График функции
-10.0 -7.5 -5.0 -2.5 0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0 5 -10
Точки пересечения с осью координат X
График функции пересекает ось X при f = 0log ( sin ( x ) ) = 0 \log{\left(\sin{\left(x \right)} \right)} = 0 log ( sin ( x ) ) = 0 Решаем это уравнение Аналитическое решение x 1 = π 2 x_{1} = \frac{\pi}{2} x 1 = 2 π Численное решение x 1 = 20.4203523413041 x_{1} = 20.4203523413041 x 1 = 20.4203523413041 x 2 = 1.57079660167231 x_{2} = 1.57079660167231 x 2 = 1.57079660167231 x 3 = − 10.9955735120111 x_{3} = -10.9955735120111 x 3 = − 10.9955735120111 x 4 = − 29.8451300946193 x_{4} = -29.8451300946193 x 4 = − 29.8451300946193 x 5 = − 98.9601678624104 x_{5} = -98.9601678624104 x 5 = − 98.9601678624104 x 6 = − 48.6946869113465 x_{6} = -48.6946869113465 x 6 = − 48.6946869113465 x 7 = 58.119464370396 x_{7} = 58.119464370396 x 7 = 58.119464370396 x 8 = − 73.8274273885517 x_{8} = -73.8274273885517 x 8 = − 73.8274273885517 x 9 = 64.4026494793175 x_{9} = 64.4026494793175 x 9 = 64.4026494793175 x 10 = − 80.1106127903609 x_{10} = -80.1106127903609 x 10 = − 80.1106127903609 x 11 = 89.5353892520407 x_{11} = 89.5353892520407 x 11 = 89.5353892520407 x 12 = 83.2522058199769 x_{12} = 83.2522058199769 x 12 = 83.2522058199769 x 13 = 14.1371668991051 x_{13} = 14.1371668991051 x 13 = 14.1371668991051 x 14 = − 48.694686949957 x_{14} = -48.694686949957 x 14 = − 48.694686949957 x 15 = 51.8362789063255 x_{15} = 51.8362789063255 x 15 = 51.8362789063255 x 16 = − 29.8451302446469 x_{16} = -29.8451302446469 x 16 = − 29.8451302446469 x 17 = − 54.9778713174705 x_{17} = -54.9778713174705 x 17 = − 54.9778713174705 x 18 = − 23.5619447959101 x_{18} = -23.5619447959101 x 18 = − 23.5619447959101 x 19 = − 36.1283154154305 x_{19} = -36.1283154154305 x 19 = − 36.1283154154305 x 20 = 58.1194640425023 x_{20} = 58.1194640425023 x 20 = 58.1194640425023 x 21 = − 54.9778719110131 x_{21} = -54.9778719110131 x 21 = − 54.9778719110131 x 22 = 51.8362786093149 x_{22} = 51.8362786093149 x 22 = 51.8362786093149 x 23 = − 54.9778705207315 x_{23} = -54.9778705207315 x 23 = − 54.9778705207315 x 24 = − 17.2787590751963 x_{24} = -17.2787590751963 x 24 = − 17.2787590751963 x 25 = − 67.5442419322395 x_{25} = -67.5442419322395 x 25 = − 67.5442419322395 x 26 = 39.2699074396221 x_{26} = 39.2699074396221 x 26 = 39.2699074396221 x 27 = 76.9690195814988 x_{27} = 76.9690195814988 x 27 = 76.9690195814988 x 28 = − 73.827427279653 x_{28} = -73.827427279653 x 28 = − 73.827427279653 x 29 = − 42.4115015246508 x_{29} = -42.4115015246508 x 29 = − 42.4115015246508 x 30 = − 29.8451302007713 x_{30} = -29.8451302007713 x 30 = − 29.8451302007713 x 31 = − 10.9955733481573 x_{31} = -10.9955733481573 x 31 = − 10.9955733481573 x 32 = 14.1371669434725 x_{32} = 14.1371669434725 x 32 = 14.1371669434725 x 33 = − 86.3937986220237 x_{33} = -86.3937986220237 x 33 = − 86.3937986220237 x 34 = 32.9867238176799 x_{34} = 32.9867238176799 x 34 = 32.9867238176799 x 35 = 76.9690208439152 x_{35} = 76.9690208439152 x 35 = 76.9690208439152 x 36 = − 10.9955741211687 x_{36} = -10.9955741211687 x 36 = − 10.9955741211687 x 37 = 26.7035372979479 x_{37} = 26.7035372979479 x 37 = 26.7035372979479 x 38 = − 67.5442421737401 x_{38} = -67.5442421737401 x 38 = − 67.5442421737401 x 39 = − 61.2610561822839 x_{39} = -61.2610561822839 x 39 = − 61.2610561822839 x 40 = − 42.4115005591739 x_{40} = -42.4115005591739 x 40 = − 42.4115005591739 x 41 = 70.6858351446748 x_{41} = 70.6858351446748 x 41 = 70.6858351446748 x 42 = 1.57079557309815 x_{42} = 1.57079557309815 x 42 = 1.57079557309815 x 43 = − 61.2610552210018 x_{43} = -61.2610552210018 x 43 = − 61.2610552210018 x 44 = 20.4203521458531 x_{44} = 20.4203521458531 x 44 = 20.4203521458531 x 45 = 83.2522046133019 x_{45} = 83.2522046133019 x 45 = 83.2522046133019 x 46 = − 86.3937977199047 x_{46} = -86.3937977199047 x 46 = − 86.3937977199047 x 47 = 58.1194640878142 x_{47} = 58.1194640878142 x 47 = 58.1194640878142 x 48 = 32.9867230918614 x_{48} = 32.9867230918614 x 48 = 32.9867230918614 x 49 = 32.9867224176774 x_{49} = 32.9867224176774 x 49 = 32.9867224176774 x 50 = − 92.676984146129 x_{50} = -92.676984146129 x 50 = − 92.676984146129 x 51 = − 98.9601690759292 x_{51} = -98.9601690759292 x 51 = − 98.9601690759292 x 52 = 26.7035380336909 x_{52} = 26.7035380336909 x 52 = 26.7035380336909 x 53 = 14.1371671149845 x_{53} = 14.1371671149845 x 53 = 14.1371671149845 x 54 = − 54.9778706878326 x_{54} = -54.9778706878326 x 54 = − 54.9778706878326 x 55 = 70.6858344584179 x_{55} = 70.6858344584179 x 55 = 70.6858344584179 x 56 = 76.9690209763474 x_{56} = 76.9690209763474 x 56 = 76.9690209763474 x 57 = − 73.8274273446888 x_{57} = -73.8274273446888 x 57 = − 73.8274273446888 x 58 = 39.2699074534854 x_{58} = 39.2699074534854 x 58 = 39.2699074534854 x 59 = 89.5353897747913 x_{59} = 89.5353897747913 x 59 = 89.5353897747913 x 60 = − 4.71238973521995 x_{60} = -4.71238973521995 x 60 = − 4.71238973521995 x 61 = − 48.6946856746846 x_{61} = -48.6946856746846 x 61 = − 48.6946856746846 x 62 = − 36.1283156556098 x_{62} = -36.1283156556098 x 62 = − 36.1283156556098 x 63 = − 98.9601685005998 x_{63} = -98.9601685005998 x 63 = − 98.9601685005998 x 64 = 7.85398147767715 x_{64} = 7.85398147767715 x 64 = 7.85398147767715 x 65 = − 4.71238851018714 x_{65} = -4.71238851018714 x 65 = − 4.71238851018714 x 66 = − 80.1106125767506 x_{66} = -80.1106125767506 x 66 = − 80.1106125767506 x 67 = − 61.2610570233699 x_{67} = -61.2610570233699 x 67 = − 61.2610570233699 x 68 = 32.9867236651869 x_{68} = 32.9867236651869 x 68 = 32.9867236651869 x 69 = 64.402649305744 x_{69} = 64.402649305744 x 69 = 64.402649305744 x 70 = 95.8185760669056 x_{70} = 95.8185760669056 x 70 = 95.8185760669056 x 71 = 83.2522055855063 x_{71} = 83.2522055855063 x 71 = 83.2522055855063 x 72 = 45.5530926724619 x_{72} = 45.5530926724619 x 72 = 45.5530926724619 x 73 = 95.8185757390985 x_{73} = 95.8185757390985 x 73 = 95.8185757390985 x 74 = 45.5530937626454 x_{74} = 45.5530937626454 x 74 = 45.5530937626454 x 75 = − 17.2787598626449 x_{75} = -17.2787598626449 x 75 = − 17.2787598626449 x 76 = 7.85398174556756 x_{76} = 7.85398174556756 x 76 = 7.85398174556756 x 77 = − 92.6769828388254 x_{77} = -92.6769828388254 x 77 = − 92.6769828388254 x 78 = − 23.5619450139675 x_{78} = -23.5619450139675 x 78 = − 23.5619450139675 x 79 = − 4.71238974981597 x_{79} = -4.71238974981597 x 79 = − 4.71238974981597 x 80 = 89.5353909237568 x_{80} = 89.5353909237568 x 80 = 89.5353909237568 x 81 = 39.2699086546913 x_{81} = 39.2699086546913 x 81 = 39.2699086546913 x 82 = − 92.6769840888992 x_{82} = -92.6769840888992 x 82 = − 92.6769840888992 x 83 = 83.2522046592344 x_{83} = 83.2522046592344 x 83 = 83.2522046592344 x 84 = − 10.9955747464857 x_{84} = -10.9955747464857 x 84 = − 10.9955747464857 x 85 = 45.5530919949765 x_{85} = 45.5530919949765 x 85 = 45.5530919949765
Точки пересечения с осью координат Y
График пересекает ось Y, когда x равняется 0:log ( sin ( 0 ) ) \log{\left(\sin{\left(0 \right)} \right)} log ( sin ( 0 ) ) f ( 0 ) = ∞ ~ f{\left(0 \right)} = \tilde{\infty} f ( 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 ( x ) sin ( x ) = 0 \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}} = 0 sin ( x ) cos ( x ) = 0 Решаем это уравнение x 1 = π 2 x_{1} = \frac{\pi}{2} x 1 = 2 π x 2 = 3 π 2 x_{2} = \frac{3 \pi}{2} x 2 = 2 3 π pi
(--, 0)
2 3*pi
(----, pi*I)
2 Интервалы возрастания и убывания функции: x 2 = π 2 x_{2} = \frac{\pi}{2} x 2 = 2 π ( − ∞ , π 2 ] \left(-\infty, \frac{\pi}{2}\right] ( − ∞ , 2 π ] [ π 2 , ∞ ) \left[\frac{\pi}{2}, \infty\right) [ 2 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 − ( 1 + cos 2 ( x ) sin 2 ( x ) ) = 0 - (1 + \frac{\cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}) = 0 − ( 1 + sin 2 ( x ) cos 2 ( x ) ) = 0 Решаем это уравнение
Горизонтальные асимптоты
Горизонтальные асимптоты найдём с помощью пределов данной функции при x->+oo и x->-oolim x → − ∞ log ( sin ( x ) ) = lim x → − ∞ log ( sin ( x ) ) \lim_{x \to -\infty} \log{\left(\sin{\left(x \right)} \right)} = \lim_{x \to -\infty} \log{\left(\sin{\left(x \right)} \right)} x → − ∞ lim log ( sin ( x ) ) = x → − ∞ lim log ( sin ( x ) ) Возьмём предел y = lim x → − ∞ log ( sin ( x ) ) y = \lim_{x \to -\infty} \log{\left(\sin{\left(x \right)} \right)} y = x → − ∞ lim log ( sin ( x ) ) lim x → ∞ log ( sin ( x ) ) = lim x → ∞ log ( sin ( x ) ) \lim_{x \to \infty} \log{\left(\sin{\left(x \right)} \right)} = \lim_{x \to \infty} \log{\left(\sin{\left(x \right)} \right)} x → ∞ lim log ( sin ( x ) ) = x → ∞ lim log ( sin ( x ) ) Возьмём предел y = lim x → ∞ log ( sin ( x ) ) y = \lim_{x \to \infty} \log{\left(\sin{\left(x \right)} \right)} y = x → ∞ lim log ( sin ( x ) )
Наклонные асимптоты
Наклонную асимптоту можно найти, подсчитав предел функции log(sin(x)), делённой на x при x->+oo и x ->-oolim x → − ∞ ( log ( sin ( x ) ) x ) = lim x → − ∞ ( log ( sin ( x ) ) x ) \lim_{x \to -\infty}\left(\frac{\log{\left(\sin{\left(x \right)} \right)}}{x}\right) = \lim_{x \to -\infty}\left(\frac{\log{\left(\sin{\left(x \right)} \right)}}{x}\right) x → − ∞ lim ( x log ( sin ( x ) ) ) = x → − ∞ lim ( x log ( sin ( x ) ) ) Возьмём предел y = x lim x → − ∞ ( log ( sin ( x ) ) x ) y = x \lim_{x \to -\infty}\left(\frac{\log{\left(\sin{\left(x \right)} \right)}}{x}\right) y = x x → − ∞ lim ( x log ( sin ( x ) ) ) lim x → ∞ ( log ( sin ( x ) ) x ) = lim x → ∞ ( log ( sin ( x ) ) x ) \lim_{x \to \infty}\left(\frac{\log{\left(\sin{\left(x \right)} \right)}}{x}\right) = \lim_{x \to \infty}\left(\frac{\log{\left(\sin{\left(x \right)} \right)}}{x}\right) x → ∞ lim ( x log ( sin ( x ) ) ) = x → ∞ lim ( x log ( sin ( x ) ) ) Возьмём предел y = x lim x → ∞ ( log ( sin ( x ) ) x ) y = x \lim_{x \to \infty}\left(\frac{\log{\left(\sin{\left(x \right)} \right)}}{x}\right) y = x x → ∞ lim ( x log ( sin ( x ) ) )
Чётность и нечётность функции
Проверим функци чётна или нечётна с помощью соотношений f = f(-x) и f = -f(-x).log ( sin ( x ) ) = log ( − sin ( x ) ) \log{\left(\sin{\left(x \right)} \right)} = \log{\left(- \sin{\left(x \right)} \right)} log ( sin ( x ) ) = log ( − sin ( x ) ) log ( sin ( x ) ) = − log ( − sin ( x ) ) \log{\left(\sin{\left(x \right)} \right)} = - \log{\left(- \sin{\left(x \right)} \right)} log ( sin ( x ) ) = − log ( − sin ( x ) ) 
