График функции
0 -90 -80 -70 -60 -50 -40 -30 -20 -10 10 20 30 2 -2
Точки пересечения с осью координат X
График функции пересекает ось X при f = 0cos ( 2 x ) = 0 \cos{\left(2 x \right)} = 0 cos ( 2 x ) = 0 Решаем это уравнение Аналитическое решение x 1 = π 4 x_{1} = \frac{\pi}{4} x 1 = 4 π x 2 = 3 π 4 x_{2} = \frac{3 \pi}{4} x 2 = 4 3 π Численное решение x 1 = 24.3473430653209 x_{1} = 24.3473430653209 x 1 = 24.3473430653209 x 2 = 38.484510006475 x_{2} = 38.484510006475 x 2 = 38.484510006475 x 3 = − 93.4623814442964 x_{3} = -93.4623814442964 x 3 = − 93.4623814442964 x 4 = 68.329640215578 x_{4} = 68.329640215578 x 4 = 68.329640215578 x 5 = − 10.2101761241668 x_{5} = -10.2101761241668 x 5 = − 10.2101761241668 x 6 = − 11.7809724509617 x_{6} = -11.7809724509617 x 6 = − 11.7809724509617 x 7 = − 54.1924732744239 x_{7} = -54.1924732744239 x 7 = − 54.1924732744239 x 8 = 66.7588438887831 x_{8} = 66.7588438887831 x 8 = 66.7588438887831 x 9 = − 79.3252145031423 x_{9} = -79.3252145031423 x 9 = − 79.3252145031423 x 10 = 91.8915851175014 x_{10} = 91.8915851175014 x 10 = 91.8915851175014 x 11 = − 3.92699081698724 x_{11} = -3.92699081698724 x 11 = − 3.92699081698724 x 12 = − 63.6172512351933 x_{12} = -63.6172512351933 x 12 = − 63.6172512351933 x 13 = 52.621676947629 x_{13} = 52.621676947629 x 13 = 52.621676947629 x 14 = − 18.0641577581413 x_{14} = -18.0641577581413 x 14 = − 18.0641577581413 x 15 = 16.4933614313464 x_{15} = 16.4933614313464 x 15 = 16.4933614313464 x 16 = 87.1791961371168 x_{16} = 87.1791961371168 x 16 = 87.1791961371168 x 17 = − 5.49778714378214 x_{17} = -5.49778714378214 x 17 = − 5.49778714378214 x 18 = − 69.9004365423729 x_{18} = -69.9004365423729 x 18 = − 69.9004365423729 x 19 = 63.6172512351933 x_{19} = 63.6172512351933 x 19 = 63.6172512351933 x 20 = − 33.7721210260903 x_{20} = -33.7721210260903 x 20 = − 33.7721210260903 x 21 = − 71.4712328691678 x_{21} = -71.4712328691678 x 21 = − 71.4712328691678 x 22 = − 35.3429173528852 x_{22} = -35.3429173528852 x 22 = − 35.3429173528852 x 23 = − 47.9092879672443 x_{23} = -47.9092879672443 x 23 = − 47.9092879672443 x 24 = − 49.4800842940392 x_{24} = -49.4800842940392 x 24 = − 49.4800842940392 x 25 = 3.92699081698724 x_{25} = 3.92699081698724 x 25 = 3.92699081698724 x 26 = − 82.4668071567321 x_{26} = -82.4668071567321 x 26 = − 82.4668071567321 x 27 = − 2.35619449019234 x_{27} = -2.35619449019234 x 27 = − 2.35619449019234 x 28 = 55.7632696012188 x_{28} = 55.7632696012188 x 28 = 55.7632696012188 x 29 = 11.7809724509617 x_{29} = 11.7809724509617 x 29 = 11.7809724509617 x 30 = − 46.3384916404494 x_{30} = -46.3384916404494 x 30 = − 46.3384916404494 x 31 = 19.6349540849362 x_{31} = 19.6349540849362 x 31 = 19.6349540849362 x 32 = − 68.329640215578 x_{32} = -68.329640215578 x 32 = − 68.329640215578 x 33 = 18.0641577581413 x_{33} = 18.0641577581413 x 33 = 18.0641577581413 x 34 = 1973.70558461779 x_{34} = 1973.70558461779 x 34 = 1973.70558461779 x 35 = − 77.7544181763474 x_{35} = -77.7544181763474 x 35 = − 77.7544181763474 x 36 = 30.6305283725005 x_{36} = 30.6305283725005 x 36 = 30.6305283725005 x 37 = 32.2013246992954 x_{37} = 32.2013246992954 x 37 = 32.2013246992954 x 38 = 40.0553063332699 x_{38} = 40.0553063332699 x 38 = 40.0553063332699 x 39 = 10.2101761241668 x_{39} = 10.2101761241668 x 39 = 10.2101761241668 x 40 = − 25.9181393921158 x_{40} = -25.9181393921158 x 40 = − 25.9181393921158 x 41 = − 91.8915851175014 x_{41} = -91.8915851175014 x 41 = − 91.8915851175014 x 42 = − 41.6261026600648 x_{42} = -41.6261026600648 x 42 = − 41.6261026600648 x 43 = − 55.7632696012188 x_{43} = -55.7632696012188 x 43 = − 55.7632696012188 x 44 = 60.4756585816035 x_{44} = 60.4756585816035 x 44 = 60.4756585816035 x 45 = 162.577419823272 x_{45} = 162.577419823272 x 45 = 162.577419823272 x 46 = 384.059701901352 x_{46} = 384.059701901352 x 46 = 384.059701901352 x 47 = − 32.2013246992954 x_{47} = -32.2013246992954 x 47 = − 32.2013246992954 x 48 = 49.4800842940392 x_{48} = 49.4800842940392 x 48 = 49.4800842940392 x 49 = 2.35619449019234 x_{49} = 2.35619449019234 x 49 = 2.35619449019234 x 50 = − 16.4933614313464 x_{50} = -16.4933614313464 x 50 = − 16.4933614313464 x 51 = − 85.6083998103219 x_{51} = -85.6083998103219 x 51 = − 85.6083998103219 x 52 = − 38.484510006475 x_{52} = -38.484510006475 x 52 = − 38.484510006475 x 53 = 90.3207887907066 x_{53} = 90.3207887907066 x 53 = 90.3207887907066 x 54 = − 24.3473430653209 x_{54} = -24.3473430653209 x 54 = − 24.3473430653209 x 55 = 46.3384916404494 x_{55} = 46.3384916404494 x 55 = 46.3384916404494 x 56 = − 27.4889357189107 x_{56} = -27.4889357189107 x 56 = − 27.4889357189107 x 57 = 82.4668071567321 x_{57} = 82.4668071567321 x 57 = 82.4668071567321 x 58 = − 40.0553063332699 x_{58} = -40.0553063332699 x 58 = − 40.0553063332699 x 59 = − 60.4756585816035 x_{59} = -60.4756585816035 x 59 = − 60.4756585816035 x 60 = 5.49778714378214 x_{60} = 5.49778714378214 x 60 = 5.49778714378214 x 61 = 74.6128255227576 x_{61} = 74.6128255227576 x 61 = 74.6128255227576 x 62 = 8.63937979737193 x_{62} = 8.63937979737193 x 62 = 8.63937979737193 x 63 = 96.6039740978861 x_{63} = 96.6039740978861 x 63 = 96.6039740978861 x 64 = 84.037603483527 x_{64} = 84.037603483527 x 64 = 84.037603483527 x 65 = 99.7455667514759 x_{65} = 99.7455667514759 x 65 = 99.7455667514759 x 66 = 62.0464549083984 x_{66} = 62.0464549083984 x 66 = 62.0464549083984 x 67 = 69.9004365423729 x_{67} = 69.9004365423729 x 67 = 69.9004365423729 x 68 = 98.174770424681 x_{68} = 98.174770424681 x 68 = 98.174770424681 x 69 = − 76.1836218495525 x_{69} = -76.1836218495525 x 69 = − 76.1836218495525 x 70 = 22.776546738526 x_{70} = 22.776546738526 x 70 = 22.776546738526 x 71 = 44.7676953136546 x_{71} = 44.7676953136546 x 71 = 44.7676953136546 x 72 = − 84.037603483527 x_{72} = -84.037603483527 x 72 = − 84.037603483527 x 73 = 85.6083998103219 x_{73} = 85.6083998103219 x 73 = 85.6083998103219 x 74 = 77.7544181763474 x_{74} = 77.7544181763474 x 74 = 77.7544181763474 x 75 = 27.4889357189107 x_{75} = 27.4889357189107 x 75 = 27.4889357189107 x 76 = − 13.3517687777566 x_{76} = -13.3517687777566 x 76 = − 13.3517687777566 x 77 = 54.1924732744239 x_{77} = 54.1924732744239 x 77 = 54.1924732744239 x 78 = 25.9181393921158 x_{78} = 25.9181393921158 x 78 = 25.9181393921158 x 79 = 76.1836218495525 x_{79} = 76.1836218495525 x 79 = 76.1836218495525 x 80 = − 98.174770424681 x_{80} = -98.174770424681 x 80 = − 98.174770424681 x 81 = 33.7721210260903 x_{81} = 33.7721210260903 x 81 = 33.7721210260903 x 82 = − 57.3340659280137 x_{82} = -57.3340659280137 x 82 = − 57.3340659280137 x 83 = − 90.3207887907066 x_{83} = -90.3207887907066 x 83 = − 90.3207887907066 x 84 = − 12461.9126586273 x_{84} = -12461.9126586273 x 84 = − 12461.9126586273 x 85 = − 99.7455667514759 x_{85} = -99.7455667514759 x 85 = − 99.7455667514759 x 86 = − 19.6349540849362 x_{86} = -19.6349540849362 x 86 = − 19.6349540849362 x 87 = 41.6261026600648 x_{87} = 41.6261026600648 x 87 = 41.6261026600648 x 88 = − 62.0464549083984 x_{88} = -62.0464549083984 x 88 = − 62.0464549083984 x 89 = 88.7499924639117 x_{89} = 88.7499924639117 x 89 = 88.7499924639117 x 90 = 47.9092879672443 x_{90} = 47.9092879672443 x 90 = 47.9092879672443
Точки пересечения с осью координат Y
График пересекает ось Y, когда x равняется 0:cos ( 2 ⋅ 0 ) \cos{\left(2 \cdot 0 \right)} cos ( 2 ⋅ 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 − 2 sin ( 2 x ) = 0 - 2 \sin{\left(2 x \right)} = 0 − 2 sin ( 2 x ) = 0 Решаем это уравнение x 1 = 0 x_{1} = 0 x 1 = 0 x 2 = π 2 x_{2} = \frac{\pi}{2} x 2 = 2 π (0, 1) pi
(--, -1)
2 Интервалы возрастания и убывания функции: x 1 = π 2 x_{1} = \frac{\pi}{2} x 1 = 2 π x 1 = 0 x_{1} = 0 x 1 = 0 ( − ∞ , 0 ] ∪ [ π 2 , ∞ ) \left(-\infty, 0\right] \cup \left[\frac{\pi}{2}, \infty\right) ( − ∞ , 0 ] ∪ [ 2 π , ∞ ) [ 0 , π 2 ] \left[0, \frac{\pi}{2}\right] [ 0 , 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 ) = 0 - 4 \cos{\left(2 x \right)} = 0 − 4 cos ( 2 x ) = 0 Решаем это уравнение x 1 = π 4 x_{1} = \frac{\pi}{4} x 1 = 4 π x 2 = 3 π 4 x_{2} = \frac{3 \pi}{4} x 2 = 4 3 π Интервалы выпуклости и вогнутости: [ π 4 , 3 π 4 ] \left[\frac{\pi}{4}, \frac{3 \pi}{4}\right] [ 4 π , 4 3 π ] ( − ∞ , π 4 ] ∪ [ 3 π 4 , ∞ ) \left(-\infty, \frac{\pi}{4}\right] \cup \left[\frac{3 \pi}{4}, \infty\right) ( − ∞ , 4 π ] ∪ [ 4 3 π , ∞ )
Горизонтальные асимптоты
Горизонтальные асимптоты найдём с помощью пределов данной функции при x->+oo и x->-oolim x → − ∞ cos ( 2 x ) = ⟨ − 1 , 1 ⟩ \lim_{x \to -\infty} \cos{\left(2 x \right)} = \left\langle -1, 1\right\rangle x → − ∞ lim cos ( 2 x ) = ⟨ − 1 , 1 ⟩ Возьмём предел y = ⟨ − 1 , 1 ⟩ y = \left\langle -1, 1\right\rangle y = ⟨ − 1 , 1 ⟩ lim x → ∞ cos ( 2 x ) = ⟨ − 1 , 1 ⟩ \lim_{x \to \infty} \cos{\left(2 x \right)} = \left\langle -1, 1\right\rangle x → ∞ lim cos ( 2 x ) = ⟨ − 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), делённой на x при x->+oo и x ->-oolim x → − ∞ ( cos ( 2 x ) x ) = 0 \lim_{x \to -\infty}\left(\frac{\cos{\left(2 x \right)}}{x}\right) = 0 x → − ∞ lim ( x cos ( 2 x ) ) = 0 Возьмём предел lim x → ∞ ( cos ( 2 x ) x ) = 0 \lim_{x \to \infty}\left(\frac{\cos{\left(2 x \right)}}{x}\right) = 0 x → ∞ lim ( x cos ( 2 x ) ) = 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 ) = cos ( 2 x ) \cos{\left(2 x \right)} = \cos{\left(2 x \right)} cos ( 2 x ) = cos ( 2 x ) cos ( 2 x ) = − cos ( 2 x ) \cos{\left(2 x \right)} = - \cos{\left(2 x \right)} cos ( 2 x ) = − cos ( 2 x ) 
