График функции
0 5 -30 -25 -20 -15 -10 -5 10 2.5 -2.5
Точки пересечения с осью координат X
График функции пересекает ось X при f = 0sin 2 ( x ) + cos ( x ) = 0 \sin^{2}{\left(x \right)} + \cos{\left(x \right)} = 0 sin 2 ( x ) + cos ( x ) = 0 Решаем это уравнение Аналитическое решение x 1 = − 2 atan ( 2 + 5 ) x_{1} = - 2 \operatorname{atan}{\left(\sqrt{2 + \sqrt{5}} \right)} x 1 = − 2 atan ( 2 + 5 ) x 2 = 2 atan ( 2 + 5 ) x_{2} = 2 \operatorname{atan}{\left(\sqrt{2 + \sqrt{5}} \right)} x 2 = 2 atan ( 2 + 5 ) Численное решение x 1 = − 14.8034063736466 x_{1} = -14.8034063736466 x 1 = − 14.8034063736466 x 2 = − 10.3293348550718 x_{2} = -10.3293348550718 x 2 = − 10.3293348550718 x 3 = − 33.6529622951853 x_{3} = -33.6529622951853 x 3 = − 33.6529622951853 x 4 = 102.768000674161 x_{4} = 102.768000674161 x 4 = 102.768000674161 x 5 = − 66.878002619688 x_{5} = -66.878002619688 x 5 = − 66.878002619688 x 6 = 85.7275585412268 x_{6} = 85.7275585412268 x 6 = 85.7275585412268 x 7 = − 4.04614954789217 x_{7} = -4.04614954789217 x 7 = − 4.04614954789217 x 8 = 33.6529622951853 x_{8} = 33.6529622951853 x 8 = 33.6529622951853 x 9 = 27.3697769880058 x_{9} = 27.3697769880058 x 9 = 27.3697769880058 x 10 = 65.0688888310833 x_{10} = 65.0688888310833 x 10 = 65.0688888310833 x 11 = − 52.5025182167241 x_{11} = -52.5025182167241 x 11 = − 52.5025182167241 x 12 = − 16.6125201622513 x_{12} = -16.6125201622513 x 12 = − 16.6125201622513 x 13 = 92.0107438484064 x_{13} = 92.0107438484064 x 13 = 92.0107438484064 x 14 = 2.23703575928741 x_{14} = 2.23703575928741 x 14 = 2.23703575928741 x 15 = 29.1788907766105 x_{15} = 29.1788907766105 x 15 = 29.1788907766105 x 16 = − 46.2193329095445 x_{16} = -46.2193329095445 x 16 = − 46.2193329095445 x 17 = − 54.3116320053289 x_{17} = -54.3116320053289 x 17 = − 54.3116320053289 x 18 = 60.5948173125085 x_{18} = 60.5948173125085 x 18 = 60.5948173125085 x 19 = 96.4848153669812 x_{19} = 96.4848153669812 x 19 = 96.4848153669812 x 20 = 14.8034063736466 x_{20} = 14.8034063736466 x 20 = 14.8034063736466 x 21 = 77.6352594454424 x_{21} = 77.6352594454424 x 21 = 77.6352594454424 x 22 = − 2.23703575928741 x_{22} = -2.23703575928741 x 22 = − 2.23703575928741 x 23 = 52.5025182167241 x_{23} = 52.5025182167241 x 23 = 52.5025182167241 x 24 = 16.6125201622513 x_{24} = 16.6125201622513 x 24 = 16.6125201622513 x 25 = − 8.520221066467 x_{25} = -8.520221066467 x 25 = − 8.520221066467 x 26 = − 102.768000674161 x_{26} = -102.768000674161 x 26 = − 102.768000674161 x 27 = 66.878002619688 x_{27} = 66.878002619688 x 27 = 66.878002619688 x 28 = − 85.7275585412268 x_{28} = -85.7275585412268 x 28 = − 85.7275585412268 x 29 = − 442.060007261858 x_{29} = -442.060007261858 x 29 = − 442.060007261858 x 30 = 4.04614954789217 x_{30} = 4.04614954789217 x 30 = 4.04614954789217 x 31 = − 96.4848153669812 x_{31} = -96.4848153669812 x 31 = − 96.4848153669812 x 32 = − 27.3697769880058 x_{32} = -27.3697769880058 x 32 = − 27.3697769880058 x 33 = 46.2193329095445 x_{33} = 46.2193329095445 x 33 = 46.2193329095445 x 34 = − 73.1611879268676 x_{34} = -73.1611879268676 x 34 = − 73.1611879268676 x 35 = 48.0284466981493 x_{35} = 48.0284466981493 x 35 = 48.0284466981493 x 36 = − 48.0284466981493 x_{36} = -48.0284466981493 x 36 = − 48.0284466981493 x 37 = − 41.7452613909697 x_{37} = -41.7452613909697 x 37 = − 41.7452613909697 x 38 = − 58.7857035239037 x_{38} = -58.7857035239037 x 38 = − 58.7857035239037 x 39 = − 98.293929155586 x_{39} = -98.293929155586 x 39 = − 98.293929155586 x 40 = 21.0865916808262 x_{40} = 21.0865916808262 x 40 = 21.0865916808262 x 41 = 39.9361476023649 x_{41} = 39.9361476023649 x 41 = 39.9361476023649 x 42 = 90.2016300598016 x_{42} = 90.2016300598016 x 42 = 90.2016300598016 x 43 = 41.7452613909697 x_{43} = 41.7452613909697 x 43 = 41.7452613909697 x 44 = − 22.8957054694309 x_{44} = -22.8957054694309 x 44 = − 22.8957054694309 x 45 = 98.293929155586 x_{45} = 98.293929155586 x 45 = 98.293929155586 x 46 = − 92.0107438484064 x_{46} = -92.0107438484064 x 46 = − 92.0107438484064 x 47 = 10.3293348550718 x_{47} = 10.3293348550718 x 47 = 10.3293348550718 x 48 = 79.4443732340472 x_{48} = 79.4443732340472 x 48 = 79.4443732340472 x 49 = 73.1611879268676 x_{49} = 73.1611879268676 x 49 = 73.1611879268676 x 50 = − 83.918444752622 x_{50} = -83.918444752622 x 50 = − 83.918444752622 x 51 = − 39.9361476023649 x_{51} = -39.9361476023649 x 51 = − 39.9361476023649 x 52 = 83.918444752622 x_{52} = 83.918444752622 x 52 = 83.918444752622 x 53 = − 35.4620760837901 x_{53} = -35.4620760837901 x 53 = − 35.4620760837901 x 54 = 8.520221066467 x_{54} = 8.520221066467 x 54 = 8.520221066467 x 55 = − 77.6352594454424 x_{55} = -77.6352594454424 x 55 = − 77.6352594454424 x 56 = 35.4620760837901 x_{56} = 35.4620760837901 x 56 = 35.4620760837901 x 57 = 54.3116320053289 x_{57} = 54.3116320053289 x 57 = 54.3116320053289 x 58 = − 90.2016300598016 x_{58} = -90.2016300598016 x 58 = − 90.2016300598016 x 59 = − 29.1788907766105 x_{59} = -29.1788907766105 x 59 = − 29.1788907766105 x 60 = − 60.5948173125085 x_{60} = -60.5948173125085 x 60 = − 60.5948173125085 x 61 = 58.7857035239037 x_{61} = 58.7857035239037 x 61 = 58.7857035239037 x 62 = − 71.3520741382629 x_{62} = -71.3520741382629 x 62 = − 71.3520741382629 x 63 = − 79.4443732340472 x_{63} = -79.4443732340472 x 63 = − 79.4443732340472 x 64 = 22.8957054694309 x_{64} = 22.8957054694309 x 64 = 22.8957054694309 x 65 = 71.3520741382629 x_{65} = 71.3520741382629 x 65 = 71.3520741382629
Точки пересечения с осью координат Y
График пересекает ось Y, когда x равняется 0:sin 2 ( 0 ) + cos ( 0 ) \sin^{2}{\left(0 \right)} + \cos{\left(0 \right)} sin 2 ( 0 ) + cos ( 0 ) 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 2 sin ( x ) cos ( x ) − sin ( x ) = 0 2 \sin{\left(x \right)} \cos{\left(x \right)} - \sin{\left(x \right)} = 0 2 sin ( x ) cos ( x ) − sin ( x ) = 0 Решаем это уравнение x 1 = 0 x_{1} = 0 x 1 = 0 x 2 = − π 3 x_{2} = - \frac{\pi}{3} x 2 = − 3 π x 3 = π 3 x_{3} = \frac{\pi}{3} x 3 = 3 π (0, 1) -pi
(----, 5/4)
3 pi
(--, 5/4)
3 Интервалы возрастания и убывания функции: x 1 = 0 x_{1} = 0 x 1 = 0 x 1 = − π 3 x_{1} = - \frac{\pi}{3} x 1 = − 3 π x 1 = π 3 x_{1} = \frac{\pi}{3} x 1 = 3 π ( − ∞ , − π 3 ] ∪ [ 0 , ∞ ) \left(-\infty, - \frac{\pi}{3}\right] \cup \left[0, \infty\right) ( − ∞ , − 3 π ] ∪ [ 0 , ∞ ) ( − ∞ , 0 ] ∪ [ π 3 , ∞ ) \left(-\infty, 0\right] \cup \left[\frac{\pi}{3}, \infty\right) ( − ∞ , 0 ] ∪ [ 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 − 2 sin 2 ( x ) + 2 cos 2 ( x ) − cos ( x ) = 0 - 2 \sin^{2}{\left(x \right)} + 2 \cos^{2}{\left(x \right)} - \cos{\left(x \right)} = 0 − 2 sin 2 ( x ) + 2 cos 2 ( x ) − cos ( x ) = 0 Решаем это уравнение x 1 = − 2 atan ( 3 6 − 33 3 ) x_{1} = - 2 \operatorname{atan}{\left(\frac{\sqrt{3} \sqrt{6 - \sqrt{33}}}{3} \right)} x 1 = − 2 atan ( 3 3 6 − 33 ) x 2 = 2 atan ( 3 6 − 33 3 ) x_{2} = 2 \operatorname{atan}{\left(\frac{\sqrt{3} \sqrt{6 - \sqrt{33}}}{3} \right)} x 2 = 2 atan ( 3 3 6 − 33 ) x 3 = − 2 atan ( 3 33 + 6 3 ) x_{3} = - 2 \operatorname{atan}{\left(\frac{\sqrt{3} \sqrt{\sqrt{33} + 6}}{3} \right)} x 3 = − 2 atan ( 3 3 33 + 6 ) x 4 = 2 atan ( 3 33 + 6 3 ) x_{4} = 2 \operatorname{atan}{\left(\frac{\sqrt{3} \sqrt{\sqrt{33} + 6}}{3} \right)} x 4 = 2 atan ( 3 3 33 + 6 ) Интервалы выпуклости и вогнутости: [ 2 atan ( 3 33 + 6 3 ) , ∞ ) \left[2 \operatorname{atan}{\left(\frac{\sqrt{3} \sqrt{\sqrt{33} + 6}}{3} \right)}, \infty\right) [ 2 atan ( 3 3 33 + 6 ) , ∞ ) ( − ∞ , − 2 atan ( 3 6 − 33 3 ) ] ∪ [ 2 atan ( 3 6 − 33 3 ) , 2 atan ( 3 33 + 6 3 ) ] \left(-\infty, - 2 \operatorname{atan}{\left(\frac{\sqrt{3} \sqrt{6 - \sqrt{33}}}{3} \right)}\right] \cup \left[2 \operatorname{atan}{\left(\frac{\sqrt{3} \sqrt{6 - \sqrt{33}}}{3} \right)}, 2 \operatorname{atan}{\left(\frac{\sqrt{3} \sqrt{\sqrt{33} + 6}}{3} \right)}\right] ( − ∞ , − 2 atan ( 3 3 6 − 33 ) ] ∪ [ 2 atan ( 3 3 6 − 33 ) , 2 atan ( 3 3 33 + 6 ) ]
Горизонтальные асимптоты
Горизонтальные асимптоты найдём с помощью пределов данной функции при x->+oo и x->-oolim x → − ∞ ( sin 2 ( x ) + cos ( x ) ) = ⟨ − 1 , 2 ⟩ \lim_{x \to -\infty}\left(\sin^{2}{\left(x \right)} + \cos{\left(x \right)}\right) = \left\langle -1, 2\right\rangle x → − ∞ lim ( sin 2 ( x ) + cos ( x ) ) = ⟨ − 1 , 2 ⟩ Возьмём предел y = ⟨ − 1 , 2 ⟩ y = \left\langle -1, 2\right\rangle y = ⟨ − 1 , 2 ⟩ lim x → ∞ ( sin 2 ( x ) + cos ( x ) ) = ⟨ − 1 , 2 ⟩ \lim_{x \to \infty}\left(\sin^{2}{\left(x \right)} + \cos{\left(x \right)}\right) = \left\langle -1, 2\right\rangle x → ∞ lim ( sin 2 ( x ) + cos ( x ) ) = ⟨ − 1 , 2 ⟩ Возьмём предел y = ⟨ − 1 , 2 ⟩ y = \left\langle -1, 2\right\rangle y = ⟨ − 1 , 2 ⟩
Наклонные асимптоты
Наклонную асимптоту можно найти, подсчитав предел функции cos(x) + sin(x)^2, делённой на x при x->+oo и x ->-oolim x → − ∞ ( sin 2 ( x ) + cos ( x ) x ) = 0 \lim_{x \to -\infty}\left(\frac{\sin^{2}{\left(x \right)} + \cos{\left(x \right)}}{x}\right) = 0 x → − ∞ lim ( x sin 2 ( x ) + cos ( x ) ) = 0 Возьмём предел lim x → ∞ ( sin 2 ( x ) + cos ( x ) x ) = 0 \lim_{x \to \infty}\left(\frac{\sin^{2}{\left(x \right)} + \cos{\left(x \right)}}{x}\right) = 0 x → ∞ lim ( x sin 2 ( 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 2 ( x ) + cos ( x ) = sin 2 ( x ) + cos ( x ) \sin^{2}{\left(x \right)} + \cos{\left(x \right)} = \sin^{2}{\left(x \right)} + \cos{\left(x \right)} sin 2 ( x ) + cos ( x ) = sin 2 ( x ) + cos ( x ) sin 2 ( x ) + cos ( x ) = − sin 2 ( x ) − cos ( x ) \sin^{2}{\left(x \right)} + \cos{\left(x \right)} = - \sin^{2}{\left(x \right)} - \cos{\left(x \right)} sin 2 ( x ) + cos ( x ) = − sin 2 ( x ) − cos ( x ) 
