График функции
0 -2000 -1500 -1000 -500 500 1000 1500 2000 2 -2
Точки пересечения с осью координат X
График функции пересекает ось X при f = 0− sin ( 3 x ) = 0 - \sin{\left (3 x \right )} = 0 − 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 = 87.9645943005 x_{1} = 87.9645943005 x 1 = 87.9645943005 x 2 = 48.171087355 x_{2} = 48.171087355 x 2 = 48.171087355 x 3 = 52.3598775598 x_{3} = 52.3598775598 x 3 = 52.3598775598 x 4 = 78.5398163397 x_{4} = 78.5398163397 x 4 = 78.5398163397 x 5 = 92.1533845053 x_{5} = 92.1533845053 x 5 = 92.1533845053 x 6 = 63.879050623 x_{6} = 63.879050623 x 6 = 63.879050623 x 7 = 56.5486677646 x_{7} = 56.5486677646 x 7 = 56.5486677646 x 8 = 26.1799387799 x_{8} = 26.1799387799 x 8 = 26.1799387799 x 9 = − 2.09439510239 x_{9} = -2.09439510239 x 9 = − 2.09439510239 x 10 = − 17.8023583703 x_{10} = -17.8023583703 x 10 = − 17.8023583703 x 11 = − 33.5103216383 x_{11} = -33.5103216383 x 11 = − 33.5103216383 x 12 = 39.7935069455 x_{12} = 39.7935069455 x 12 = 39.7935069455 x 13 = 32.4631240871 x_{13} = 32.4631240871 x 13 = 32.4631240871 x 14 = 68.0678408278 x_{14} = 68.0678408278 x 14 = 68.0678408278 x 15 = − 92.1533845053 x_{15} = -92.1533845053 x 15 = − 92.1533845053 x 16 = − 19.8967534727 x_{16} = -19.8967534727 x 16 = − 19.8967534727 x 17 = − 90.0589894029 x_{17} = -90.0589894029 x 17 = − 90.0589894029 x 18 = 17.8023583703 x_{18} = 17.8023583703 x 18 = 17.8023583703 x 19 = − 4.18879020479 x_{19} = -4.18879020479 x 19 = − 4.18879020479 x 20 = − 55.5014702134 x_{20} = -55.5014702134 x 20 = − 55.5014702134 x 21 = 30.3687289847 x_{21} = 30.3687289847 x 21 = 30.3687289847 x 22 = 41.8879020479 x_{22} = 41.8879020479 x 22 = 41.8879020479 x 23 = 83.7758040957 x_{23} = 83.7758040957 x 23 = 83.7758040957 x 24 = − 13.6135681656 x_{24} = -13.6135681656 x 24 = − 13.6135681656 x 25 = 61.7846555206 x_{25} = 61.7846555206 x 25 = 61.7846555206 x 26 = − 61.7846555206 x_{26} = -61.7846555206 x 26 = − 61.7846555206 x 27 = − 77.4926187885 x_{27} = -77.4926187885 x 27 = − 77.4926187885 x 28 = 8.37758040957 x_{28} = 8.37758040957 x 28 = 8.37758040957 x 29 = 54.4542726622 x_{29} = 54.4542726622 x 29 = 54.4542726622 x 30 = − 46.0766922527 x_{30} = -46.0766922527 x 30 = − 46.0766922527 x 31 = 43.9822971503 x_{31} = 43.9822971503 x 31 = 43.9822971503 x 32 = 59.6902604182 x_{32} = 59.6902604182 x 32 = 59.6902604182 x 33 = − 21.9911485751 x_{33} = -21.9911485751 x 33 = − 21.9911485751 x 34 = − 37.6991118431 x_{34} = -37.6991118431 x 34 = − 37.6991118431 x 35 = 21.9911485751 x_{35} = 21.9911485751 x 35 = 21.9911485751 x 36 = 70.1622359302 x_{36} = 70.1622359302 x 36 = 70.1622359302 x 37 = 19.8967534727 x_{37} = 19.8967534727 x 37 = 19.8967534727 x 38 = 0 x_{38} = 0 x 38 = 0 x 39 = − 94.2477796077 x_{39} = -94.2477796077 x 39 = − 94.2477796077 x 40 = 90.0589894029 x_{40} = 90.0589894029 x 40 = 90.0589894029 x 41 = − 10.471975512 x_{41} = -10.471975512 x 41 = − 10.471975512 x 42 = − 68.0678408278 x_{42} = -68.0678408278 x 42 = − 68.0678408278 x 43 = 15.7079632679 x_{43} = 15.7079632679 x 43 = 15.7079632679 x 44 = − 57.5958653158 x_{44} = -57.5958653158 x 44 = − 57.5958653158 x 45 = 50.2654824574 x_{45} = 50.2654824574 x 45 = 50.2654824574 x 46 = − 63.879050623 x_{46} = -63.879050623 x 46 = − 63.879050623 x 47 = − 59.6902604182 x_{47} = -59.6902604182 x 47 = − 59.6902604182 x 48 = 28.2743338823 x_{48} = 28.2743338823 x 48 = 28.2743338823 x 49 = − 43.9822971503 x_{49} = -43.9822971503 x 49 = − 43.9822971503 x 50 = − 41.8879020479 x_{50} = -41.8879020479 x 50 = − 41.8879020479 x 51 = − 81.6814089933 x_{51} = -81.6814089933 x 51 = − 81.6814089933 x 52 = 72.2566310326 x_{52} = 72.2566310326 x 52 = 72.2566310326 x 53 = − 6.28318530718 x_{53} = -6.28318530718 x 53 = − 6.28318530718 x 54 = 109.955742876 x_{54} = 109.955742876 x 54 = 109.955742876 x 55 = 46.0766922527 x_{55} = 46.0766922527 x 55 = 46.0766922527 x 56 = − 65.9734457254 x_{56} = -65.9734457254 x 56 = − 65.9734457254 x 57 = − 72.2566310326 x_{57} = -72.2566310326 x 57 = − 72.2566310326 x 58 = 569.675467851 x_{58} = 569.675467851 x 58 = 569.675467851 x 59 = − 11.5191730632 x_{59} = -11.5191730632 x 59 = − 11.5191730632 x 60 = 65.9734457254 x_{60} = 65.9734457254 x 60 = 65.9734457254 x 61 = 85.8701991981 x_{61} = 85.8701991981 x 61 = 85.8701991981 x 62 = 96.3421747101 x_{62} = 96.3421747101 x 62 = 96.3421747101 x 63 = − 98.4365698125 x_{63} = -98.4365698125 x 63 = − 98.4365698125 x 64 = − 24.0855436775 x_{64} = -24.0855436775 x 64 = − 24.0855436775 x 65 = 94.2477796077 x_{65} = 94.2477796077 x 65 = 94.2477796077 x 66 = − 15.7079632679 x_{66} = -15.7079632679 x 66 = − 15.7079632679 x 67 = 10.471975512 x_{67} = 10.471975512 x 67 = 10.471975512 x 68 = − 70.1622359302 x_{68} = -70.1622359302 x 68 = − 70.1622359302 x 69 = 81.6814089933 x_{69} = 81.6814089933 x 69 = 81.6814089933 x 70 = 98.4365698125 x_{70} = 98.4365698125 x 70 = 98.4365698125 x 71 = − 8.37758040957 x_{71} = -8.37758040957 x 71 = − 8.37758040957 x 72 = 100.530964915 x_{72} = 100.530964915 x 72 = 100.530964915 x 73 = 74.351026135 x_{73} = 74.351026135 x 73 = 74.351026135 x 74 = − 29.3215314335 x_{74} = -29.3215314335 x 74 = − 29.3215314335 x 75 = − 79.5870138909 x_{75} = -79.5870138909 x 75 = − 79.5870138909 x 76 = − 54.4542726622 x_{76} = -54.4542726622 x 76 = − 54.4542726622 x 77 = − 83.7758040957 x_{77} = -83.7758040957 x 77 = − 83.7758040957 x 78 = − 39.7935069455 x_{78} = -39.7935069455 x 78 = − 39.7935069455 x 79 = − 31.4159265359 x_{79} = -31.4159265359 x 79 = − 31.4159265359 x 80 = 2.09439510239 x_{80} = 2.09439510239 x 80 = 2.09439510239 x 81 = − 85.8701991981 x_{81} = -85.8701991981 x 81 = − 85.8701991981 x 82 = 4.18879020479 x_{82} = 4.18879020479 x 82 = 4.18879020479 x 83 = 6.28318530718 x_{83} = 6.28318530718 x 83 = 6.28318530718 x 84 = − 87.9645943005 x_{84} = -87.9645943005 x 84 = − 87.9645943005 x 85 = 76.4454212374 x_{85} = 76.4454212374 x 85 = 76.4454212374 x 86 = − 99.4837673637 x_{86} = -99.4837673637 x 86 = − 99.4837673637 x 87 = − 50.2654824574 x_{87} = -50.2654824574 x 87 = − 50.2654824574 x 88 = 746.651854003 x_{88} = 746.651854003 x 88 = 746.651854003 x 89 = − 48.171087355 x_{89} = -48.171087355 x 89 = − 48.171087355 x 90 = − 26.1799387799 x_{90} = -26.1799387799 x 90 = − 26.1799387799 x 91 = − 35.6047167407 x_{91} = -35.6047167407 x 91 = − 35.6047167407 x 92 = 24.0855436775 x_{92} = 24.0855436775 x 92 = 24.0855436775 x 93 = 80.6342114421 x_{93} = 80.6342114421 x 93 = 80.6342114421 x 94 = 34.5575191895 x_{94} = 34.5575191895 x 94 = 34.5575191895 x 95 = 37.6991118431 x_{95} = 37.6991118431 x 95 = 37.6991118431 x 96 = − 28.2743338823 x_{96} = -28.2743338823 x 96 = − 28.2743338823
Точки пересечения с осью координат Y
График пересекает ось Y, когда x равняется 0:− sin ( 0 ⋅ 3 ) - \sin{\left (0 \cdot 3 \right )} − sin ( 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 − 3 cos ( 3 x ) = 0 - 3 \cos{\left (3 x \right )} = 0 − 3 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
(--, -1)
6 pi
(--, 1)
2 Интервалы возрастания и убывания функции: 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 [pi/6, pi/2] (-oo, pi/6] U [pi/2, oo)
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 9 sin ( 3 x ) = 0 9 \sin{\left (3 x \right )} = 0 9 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, pi/3] (-oo, 0] U [pi/3, oo)
Горизонтальные асимптоты
Горизонтальные асимптоты найдём с помощью пределов данной функции при x->+oo и x->-oolim x → − ∞ ( − sin ( 3 x ) ) = ⟨ − 1 , 1 ⟩ \lim_{x \to -\infty}\left(- \sin{\left (3 x \right )}\right) = \langle -1, 1\rangle x → − ∞ lim ( − sin ( 3 x ) ) = ⟨ − 1 , 1 ⟩ Возьмём предел y = ⟨ − 1 , 1 ⟩ y = \langle -1, 1\rangle y = ⟨ − 1 , 1 ⟩ lim x → ∞ ( − sin ( 3 x ) ) = ⟨ − 1 , 1 ⟩ \lim_{x \to \infty}\left(- \sin{\left (3 x \right )}\right) = \langle -1, 1\rangle x → ∞ lim ( − sin ( 3 x ) ) = ⟨ − 1 , 1 ⟩ Возьмём предел y = ⟨ − 1 , 1 ⟩ y = \langle -1, 1\rangle y = ⟨ − 1 , 1 ⟩
Наклонные асимптоты
Наклонную асимптоту можно найти, подсчитав предел функции -sin(3*x), делённой на x при x->+oo и x ->-oolim x → − ∞ ( − 1 x sin ( 3 x ) ) = 0 \lim_{x \to -\infty}\left(- \frac{1}{x} \sin{\left (3 x \right )}\right) = 0 x → − ∞ lim ( − x 1 sin ( 3 x ) ) = 0 Возьмём предел lim x → ∞ ( − 1 x sin ( 3 x ) ) = 0 \lim_{x \to \infty}\left(- \frac{1}{x} \sin{\left (3 x \right )}\right) = 0 x → ∞ lim ( − x 1 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).− sin ( 3 x ) = sin ( 3 x ) - \sin{\left (3 x \right )} = \sin{\left (3 x \right )} − sin ( 3 x ) = sin ( 3 x ) − sin ( 3 x ) = − sin ( 3 x ) - \sin{\left (3 x \right )} = - \sin{\left (3 x \right )} − sin ( 3 x ) = − sin ( 3 x ) 