График функции
0 -20000 -15000 -10000 -5000 5000 10000 15000 20000 5 -5
Точки пересечения с осью координат X
График функции пересекает ось X при f = 0sin ( x ) + 1 2 sin ( 2 x ) + 1 3 sin ( 3 x ) = 0 \sin{\left (x \right )} + \frac{1}{2} \sin{\left (2 x \right )} + \frac{1}{3} \sin{\left (3 x \right )} = 0 sin ( x ) + 2 1 sin ( 2 x ) + 3 1 sin ( 3 x ) = 0 Решаем это уравнение Аналитическое решение x 1 = 0 x_{1} = 0 x 1 = 0 x 2 = π x_{2} = \pi x 2 = π Численное решение x 1 = − 94.2477796077 x_{1} = -94.2477796077 x 1 = − 94.2477796077 x 2 = 81.6814089933 x_{2} = 81.6814089933 x 2 = 81.6814089933 x 3 = 84.8230016469 x_{3} = 84.8230016469 x 3 = 84.8230016469 x 4 = − 3.14159265359 x_{4} = -3.14159265359 x 4 = − 3.14159265359 x 5 = 179.070781255 x_{5} = 179.070781255 x 5 = 179.070781255 x 6 = 65.9734457254 x_{6} = 65.9734457254 x 6 = 65.9734457254 x 7 = 15.7079632679 x_{7} = 15.7079632679 x 7 = 15.7079632679 x 8 = 100.530964915 x_{8} = 100.530964915 x 8 = 100.530964915 x 9 = 50.2654824574 x_{9} = 50.2654824574 x 9 = 50.2654824574 x 10 = − 53.407075111 x_{10} = -53.407075111 x 10 = − 53.407075111 x 11 = 40.8407044967 x_{11} = 40.8407044967 x 11 = 40.8407044967 x 12 = − 59.6902604182 x_{12} = -59.6902604182 x 12 = − 59.6902604182 x 13 = 97.3893722613 x_{13} = 97.3893722613 x 13 = 97.3893722613 x 14 = 28.2743338823 x_{14} = 28.2743338823 x 14 = 28.2743338823 x 15 = − 43.9822971503 x_{15} = -43.9822971503 x 15 = − 43.9822971503 x 16 = 25.1327412287 x_{16} = 25.1327412287 x 16 = 25.1327412287 x 17 = − 81.6814089933 x_{17} = -81.6814089933 x 17 = − 81.6814089933 x 18 = − 267.035375555 x_{18} = -267.035375555 x 18 = − 267.035375555 x 19 = 87.9645943005 x_{19} = 87.9645943005 x 19 = 87.9645943005 x 20 = 69.115038379 x_{20} = 69.115038379 x 20 = 69.115038379 x 21 = − 34.5575191895 x_{21} = -34.5575191895 x 21 = − 34.5575191895 x 22 = 78.5398163397 x_{22} = 78.5398163397 x 22 = 78.5398163397 x 23 = − 31.4159265359 x_{23} = -31.4159265359 x 23 = − 31.4159265359 x 24 = − 103.672557568 x_{24} = -103.672557568 x 24 = − 103.672557568 x 25 = 72.2566310326 x_{25} = 72.2566310326 x 25 = 72.2566310326 x 26 = 56.5486677646 x_{26} = 56.5486677646 x 26 = 56.5486677646 x 27 = − 75.3982236862 x_{27} = -75.3982236862 x 27 = − 75.3982236862 x 28 = − 6.28318530718 x_{28} = -6.28318530718 x 28 = − 6.28318530718 x 29 = − 9.42477796077 x_{29} = -9.42477796077 x 29 = − 9.42477796077 x 30 = 6.28318530718 x_{30} = 6.28318530718 x 30 = 6.28318530718 x 31 = − 65.9734457254 x_{31} = -65.9734457254 x 31 = − 65.9734457254 x 32 = − 87.9645943005 x_{32} = -87.9645943005 x 32 = − 87.9645943005 x 33 = − 72.2566310326 x_{33} = -72.2566310326 x 33 = − 72.2566310326 x 34 = 9.42477796077 x_{34} = 9.42477796077 x 34 = 9.42477796077 x 35 = − 50.2654824574 x_{35} = -50.2654824574 x 35 = − 50.2654824574 x 36 = 0 x_{36} = 0 x 36 = 0 x 37 = 91.1061869541 x_{37} = 91.1061869541 x 37 = 91.1061869541 x 38 = 59.6902604182 x_{38} = 59.6902604182 x 38 = 59.6902604182 x 39 = − 47.1238898038 x_{39} = -47.1238898038 x 39 = − 47.1238898038 x 40 = 12.5663706144 x_{40} = 12.5663706144 x 40 = 12.5663706144 x 41 = − 21.9911485751 x_{41} = -21.9911485751 x 41 = − 21.9911485751 x 42 = − 37.6991118431 x_{42} = -37.6991118431 x 42 = − 37.6991118431 x 43 = − 97.3893722613 x_{43} = -97.3893722613 x 43 = − 97.3893722613 x 44 = 94.2477796077 x_{44} = 94.2477796077 x 44 = 94.2477796077 x 45 = 34.5575191895 x_{45} = 34.5575191895 x 45 = 34.5575191895 x 46 = − 91.1061869541 x_{46} = -91.1061869541 x 46 = − 91.1061869541 x 47 = 21.9911485751 x_{47} = 21.9911485751 x 47 = 21.9911485751 x 48 = 37.6991118431 x_{48} = 37.6991118431 x 48 = 37.6991118431 x 49 = 53.407075111 x_{49} = 53.407075111 x 49 = 53.407075111 x 50 = − 78.5398163397 x_{50} = -78.5398163397 x 50 = − 78.5398163397 x 51 = 22892.7856667 x_{51} = 22892.7856667 x 51 = 22892.7856667 x 52 = 43.9822971503 x_{52} = 43.9822971503 x 52 = 43.9822971503 x 53 = − 15.7079632679 x_{53} = -15.7079632679 x 53 = − 15.7079632679 x 54 = − 28.2743338823 x_{54} = -28.2743338823 x 54 = − 28.2743338823
Точки пересечения с осью координат Y
График пересекает ось Y, когда x равняется 0:sin ( 0 ) + 1 2 sin ( 0 ⋅ 2 ) + 1 3 sin ( 0 ⋅ 3 ) \sin{\left (0 \right )} + \frac{1}{2} \sin{\left (0 \cdot 2 \right )} + \frac{1}{3} \sin{\left (0 \cdot 3 \right )} sin ( 0 ) + 2 1 sin ( 0 ⋅ 2 ) + 3 1 sin ( 0 ⋅ 3 ) f ( 0 ) = 0 f{\left (0 \right )} = 0 f ( 0 ) = 0 (0, 0)
Горизонтальные асимптоты
Горизонтальные асимптоты найдём с помощью пределов данной функции при x->+oo и x->-oolim x → − ∞ ( sin ( x ) + 1 2 sin ( 2 x ) + 1 3 sin ( 3 x ) ) = ⟨ − 11 6 , 11 6 ⟩ \lim_{x \to -\infty}\left(\sin{\left (x \right )} + \frac{1}{2} \sin{\left (2 x \right )} + \frac{1}{3} \sin{\left (3 x \right )}\right) = \langle - \frac{11}{6}, \frac{11}{6}\rangle x → − ∞ lim ( sin ( x ) + 2 1 sin ( 2 x ) + 3 1 sin ( 3 x ) ) = ⟨ − 6 11 , 6 11 ⟩ Возьмём предел y = ⟨ − 11 6 , 11 6 ⟩ y = \langle - \frac{11}{6}, \frac{11}{6}\rangle y = ⟨ − 6 11 , 6 11 ⟩ lim x → ∞ ( sin ( x ) + 1 2 sin ( 2 x ) + 1 3 sin ( 3 x ) ) = ⟨ − 11 6 , 11 6 ⟩ \lim_{x \to \infty}\left(\sin{\left (x \right )} + \frac{1}{2} \sin{\left (2 x \right )} + \frac{1}{3} \sin{\left (3 x \right )}\right) = \langle - \frac{11}{6}, \frac{11}{6}\rangle x → ∞ lim ( sin ( x ) + 2 1 sin ( 2 x ) + 3 1 sin ( 3 x ) ) = ⟨ − 6 11 , 6 11 ⟩ Возьмём предел y = ⟨ − 11 6 , 11 6 ⟩ y = \langle - \frac{11}{6}, \frac{11}{6}\rangle y = ⟨ − 6 11 , 6 11 ⟩
Наклонные асимптоты
Наклонную асимптоту можно найти, подсчитав предел функции sin(x) + sin(2*x)/2 + sin(3*x)/3, делённой на x при x->+oo и x ->-oolim x → − ∞ ( 1 x ( sin ( x ) + 1 2 sin ( 2 x ) + 1 3 sin ( 3 x ) ) ) = 0 \lim_{x \to -\infty}\left(\frac{1}{x} \left(\sin{\left (x \right )} + \frac{1}{2} \sin{\left (2 x \right )} + \frac{1}{3} \sin{\left (3 x \right )}\right)\right) = 0 x → − ∞ lim ( x 1 ( sin ( x ) + 2 1 sin ( 2 x ) + 3 1 sin ( 3 x ) ) ) = 0 Возьмём предел lim x → ∞ ( 1 x ( sin ( x ) + 1 2 sin ( 2 x ) + 1 3 sin ( 3 x ) ) ) = 0 \lim_{x \to \infty}\left(\frac{1}{x} \left(\sin{\left (x \right )} + \frac{1}{2} \sin{\left (2 x \right )} + \frac{1}{3} \sin{\left (3 x \right )}\right)\right) = 0 x → ∞ lim ( x 1 ( sin ( x ) + 2 1 sin ( 2 x ) + 3 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 ( x ) + 1 2 sin ( 2 x ) + 1 3 sin ( 3 x ) = − sin ( x ) − 1 2 sin ( 2 x ) − 1 3 sin ( 3 x ) \sin{\left (x \right )} + \frac{1}{2} \sin{\left (2 x \right )} + \frac{1}{3} \sin{\left (3 x \right )} = - \sin{\left (x \right )} - \frac{1}{2} \sin{\left (2 x \right )} - \frac{1}{3} \sin{\left (3 x \right )} sin ( x ) + 2 1 sin ( 2 x ) + 3 1 sin ( 3 x ) = − sin ( x ) − 2 1 sin ( 2 x ) − 3 1 sin ( 3 x ) sin ( x ) + 1 2 sin ( 2 x ) + 1 3 sin ( 3 x ) = − − 1 sin ( x ) − − 1 2 sin ( 2 x ) − − 1 3 sin ( 3 x ) \sin{\left (x \right )} + \frac{1}{2} \sin{\left (2 x \right )} + \frac{1}{3} \sin{\left (3 x \right )} = - -1 \sin{\left (x \right )} - - \frac{1}{2} \sin{\left (2 x \right )} - - \frac{1}{3} \sin{\left (3 x \right )} sin ( x ) + 2 1 sin ( 2 x ) + 3 1 sin ( 3 x ) = − − 1 sin ( x ) − − 2 1 sin ( 2 x ) − − 3 1 sin ( 3 x ) 