График функции
0 5 -65 -60 -55 -50 -45 -40 -35 -30 -25 -20 -15 -10 -5 10 5 -5
Точки пересечения с осью координат X
График функции пересекает ось X при f = 02 sin ( x ) − 2 = 0 2 \sin{\left(x \right)} - 2 = 0 2 sin ( x ) − 2 = 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 = − 80.1106124650157 x_{6} = -80.1106124650157 x 6 = − 80.1106124650157 x 7 = 39.2699077336963 x_{7} = 39.2699077336963 x 7 = 39.2699077336963 x 8 = − 36.1283153448593 x_{8} = -36.1283153448593 x 8 = − 36.1283153448593 x 9 = 64.4026493072124 x_{9} = 64.4026493072124 x 9 = 64.4026493072124 x 10 = 20.420352160156 x_{10} = 20.420352160156 x 10 = 20.420352160156 x 11 = − 48.6946865760795 x_{11} = -48.6946865760795 x 11 = − 48.6946865760795 x 12 = 45.553093730794 x_{12} = 45.553093730794 x 12 = 45.553093730794 x 13 = − 36.1283154173375 x_{13} = -36.1283154173375 x 13 = − 36.1283154173375 x 14 = − 80.1106125781572 x_{14} = -80.1106125781572 x 14 = − 80.1106125781572 x 15 = 51.8362782775539 x_{15} = 51.8362782775539 x 15 = 51.8362782775539 x 16 = − 36.1283160197768 x_{16} = -36.1283160197768 x 16 = − 36.1283160197768 x 17 = 95.8185764110282 x_{17} = 95.8185764110282 x 17 = 95.8185764110282 x 18 = − 61.2610562447228 x_{18} = -61.2610562447228 x 18 = − 61.2610562447228 x 19 = 14.1371671100222 x_{19} = 14.1371671100222 x 19 = 14.1371671100222 x 20 = − 92.6769829355125 x_{20} = -92.6769829355125 x 20 = − 92.6769829355125 x 21 = − 10.9955746401247 x_{21} = -10.9955746401247 x 21 = − 10.9955746401247 x 22 = − 42.4115013226904 x_{22} = -42.4115013226904 x 22 = − 42.4115013226904 x 23 = 64.4026499096387 x_{23} = 64.4026499096387 x 23 = 64.4026499096387 x 24 = 51.8362788867584 x_{24} = 51.8362788867584 x 24 = 51.8362788867584 x 25 = − 73.8274269047688 x_{25} = -73.8274269047688 x 25 = − 73.8274269047688 x 26 = − 42.4115005850814 x_{26} = -42.4115005850814 x 26 = − 42.4115005850814 x 27 = 89.535390888605 x_{27} = 89.535390888605 x 27 = 89.535390888605 x 28 = 32.9867225164981 x_{28} = 32.9867225164981 x 28 = 32.9867225164981 x 29 = 76.9690204681432 x_{29} = 76.9690204681432 x 29 = 76.9690204681432 x 30 = 14.1371673791846 x_{30} = 14.1371673791846 x 30 = 14.1371673791846 x 31 = − 73.8274272798455 x_{31} = -73.8274272798455 x 31 = − 73.8274272798455 x 32 = − 17.2787598356363 x_{32} = -17.2787598356363 x 32 = − 17.2787598356363 x 33 = 26.703537322248 x_{33} = 26.703537322248 x 33 = 26.703537322248 x 34 = 1.57079525114023 x_{34} = 1.57079525114023 x 34 = 1.57079525114023 x 35 = 95.8185754266891 x_{35} = 95.8185754266891 x 35 = 95.8185754266891 x 36 = − 42.4115017818136 x_{36} = -42.4115017818136 x 36 = − 42.4115017818136 x 37 = 83.252204888767 x_{37} = 83.252204888767 x 37 = 83.252204888767 x 38 = − 48.6946857788076 x_{38} = -48.6946857788076 x 38 = − 48.6946857788076 x 39 = − 86.3937977431483 x_{39} = -86.3937977431483 x 39 = − 86.3937977431483 x 40 = − 17.2787590920677 x_{40} = -17.2787590920677 x 40 = − 17.2787590920677 x 41 = 58.119464520069 x_{41} = 58.119464520069 x 41 = 58.119464520069 x 42 = − 98.9601689530982 x_{42} = -98.9601689530982 x 42 = − 98.9601689530982 x 43 = 45.5530929823099 x_{43} = 45.5530929823099 x 43 = 45.5530929823099 x 44 = 14.1371665172699 x_{44} = 14.1371665172699 x 44 = 14.1371665172699 x 45 = − 86.3937984749131 x_{45} = -86.3937984749131 x 45 = − 86.3937984749131 x 46 = − 73.8274277616689 x_{46} = -73.8274277616689 x 46 = − 73.8274277616689 x 47 = − 4.71238862219396 x_{47} = -4.71238862219396 x 47 = − 4.71238862219396 x 48 = − 67.5442421706656 x_{48} = -67.5442421706656 x 48 = − 67.5442421706656 x 49 = 26.7035380604159 x_{49} = 26.7035380604159 x 49 = 26.7035380604159 x 50 = 58.1194643979608 x_{50} = 58.1194643979608 x 50 = 58.1194643979608 x 51 = 7.85398174307326 x_{51} = 7.85398174307326 x 51 = 7.85398174307326 x 52 = 1.57079657289894 x_{52} = 1.57079657289894 x 52 = 1.57079657289894 x 53 = − 92.6769837307794 x_{53} = -92.6769837307794 x 53 = − 92.6769837307794 x 54 = − 61.2610569934486 x_{54} = -61.2610569934486 x 54 = − 61.2610569934486 x 55 = 102.101760799573 x_{55} = 102.101760799573 x 55 = 102.101760799573 x 56 = 51.8362789031518 x_{56} = 51.8362789031518 x 56 = 51.8362789031518 x 57 = − 23.5619450115115 x_{57} = -23.5619450115115 x 57 = − 23.5619450115115 x 58 = 89.5353901350773 x_{58} = 89.5353901350773 x 58 = 89.5353901350773 x 59 = 70.6858352127237 x_{59} = 70.6858352127237 x 59 = 70.6858352127237 x 60 = − 80.1106131679426 x_{60} = -80.1106131679426 x 60 = − 80.1106131679426 x 61 = 20.4203521477756 x_{61} = 20.4203521477756 x 61 = 20.4203521477756 x 62 = 83.2522056907544 x_{62} = 83.2522056907544 x 62 = 83.2522056907544 x 63 = − 98.9601681513438 x_{63} = -98.9601681513438 x 63 = − 98.9601681513438 x 64 = 7.85398177249874 x_{64} = 7.85398177249874 x 64 = 7.85398177249874 x 65 = − 23.5619443878998 x_{65} = -23.5619443878998 x 65 = − 23.5619443878998 x 66 = − 54.9778709962906 x_{66} = -54.9778709962906 x 66 = − 54.9778709962906 x 67 = − 4.71238942125338 x_{67} = -4.71238942125338 x 67 = − 4.71238942125338 x 68 = − 67.5442415371049 x_{68} = -67.5442415371049 x 68 = − 67.5442415371049 x 69 = 70.6858358251975 x_{69} = 70.6858358251975 x 69 = 70.6858358251975 x 70 = 95.8185760629547 x_{70} = 95.8185760629547 x 70 = 95.8185760629547 x 71 = 7.85398112872719 x_{71} = 7.85398112872719 x 71 = 7.85398112872719 x 72 = 58.1194636580315 x_{72} = 58.1194636580315 x 72 = 58.1194636580315 x 73 = 95.8185759975842 x_{73} = 95.8185759975842 x 73 = 95.8185759975842 x 74 = 70.6858344802043 x_{74} = 70.6858344802043 x 74 = 70.6858344802043 x 75 = 20.4203527610188 x_{75} = 20.4203527610188 x 75 = 20.4203527610188 x 76 = − 54.9778717966574 x_{76} = -54.9778717966574 x 76 = − 54.9778717966574 x 77 = − 29.8451300954883 x_{77} = -29.8451300954883 x 77 = − 29.8451300954883 x 78 = 39.2699085343272 x_{78} = 39.2699085343272 x 78 = 39.2699085343272 x 79 = 64.4026492731727 x_{79} = 64.4026492731727 x 79 = 64.4026492731727 x 80 = 32.9867233134552 x_{80} = 32.9867233134552 x 80 = 32.9867233134552 x 81 = − 29.8451306226524 x_{81} = -29.8451306226524 x 81 = − 29.8451306226524 x 82 = 76.9690196732095 x_{82} = 76.9690196732095 x 82 = 76.9690196732095 x 83 = − 86.3937988139119 x_{83} = -86.3937988139119 x 83 = − 86.3937988139119
Точки пересечения с осью координат Y
График пересекает ось Y, когда x равняется 0:( − 1 ) 2 + 2 sin ( 0 ) \left(-1\right) 2 + 2 \sin{\left(0 \right)} ( − 1 ) 2 + 2 sin ( 0 ) f ( 0 ) = − 2 f{\left(0 \right)} = -2 f ( 0 ) = − 2 (0, -2)
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 cos ( x ) = 0 2 \cos{\left(x \right)} = 0 2 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
(--, 2 - 2)
2 3*pi
(----, -2 - 2)
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 − 2 sin ( x ) = 0 - 2 \sin{\left(x \right)} = 0 − 2 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 → − ∞ ( 2 sin ( x ) − 2 ) = ⟨ − 4 , 0 ⟩ \lim_{x \to -\infty}\left(2 \sin{\left(x \right)} - 2\right) = \left\langle -4, 0\right\rangle x → − ∞ lim ( 2 sin ( x ) − 2 ) = ⟨ − 4 , 0 ⟩ Возьмём предел y = ⟨ − 4 , 0 ⟩ y = \left\langle -4, 0\right\rangle y = ⟨ − 4 , 0 ⟩ lim x → ∞ ( 2 sin ( x ) − 2 ) = ⟨ − 4 , 0 ⟩ \lim_{x \to \infty}\left(2 \sin{\left(x \right)} - 2\right) = \left\langle -4, 0\right\rangle x → ∞ lim ( 2 sin ( x ) − 2 ) = ⟨ − 4 , 0 ⟩ Возьмём предел y = ⟨ − 4 , 0 ⟩ y = \left\langle -4, 0\right\rangle y = ⟨ − 4 , 0 ⟩
Наклонные асимптоты
Наклонную асимптоту можно найти, подсчитав предел функции 2*sin(x) - 1*2, делённой на x при x->+oo и x ->-oolim x → − ∞ ( 2 sin ( x ) − 2 x ) = 0 \lim_{x \to -\infty}\left(\frac{2 \sin{\left(x \right)} - 2}{x}\right) = 0 x → − ∞ lim ( x 2 sin ( x ) − 2 ) = 0 Возьмём предел lim x → ∞ ( 2 sin ( x ) − 2 x ) = 0 \lim_{x \to \infty}\left(\frac{2 \sin{\left(x \right)} - 2}{x}\right) = 0 x → ∞ lim ( x 2 sin ( x ) − 2 ) = 0 Возьмём предел
Чётность и нечётность функции
Проверим функци чётна или нечётна с помощью соотношений f = f(-x) и f = -f(-x).2 sin ( x ) − 2 = − 2 sin ( x ) − 2 2 \sin{\left(x \right)} - 2 = - 2 \sin{\left(x \right)} - 2 2 sin ( x ) − 2 = − 2 sin ( x ) − 2 2 sin ( x ) − 2 = 2 sin ( x ) + 2 2 \sin{\left(x \right)} - 2 = 2 \sin{\left(x \right)} + 2 2 sin ( x ) − 2 = 2 sin ( x ) + 2 
