График функции
0 -90 -80 -70 -60 -50 -40 -30 -20 -10 10 2 -2
Точки пересечения с осью координат X
График функции пересекает ось X при f = 0− sin ( 2 x ) = 0 - \sin{\left(2 x \right)} = 0 − 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 = − 45.553093477052 x_{1} = -45.553093477052 x 1 = − 45.553093477052 x 2 = − 86.3937979737193 x_{2} = -86.3937979737193 x 2 = − 86.3937979737193 x 3 = − 97.3893722612836 x_{3} = -97.3893722612836 x 3 = − 97.3893722612836 x 4 = − 72.2566310325652 x_{4} = -72.2566310325652 x 4 = − 72.2566310325652 x 5 = 31.4159265358979 x_{5} = 31.4159265358979 x 5 = 31.4159265358979 x 6 = 86.3937979737193 x_{6} = 86.3937979737193 x 6 = 86.3937979737193 x 7 = 50.2654824574367 x_{7} = 50.2654824574367 x 7 = 50.2654824574367 x 8 = 1.5707963267949 x_{8} = 1.5707963267949 x 8 = 1.5707963267949 x 9 = 36.1283155162826 x_{9} = 36.1283155162826 x 9 = 36.1283155162826 x 10 = − 28.2743338823081 x_{10} = -28.2743338823081 x 10 = − 28.2743338823081 x 11 = 4.71238898038469 x_{11} = 4.71238898038469 x 11 = 4.71238898038469 x 12 = 95.8185759344887 x_{12} = 95.8185759344887 x 12 = 95.8185759344887 x 13 = − 7.85398163397448 x_{13} = -7.85398163397448 x 13 = − 7.85398163397448 x 14 = − 73.8274273593601 x_{14} = -73.8274273593601 x 14 = − 73.8274273593601 x 15 = − 23.5619449019235 x_{15} = -23.5619449019235 x 15 = − 23.5619449019235 x 16 = − 65.9734457253857 x_{16} = -65.9734457253857 x 16 = − 65.9734457253857 x 17 = − 48.6946861306418 x_{17} = -48.6946861306418 x 17 = − 48.6946861306418 x 18 = 87.9645943005142 x_{18} = 87.9645943005142 x 18 = 87.9645943005142 x 19 = − 37.6991118430775 x_{19} = -37.6991118430775 x 19 = − 37.6991118430775 x 20 = − 21.9911485751286 x_{20} = -21.9911485751286 x 20 = − 21.9911485751286 x 21 = − 42.4115008234622 x_{21} = -42.4115008234622 x 21 = − 42.4115008234622 x 22 = 100.530964914873 x_{22} = 100.530964914873 x 22 = 100.530964914873 x 23 = 59.6902604182061 x_{23} = 59.6902604182061 x 23 = 59.6902604182061 x 24 = − 36.1283155162826 x_{24} = -36.1283155162826 x 24 = − 36.1283155162826 x 25 = − 58.1194640914112 x_{25} = -58.1194640914112 x 25 = − 58.1194640914112 x 26 = 42.4115008234622 x_{26} = 42.4115008234622 x 26 = 42.4115008234622 x 27 = − 6.28318530717959 x_{27} = -6.28318530717959 x 27 = − 6.28318530717959 x 28 = 113.097335529233 x_{28} = 113.097335529233 x 28 = 113.097335529233 x 29 = 28.2743338823081 x_{29} = 28.2743338823081 x 29 = 28.2743338823081 x 30 = − 14.1371669411541 x_{30} = -14.1371669411541 x 30 = − 14.1371669411541 x 31 = 7.85398163397448 x_{31} = 7.85398163397448 x 31 = 7.85398163397448 x 32 = − 61.261056745001 x_{32} = -61.261056745001 x 32 = − 61.261056745001 x 33 = 590.619418874881 x_{33} = 590.619418874881 x 33 = 590.619418874881 x 34 = − 15.707963267949 x_{34} = -15.707963267949 x 34 = − 15.707963267949 x 35 = 20.4203522483337 x_{35} = 20.4203522483337 x 35 = 20.4203522483337 x 36 = 26.7035375555132 x_{36} = 26.7035375555132 x 36 = 26.7035375555132 x 37 = 78.5398163397448 x_{37} = 78.5398163397448 x 37 = 78.5398163397448 x 38 = − 51.8362787842316 x_{38} = -51.8362787842316 x 38 = − 51.8362787842316 x 39 = 65.9734457253857 x_{39} = 65.9734457253857 x 39 = 65.9734457253857 x 40 = − 94.2477796076938 x_{40} = -94.2477796076938 x 40 = − 94.2477796076938 x 41 = − 50.2654824574367 x_{41} = -50.2654824574367 x 41 = − 50.2654824574367 x 42 = 14.1371669411541 x_{42} = 14.1371669411541 x 42 = 14.1371669411541 x 43 = − 40.8407044966673 x_{43} = -40.8407044966673 x 43 = − 40.8407044966673 x 44 = − 31.4159265358979 x_{44} = -31.4159265358979 x 44 = − 31.4159265358979 x 45 = − 20.4203522483337 x_{45} = -20.4203522483337 x 45 = − 20.4203522483337 x 46 = − 83.2522053201295 x_{46} = -83.2522053201295 x 46 = − 83.2522053201295 x 47 = − 67.5442420521806 x_{47} = -67.5442420521806 x 47 = − 67.5442420521806 x 48 = 72.2566310325652 x_{48} = 72.2566310325652 x 48 = 72.2566310325652 x 49 = − 80.1106126665397 x_{49} = -80.1106126665397 x 49 = − 80.1106126665397 x 50 = − 119.380520836412 x_{50} = -119.380520836412 x 50 = − 119.380520836412 x 51 = 48.6946861306418 x_{51} = 48.6946861306418 x 51 = 48.6946861306418 x 52 = − 1.5707963267949 x_{52} = -1.5707963267949 x 52 = − 1.5707963267949 x 53 = 0 x_{53} = 0 x 53 = 0 x 54 = 23.5619449019235 x_{54} = 23.5619449019235 x 54 = 23.5619449019235 x 55 = 43.9822971502571 x_{55} = 43.9822971502571 x 55 = 43.9822971502571 x 56 = − 75.398223686155 x_{56} = -75.398223686155 x 56 = − 75.398223686155 x 57 = − 17.2787595947439 x_{57} = -17.2787595947439 x 57 = − 17.2787595947439 x 58 = 80.1106126665397 x_{58} = 80.1106126665397 x 58 = 80.1106126665397 x 59 = 56.5486677646163 x_{59} = 56.5486677646163 x 59 = 56.5486677646163 x 60 = 21.9911485751286 x_{60} = 21.9911485751286 x 60 = 21.9911485751286 x 61 = 64.4026493985908 x_{61} = 64.4026493985908 x 61 = 64.4026493985908 x 62 = 89.5353906273091 x_{62} = 89.5353906273091 x 62 = 89.5353906273091 x 63 = − 43.9822971502571 x_{63} = -43.9822971502571 x 63 = − 43.9822971502571 x 64 = 58.1194640914112 x_{64} = 58.1194640914112 x 64 = 58.1194640914112 x 65 = 12.5663706143592 x_{65} = 12.5663706143592 x 65 = 12.5663706143592 x 66 = − 64.4026493985908 x_{66} = -64.4026493985908 x 66 = − 64.4026493985908 x 67 = 67.5442420521806 x_{67} = 67.5442420521806 x 67 = 67.5442420521806 x 68 = 92.6769832808989 x_{68} = 92.6769832808989 x 68 = 92.6769832808989 x 69 = 6.28318530717959 x_{69} = 6.28318530717959 x 69 = 6.28318530717959 x 70 = 34.5575191894877 x_{70} = 34.5575191894877 x 70 = 34.5575191894877 x 71 = − 9.42477796076938 x_{71} = -9.42477796076938 x 71 = − 9.42477796076938 x 72 = 81.6814089933346 x_{72} = 81.6814089933346 x 72 = 81.6814089933346 x 73 = 73.8274273593601 x_{73} = 73.8274273593601 x 73 = 73.8274273593601 x 74 = − 95.8185759344887 x_{74} = -95.8185759344887 x 74 = − 95.8185759344887 x 75 = − 89.5353906273091 x_{75} = -89.5353906273091 x 75 = − 89.5353906273091 x 76 = − 59.6902604182061 x_{76} = -59.6902604182061 x 76 = − 59.6902604182061 x 77 = 37.6991118430775 x_{77} = 37.6991118430775 x 77 = 37.6991118430775 x 78 = 29.845130209103 x_{78} = 29.845130209103 x 78 = 29.845130209103 x 79 = 45.553093477052 x_{79} = 45.553093477052 x 79 = 45.553093477052 x 80 = − 53.4070751110265 x_{80} = -53.4070751110265 x 80 = − 53.4070751110265 x 81 = − 87.9645943005142 x_{81} = -87.9645943005142 x 81 = − 87.9645943005142 x 82 = 15.707963267949 x_{82} = 15.707963267949 x 82 = 15.707963267949 x 83 = − 29.845130209103 x_{83} = -29.845130209103 x 83 = − 29.845130209103 x 84 = − 39.2699081698724 x_{84} = -39.2699081698724 x 84 = − 39.2699081698724 x 85 = 70.6858347057703 x_{85} = 70.6858347057703 x 85 = 70.6858347057703 x 86 = − 483.805268652828 x_{86} = -483.805268652828 x 86 = − 483.805268652828 x 87 = 94.2477796076938 x_{87} = 94.2477796076938 x 87 = 94.2477796076938 x 88 = − 81.6814089933346 x_{88} = -81.6814089933346 x 88 = − 81.6814089933346 x 89 = 51.8362787842316 x_{89} = 51.8362787842316 x 89 = 51.8362787842316
Точки пересечения с осью координат Y
График пересекает ось Y, когда x равняется 0:− sin ( 2 ⋅ 0 ) - \sin{\left(2 \cdot 0 \right)} − sin ( 2 ⋅ 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 − 2 cos ( 2 x ) = 0 - 2 \cos{\left(2 x \right)} = 0 − 2 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 3*pi
(----, 1)
4 Интервалы возрастания и убывания функции: x 1 = π 4 x_{1} = \frac{\pi}{4} x 1 = 4 π x 1 = 3 π 4 x_{1} = \frac{3 \pi}{4} x 1 = 4 3 π [ π 4 , 3 π 4 ] \left[\frac{\pi}{4}, \frac{3 \pi}{4}\right] [ 4 π , 4 3 π ] ( − ∞ , π 4 ] ∪ [ 3 π 4 , ∞ ) \left(-\infty, \frac{\pi}{4}\right] \cup \left[\frac{3 \pi}{4}, \infty\right) ( − ∞ , 4 π ] ∪ [ 4 3 π , ∞ )
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 4 sin ( 2 x ) = 0 4 \sin{\left(2 x \right)} = 0 4 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: [ 0 , π 2 ] \left[0, \frac{\pi}{2}\right] [ 0 , 2 π ] ( − ∞ , 0 ] ∪ [ π 2 , ∞ ) \left(-\infty, 0\right] \cup \left[\frac{\pi}{2}, \infty\right) ( − ∞ , 0 ] ∪ [ 2 π , ∞ )
Горизонтальные асимптоты
Горизонтальные асимптоты найдём с помощью пределов данной функции при x->+oo и x->-oolim x → − ∞ ( − sin ( 2 x ) ) = ⟨ − 1 , 1 ⟩ \lim_{x \to -\infty}\left(- \sin{\left(2 x \right)}\right) = \left\langle -1, 1\right\rangle x → − ∞ lim ( − sin ( 2 x ) ) = ⟨ − 1 , 1 ⟩ Возьмём предел y = ⟨ − 1 , 1 ⟩ y = \left\langle -1, 1\right\rangle y = ⟨ − 1 , 1 ⟩ lim x → ∞ ( − sin ( 2 x ) ) = ⟨ − 1 , 1 ⟩ \lim_{x \to \infty}\left(- \sin{\left(2 x \right)}\right) = \left\langle -1, 1\right\rangle x → ∞ lim ( − sin ( 2 x ) ) = ⟨ − 1 , 1 ⟩ Возьмём предел y = ⟨ − 1 , 1 ⟩ y = \left\langle -1, 1\right\rangle y = ⟨ − 1 , 1 ⟩
Наклонные асимптоты
Наклонную асимптоту можно найти, подсчитав предел функции -sin(2*x), делённой на x при x->+oo и x ->-oolim x → − ∞ ( − sin ( 2 x ) x ) = 0 \lim_{x \to -\infty}\left(- \frac{\sin{\left(2 x \right)}}{x}\right) = 0 x → − ∞ lim ( − x sin ( 2 x ) ) = 0 Возьмём предел lim x → ∞ ( − sin ( 2 x ) x ) = 0 \lim_{x \to \infty}\left(- \frac{\sin{\left(2 x \right)}}{x}\right) = 0 x → ∞ lim ( − x 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).− sin ( 2 x ) = sin ( 2 x ) - \sin{\left(2 x \right)} = \sin{\left(2 x \right)} − sin ( 2 x ) = sin ( 2 x ) − sin ( 2 x ) = − sin ( 2 x ) - \sin{\left(2 x \right)} = - \sin{\left(2 x \right)} − sin ( 2 x ) = − sin ( 2 x ) 
