График функции
0 -90 -80 -70 -60 -50 -40 -30 -20 -10 10 0 2
Точки пересечения с осью координат X
График функции пересекает ось X при f = 0sin 2 ( 3 x ) = 0 \sin^{2}{\left(3 x \right)} = 0 sin 2 ( 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 = − 90.0589893540864 x_{1} = -90.0589893540864 x 1 = − 90.0589893540864 x 2 = − 68.0678407742199 x_{2} = -68.0678407742199 x 2 = − 68.0678407742199 x 3 = − 52.3598773534304 x_{3} = -52.3598773534304 x 3 = − 52.3598773534304 x 4 = − 458.672527220906 x_{4} = -458.672527220906 x 4 = − 458.672527220906 x 5 = 52.3598775005845 x_{5} = 52.3598775005845 x 5 = 52.3598775005845 x 6 = 10.4719754514552 x_{6} = 10.4719754514552 x 6 = 10.4719754514552 x 7 = − 79.587013962129 x_{7} = -79.587013962129 x 7 = − 79.587013962129 x 8 = 98.4365697412614 x_{8} = 98.4365697412614 x 8 = 98.4365697412614 x 9 = − 57.5958653860056 x_{9} = -57.5958653860056 x 9 = − 57.5958653860056 x 10 = 92.153384609279 x_{10} = 92.153384609279 x 10 = 92.153384609279 x 11 = 28.2743338654616 x_{11} = 28.2743338654616 x 11 = 28.2743338654616 x 12 = 74.351026078903 x_{12} = 74.351026078903 x 12 = 74.351026078903 x 13 = 30.3687289225719 x_{13} = 30.3687289225719 x 13 = 30.3687289225719 x 14 = 15.7079631676307 x_{14} = 15.7079631676307 x 14 = 15.7079631676307 x 15 = − 87.9645943606119 x_{15} = -87.9645943606119 x 15 = − 87.9645943606119 x 16 = − 28.2743339027943 x_{16} = -28.2743339027943 x 16 = − 28.2743339027943 x 17 = 6.2831852847685 x_{17} = 6.2831852847685 x 17 = 6.2831852847685 x 18 = 83.7758041555675 x_{18} = 83.7758041555675 x 18 = 83.7758041555675 x 19 = − 65.9734457660819 x_{19} = -65.9734457660819 x 19 = − 65.9734457660819 x 20 = 76.445421167341 x_{20} = 76.445421167341 x 20 = 76.445421167341 x 21 = − 33.5103216848946 x_{21} = -33.5103216848946 x 21 = − 33.5103216848946 x 22 = 90.0589892608867 x_{22} = 90.0589892608867 x 22 = 90.0589892608867 x 23 = 100.530964959912 x_{23} = 100.530964959912 x 23 = 100.530964959912 x 24 = 34.5575193908685 x_{24} = 34.5575193908685 x 24 = 34.5575193908685 x 25 = 43.9822971689772 x_{25} = 43.9822971689772 x 25 = 43.9822971689772 x 26 = 48.1710874341111 x_{26} = 48.1710874341111 x 26 = 48.1710874341111 x 27 = 94.2477796093541 x_{27} = 94.2477796093541 x 27 = 94.2477796093541 x 28 = 70.1622360218137 x_{28} = 70.1622360218137 x 28 = 70.1622360218137 x 29 = − 4.18879013064742 x_{29} = -4.18879013064742 x 29 = − 4.18879013064742 x 30 = 65.973445751897 x_{30} = 65.973445751897 x 30 = 65.973445751897 x 31 = 19.896753540642 x_{31} = 19.896753540642 x 31 = 19.896753540642 x 32 = − 48.1710872785046 x_{32} = -48.1710872785046 x 32 = − 48.1710872785046 x 33 = 63.8790506955489 x_{33} = 63.8790506955489 x 33 = 63.8790506955489 x 34 = 54.4542725942726 x_{34} = 54.4542725942726 x 34 = 54.4542725942726 x 35 = − 35.6047168094363 x_{35} = -35.6047168094363 x 35 = − 35.6047168094363 x 36 = 83.7758041420923 x_{36} = 83.7758041420923 x 36 = 83.7758041420923 x 37 = − 59.6902604556648 x_{37} = -59.6902604556648 x 37 = − 59.6902604556648 x 38 = − 46.0766921948394 x_{38} = -46.0766921948394 x 38 = − 46.0766921948394 x 39 = − 99.4837673664303 x_{39} = -99.4837673664303 x 39 = − 99.4837673664303 x 40 = − 61.7846554917911 x_{40} = -61.7846554917911 x 40 = − 61.7846554917911 x 41 = 59.6902602103008 x_{41} = 59.6902602103008 x 41 = 59.6902602103008 x 42 = − 71.2094334160924 x_{42} = -71.2094334160924 x 42 = − 71.2094334160924 x 43 = 0 x_{43} = 0 x 43 = 0 x 44 = − 81.6814090352758 x_{44} = -81.6814090352758 x 44 = − 81.6814090352758 x 45 = 12.566370486226 x_{45} = 12.566370486226 x 45 = 12.566370486226 x 46 = 32.463124022232 x_{46} = 32.463124022232 x 46 = 32.463124022232 x 47 = − 92.1533844285198 x_{47} = -92.1533844285198 x 47 = − 92.1533844285198 x 48 = 39.7935070144417 x_{48} = 39.7935070144417 x 48 = 39.7935070144417 x 49 = − 43.9822971750521 x_{49} = -43.9822971750521 x 49 = − 43.9822971750521 x 50 = 26.1799388472337 x_{50} = 26.1799388472337 x 50 = 26.1799388472337 x 51 = 56.5486679087207 x_{51} = 56.5486679087207 x 51 = 56.5486679087207 x 52 = − 21.9911485865686 x_{52} = -21.9911485865686 x 52 = − 21.9911485865686 x 53 = 17.8023584422991 x_{53} = 17.8023584422991 x 53 = 17.8023584422991 x 54 = 81.6814093004925 x_{54} = 81.6814093004925 x 54 = 81.6814093004925 x 55 = 78.5398164276763 x_{55} = 78.5398164276763 x 55 = 78.5398164276763 x 56 = 37.699111685071 x_{56} = 37.699111685071 x 56 = 37.699111685071 x 57 = − 94.2477795706448 x_{57} = -94.2477795706448 x 57 = − 94.2477795706448 x 58 = 21.9911485850825 x_{58} = 21.9911485850825 x 58 = 21.9911485850825 x 59 = 61.7846555856111 x_{59} = 61.7846555856111 x 59 = 61.7846555856111 x 60 = − 11.5191731177358 x_{60} = -11.5191731177358 x 60 = − 11.5191731177358 x 61 = − 83.7758040749281 x_{61} = -83.7758040749281 x 61 = − 83.7758040749281 x 62 = − 13.6135682324625 x_{62} = -13.6135682324625 x 62 = − 13.6135682324625 x 63 = − 17.8023583237749 x_{63} = -17.8023583237749 x 63 = − 17.8023583237749 x 64 = − 39.7935069081029 x_{64} = -39.7935069081029 x 64 = − 39.7935069081029 x 65 = 68.0678406435528 x_{65} = 68.0678406435528 x 65 = 68.0678406435528 x 66 = − 37.6991118757886 x_{66} = -37.6991118757886 x 66 = − 37.6991118757886 x 67 = 8.37758034488241 x_{67} = 8.37758034488241 x 67 = 8.37758034488241 x 68 = − 77.4926188103359 x_{68} = -77.4926188103359 x 68 = − 77.4926188103359 x 69 = 72.2566310277355 x_{69} = 72.2566310277355 x 69 = 72.2566310277355 x 70 = 50.2654824464452 x_{70} = 50.2654824464452 x 70 = 50.2654824464452 x 71 = 41.8879021182991 x_{71} = 41.8879021182991 x 71 = 41.8879021182991 x 72 = − 2.09439503742649 x_{72} = -2.09439503742649 x 72 = − 2.09439503742649 x 73 = − 55.5014702493934 x_{73} = -55.5014702493934 x 73 = − 55.5014702493934 x 74 = 4.18879026129896 x_{74} = 4.18879026129896 x 74 = 4.18879026129896 x 75 = − 70.1622358532669 x_{75} = -70.1622358532669 x 75 = − 70.1622358532669 x 76 = − 6.28318536172932 x_{76} = -6.28318536172932 x 76 = − 6.28318536172932 x 77 = − 15.7079632956456 x_{77} = -15.7079632956456 x 77 = − 15.7079632956456 x 78 = − 26.1799387042774 x_{78} = -26.1799387042774 x 78 = − 26.1799387042774 x 79 = 87.9645943339958 x_{79} = 87.9645943339958 x 79 = 87.9645943339958 x 80 = − 72.2566310094027 x_{80} = -72.2566310094027 x 80 = − 72.2566310094027 x 81 = 96.3421746575137 x_{81} = 96.3421746575137 x 81 = 96.3421746575137 x 82 = − 24.0855436159154 x_{82} = -24.0855436159154 x 82 = − 24.0855436159154 x 83 = 85.8701992723906 x_{83} = 85.8701992723906 x 83 = 85.8701992723906 x 84 = − 50.2654824528405 x_{84} = -50.2654824528405 x 84 = − 50.2654824528405
Точки пересечения с осью координат Y
График пересекает ось Y, когда x равняется 0:sin 2 ( 3 ⋅ 0 ) \sin^{2}{\left(3 \cdot 0 \right)} sin 2 ( 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 sin ( 3 x ) cos ( 3 x ) = 0 6 \sin{\left(3 x \right)} \cos{\left(3 x \right)} = 0 6 sin ( 3 x ) cos ( 3 x ) = 0 Решаем это уравнение x 1 = 0 x_{1} = 0 x 1 = 0 x 2 = π 6 x_{2} = \frac{\pi}{6} x 2 = 6 π x 3 = π 3 x_{3} = \frac{\pi}{3} x 3 = 3 π x 4 = π 2 x_{4} = \frac{\pi}{2} x 4 = 2 π (0, 0) pi
(--, 1)
6 pi
(--, 0)
3 pi
(--, 1)
2 Интервалы возрастания и убывания функции: x 1 = 0 x_{1} = 0 x 1 = 0 x 2 = π 3 x_{2} = \frac{\pi}{3} x 2 = 3 π x 2 = π 6 x_{2} = \frac{\pi}{6} x 2 = 6 π x 2 = π 2 x_{2} = \frac{\pi}{2} x 2 = 2 π [ 0 , π 6 ] ∪ [ π 3 , ∞ ) \left[0, \frac{\pi}{6}\right] \cup \left[\frac{\pi}{3}, \infty\right) [ 0 , 6 π ] ∪ [ 3 π , ∞ ) ( − ∞ , 0 ] \left(-\infty, 0\right] ( − ∞ , 0 ]
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 2 ( 3 x ) + cos 2 ( 3 x ) ) = 0 18 \left(- \sin^{2}{\left(3 x \right)} + \cos^{2}{\left(3 x \right)}\right) = 0 18 ( − sin 2 ( 3 x ) + cos 2 ( 3 x ) ) = 0 Решаем это уравнение x 1 = − π 12 x_{1} = - \frac{\pi}{12} x 1 = − 12 π x 2 = π 12 x_{2} = \frac{\pi}{12} x 2 = 12 π Интервалы выпуклости и вогнутости: [ − π 12 , π 12 ] \left[- \frac{\pi}{12}, \frac{\pi}{12}\right] [ − 12 π , 12 π ] ( − ∞ , − π 12 ] ∪ [ π 12 , ∞ ) \left(-\infty, - \frac{\pi}{12}\right] \cup \left[\frac{\pi}{12}, \infty\right) ( − ∞ , − 12 π ] ∪ [ 12 π , ∞ )
Горизонтальные асимптоты
Горизонтальные асимптоты найдём с помощью пределов данной функции при x->+oo и x->-oolim x → − ∞ sin 2 ( 3 x ) = ⟨ 0 , 1 ⟩ \lim_{x \to -\infty} \sin^{2}{\left(3 x \right)} = \left\langle 0, 1\right\rangle x → − ∞ lim sin 2 ( 3 x ) = ⟨ 0 , 1 ⟩ Возьмём предел y = ⟨ 0 , 1 ⟩ y = \left\langle 0, 1\right\rangle y = ⟨ 0 , 1 ⟩ lim x → ∞ sin 2 ( 3 x ) = ⟨ 0 , 1 ⟩ \lim_{x \to \infty} \sin^{2}{\left(3 x \right)} = \left\langle 0, 1\right\rangle x → ∞ lim sin 2 ( 3 x ) = ⟨ 0 , 1 ⟩ Возьмём предел y = ⟨ 0 , 1 ⟩ y = \left\langle 0, 1\right\rangle y = ⟨ 0 , 1 ⟩
Наклонные асимптоты
Наклонную асимптоту можно найти, подсчитав предел функции sin(3*x)^2, делённой на x при x->+oo и x ->-oolim x → − ∞ ( sin 2 ( 3 x ) x ) = 0 \lim_{x \to -\infty}\left(\frac{\sin^{2}{\left(3 x \right)}}{x}\right) = 0 x → − ∞ lim ( x sin 2 ( 3 x ) ) = 0 Возьмём предел lim x → ∞ ( sin 2 ( 3 x ) x ) = 0 \lim_{x \to \infty}\left(\frac{\sin^{2}{\left(3 x \right)}}{x}\right) = 0 x → ∞ lim ( x sin 2 ( 3 x ) ) = 0 Возьмём предел
Чётность и нечётность функции
Проверим функци чётна или нечётна с помощью соотношений f = f(-x) и f = -f(-x).sin 2 ( 3 x ) = sin 2 ( 3 x ) \sin^{2}{\left(3 x \right)} = \sin^{2}{\left(3 x \right)} sin 2 ( 3 x ) = sin 2 ( 3 x ) sin 2 ( 3 x ) = − sin 2 ( 3 x ) \sin^{2}{\left(3 x \right)} = - \sin^{2}{\left(3 x \right)} sin 2 ( 3 x ) = − sin 2 ( 3 x ) 
