График функции
0 5 -65 -60 -55 -50 -45 -40 -35 -30 -25 -20 -15 -10 -5 10 0 4
Точки пересечения с осью координат X
График функции пересекает ось X при f = 01 − sin ( x ) = 0 1 - \sin{\left(x \right)} = 0 1 − 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 = − 67.5442420547782 x_{1} = -67.5442420547782 x 1 = − 67.5442420547782 x 2 = 1.57079582971902 x_{2} = 1.57079582971902 x 2 = 1.57079582971902 x 3 = − 10.9955738413568 x_{3} = -10.9955738413568 x 3 = − 10.9955738413568 x 4 = − 23.5619449492902 x_{4} = -23.5619449492902 x 4 = − 23.5619449492902 x 5 = − 29.8451297624452 x_{5} = -29.8451297624452 x 5 = − 29.8451297624452 x 6 = 89.5353893728458 x_{6} = 89.5353893728458 x 6 = 89.5353893728458 x 7 = − 80.1106124650157 x_{7} = -80.1106124650157 x 7 = − 80.1106124650157 x 8 = 39.2699077336963 x_{8} = 39.2699077336963 x 8 = 39.2699077336963 x 9 = − 36.1283153448593 x_{9} = -36.1283153448593 x 9 = − 36.1283153448593 x 10 = 64.4026493072124 x_{10} = 64.4026493072124 x 10 = 64.4026493072124 x 11 = 20.420352160156 x_{11} = 20.420352160156 x 11 = 20.420352160156 x 12 = − 48.6946865760795 x_{12} = -48.6946865760795 x 12 = − 48.6946865760795 x 13 = 45.553093730794 x_{13} = 45.553093730794 x 13 = 45.553093730794 x 14 = − 36.1283154173375 x_{14} = -36.1283154173375 x 14 = − 36.1283154173375 x 15 = − 80.1106125781572 x_{15} = -80.1106125781572 x 15 = − 80.1106125781572 x 16 = 51.8362782775539 x_{16} = 51.8362782775539 x 16 = 51.8362782775539 x 17 = − 36.1283160197768 x_{17} = -36.1283160197768 x 17 = − 36.1283160197768 x 18 = 95.8185764110282 x_{18} = 95.8185764110282 x 18 = 95.8185764110282 x 19 = − 61.2610562447228 x_{19} = -61.2610562447228 x 19 = − 61.2610562447228 x 20 = 14.1371671100222 x_{20} = 14.1371671100222 x 20 = 14.1371671100222 x 21 = − 92.6769829355125 x_{21} = -92.6769829355125 x 21 = − 92.6769829355125 x 22 = − 10.9955746401247 x_{22} = -10.9955746401247 x 22 = − 10.9955746401247 x 23 = − 42.4115013226904 x_{23} = -42.4115013226904 x 23 = − 42.4115013226904 x 24 = 64.4026499096387 x_{24} = 64.4026499096387 x 24 = 64.4026499096387 x 25 = 51.8362788867584 x_{25} = 51.8362788867584 x 25 = 51.8362788867584 x 26 = − 73.8274269047688 x_{26} = -73.8274269047688 x 26 = − 73.8274269047688 x 27 = − 42.4115005850814 x_{27} = -42.4115005850814 x 27 = − 42.4115005850814 x 28 = 89.535390888605 x_{28} = 89.535390888605 x 28 = 89.535390888605 x 29 = 32.9867225164981 x_{29} = 32.9867225164981 x 29 = 32.9867225164981 x 30 = 76.9690204681432 x_{30} = 76.9690204681432 x 30 = 76.9690204681432 x 31 = 14.1371673791846 x_{31} = 14.1371673791846 x 31 = 14.1371673791846 x 32 = − 73.8274272798455 x_{32} = -73.8274272798455 x 32 = − 73.8274272798455 x 33 = − 17.2787598356363 x_{33} = -17.2787598356363 x 33 = − 17.2787598356363 x 34 = 26.703537322248 x_{34} = 26.703537322248 x 34 = 26.703537322248 x 35 = − 17.2787583315643 x_{35} = -17.2787583315643 x 35 = − 17.2787583315643 x 36 = 1.57079525114023 x_{36} = 1.57079525114023 x 36 = 1.57079525114023 x 37 = 95.8185754266891 x_{37} = 95.8185754266891 x 37 = 95.8185754266891 x 38 = − 42.4115017818136 x_{38} = -42.4115017818136 x 38 = − 42.4115017818136 x 39 = 83.252204888767 x_{39} = 83.252204888767 x 39 = 83.252204888767 x 40 = 1.57079769954017 x_{40} = 1.57079769954017 x 40 = 1.57079769954017 x 41 = − 48.6946857788076 x_{41} = -48.6946857788076 x 41 = − 48.6946857788076 x 42 = − 86.3937977431483 x_{42} = -86.3937977431483 x 42 = − 86.3937977431483 x 43 = − 17.2787590920677 x_{43} = -17.2787590920677 x 43 = − 17.2787590920677 x 44 = 45.5530922954328 x_{44} = 45.5530922954328 x 44 = 45.5530922954328 x 45 = 58.119464520069 x_{45} = 58.119464520069 x 45 = 58.119464520069 x 46 = − 98.9601689530982 x_{46} = -98.9601689530982 x 46 = − 98.9601689530982 x 47 = 45.5530929823099 x_{47} = 45.5530929823099 x 47 = 45.5530929823099 x 48 = 14.1371665172699 x_{48} = 14.1371665172699 x 48 = 14.1371665172699 x 49 = − 86.3937984749131 x_{49} = -86.3937984749131 x 49 = − 86.3937984749131 x 50 = − 73.8274277616689 x_{50} = -73.8274277616689 x 50 = − 73.8274277616689 x 51 = − 4.71238862219396 x_{51} = -4.71238862219396 x 51 = − 4.71238862219396 x 52 = − 61.2610555612794 x_{52} = -61.2610555612794 x 52 = − 61.2610555612794 x 53 = − 67.5442421706656 x_{53} = -67.5442421706656 x 53 = − 67.5442421706656 x 54 = 26.7035380604159 x_{54} = 26.7035380604159 x 54 = 26.7035380604159 x 55 = 58.1194643979608 x_{55} = 58.1194643979608 x 55 = 58.1194643979608 x 56 = 3017.49974516717 x_{56} = 3017.49974516717 x 56 = 3017.49974516717 x 57 = 7.85398174307326 x_{57} = 7.85398174307326 x 57 = 7.85398174307326 x 58 = 1.57079657289894 x_{58} = 1.57079657289894 x 58 = 1.57079657289894 x 59 = − 92.6769837307794 x_{59} = -92.6769837307794 x 59 = − 92.6769837307794 x 60 = − 61.2610569934486 x_{60} = -61.2610569934486 x 60 = − 61.2610569934486 x 61 = 102.101760799573 x_{61} = 102.101760799573 x 61 = 102.101760799573 x 62 = 51.8362789031518 x_{62} = 51.8362789031518 x 62 = 51.8362789031518 x 63 = − 23.5619450115115 x_{63} = -23.5619450115115 x 63 = − 23.5619450115115 x 64 = 89.5353901350773 x_{64} = 89.5353901350773 x 64 = 89.5353901350773 x 65 = 70.6858352127237 x_{65} = 70.6858352127237 x 65 = 70.6858352127237 x 66 = − 80.1106131679426 x_{66} = -80.1106131679426 x 66 = − 80.1106131679426 x 67 = 20.4203521477756 x_{67} = 20.4203521477756 x 67 = 20.4203521477756 x 68 = 83.2522056907544 x_{68} = 83.2522056907544 x 68 = 83.2522056907544 x 69 = − 98.9601681513438 x_{69} = -98.9601681513438 x 69 = − 98.9601681513438 x 70 = 7.85398177249874 x_{70} = 7.85398177249874 x 70 = 7.85398177249874 x 71 = − 23.5619443878998 x_{71} = -23.5619443878998 x 71 = − 23.5619443878998 x 72 = − 54.9778709962906 x_{72} = -54.9778709962906 x 72 = − 54.9778709962906 x 73 = − 4.71238942125338 x_{73} = -4.71238942125338 x 73 = − 4.71238942125338 x 74 = − 67.5442415371049 x_{74} = -67.5442415371049 x 74 = − 67.5442415371049 x 75 = 70.6858358251975 x_{75} = 70.6858358251975 x 75 = 70.6858358251975 x 76 = 95.8185760629547 x_{76} = 95.8185760629547 x 76 = 95.8185760629547 x 77 = 26.7035387715281 x_{77} = 26.7035387715281 x 77 = 26.7035387715281 x 78 = 7.85398112872719 x_{78} = 7.85398112872719 x 78 = 7.85398112872719 x 79 = 58.1194636580315 x_{79} = 58.1194636580315 x 79 = 58.1194636580315 x 80 = 95.8185759975842 x_{80} = 95.8185759975842 x 80 = 95.8185759975842 x 81 = 70.6858344802043 x_{81} = 70.6858344802043 x 81 = 70.6858344802043 x 82 = 20.4203527610188 x_{82} = 20.4203527610188 x 82 = 20.4203527610188 x 83 = − 77664.8827844698 x_{83} = -77664.8827844698 x 83 = − 77664.8827844698 x 84 = − 54.9778717966574 x_{84} = -54.9778717966574 x 84 = − 54.9778717966574 x 85 = − 29.8451300954883 x_{85} = -29.8451300954883 x 85 = − 29.8451300954883 x 86 = 39.2699085343272 x_{86} = 39.2699085343272 x 86 = 39.2699085343272 x 87 = 64.4026492731727 x_{87} = 64.4026492731727 x 87 = 64.4026492731727 x 88 = 32.9867233134552 x_{88} = 32.9867233134552 x 88 = 32.9867233134552 x 89 = − 29.8451306226524 x_{89} = -29.8451306226524 x 89 = − 29.8451306226524 x 90 = 76.9690196732095 x_{90} = 76.9690196732095 x 90 = 76.9690196732095 x 91 = − 86.3937988139119 x_{91} = -86.3937988139119 x 91 = − 86.3937988139119
Точки пересечения с осью координат Y
График пересекает ось Y, когда x равняется 0:1 − sin ( 0 ) 1 - \sin{\left(0 \right)} 1 − sin ( 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 − cos ( x ) = 0 - \cos{\left(x \right)} = 0 − 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
(----, 2)
2 Интервалы возрастания и убывания функции: x 1 = π 2 x_{1} = \frac{\pi}{2} x 1 = 2 π x 1 = 3 π 2 x_{1} = \frac{3 \pi}{2} x 1 = 2 3 π [ π 2 , 3 π 2 ] \left[\frac{\pi}{2}, \frac{3 \pi}{2}\right] [ 2 π , 2 3 π ] ( − ∞ , π 2 ] ∪ [ 3 π 2 , ∞ ) \left(-\infty, \frac{\pi}{2}\right] \cup \left[\frac{3 \pi}{2}, \infty\right) ( − ∞ , 2 π ] ∪ [ 2 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 sin ( x ) = 0 \sin{\left(x \right)} = 0 sin ( x ) = 0 Решаем это уравнение x 1 = 0 x_{1} = 0 x 1 = 0 x 2 = π x_{2} = \pi x 2 = π Интервалы выпуклости и вогнутости: [ 0 , π ] \left[0, \pi\right] [ 0 , π ] ( − ∞ , 0 ] ∪ [ π , ∞ ) \left(-\infty, 0\right] \cup \left[\pi, \infty\right) ( − ∞ , 0 ] ∪ [ π , ∞ )
Горизонтальные асимптоты
Горизонтальные асимптоты найдём с помощью пределов данной функции при x->+oo и x->-oolim x → − ∞ ( 1 − sin ( x ) ) = ⟨ 0 , 2 ⟩ \lim_{x \to -\infty}\left(1 - \sin{\left(x \right)}\right) = \left\langle 0, 2\right\rangle x → − ∞ lim ( 1 − sin ( x ) ) = ⟨ 0 , 2 ⟩ Возьмём предел y = ⟨ 0 , 2 ⟩ y = \left\langle 0, 2\right\rangle y = ⟨ 0 , 2 ⟩ lim x → ∞ ( 1 − sin ( x ) ) = ⟨ 0 , 2 ⟩ \lim_{x \to \infty}\left(1 - \sin{\left(x \right)}\right) = \left\langle 0, 2\right\rangle x → ∞ lim ( 1 − sin ( x ) ) = ⟨ 0 , 2 ⟩ Возьмём предел y = ⟨ 0 , 2 ⟩ y = \left\langle 0, 2\right\rangle y = ⟨ 0 , 2 ⟩
Наклонные асимптоты
Наклонную асимптоту можно найти, подсчитав предел функции 1 - sin(x), делённой на x при x->+oo и x ->-oolim x → − ∞ ( 1 − sin ( x ) x ) = 0 \lim_{x \to -\infty}\left(\frac{1 - \sin{\left(x \right)}}{x}\right) = 0 x → − ∞ lim ( x 1 − sin ( x ) ) = 0 Возьмём предел lim x → ∞ ( 1 − sin ( x ) x ) = 0 \lim_{x \to \infty}\left(\frac{1 - \sin{\left(x \right)}}{x}\right) = 0 x → ∞ lim ( x 1 − sin ( x ) ) = 0 Возьмём предел
Чётность и нечётность функции
Проверим функци чётна или нечётна с помощью соотношений f = f(-x) и f = -f(-x).1 − sin ( x ) = sin ( x ) + 1 1 - \sin{\left(x \right)} = \sin{\left(x \right)} + 1 1 − sin ( x ) = sin ( x ) + 1 1 − sin ( x ) = − sin ( x ) − 1 1 - \sin{\left(x \right)} = - \sin{\left(x \right)} - 1 1 − sin ( x ) = − sin ( x ) − 1 
