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