x_{23} = 59.6902604182061 x 23 = 59.6902604182061 x 24 = 2.0943951023932 x_{24} = 2.0943951023932 x 24 = 2.0943951023932 x 25 = 85.870199198121 x_{25} = 85.870199198121 x 25 = 85.870199198121 x 26 = 83.7758040957278 x_{26} = 83.7758040957278 x 26 = 83.7758040957278 x 27 = − 6.28318530717959 x_{27} = -6.28318530717959 x 27 = − 6.28318530717959 x 28 = − 4.18879020478639 x_{28} = -4.18879020478639 x 28 = − 4.18879020478639 x 29 = 8.37758040957278 x_{29} = 8.37758040957278 x 29 = 8.37758040957278 x 30 = 28.2743338823081 x_{30} = 28.2743338823081 x 30 = 28.2743338823081 x 31 = − 17.8023583703422 x_{31} = -17.8023583703422 x 31 = − 17.8023583703422 x 32 = 30.3687289847013 x_{32} = 30.3687289847013 x 32 = 30.3687289847013 x 33 = 10.471975511966 x_{33} = 10.471975511966 x 33 = 10.471975511966 x 34 = − 15.707963267949 x_{34} = -15.707963267949 x 34 = − 15.707963267949 x 35 = 78.5398163397448 x_{35} = 78.5398163397448 x 35 = 78.5398163397448 x 36 = 65.9734457253857 x_{36} = 65.9734457253857 x 36 = 65.9734457253857 x 37 = − 94.2477796076938 x_{37} = -94.2477796076938 x 37 = − 94.2477796076938 x 38 = − 50.2654824574367 x_{38} = -50.2654824574367 x 38 = − 50.2654824574367 x 39 = 98.4365698124802 x_{39} = 98.4365698124802 x 39 = 98.4365698124802 x 40 = − 61.7846555205993 x_{40} = -61.7846555205993 x 40 = − 61.7846555205993 x 41 = 68.0678408277789 x_{41} = 68.0678408277789 x 41 = 68.0678408277789 x 42 = − 31.4159265358979 x_{42} = -31.4159265358979 x 42 = − 31.4159265358979 x 43 = − 57.5958653158129 x_{43} = -57.5958653158129 x 43 = − 57.5958653158129 x 44 = − 79.5870138909414 x_{44} = -79.5870138909414 x 44 = − 79.5870138909414 x 45 = 72.2566310325652 x_{45} = 72.2566310325652 x 45 = 72.2566310325652 x 46 = − 19.8967534727354 x_{46} = -19.8967534727354 x 46 = − 19.8967534727354 x 47 = 4.18879020478639 x_{47} = 4.18879020478639 x 47 = 4.18879020478639 x 48 = 0 x_{48} = 0 x 48 = 0 x 49 = 43.9822971502571 x_{49} = 43.9822971502571 x 49 = 43.9822971502571 x 50 = − 85.870199198121 x_{50} = -85.870199198121 x 50 = − 85.870199198121 x 51 = 74.3510261349584 x_{51} = 74.3510261349584 x 51 = 74.3510261349584 x 52 = 56.5486677646163 x_{52} = 56.5486677646163 x 52 = 56.5486677646163 x 53 = 32.4631240870945 x_{53} = 32.4631240870945 x 53 = 32.4631240870945 x 54 = − 70.162235930172 x_{54} = -70.162235930172 x 54 = − 70.162235930172 x 55 = 21.9911485751286 x_{55} = 21.9911485751286 x 55 = 21.9911485751286 x 56 = 17.8023583703422 x_{56} = 17.8023583703422 x 56 = 17.8023583703422 x 57 = 52.3598775598299 x_{57} = 52.3598775598299 x 57 = 52.3598775598299 x 58 = − 8.37758040957278 x_{58} = -8.37758040957278 x 58 = − 8.37758040957278 x 59 = − 43.9822971502571 x_{59} = -43.9822971502571 x 59 = − 43.9822971502571 x 60 = 46.0766922526503 x_{60} = 46.0766922526503 x 60 = 46.0766922526503 x 61 = − 54.4542726622231 x_{61} = -54.4542726622231 x 61 = − 54.4542726622231 x 62 = − 33.5103216382911 x_{62} = -33.5103216382911 x 62 = − 33.5103216382911 x 63 = 61.7846555205993 x_{63} = 61.7846555205993 x 63 = 61.7846555205993 x 64 = − 55.5014702134197 x_{64} = -55.5014702134197 x 64 = − 55.5014702134197 x 65 = − 39.7935069454707 x_{65} = -39.7935069454707 x 65 = − 39.7935069454707 x 66 = 54.4542726622231 x_{66} = 54.4542726622231 x 66 = 54.4542726622231 x 67 = − 98.4365698124802 x_{67} = -98.4365698124802 x 67 = − 98.4365698124802 x 68 = − 26.1799387799149 x_{68} = -26.1799387799149 x 68 = − 26.1799387799149 x 69 = 6.28318530717959 x_{69} = 6.28318530717959 x 69 = 6.28318530717959 x 70 = 34.5575191894877 x_{70} = 34.5575191894877 x 70 = 34.5575191894877 x 71 = − 13.6135681655558 x_{71} = -13.6135681655558 x 71 = − 13.6135681655558 x 72 = 63.8790506229925 x_{72} = 63.8790506229925 x 72 = 63.8790506229925 x 73 = 81.6814089933346 x_{73} = 81.6814089933346 x 73 = 81.6814089933346 x 74 = 96.342174710087 x_{74} = 96.342174710087 x 74 = 96.342174710087 x 75 = 19.8967534727354 x_{75} = 19.8967534727354 x 75 = 19.8967534727354 x 76 = − 90.0589894029074 x_{76} = -90.0589894029074 x 76 = − 90.0589894029074 x 77 = 746.651854003174 x_{77} = 746.651854003174 x 77 = 746.651854003174 x 78 = − 59.6902604182061 x_{78} = -59.6902604182061 x 78 = − 59.6902604182061 x 79 = 37.6991118430775 x_{79} = 37.6991118430775 x 79 = 37.6991118430775 x 80 = 76.4454212373516 x_{80} = 76.4454212373516 x 80 = 76.4454212373516 x 81 = − 63.8790506229925 x_{81} = -63.8790506229925 x 81 = − 63.8790506229925 x 82 = − 48.1710873550435 x_{82} = -48.1710873550435 x 82 = − 48.1710873550435 x 83 = 39.7935069454707 x_{83} = 39.7935069454707 x 83 = 39.7935069454707 x 84 = 24.0855436775217 x_{84} = 24.0855436775217 x 84 = 24.0855436775217 x 85 = 70.162235930172 x_{85} = 70.162235930172 x 85 = 70.162235930172 x 86 = 109.955742875643 x_{86} = 109.955742875643 x 86 = 109.955742875643 x 87 = − 35.6047167406843 x_{87} = -35.6047167406843 x 87 = − 35.6047167406843 x 88 = − 68.0678408277789 x_{88} = -68.0678408277789 x 88 = − 68.0678408277789 x 89 = − 87.9645943005142 x_{89} = -87.9645943005142 x 89 = − 87.9645943005142 x 90 = − 24.0855436775217 x_{90} = -24.0855436775217 x 90 = − 24.0855436775217 x 91 = 15.707963267949 x_{91} = 15.707963267949 x 91 = 15.707963267949 x 92 = − 11.5191730631626 x_{92} = -11.5191730631626 x 92 = − 11.5191730631626 x 93 = − 41.8879020478639 x_{93} = -41.8879020478639 x 93 = − 41.8879020478639 x 94 = − 2.0943951023932 x_{94} = -2.0943951023932 x 94 = − 2.0943951023932 x 95 = 94.2477796076938 x_{95} = 94.2477796076938 x 95 = 94.2477796076938 x 96 = − 81.6814089933346 x_{96} = -81.6814089933346 x 96 = − 81.6814089933346
Точки пересечения с осью координат Y
График пересекает ось Y, когда x равняется 0:2 sin ( 3 ⋅ 0 ) 2 \sin{\left(3 \cdot 0 \right)} 2 sin ( 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 6 cos ( 3 x ) = 0 6 \cos{\left(3 x \right)} = 0 6 cos ( 3 x ) = 0 Решаем это уравнение x 1 = π 6 x_{1} = \frac{\pi}{6} x 1 = 6 π x 2 = π 2 x_{2} = \frac{\pi}{2} x 2 = 2 π pi
(--, 2)
6 pi
(--, -2)
2 Интервалы возрастания и убывания функции: x 1 = π 2 x_{1} = \frac{\pi}{2} x 1 = 2 π x 1 = π 6 x_{1} = \frac{\pi}{6} x 1 = 6 π ( − ∞ , π 6 ] ∪ [ π 2 , ∞ ) \left(-\infty, \frac{\pi}{6}\right] \cup \left[\frac{\pi}{2}, \infty\right) ( − ∞ , 6 π ] ∪ [ 2 π , ∞ ) [ π 6 , π 2 ] \left[\frac{\pi}{6}, \frac{\pi}{2}\right] [ 6 π , 2 π ]
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 sin ( 3 x ) = 0 - 18 \sin{\left(3 x \right)} = 0 − 18 sin ( 3 x ) = 0 Решаем это уравнение x 1 = 0 x_{1} = 0 x 1 = 0 x 2 = π 3 x_{2} = \frac{\pi}{3} x 2 = 3 π Интервалы выпуклости и вогнутости: ( − ∞ , 0 ] ∪ [ π 3 , ∞ ) \left(-\infty, 0\right] \cup \left[\frac{\pi}{3}, \infty\right) ( − ∞ , 0 ] ∪ [ 3 π , ∞ ) [ 0 , π 3 ] \left[0, \frac{\pi}{3}\right] [ 0 , 3 π ]
Горизонтальные асимптоты
Горизонтальные асимптоты найдём с помощью пределов данной функции при x->+oo и x->-oolim x → − ∞ ( 2 sin ( 3 x ) ) = ⟨ − 2 , 2 ⟩ \lim_{x \to -\infty}\left(2 \sin{\left(3 x \right)}\right) = \left\langle -2, 2\right\rangle x → − ∞ lim ( 2 sin ( 3 x ) ) = ⟨ − 2 , 2 ⟩ Возьмём предел y = ⟨ − 2 , 2 ⟩ y = \left\langle -2, 2\right\rangle y = ⟨ − 2 , 2 ⟩ lim x → ∞ ( 2 sin ( 3 x ) ) = ⟨ − 2 , 2 ⟩ \lim_{x \to \infty}\left(2 \sin{\left(3 x \right)}\right) = \left\langle -2, 2\right\rangle x → ∞ lim ( 2 sin ( 3 x ) ) = ⟨ − 2 , 2 ⟩ Возьмём предел y = ⟨ − 2 , 2 ⟩ y = \left\langle -2, 2\right\rangle y = ⟨ − 2 , 2 ⟩
Наклонные асимптоты
Наклонную асимптоту можно найти, подсчитав предел функции 2*sin(3*x), делённой на x при x->+oo и x ->-oolim x → − ∞ ( 2 sin ( 3 x ) x ) = 0 \lim_{x \to -\infty}\left(\frac{2 \sin{\left(3 x \right)}}{x}\right) = 0 x → − ∞ lim ( x 2 sin ( 3 x ) ) = 0 Возьмём предел lim x → ∞ ( 2 sin ( 3 x ) x ) = 0 \lim_{x \to \infty}\left(\frac{2 \sin{\left(3 x \right)}}{x}\right) = 0 x → ∞ lim ( x 2 sin ( 3 x ) ) = 0 Возьмём предел
Чётность и нечётность функции
Проверим функци чётна или нечётна с помощью соотношений f = f(-x) и f = -f(-x).2 sin ( 3 x ) = − 2 sin ( 3 x ) 2 \sin{\left(3 x \right)} = - 2 \sin{\left(3 x \right)} 2 sin ( 3 x ) = − 2 sin ( 3 x ) 2 sin ( 3 x ) = 2 sin ( 3 x ) 2 \sin{\left(3 x \right)} = 2 \sin{\left(3 x \right)} 2 sin ( 3 x ) = 2 sin ( 3 x ) 
