График функции
0 -50 -40 -30 -20 -10 10 20 30 40 50 60 2 -2
Точки пересечения с осью координат X
График функции пересекает ось X при f = 0cos ( 2 x + 45 ) = 0 \cos{\left(2 x + 45 \right)} = 0 cos ( 2 x + 45 ) = 0 Решаем это уравнение Аналитическое решение x 1 = − 45 2 + π 4 x_{1} = - \frac{45}{2} + \frac{\pi}{4} x 1 = − 2 45 + 4 π x 2 = − 45 2 + 3 π 4 x_{2} = - \frac{45}{2} + \frac{3 \pi}{4} x 2 = − 2 45 + 4 3 π Численное решение x 1 = − 51.5597320457056 x_{1} = -51.5597320457056 x 1 = − 51.5597320457056 x 2 = 6.55973204570559 x_{2} = 6.55973204570559 x 2 = 6.55973204570559 x 3 = 66.2499924639117 x_{3} = 66.2499924639117 x 3 = 66.2499924639117 x 4 = − 45.276546738526 x_{4} = -45.276546738526 x 4 = − 45.276546738526 x 5 = 36.4048622548086 x_{5} = 36.4048622548086 x 5 = 36.4048622548086 x 6 = 75.674770424681 x_{6} = 75.674770424681 x 6 = 75.674770424681 x 7 = − 4.43584224185869 x_{7} = -4.43584224185869 x 7 = − 4.43584224185869 x 8 = − 35.8517687777566 x_{8} = -35.8517687777566 x 8 = − 35.8517687777566 x 9 = 80.3871594050657 x_{9} = 80.3871594050657 x 9 = 80.3871594050657 x 10 = 37.9756585816035 x_{10} = 37.9756585816035 x 10 = 37.9756585816035 x 11 = − 78.2632696012188 x_{11} = -78.2632696012188 x 11 = − 78.2632696012188 x 12 = 58.3960108299372 x_{12} = 58.3960108299372 x 12 = 58.3960108299372 x 13 = 1.8473430653209 x_{13} = 1.8473430653209 x 13 = 1.8473430653209 x 14 = 61.537603483527 x_{14} = 61.537603483527 x 14 = 61.537603483527 x 15 = − 81.4048622548086 x_{15} = -81.4048622548086 x 15 = − 81.4048622548086 x 16 = − 70.4092879672444 x_{16} = -70.4092879672444 x 16 = − 70.4092879672444 x 17 = 39.5464549083984 x_{17} = 39.5464549083984 x 17 = 39.5464549083984 x 18 = − 12.2898238758332 x_{18} = -12.2898238758332 x 18 = − 12.2898238758332 x 19 = − 20.1438055098077 x_{19} = -20.1438055098077 x 19 = − 20.1438055098077 x 20 = 15.984510006475 x_{20} = 15.984510006475 x 20 = 15.984510006475 x 21 = 86.6703447122453 x_{21} = 86.6703447122453 x 21 = 86.6703447122453 x 22 = − 57.8429173528852 x_{22} = -57.8429173528852 x 22 = − 57.8429173528852 x 23 = − 31.1393797973719 x_{23} = -31.1393797973719 x 23 = − 31.1393797973719 x 24 = − 7.57743489544848 x_{24} = -7.57743489544848 x 24 = − 7.57743489544848 x 25 = 81.9579557318606 x_{25} = 81.9579557318606 x 25 = 81.9579557318606 x 26 = − 23.2853981633974 x_{26} = -23.2853981633974 x 26 = − 23.2853981633974 x 27 = 8.13052837250048 x_{27} = 8.13052837250048 x 27 = 8.13052837250048 x 28 = 52.1128255227576 x_{28} = 52.1128255227576 x 28 = 52.1128255227576 x 29 = 23.8384916404495 x_{29} = 23.8384916404495 x 29 = 23.8384916404495 x 30 = − 87.6880475619882 x_{30} = -87.6880475619882 x 30 = − 87.6880475619882 x 31 = − 89.2588438887831 x_{31} = -89.2588438887831 x 31 = − 89.2588438887831 x 32 = 72.5331777710912 x_{32} = 72.5331777710912 x 32 = 72.5331777710912 x 33 = 17.5553063332699 x_{33} = 17.5553063332699 x 33 = 17.5553063332699 x 34 = − 42.1349540849362 x_{34} = -42.1349540849362 x 34 = − 42.1349540849362 x 35 = − 59.4137136796801 x_{35} = -59.4137136796801 x 35 = − 59.4137136796801 x 36 = 64.6791961371168 x_{36} = 64.6791961371168 x 36 = 64.6791961371168 x 37 = − 27.9977871437821 x_{37} = -27.9977871437821 x 37 = − 27.9977871437821 x 38 = 67.8207887907066 x_{38} = 67.8207887907066 x 38 = 67.8207887907066 x 39 = − 13.8606202026281 x_{39} = -13.8606202026281 x 39 = − 13.8606202026281 x 40 = 20.6968989868597 x_{40} = 20.6968989868597 x 40 = 20.6968989868597 x 41 = 30.121676947629 x_{41} = 30.121676947629 x 41 = 30.121676947629 x 42 = 96.0951226730147 x_{42} = 96.0951226730147 x 42 = 96.0951226730147 x 43 = − 73.5508806208341 x_{43} = -73.5508806208341 x 43 = − 73.5508806208341 x 44 = − 86.1172512351933 x_{44} = -86.1172512351933 x 44 = − 86.1172512351933 x 45 = − 71.9800842940392 x_{45} = -71.9800842940392 x 45 = − 71.9800842940392 x 46 = 89.8119373658351 x_{46} = 89.8119373658351 x 46 = 89.8119373658351 x 47 = 94.5243263462198 x_{47} = 94.5243263462198 x 47 = 94.5243263462198 x 48 = 74.1039740978861 x_{48} = 74.1039740978861 x 48 = 74.1039740978861 x 49 = − 15.431416529423 x_{49} = -15.431416529423 x 49 = − 15.431416529423 x 50 = 45.829640215578 x_{50} = 45.829640215578 x 50 = 45.829640215578 x 51 = − 65.6968989868597 x_{51} = -65.6968989868597 x 51 = − 65.6968989868597 x 52 = − 37.4225651045515 x_{52} = -37.4225651045515 x 52 = − 37.4225651045515 x 53 = − 43.7057504117311 x_{53} = -43.7057504117311 x 53 = − 43.7057504117311 x 54 = − 26.4269908169872 x_{54} = -26.4269908169872 x 54 = − 26.4269908169872 x 55 = 28.5508806208341 x_{55} = 28.5508806208341 x 55 = 28.5508806208341 x 56 = − 67.2676953136546 x_{56} = -67.2676953136546 x 56 = − 67.2676953136546 x 57 = 44.2588438887831 x_{57} = 44.2588438887831 x 57 = 44.2588438887831 x 58 = − 92.4004365423729 x_{58} = -92.4004365423729 x 58 = − 92.4004365423729 x 59 = 83.5287520586555 x_{59} = 83.5287520586555 x 59 = 83.5287520586555 x 60 = 31.6924732744239 x_{60} = 31.6924732744239 x 60 = 31.6924732744239 x 61 = 50.5420291959627 x_{61} = 50.5420291959627 x 61 = 50.5420291959627 x 62 = − 34.2809724509617 x_{62} = -34.2809724509617 x 62 = − 34.2809724509617 x 63 = 59.9668071567321 x_{63} = 59.9668071567321 x 63 = 59.9668071567321 x 64 = 14.4137136796801 x_{64} = 14.4137136796801 x 64 = 14.4137136796801 x 65 = − 6.00663856865359 x_{65} = -6.00663856865359 x 65 = − 6.00663856865359 x 66 = − 56.2721210260903 x_{66} = -56.2721210260903 x 66 = − 56.2721210260903 x 67 = − 29.568583470577 x_{67} = -29.568583470577 x 67 = − 29.568583470577 x 68 = − 21.7146018366026 x_{68} = -21.7146018366026 x 68 = − 21.7146018366026 x 69 = − 1.2942495882689 x_{69} = -1.2942495882689 x 69 = − 1.2942495882689 x 70 = 53.6836218495525 x_{70} = 53.6836218495525 x 70 = 53.6836218495525 x 71 = − 49.9889357189107 x_{71} = -49.9889357189107 x 71 = − 49.9889357189107 x 72 = − 48.4181393921158 x_{72} = -48.4181393921158 x 72 = − 48.4181393921158 x 73 = 0.276546738526001 x_{73} = 0.276546738526001 x 73 = 0.276546738526001 x 74 = 22.2676953136546 x_{74} = 22.2676953136546 x 74 = 22.2676953136546 x 75 = − 100.254418176347 x_{75} = -100.254418176347 x 75 = − 100.254418176347 x 76 = 42.6880475619882 x_{76} = 42.6880475619882 x 76 = 42.6880475619882 x 77 = 9.70132469929538 x_{77} = 9.70132469929538 x 77 = 9.70132469929538 x 78 = 88.2411410390402 x_{78} = 88.2411410390402 x 78 = 88.2411410390402 x 79 = 102.378307980194 x_{79} = 102.378307980194 x 79 = 102.378307980194 x 80 = 85.0995483854504 x_{80} = 85.0995483854504 x 80 = 85.0995483854504 x 81 = 97.6659189998096 x_{81} = 97.6659189998096 x 81 = 97.6659189998096 x 82 = − 79.8340659280137 x_{82} = -79.8340659280137 x 82 = − 79.8340659280137 x 83 = − 93.9712328691678 x_{83} = -93.9712328691678 x 83 = − 93.9712328691678 x 84 = − 95.5420291959627 x_{84} = -95.5420291959627 x 84 = − 95.5420291959627 x 85 = − 64.1261026600648 x_{85} = -64.1261026600648 x 85 = − 64.1261026600648
Точки пересечения с осью координат Y
График пересекает ось Y, когда x равняется 0:cos ( 2 ⋅ 0 + 45 ) \cos{\left(2 \cdot 0 + 45 \right)} cos ( 2 ⋅ 0 + 45 ) f ( 0 ) = cos ( 45 ) f{\left(0 \right)} = \cos{\left(45 \right)} f ( 0 ) = cos ( 45 ) (0, cos(45))
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 ( 2 x + 45 ) = 0 - 2 \sin{\left(2 x + 45 \right)} = 0 − 2 sin ( 2 x + 45 ) = 0 Решаем это уравнение x 1 = − 45 2 x_{1} = - \frac{45}{2} x 1 = − 2 45 x 2 = − 45 2 + π 2 x_{2} = - \frac{45}{2} + \frac{\pi}{2} x 2 = − 2 45 + 2 π (-45/2, 1) 45 pi
(- -- + --, -1)
2 2 Интервалы возрастания и убывания функции: x 1 = − 45 2 + π 2 x_{1} = - \frac{45}{2} + \frac{\pi}{2} x 1 = − 2 45 + 2 π x 1 = − 45 2 x_{1} = - \frac{45}{2} x 1 = − 2 45 ( − ∞ , − 45 2 ] ∪ [ − 45 2 + π 2 , ∞ ) \left(-\infty, - \frac{45}{2}\right] \cup \left[- \frac{45}{2} + \frac{\pi}{2}, \infty\right) ( − ∞ , − 2 45 ] ∪ [ − 2 45 + 2 π , ∞ ) [ − 45 2 , − 45 2 + π 2 ] \left[- \frac{45}{2}, - \frac{45}{2} + \frac{\pi}{2}\right] [ − 2 45 , − 2 45 + 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 − 4 cos ( 2 x + 45 ) = 0 - 4 \cos{\left(2 x + 45 \right)} = 0 − 4 cos ( 2 x + 45 ) = 0 Решаем это уравнение x 1 = − 45 2 + π 4 x_{1} = - \frac{45}{2} + \frac{\pi}{4} x 1 = − 2 45 + 4 π x 2 = − 45 2 + 3 π 4 x_{2} = - \frac{45}{2} + \frac{3 \pi}{4} x 2 = − 2 45 + 4 3 π Интервалы выпуклости и вогнутости: [ − 45 2 + π 4 , − 45 2 + 3 π 4 ] \left[- \frac{45}{2} + \frac{\pi}{4}, - \frac{45}{2} + \frac{3 \pi}{4}\right] [ − 2 45 + 4 π , − 2 45 + 4 3 π ] ( − ∞ , − 45 2 + π 4 ] ∪ [ − 45 2 + 3 π 4 , ∞ ) \left(-\infty, - \frac{45}{2} + \frac{\pi}{4}\right] \cup \left[- \frac{45}{2} + \frac{3 \pi}{4}, \infty\right) ( − ∞ , − 2 45 + 4 π ] ∪ [ − 2 45 + 4 3 π , ∞ )
Горизонтальные асимптоты
Горизонтальные асимптоты найдём с помощью пределов данной функции при x->+oo и x->-oolim x → − ∞ cos ( 2 x + 45 ) = ⟨ − 1 , 1 ⟩ \lim_{x \to -\infty} \cos{\left(2 x + 45 \right)} = \left\langle -1, 1\right\rangle x → − ∞ lim cos ( 2 x + 45 ) = ⟨ − 1 , 1 ⟩ Возьмём предел y = ⟨ − 1 , 1 ⟩ y = \left\langle -1, 1\right\rangle y = ⟨ − 1 , 1 ⟩ lim x → ∞ cos ( 2 x + 45 ) = ⟨ − 1 , 1 ⟩ \lim_{x \to \infty} \cos{\left(2 x + 45 \right)} = \left\langle -1, 1\right\rangle x → ∞ lim cos ( 2 x + 45 ) = ⟨ − 1 , 1 ⟩ Возьмём предел y = ⟨ − 1 , 1 ⟩ y = \left\langle -1, 1\right\rangle y = ⟨ − 1 , 1 ⟩
Наклонные асимптоты
Наклонную асимптоту можно найти, подсчитав предел функции cos(2*x + 45), делённой на x при x->+oo и x ->-oolim x → − ∞ ( cos ( 2 x + 45 ) x ) = 0 \lim_{x \to -\infty}\left(\frac{\cos{\left(2 x + 45 \right)}}{x}\right) = 0 x → − ∞ lim ( x cos ( 2 x + 45 ) ) = 0 Возьмём предел lim x → ∞ ( cos ( 2 x + 45 ) x ) = 0 \lim_{x \to \infty}\left(\frac{\cos{\left(2 x + 45 \right)}}{x}\right) = 0 x → ∞ lim ( x cos ( 2 x + 45 ) ) = 0 Возьмём предел
Чётность и нечётность функции
Проверим функци чётна или нечётна с помощью соотношений f = f(-x) и f = -f(-x).cos ( 2 x + 45 ) = cos ( 2 x − 45 ) \cos{\left(2 x + 45 \right)} = \cos{\left(2 x - 45 \right)} cos ( 2 x + 45 ) = cos ( 2 x − 45 ) cos ( 2 x + 45 ) = − cos ( 2 x − 45 ) \cos{\left(2 x + 45 \right)} = - \cos{\left(2 x - 45 \right)} cos ( 2 x + 45 ) = − cos ( 2 x − 45 ) 
