График функции
0 -80 -70 -60 -50 -40 -30 -20 -10 10 20 5 -5
Точки пересечения с осью координат X
График функции пересекает ось X при f = 03 sin ( 3 x ) = 0 3 \sin{\left(3 x \right)} = 0 3 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 = 17.8023583703422 x_{1} = 17.8023583703422 x 1 = 17.8023583703422 x 2 = − 81.6814089933346 x_{2} = -81.6814089933346 x 2 = − 81.6814089933346 x 3 = 26.1799387799149 x_{3} = 26.1799387799149 x 3 = 26.1799387799149 x 4 = − 54.4542726622231 x_{4} = -54.4542726622231 x 4 = − 54.4542726622231 x 5 = 19.8967534727354 x_{5} = 19.8967534727354 x 5 = 19.8967534727354 x 6 = − 48.1710873550435 x_{6} = -48.1710873550435 x 6 = − 48.1710873550435 x 7 = − 10.471975511966 x_{7} = -10.471975511966 x 7 = − 10.471975511966 x 8 = 78.5398163397448 x_{8} = 78.5398163397448 x 8 = 78.5398163397448 x 9 = − 13.6135681655558 x_{9} = -13.6135681655558 x 9 = − 13.6135681655558 x 10 = 39.7935069454707 x_{10} = 39.7935069454707 x 10 = 39.7935069454707 x 11 = 65.9734457253857 x_{11} = 65.9734457253857 x 11 = 65.9734457253857 x 12 = 92.1533845053006 x_{12} = 92.1533845053006 x 12 = 92.1533845053006 x 13 = 10.471975511966 x_{13} = 10.471975511966 x 13 = 10.471975511966 x 14 = 90.0589894029074 x_{14} = 90.0589894029074 x 14 = 90.0589894029074 x 15 = 59.6902604182061 x_{15} = 59.6902604182061 x 15 = 59.6902604182061 x 16 = − 43.9822971502571 x_{16} = -43.9822971502571 x 16 = − 43.9822971502571 x 17 = 109.955742875643 x_{17} = 109.955742875643 x 17 = 109.955742875643 x 18 = 52.3598775598299 x_{18} = 52.3598775598299 x 18 = 52.3598775598299 x 19 = 2491.28297429671 x_{19} = 2491.28297429671 x 19 = 2491.28297429671 x 20 = 8.37758040957278 x_{20} = 8.37758040957278 x 20 = 8.37758040957278 x 21 = 85.870199198121 x_{21} = 85.870199198121 x 21 = 85.870199198121 x 22 = 48.1710873550435 x_{22} = 48.1710873550435 x 22 = 48.1710873550435 x 23 = − 31.4159265358979 x_{23} = -31.4159265358979 x 23 = − 31.4159265358979 x 24 = 2.0943951023932 x_{24} = 2.0943951023932 x 24 = 2.0943951023932 x 25 = − 4.18879020478639 x_{25} = -4.18879020478639 x 25 = − 4.18879020478639 x 26 = − 2.0943951023932 x_{26} = -2.0943951023932 x 26 = − 2.0943951023932 x 27 = 83.7758040957278 x_{27} = 83.7758040957278 x 27 = 83.7758040957278 x 28 = 96.342174710087 x_{28} = 96.342174710087 x 28 = 96.342174710087 x 29 = − 90.0589894029074 x_{29} = -90.0589894029074 x 29 = − 90.0589894029074 x 30 = − 59.6902604182061 x_{30} = -59.6902604182061 x 30 = − 59.6902604182061 x 31 = − 83.7758040957278 x_{31} = -83.7758040957278 x 31 = − 83.7758040957278 x 32 = − 63.8790506229925 x_{32} = -63.8790506229925 x 32 = − 63.8790506229925 x 33 = − 11.5191730631626 x_{33} = -11.5191730631626 x 33 = − 11.5191730631626 x 34 = 94.2477796076938 x_{34} = 94.2477796076938 x 34 = 94.2477796076938 x 35 = 690.103186238558 x_{35} = 690.103186238558 x 35 = 690.103186238558 x 36 = − 21.9911485751286 x_{36} = -21.9911485751286 x 36 = − 21.9911485751286 x 37 = − 98.4365698124802 x_{37} = -98.4365698124802 x 37 = − 98.4365698124802 x 38 = − 92.1533845053006 x_{38} = -92.1533845053006 x 38 = − 92.1533845053006 x 39 = − 99.4837673636768 x_{39} = -99.4837673636768 x 39 = − 99.4837673636768 x 40 = 81.6814089933346 x_{40} = 81.6814089933346 x 40 = 81.6814089933346 x 41 = − 68.0678408277789 x_{41} = -68.0678408277789 x 41 = − 68.0678408277789 x 42 = − 77.4926187885482 x_{42} = -77.4926187885482 x 42 = − 77.4926187885482 x 43 = 98.4365698124802 x_{43} = 98.4365698124802 x 43 = 98.4365698124802 x 44 = − 17.8023583703422 x_{44} = -17.8023583703422 x 44 = − 17.8023583703422 x 45 = 6.28318530717959 x_{45} = 6.28318530717959 x 45 = 6.28318530717959 x 46 = 24.0855436775217 x_{46} = 24.0855436775217 x 46 = 24.0855436775217 x 47 = − 65.9734457253857 x_{47} = -65.9734457253857 x 47 = − 65.9734457253857 x 48 = − 46.0766922526503 x_{48} = -46.0766922526503 x 48 = − 46.0766922526503 x 49 = 37.6991118430775 x_{49} = 37.6991118430775 x 49 = 37.6991118430775 x 50 = − 26.1799387799149 x_{50} = -26.1799387799149 x 50 = − 26.1799387799149 x 51 = 21.9911485751286 x_{51} = 21.9911485751286 x 51 = 21.9911485751286 x 52 = − 33.5103216382911 x_{52} = -33.5103216382911 x 52 = − 33.5103216382911 x 53 = − 24.0855436775217 x_{53} = -24.0855436775217 x 53 = − 24.0855436775217 x 54 = − 41.8879020478639 x_{54} = -41.8879020478639 x 54 = − 41.8879020478639 x 55 = − 94.2477796076938 x_{55} = -94.2477796076938 x 55 = − 94.2477796076938 x 56 = 76.4454212373516 x_{56} = 76.4454212373516 x 56 = 76.4454212373516 x 57 = − 35.6047167406843 x_{57} = -35.6047167406843 x 57 = − 35.6047167406843 x 58 = 41.8879020478639 x_{58} = 41.8879020478639 x 58 = 41.8879020478639 x 59 = − 28.2743338823081 x_{59} = -28.2743338823081 x 59 = − 28.2743338823081 x 60 = − 55.5014702134197 x_{60} = -55.5014702134197 x 60 = − 55.5014702134197 x 61 = 746.651854003174 x_{61} = 746.651854003174 x 61 = 746.651854003174 x 62 = − 85.870199198121 x_{62} = -85.870199198121 x 62 = − 85.870199198121 x 63 = − 87.9645943005142 x_{63} = -87.9645943005142 x 63 = − 87.9645943005142 x 64 = 30.3687289847013 x_{64} = 30.3687289847013 x 64 = 30.3687289847013 x 65 = − 70.162235930172 x_{65} = -70.162235930172 x 65 = − 70.162235930172 x 66 = 100.530964914873 x_{66} = 100.530964914873 x 66 = 100.530964914873 x 67 = 80.634211442138 x_{67} = 80.634211442138 x 67 = 80.634211442138 x 68 = 28.2743338823081 x_{68} = 28.2743338823081 x 68 = 28.2743338823081 x 69 = 63.8790506229925 x_{69} = 63.8790506229925 x 69 = 63.8790506229925 x 70 = − 8.37758040957278 x_{70} = -8.37758040957278 x 70 = − 8.37758040957278 x 71 = 56.5486677646163 x_{71} = 56.5486677646163 x 71 = 56.5486677646163 x 72 = − 57.5958653158129 x_{72} = -57.5958653158129 x 72 = − 57.5958653158129 x 73 = 34.5575191894877 x_{73} = 34.5575191894877 x 73 = 34.5575191894877 x 74 = − 19.8967534727354 x_{74} = -19.8967534727354 x 74 = − 19.8967534727354 x 75 = 61.7846555205993 x_{75} = 61.7846555205993 x 75 = 61.7846555205993 x 76 = 72.2566310325652 x_{76} = 72.2566310325652 x 76 = 72.2566310325652 x 77 = − 6.28318530717959 x_{77} = -6.28318530717959 x 77 = − 6.28318530717959 x 78 = − 50.2654824574367 x_{78} = -50.2654824574367 x 78 = − 50.2654824574367 x 79 = 50.2654824574367 x_{79} = 50.2654824574367 x 79 = 50.2654824574367 x 80 = − 61.7846555205993 x_{80} = -61.7846555205993 x 80 = − 61.7846555205993 x 81 = − 29.3215314335047 x_{81} = -29.3215314335047 x 81 = − 29.3215314335047 x 82 = 54.4542726622231 x_{82} = 54.4542726622231 x 82 = 54.4542726622231 x 83 = 15.707963267949 x_{83} = 15.707963267949 x 83 = 15.707963267949 x 84 = 87.9645943005142 x_{84} = 87.9645943005142 x 84 = 87.9645943005142 x 85 = − 39.7935069454707 x_{85} = -39.7935069454707 x 85 = − 39.7935069454707 x 86 = − 79.5870138909414 x_{86} = -79.5870138909414 x 86 = − 79.5870138909414 x 87 = 68.0678408277789 x_{87} = 68.0678408277789 x 87 = 68.0678408277789 x 88 = 0 x_{88} = 0 x 88 = 0 x 89 = 70.162235930172 x_{89} = 70.162235930172 x 89 = 70.162235930172 x 90 = − 72.2566310325652 x_{90} = -72.2566310325652 x 90 = − 72.2566310325652 x 91 = 32.4631240870945 x_{91} = 32.4631240870945 x 91 = 32.4631240870945 x 92 = − 37.6991118430775 x_{92} = -37.6991118430775 x 92 = − 37.6991118430775 x 93 = 46.0766922526503 x_{93} = 46.0766922526503 x 93 = 46.0766922526503 x 94 = 43.9822971502571 x_{94} = 43.9822971502571 x 94 = 43.9822971502571 x 95 = − 15.707963267949 x_{95} = -15.707963267949 x 95 = − 15.707963267949 x 96 = 4.18879020478639 x_{96} = 4.18879020478639 x 96 = 4.18879020478639 x 97 = 74.3510261349584 x_{97} = 74.3510261349584 x 97 = 74.3510261349584
Точки пересечения с осью координат Y
График пересекает ось Y, когда x равняется 0:3 sin ( 3 ⋅ 0 ) 3 \sin{\left(3 \cdot 0 \right)} 3 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 9 cos ( 3 x ) = 0 9 \cos{\left(3 x \right)} = 0 9 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
(--, 3)
6 pi
(--, -3)
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 − 27 sin ( 3 x ) = 0 - 27 \sin{\left(3 x \right)} = 0 − 27 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 → − ∞ ( 3 sin ( 3 x ) ) = ⟨ − 3 , 3 ⟩ \lim_{x \to -\infty}\left(3 \sin{\left(3 x \right)}\right) = \left\langle -3, 3\right\rangle x → − ∞ lim ( 3 sin ( 3 x ) ) = ⟨ − 3 , 3 ⟩ Возьмём предел y = ⟨ − 3 , 3 ⟩ y = \left\langle -3, 3\right\rangle y = ⟨ − 3 , 3 ⟩ lim x → ∞ ( 3 sin ( 3 x ) ) = ⟨ − 3 , 3 ⟩ \lim_{x \to \infty}\left(3 \sin{\left(3 x \right)}\right) = \left\langle -3, 3\right\rangle x → ∞ lim ( 3 sin ( 3 x ) ) = ⟨ − 3 , 3 ⟩ Возьмём предел y = ⟨ − 3 , 3 ⟩ y = \left\langle -3, 3\right\rangle y = ⟨ − 3 , 3 ⟩
Наклонные асимптоты
Наклонную асимптоту можно найти, подсчитав предел функции 3*sin(3*x), делённой на x при x->+oo и x ->-oolim x → − ∞ ( 3 sin ( 3 x ) x ) = 0 \lim_{x \to -\infty}\left(\frac{3 \sin{\left(3 x \right)}}{x}\right) = 0 x → − ∞ lim ( x 3 sin ( 3 x ) ) = 0 Возьмём предел lim x → ∞ ( 3 sin ( 3 x ) x ) = 0 \lim_{x \to \infty}\left(\frac{3 \sin{\left(3 x \right)}}{x}\right) = 0 x → ∞ lim ( x 3 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).3 sin ( 3 x ) = − 3 sin ( 3 x ) 3 \sin{\left(3 x \right)} = - 3 \sin{\left(3 x \right)} 3 sin ( 3 x ) = − 3 sin ( 3 x ) 3 sin ( 3 x ) = 3 sin ( 3 x ) 3 \sin{\left(3 x \right)} = 3 \sin{\left(3 x \right)} 3 sin ( 3 x ) = 3 sin ( 3 x ) 
