График функции
0 -2000 -1500 -1000 -500 500 1000 1500 2000 0 4
Точки пересечения с осью координат X
График функции пересекает ось X при f = 0sin ( 4 x ) + 1 = 0 \sin{\left (4 x \right )} + 1 = 0 sin ( 4 x ) + 1 = 0 Решаем это уравнение Аналитическое решение x 1 = − π 8 x_{1} = - \frac{\pi}{8} x 1 = − 8 π x 2 = 3 π 8 x_{2} = \frac{3 \pi}{8} x 2 = 8 3 π Численное решение x 1 = − 66.3661446973 x_{1} = -66.3661446973 x 1 = − 66.3661446973 x 2 = − 33.3794220507 x_{2} = -33.3794220507 x 2 = − 33.3794220507 x 3 = 98.567469389 x_{3} = 98.567469389 x 3 = 98.567469389 x 4 = − 60.0829594332 x_{4} = -60.0829594332 x 4 = − 60.0829594332 x 5 = − 61.653755885 x_{5} = -61.653755885 x 5 = − 61.653755885 x 6 = − 77.3617191923 x_{6} = -77.3617191923 x 6 = − 77.3617191923 x 7 = − 53.799774149 x_{7} = -53.799774149 x 7 = − 53.799774149 x 8 = 68.7223393336 x_{8} = 68.7223393336 x 8 = 68.7223393336 x 9 = − 99.3528677616 x_{9} = -99.3528677616 x 9 = − 99.3528677616 x 10 = 87.57189528 x_{10} = 87.57189528 x 10 = 87.57189528 x 11 = − 89.9280899396 x_{11} = -89.9280899396 x 11 = − 89.9280899396 x 12 = 20.0276531245 x_{12} = 20.0276531245 x 12 = 20.0276531245 x 13 = − 58.5121634186 x_{13} = -58.5121634186 x 13 = − 58.5121634186 x 14 = 76.5763208145 x_{14} = 76.5763208145 x 14 = 76.5763208145 x 15 = 2.74889373159 x_{15} = 2.74889373159 x 15 = 2.74889373159 x 16 = 49.8727834643 x_{16} = 49.8727834643 x 16 = 49.8727834643 x 17 = 48.3019870147 x_{17} = 48.3019870147 x 17 = 48.3019870147 x 18 = 92.2842841758 x_{18} = 92.2842841758 x 18 = 92.2842841758 x 19 = − 82.0741080135 x_{19} = -82.0741080135 x 19 = − 82.0741080135 x 20 = − 8.24668059419 x_{20} = -8.24668059419 x 20 = − 8.24668059419 x 21 = 84.4303024715 x_{21} = 84.4303024715 x 21 = 84.4303024715 x 22 = 78.1471173828 x_{22} = 78.1471173828 x 22 = 78.1471173828 x 23 = − 97.7820713167 x_{23} = -97.7820713167 x 23 = − 97.7820713167 x 24 = − 0.392698986135 x_{24} = -0.392698986135 x 24 = − 0.392698986135 x 25 = − 52.228977747 x_{25} = -52.228977747 x 25 = − 52.228977747 x 26 = 43.5895981498 x_{26} = 43.5895981498 x 26 = 43.5895981498 x 27 = − 47.5165889945 x_{27} = -47.5165889945 x 27 = − 47.5165889945 x 28 = − 63.2245518402 x_{28} = -63.2245518402 x 28 = − 63.2245518402 x 29 = 10.6028750937 x_{29} = 10.6028750937 x 29 = 10.6028750937 x 30 = − 22.3838475557 x_{30} = -22.3838475557 x 30 = − 22.3838475557 x 31 = 21.5984495824 x_{31} = 21.5984495824 x 31 = 21.5984495824 x 32 = 34.1648201986 x_{32} = 34.1648201986 x 32 = 34.1648201986 x 33 = − 31.808625564 x_{33} = -31.808625564 x 33 = − 31.808625564 x 34 = 46.731190792 x_{34} = 46.731190792 x 34 = 46.731190792 x 35 = − 80.5033119348 x_{35} = -80.5033119348 x 35 = − 80.5033119348 x 36 = 71.8639320431 x_{36} = 71.8639320431 x 36 = 71.8639320431 x 37 = 84.4303027206 x_{37} = 84.4303027206 x 37 = 84.4303027206 x 38 = − 41.2334033016 x_{38} = -41.2334033016 x 38 = − 41.2334033016 x 39 = 13.7444679819 x_{39} = 13.7444679819 x 39 = 13.7444679819 x 40 = − 23.9546440699 x_{40} = -23.9546440699 x 40 = − 23.9546440699 x 41 = − 69.5077375715 x_{41} = -69.5077375715 x 41 = − 69.5077375715 x 42 = 32.5940236668 x_{42} = 32.5940236668 x 42 = 32.5940236668 x 43 = − 25.5254404172 x_{43} = -25.5254404172 x 43 = − 25.5254404172 x 44 = − 36.5210148981 x_{44} = -36.5210148981 x 44 = − 36.5210148981 x 45 = − 9.81747697788 x_{45} = -9.81747697788 x 45 = − 9.81747697788 x 46 = 42.0188017155 x_{46} = 42.0188017155 x 46 = 42.0188017155 x 47 = 64.009950304 x_{47} = 64.009950304 x 47 = 64.009950304 x 48 = − 88.3572932691 x_{48} = -88.3572932691 x 48 = − 88.3572932691 x 49 = − 1.96349547006 x_{49} = -1.96349547006 x 49 = − 1.96349547006 x 50 = − 44.3749961262 x_{50} = -44.3749961262 x 50 = − 44.3749961262 x 51 = 79.7179137087 x_{51} = 79.7179137087 x 51 = 79.7179137087 x 52 = 4.31968985435 x_{52} = 4.31968985435 x 52 = 4.31968985435 x 53 = 57.7267651334 x_{53} = 57.7267651334 x 53 = 57.7267651334 x 54 = − 367.959039668 x_{54} = -367.959039668 x 54 = − 367.959039668 x 55 = − 3.53429183982 x_{55} = -3.53429183982 x 55 = − 3.53429183982 x 56 = 86.0010988907 x_{56} = 86.0010988907 x 56 = 86.0010988907 x 57 = − 55.3705706219 x_{57} = -55.3705706219 x 57 = − 55.3705706219 x 58 = 70.2931355951 x_{58} = 70.2931355951 x 58 = 70.2931355951 x 59 = − 91.4988861483 x_{59} = -91.4988861483 x 59 = − 91.4988861483 x 60 = 18.4568567378 x_{60} = 18.4568567378 x 60 = 18.4568567378 x 61 = 62.4391538934 x_{61} = 62.4391538934 x 61 = 62.4391538934 x 62 = 5.89048630602 x_{62} = 5.89048630602 x 62 = 5.89048630602 x 63 = 48.3019869823 x_{63} = 48.3019869823 x 63 = 48.3019869823 x 64 = 27.8816348853 x_{64} = 27.8816348853 x 64 = 27.8816348853 x 65 = − 14.5298663256 x_{65} = -14.5298663256 x 65 = − 14.5298663256 x 66 = 40.4480053155 x_{66} = 40.4480053155 x 66 = 40.4480053155 x 67 = − 11.3882734788 x_{67} = -11.3882734788 x 67 = − 11.3882734788 x 68 = − 96.2112749007 x_{68} = -96.2112749007 x 68 = − 96.2112749007 x 69 = − 83.6449044644 x_{69} = -83.6449044644 x 69 = − 83.6449044644 x 70 = − 17.6714587256 x_{70} = -17.6714587256 x 70 = − 17.6714587256 x 71 = − 39.6626073054 x_{71} = -39.6626073054 x 71 = − 39.6626073054 x 72 = 54.5851722405 x_{72} = 54.5851722405 x 72 = 54.5851722405 x 73 = − 85.2157006171 x_{73} = -85.2157006171 x 73 = − 85.2157006171 x 74 = 35.7356165577 x_{74} = 35.7356165577 x 74 = 35.7356165577 x 75 = − 16.1006622735 x_{75} = -16.1006622735 x 75 = − 16.1006622735 x 76 = − 45.945792678 x_{76} = -45.945792678 x 76 = − 45.945792678 x 77 = 90.7134878812 x_{77} = 90.7134878812 x 77 = 90.7134878812 x 78 = 56.1559687898 x_{78} = 56.1559687898 x 78 = 56.1559687898 x 79 = 24.7400422577 x_{79} = 24.7400422577 x 79 = 24.7400422577 x 80 = 26.3108384344 x_{80} = 26.3108384344 x 80 = 26.3108384344 x 81 = − 19.2422547864 x_{81} = -19.2422547864 x 81 = − 19.2422547864 x 82 = 100.138265971 x_{82} = 100.138265971 x 82 = 100.138265971 x 83 = − 67.9369412996 x_{83} = -67.9369412996 x 83 = − 67.9369412996 x 84 = 93.8550806217 x_{84} = 93.8550806217 x 84 = 93.8550806217 x 85 = 27.8816346273 x_{85} = 27.8816346273 x 85 = 27.8816346273 x 86 = − 38.0918108532 x_{86} = -38.0918108532 x 86 = − 38.0918108532 x 87 = − 30.2378291705 x_{87} = -30.2378291705 x 87 = − 30.2378291705 x 88 = − 74.2201263237 x_{88} = -74.2201263237 x 88 = − 74.2201263237 x 89 = 65.5807467158 x_{89} = 65.5807467158 x 89 = 65.5807467158 x 90 = − 75.7909227332 x_{90} = -75.7909227332 x 90 = − 75.7909227332 x 91 = 12.1736716095 x_{91} = 12.1736716095 x 91 = 12.1736716095
Точки пересечения с осью координат Y
График пересекает ось Y, когда x равняется 0:sin ( 0 ⋅ 4 ) + 1 \sin{\left (0 \cdot 4 \right )} + 1 sin ( 0 ⋅ 4 ) + 1 f ( 0 ) = 1 f{\left (0 \right )} = 1 f ( 0 ) = 1 (0, 1)
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 4 cos ( 4 x ) = 0 4 \cos{\left (4 x \right )} = 0 4 cos ( 4 x ) = 0 Решаем это уравнение x 1 = π 8 x_{1} = \frac{\pi}{8} x 1 = 8 π x 2 = 3 π 8 x_{2} = \frac{3 \pi}{8} x 2 = 8 3 π pi
(--, 2)
8 3*pi
(----, 0)
8 Интервалы возрастания и убывания функции: x 2 = 3 π 8 x_{2} = \frac{3 \pi}{8} x 2 = 8 3 π x 2 = π 8 x_{2} = \frac{\pi}{8} x 2 = 8 π (-oo, pi/8] U [3*pi/8, oo) [pi/8, 3*pi/8]
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 − 16 sin ( 4 x ) = 0 - 16 \sin{\left (4 x \right )} = 0 − 16 sin ( 4 x ) = 0 Решаем это уравнение x 1 = 0 x_{1} = 0 x 1 = 0 x 2 = π 4 x_{2} = \frac{\pi}{4} x 2 = 4 π Интервалы выпуклости и вогнутости: (-oo, 0] U [pi/4, oo) [0, pi/4]
Горизонтальные асимптоты
Горизонтальные асимптоты найдём с помощью пределов данной функции при x->+oo и x->-oolim x → − ∞ ( sin ( 4 x ) + 1 ) = ⟨ 0 , 2 ⟩ \lim_{x \to -\infty}\left(\sin{\left (4 x \right )} + 1\right) = \langle 0, 2\rangle x → − ∞ lim ( sin ( 4 x ) + 1 ) = ⟨ 0 , 2 ⟩ Возьмём предел y = ⟨ 0 , 2 ⟩ y = \langle 0, 2\rangle y = ⟨ 0 , 2 ⟩ lim x → ∞ ( sin ( 4 x ) + 1 ) = ⟨ 0 , 2 ⟩ \lim_{x \to \infty}\left(\sin{\left (4 x \right )} + 1\right) = \langle 0, 2\rangle x → ∞ lim ( sin ( 4 x ) + 1 ) = ⟨ 0 , 2 ⟩ Возьмём предел y = ⟨ 0 , 2 ⟩ y = \langle 0, 2\rangle y = ⟨ 0 , 2 ⟩
Наклонные асимптоты
Наклонную асимптоту можно найти, подсчитав предел функции sin(4*x) + 1, делённой на x при x->+oo и x ->-oolim x → − ∞ ( 1 x ( sin ( 4 x ) + 1 ) ) = 0 \lim_{x \to -\infty}\left(\frac{1}{x} \left(\sin{\left (4 x \right )} + 1\right)\right) = 0 x → − ∞ lim ( x 1 ( sin ( 4 x ) + 1 ) ) = 0 Возьмём предел lim x → ∞ ( 1 x ( sin ( 4 x ) + 1 ) ) = 0 \lim_{x \to \infty}\left(\frac{1}{x} \left(\sin{\left (4 x \right )} + 1\right)\right) = 0 x → ∞ lim ( x 1 ( sin ( 4 x ) + 1 ) ) = 0 Возьмём предел
Чётность и нечётность функции
Проверим функци чётна или нечётна с помощью соотношений f = f(-x) и f = -f(-x).sin ( 4 x ) + 1 = − sin ( 4 x ) + 1 \sin{\left (4 x \right )} + 1 = - \sin{\left (4 x \right )} + 1 sin ( 4 x ) + 1 = − sin ( 4 x ) + 1 sin ( 4 x ) + 1 = − − 1 sin ( 4 x ) − 1 \sin{\left (4 x \right )} + 1 = - -1 \sin{\left (4 x \right )} - 1 sin ( 4 x ) + 1 = − − 1 sin ( 4 x ) − 1 