График функции
0 -150000 -125000 -100000 -75000 -50000 -25000 25000 50000 75000 100000 125000 1 -1
Точки пересечения с осью координат X
График функции пересекает ось X при f = 01 2 sin ( 2 x ) = 0 \frac{1}{2} \sin{\left (2 x \right )} = 0 2 1 sin ( 2 x ) = 0 Решаем это уравнение Аналитическое решение x 1 = 0 x_{1} = 0 x 1 = 0 x 2 = π 2 x_{2} = \frac{\pi}{2} x 2 = 2 π Численное решение x 1 = − 95.8185759345 x_{1} = -95.8185759345 x 1 = − 95.8185759345 x 2 = 78.5398163397 x_{2} = 78.5398163397 x 2 = 78.5398163397 x 3 = 73.8274273594 x_{3} = 73.8274273594 x 3 = 73.8274273594 x 4 = 36.1283155163 x_{4} = 36.1283155163 x 4 = 36.1283155163 x 5 = 56.5486677646 x_{5} = 56.5486677646 x 5 = 56.5486677646 x 6 = 23.5619449019 x_{6} = 23.5619449019 x 6 = 23.5619449019 x 7 = 51.8362787842 x_{7} = 51.8362787842 x 7 = 51.8362787842 x 8 = 14.1371669412 x_{8} = 14.1371669412 x 8 = 14.1371669412 x 9 = − 51.8362787842 x_{9} = -51.8362787842 x 9 = − 51.8362787842 x 10 = 12.5663706144 x_{10} = 12.5663706144 x 10 = 12.5663706144 x 11 = − 97.3893722613 x_{11} = -97.3893722613 x 11 = − 97.3893722613 x 12 = − 7.85398163397 x_{12} = -7.85398163397 x 12 = − 7.85398163397 x 13 = 89.5353906273 x_{13} = 89.5353906273 x 13 = 89.5353906273 x 14 = − 23.5619449019 x_{14} = -23.5619449019 x 14 = − 23.5619449019 x 15 = − 119.380520836 x_{15} = -119.380520836 x 15 = − 119.380520836 x 16 = 48.6946861306 x_{16} = 48.6946861306 x 16 = 48.6946861306 x 17 = 29.8451302091 x_{17} = 29.8451302091 x 17 = 29.8451302091 x 18 = 43.9822971503 x_{18} = 43.9822971503 x 18 = 43.9822971503 x 19 = 45.5530934771 x_{19} = 45.5530934771 x 19 = 45.5530934771 x 20 = 113.097335529 x_{20} = 113.097335529 x 20 = 113.097335529 x 21 = 92.6769832809 x_{21} = 92.6769832809 x 21 = 92.6769832809 x 22 = − 483.805268653 x_{22} = -483.805268653 x 22 = − 483.805268653 x 23 = − 80.1106126665 x_{23} = -80.1106126665 x 23 = − 80.1106126665 x 24 = − 58.1194640914 x_{24} = -58.1194640914 x 24 = − 58.1194640914 x 25 = 87.9645943005 x_{25} = 87.9645943005 x 25 = 87.9645943005 x 26 = 64.4026493986 x_{26} = 64.4026493986 x 26 = 64.4026493986 x 27 = 26.7035375555 x_{27} = 26.7035375555 x 27 = 26.7035375555 x 28 = − 83.2522053201 x_{28} = -83.2522053201 x 28 = − 83.2522053201 x 29 = 42.4115008235 x_{29} = 42.4115008235 x 29 = 42.4115008235 x 30 = 59.6902604182 x_{30} = 59.6902604182 x 30 = 59.6902604182 x 31 = − 67.5442420522 x_{31} = -67.5442420522 x 31 = − 67.5442420522 x 32 = − 86.3937979737 x_{32} = -86.3937979737 x 32 = − 86.3937979737 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 = 31.4159265359 x_{35} = 31.4159265359 x 35 = 31.4159265359 x 36 = 21.9911485751 x_{36} = 21.9911485751 x 36 = 21.9911485751 x 37 = − 1.57079632679 x_{37} = -1.57079632679 x 37 = − 1.57079632679 x 38 = − 29.8451302091 x_{38} = -29.8451302091 x 38 = − 29.8451302091 x 39 = 0 x_{39} = 0 x 39 = 0 x 40 = − 14.1371669412 x_{40} = -14.1371669412 x 40 = − 14.1371669412 x 41 = − 94.2477796077 x_{41} = -94.2477796077 x 41 = − 94.2477796077 x 42 = 67.5442420522 x_{42} = 67.5442420522 x 42 = 67.5442420522 x 43 = 86.3937979737 x_{43} = 86.3937979737 x 43 = 86.3937979737 x 44 = − 17.2787595947 x_{44} = -17.2787595947 x 44 = − 17.2787595947 x 45 = 15.7079632679 x_{45} = 15.7079632679 x 45 = 15.7079632679 x 46 = 50.2654824574 x_{46} = 50.2654824574 x 46 = 50.2654824574 x 47 = − 53.407075111 x_{47} = -53.407075111 x 47 = − 53.407075111 x 48 = − 59.6902604182 x_{48} = -59.6902604182 x 48 = − 59.6902604182 x 49 = 28.2743338823 x_{49} = 28.2743338823 x 49 = 28.2743338823 x 50 = − 43.9822971503 x_{50} = -43.9822971503 x 50 = − 43.9822971503 x 51 = − 48.6946861306 x_{51} = -48.6946861306 x 51 = − 48.6946861306 x 52 = − 81.6814089933 x_{52} = -81.6814089933 x 52 = − 81.6814089933 x 53 = 72.2566310326 x_{53} = 72.2566310326 x 53 = 72.2566310326 x 54 = 4.71238898038 x_{54} = 4.71238898038 x 54 = 4.71238898038 x 55 = − 6.28318530718 x_{55} = -6.28318530718 x 55 = − 6.28318530718 x 56 = 7.85398163397 x_{56} = 7.85398163397 x 56 = 7.85398163397 x 57 = − 39.2699081699 x_{57} = -39.2699081699 x 57 = − 39.2699081699 x 58 = − 72.2566310326 x_{58} = -72.2566310326 x 58 = − 72.2566310326 x 59 = − 73.8274273594 x_{59} = -73.8274273594 x 59 = − 73.8274273594 x 60 = − 45.5530934771 x_{60} = -45.5530934771 x 60 = − 45.5530934771 x 61 = 80.1106126665 x_{61} = 80.1106126665 x 61 = 80.1106126665 x 62 = 590.619418875 x_{62} = 590.619418875 x 62 = 590.619418875 x 63 = − 61.261056745 x_{63} = -61.261056745 x 63 = − 61.261056745 x 64 = 94.2477796077 x_{64} = 94.2477796077 x 64 = 94.2477796077 x 65 = 20.4203522483 x_{65} = 20.4203522483 x 65 = 20.4203522483 x 66 = − 15.7079632679 x_{66} = -15.7079632679 x 66 = − 15.7079632679 x 67 = − 65.9734457254 x_{67} = -65.9734457254 x 67 = − 65.9734457254 x 68 = 81.6814089933 x_{68} = 81.6814089933 x 68 = 81.6814089933 x 69 = 65.9734457254 x_{69} = 65.9734457254 x 69 = 65.9734457254 x 70 = 100.530964915 x_{70} = 100.530964915 x 70 = 100.530964915 x 71 = − 89.5353906273 x_{71} = -89.5353906273 x 71 = − 89.5353906273 x 72 = − 42.4115008235 x_{72} = -42.4115008235 x 72 = − 42.4115008235 x 73 = 58.1194640914 x_{73} = 58.1194640914 x 73 = 58.1194640914 x 74 = 95.8185759345 x_{74} = 95.8185759345 x 74 = 95.8185759345 x 75 = − 36.1283155163 x_{75} = -36.1283155163 x 75 = − 36.1283155163 x 76 = − 31.4159265359 x_{76} = -31.4159265359 x 76 = − 31.4159265359 x 77 = − 75.3982236862 x_{77} = -75.3982236862 x 77 = − 75.3982236862 x 78 = − 9.42477796077 x_{78} = -9.42477796077 x 78 = − 9.42477796077 x 79 = 6.28318530718 x_{79} = 6.28318530718 x 79 = 6.28318530718 x 80 = − 87.9645943005 x_{80} = -87.9645943005 x 80 = − 87.9645943005 x 81 = − 50.2654824574 x_{81} = -50.2654824574 x 81 = − 50.2654824574 x 82 = − 64.4026493986 x_{82} = -64.4026493986 x 82 = − 64.4026493986 x 83 = − 20.4203522483 x_{83} = -20.4203522483 x 83 = − 20.4203522483 x 84 = 34.5575191895 x_{84} = 34.5575191895 x 84 = 34.5575191895 x 85 = 37.6991118431 x_{85} = 37.6991118431 x 85 = 37.6991118431 x 86 = 70.6858347058 x_{86} = 70.6858347058 x 86 = 70.6858347058 x 87 = − 40.8407044967 x_{87} = -40.8407044967 x 87 = − 40.8407044967 x 88 = 1.57079632679 x_{88} = 1.57079632679 x 88 = 1.57079632679 x 89 = − 28.2743338823 x_{89} = -28.2743338823 x 89 = − 28.2743338823
Точки пересечения с осью координат Y
График пересекает ось Y, когда x равняется 0:1 2 sin ( 0 ⋅ 2 ) \frac{1}{2} \sin{\left (0 \cdot 2 \right )} 2 1 sin ( 0 ⋅ 2 ) 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 cos ( 2 x ) = 0 \cos{\left (2 x \right )} = 0 cos ( 2 x ) = 0 Решаем это уравнение x 1 = π 4 x_{1} = \frac{\pi}{4} x 1 = 4 π x 2 = 3 π 4 x_{2} = \frac{3 \pi}{4} x 2 = 4 3 π pi 1
(--, -)
4 2 3*pi
(----, -1/2)
4 Интервалы возрастания и убывания функции: x 2 = 3 π 4 x_{2} = \frac{3 \pi}{4} x 2 = 4 3 π x 2 = π 4 x_{2} = \frac{\pi}{4} x 2 = 4 π (-oo, pi/4] U [3*pi/4, oo) [pi/4, 3*pi/4]
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 − 2 sin ( 2 x ) = 0 - 2 \sin{\left (2 x \right )} = 0 − 2 sin ( 2 x ) = 0 Решаем это уравнение x 1 = 0 x_{1} = 0 x 1 = 0 x 2 = π 2 x_{2} = \frac{\pi}{2} x 2 = 2 π Интервалы выпуклости и вогнутости: (-oo, 0] U [pi/2, oo) [0, pi/2]
Горизонтальные асимптоты
Горизонтальные асимптоты найдём с помощью пределов данной функции при x->+oo и x->-oolim x → − ∞ ( 1 2 sin ( 2 x ) ) = ⟨ − 1 2 , 1 2 ⟩ \lim_{x \to -\infty}\left(\frac{1}{2} \sin{\left (2 x \right )}\right) = \langle - \frac{1}{2}, \frac{1}{2}\rangle x → − ∞ lim ( 2 1 sin ( 2 x ) ) = ⟨ − 2 1 , 2 1 ⟩ Возьмём предел y = ⟨ − 1 2 , 1 2 ⟩ y = \langle - \frac{1}{2}, \frac{1}{2}\rangle y = ⟨ − 2 1 , 2 1 ⟩ lim x → ∞ ( 1 2 sin ( 2 x ) ) = ⟨ − 1 2 , 1 2 ⟩ \lim_{x \to \infty}\left(\frac{1}{2} \sin{\left (2 x \right )}\right) = \langle - \frac{1}{2}, \frac{1}{2}\rangle x → ∞ lim ( 2 1 sin ( 2 x ) ) = ⟨ − 2 1 , 2 1 ⟩ Возьмём предел y = ⟨ − 1 2 , 1 2 ⟩ y = \langle - \frac{1}{2}, \frac{1}{2}\rangle y = ⟨ − 2 1 , 2 1 ⟩
Наклонные асимптоты
Наклонную асимптоту можно найти, подсчитав предел функции sin(2*x)/2, делённой на x при x->+oo и x ->-oolim x → − ∞ ( 1 2 x sin ( 2 x ) ) = 0 \lim_{x \to -\infty}\left(\frac{1}{2 x} \sin{\left (2 x \right )}\right) = 0 x → − ∞ lim ( 2 x 1 sin ( 2 x ) ) = 0 Возьмём предел lim x → ∞ ( 1 2 x sin ( 2 x ) ) = 0 \lim_{x \to \infty}\left(\frac{1}{2 x} \sin{\left (2 x \right )}\right) = 0 x → ∞ lim ( 2 x 1 sin ( 2 x ) ) = 0 Возьмём предел
Чётность и нечётность функции
Проверим функци чётна или нечётна с помощью соотношений f = f(-x) и f = -f(-x).1 2 sin ( 2 x ) = − 1 2 sin ( 2 x ) \frac{1}{2} \sin{\left (2 x \right )} = - \frac{1}{2} \sin{\left (2 x \right )} 2 1 sin ( 2 x ) = − 2 1 sin ( 2 x ) 1 2 sin ( 2 x ) = − 1 2 ( − 1 sin ( 2 x ) ) \frac{1}{2} \sin{\left (2 x \right )} = - \frac{1}{2} \left(-1 \sin{\left (2 x \right )}\right) 2 1 sin ( 2 x ) = − 2 1 ( − 1 sin ( 2 x ) ) 