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