График функции
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Точки пересечения с осью координат X
График функции пересекает ось X при f = 02 cos 2 ( x ) = 0 2 \cos^{2}{\left (x \right )} = 0 2 cos 2 ( x ) = 0 Решаем это уравнение Аналитическое решение 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 Численное решение x 1 = 48.6946859239 x_{1} = 48.6946859239 x 1 = 48.6946859239 x 2 = − 10.995574535 x_{2} = -10.995574535 x 2 = − 10.995574535 x 3 = 92.6769830795 x_{3} = 92.6769830795 x 3 = 92.6769830795 x 4 = − 48.694686092 x_{4} = -48.694686092 x 4 = − 48.694686092 x 5 = − 54.9778713137 x_{5} = -54.9778713137 x 5 = − 54.9778713137 x 6 = − 20.4203520322 x_{6} = -20.4203520322 x 6 = − 20.4203520322 x 7 = − 92.6769831824 x_{7} = -92.6769831824 x 7 = − 92.6769831824 x 8 = 4.71238876848 x_{8} = 4.71238876848 x 8 = 4.71238876848 x 9 = − 17.2787598091 x_{9} = -17.2787598091 x 9 = − 17.2787598091 x 10 = 1.57079654544 x_{10} = 1.57079654544 x 10 = 1.57079654544 x 11 = − 98.9601688304 x_{11} = -98.9601688304 x 11 = − 98.9601688304 x 12 = − 76.9690198771 x_{12} = -76.9690198771 x 12 = − 76.9690198771 x 13 = 76.9690197632 x_{13} = 76.9690197632 x 13 = 76.9690197632 x 14 = 83.2522052341 x_{14} = 83.2522052341 x 14 = 83.2522052341 x 15 = 70.6858345017 x_{15} = 70.6858345017 x 15 = 70.6858345017 x 16 = 17.2787595624 x_{16} = 17.2787595624 x 16 = 17.2787595624 x 17 = 32.9867229281 x_{17} = 32.9867229281 x 17 = 32.9867229281 x 18 = 26.7035373461 x_{18} = 26.7035373461 x 18 = 26.7035373461 x 19 = 36.1283156002 x_{19} = 36.1283156002 x 19 = 36.1283156002 x 20 = 42.4115007292 x_{20} = 42.4115007292 x 20 = 42.4115007292 x 21 = − 98.9601686845 x_{21} = -98.9601686845 x 21 = − 98.9601686845 x 22 = − 58.1194639993 x_{22} = -58.1194639993 x 22 = − 58.1194639993 x 23 = − 39.2699083866 x_{23} = -39.2699083866 x 23 = − 39.2699083866 x 24 = 98.9601685932 x_{24} = 98.9601685932 x 24 = 98.9601685932 x 25 = − 95.8185758681 x_{25} = -95.8185758681 x 25 = − 95.8185758681 x 26 = 76.9690207492 x_{26} = 76.9690207492 x 26 = 76.9690207492 x 27 = − 89.5353907468 x_{27} = -89.5353907468 x 27 = − 89.5353907468 x 28 = 89.5353908553 x_{28} = 89.5353908553 x 28 = 89.5353908553 x 29 = − 45.5530935883 x_{29} = -45.5530935883 x 29 = − 45.5530935883 x 30 = − 51.8362786897 x_{30} = -51.8362786897 x 30 = − 51.8362786897 x 31 = 32.9867226138 x_{31} = 32.9867226138 x 31 = 32.9867226138 x 32 = − 61.2610562243 x_{32} = -61.2610562243 x 32 = − 61.2610562243 x 33 = 14.1371671048 x_{33} = 14.1371671048 x 33 = 14.1371671048 x 34 = − 64.4026491876 x_{34} = -64.4026491876 x 34 = − 64.4026491876 x 35 = 61.2610566753 x_{35} = 61.2610566753 x 35 = 61.2610566753 x 36 = 86.3937978883 x_{36} = 86.3937978883 x 36 = 86.3937978883 x 37 = 7.85398174059 x_{37} = 7.85398174059 x 37 = 7.85398174059 x 38 = 73.8274274796 x_{38} = 73.8274274796 x 38 = 73.8274274796 x 39 = − 92.6769830239 x_{39} = -92.6769830239 x 39 = − 92.6769830239 x 40 = 95.818576059 x_{40} = 95.818576059 x 40 = 95.818576059 x 41 = − 26.7035375428 x_{41} = -26.7035375428 x 41 = − 26.7035375428 x 42 = − 70.6858346386 x_{42} = -70.6858346386 x 42 = − 70.6858346386 x 43 = 20.4203521497 x_{43} = 20.4203521497 x 43 = 20.4203521497 x 44 = − 10.9955741902 x_{44} = -10.9955741902 x 44 = − 10.9955741902 x 45 = − 26.703537299 x_{45} = -26.703537299 x 45 = − 26.703537299 x 46 = − 39.2699081529 x_{46} = -39.2699081529 x 46 = − 39.2699081529 x 47 = − 32.9867231092 x_{47} = -32.9867231092 x 47 = − 32.9867231092 x 48 = 541.92473289 x_{48} = 541.92473289 x 48 = 541.92473289 x 49 = 23.5619449395 x_{49} = 23.5619449395 x 49 = 23.5619449395 x 50 = 98.9601683381 x_{50} = 98.9601683381 x 50 = 98.9601683381 x 51 = − 48.6946858739 x_{51} = -48.6946858739 x 51 = − 48.6946858739 x 52 = − 29.8451300964 x_{52} = -29.8451300964 x 52 = − 29.8451300964 x 53 = − 32.9867227514 x_{53} = -32.9867227514 x 53 = − 32.9867227514 x 54 = 51.8362789 x_{54} = 51.8362789 x 54 = 51.8362789 x 55 = − 83.2522055415 x_{55} = -83.2522055415 x 55 = − 83.2522055415 x 56 = 83.2522055731 x_{56} = 83.2522055731 x 56 = 83.2522055731 x 57 = − 54.9778716831 x_{57} = -54.9778716831 x 57 = − 54.9778716831 x 58 = 61.261056999 x_{58} = 61.261056999 x 58 = 61.261056999 x 59 = 10.9955740393 x_{59} = 10.9955740393 x 59 = 10.9955740393 x 60 = − 14.1371668393 x_{60} = -14.1371668393 x 60 = − 14.1371668393 x 61 = − 70.6858344488 x_{61} = -70.6858344488 x 61 = − 70.6858344488 x 62 = − 1.57079642969 x_{62} = -1.57079642969 x 62 = − 1.57079642969 x 63 = − 67.5442421676 x_{63} = -67.5442421676 x 63 = − 67.5442421676 x 64 = − 80.1106125796 x_{64} = -80.1106125796 x 64 = − 80.1106125796 x 65 = 29.8451303203 x_{65} = 29.8451303203 x 65 = 29.8451303203 x 66 = 10.9955743697 x_{66} = 10.9955743697 x 66 = 10.9955743697 x 67 = − 4.71238872431 x_{67} = -4.71238872431 x 67 = − 4.71238872431 x 68 = − 7.85398149857 x_{68} = -7.85398149857 x 68 = − 7.85398149857 x 69 = 39.269908118 x_{69} = 39.269908118 x 69 = 39.269908118 x 70 = − 76.9690202569 x_{70} = -76.9690202569 x 70 = − 76.9690202569 x 71 = 23.561945123 x_{71} = 23.561945123 x 71 = 23.561945123 x 72 = − 23.561945009 x_{72} = -23.561945009 x 72 = − 23.561945009 x 73 = 80.1106131435 x_{73} = 80.1106131435 x 73 = 80.1106131435 x 74 = 45.5530937005 x_{74} = 45.5530937005 x 74 = 45.5530937005 x 75 = 54.977871485 x_{75} = 54.977871485 x 75 = 54.977871485 x 76 = − 61.2610569641 x_{76} = -61.2610569641 x 76 = − 61.2610569641 x 77 = 17.2787598503 x_{77} = 17.2787598503 x 77 = 17.2787598503 x 78 = − 42.4115006099 x_{78} = -42.4115006099 x 78 = − 42.4115006099 x 79 = − 98.9601684415 x_{79} = -98.9601684415 x 79 = − 98.9601684415 x 80 = − 73.82742728 x_{80} = -73.82742728 x 80 = − 73.82742728 x 81 = − 4.71238899124 x_{81} = -4.71238899124 x 81 = − 4.71238899124 x 82 = 64.4026493087 x_{82} = 64.4026493087 x 82 = 64.4026493087 x 83 = 76.9690200401 x_{83} = 76.9690200401 x 83 = 76.9690200401 x 84 = 39.2699084247 x_{84} = 39.2699084247 x 84 = 39.2699084247 x 85 = − 36.1283154192 x_{85} = -36.1283154192 x 85 = − 36.1283154192 x 86 = − 86.3937977655 x_{86} = -86.3937977655 x 86 = − 86.3937977655 x 87 = 67.5442422779 x_{87} = 67.5442422779 x 87 = 67.5442422779 x 88 = 54.9778711884 x_{88} = 54.9778711884 x 88 = 54.9778711884 x 89 = 58.119464438 x_{89} = 58.119464438 x 89 = 58.119464438 x 90 = − 17.2787590277 x_{90} = -17.2787590277 x 90 = − 17.2787590277 x 91 = 80.1106126772 x_{91} = 80.1106126772 x 91 = 80.1106126772
Точки пересечения с осью координат Y
График пересекает ось Y, когда x равняется 0:2 cos 2 ( 0 ) 2 \cos^{2}{\left (0 \right )} 2 cos 2 ( 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 − 4 sin ( x ) cos ( x ) = 0 - 4 \sin{\left (x \right )} \cos{\left (x \right )} = 0 − 4 sin ( x ) cos ( 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 x 3 = π x_{3} = \pi x 3 = π x 4 = 3 π 2 x_{4} = \frac{3 \pi}{2} x 4 = 2 3 π (0, 2) pi
(--, 0)
2 (pi, 2) 3*pi
(----, 0)
2 Интервалы возрастания и убывания функции: x 4 = π 2 x_{4} = \frac{\pi}{2} x 4 = 2 π x 4 = 3 π 2 x_{4} = \frac{3 \pi}{2} x 4 = 2 3 π x 4 = 0 x_{4} = 0 x 4 = 0 x 4 = π x_{4} = \pi x 4 = π [3*pi/2, oo) (-oo, pi/2] U [pi, 3*pi/2]
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 ( sin 2 ( x ) − cos 2 ( x ) ) = 0 4 \left(\sin^{2}{\left (x \right )} - \cos^{2}{\left (x \right )}\right) = 0 4 ( sin 2 ( x ) − cos 2 ( x ) ) = 0 Решаем это уравнение x 1 = − 3 π 4 x_{1} = - \frac{3 \pi}{4} x 1 = − 4 3 π x 2 = − π 4 x_{2} = - \frac{\pi}{4} x 2 = − 4 π x 3 = π 4 x_{3} = \frac{\pi}{4} x 3 = 4 π x 4 = 3 π 4 x_{4} = \frac{3 \pi}{4} x 4 = 4 3 π Интервалы выпуклости и вогнутости: [-3*pi/4, -pi/4] U [pi/4, oo) (-oo, -3*pi/4]
Горизонтальные асимптоты
Горизонтальные асимптоты найдём с помощью пределов данной функции при x->+oo и x->-oolim x → − ∞ ( 2 cos 2 ( x ) ) = ⟨ 0 , 2 ⟩ \lim_{x \to -\infty}\left(2 \cos^{2}{\left (x \right )}\right) = \langle 0, 2\rangle x → − ∞ lim ( 2 cos 2 ( x ) ) = ⟨ 0 , 2 ⟩ Возьмём предел y = ⟨ 0 , 2 ⟩ y = \langle 0, 2\rangle y = ⟨ 0 , 2 ⟩ lim x → ∞ ( 2 cos 2 ( x ) ) = ⟨ 0 , 2 ⟩ \lim_{x \to \infty}\left(2 \cos^{2}{\left (x \right )}\right) = \langle 0, 2\rangle x → ∞ lim ( 2 cos 2 ( x ) ) = ⟨ 0 , 2 ⟩ Возьмём предел y = ⟨ 0 , 2 ⟩ y = \langle 0, 2\rangle y = ⟨ 0 , 2 ⟩
Наклонные асимптоты
Наклонную асимптоту можно найти, подсчитав предел функции 2*cos(x)^2, делённой на x при x->+oo и x ->-oolim x → − ∞ ( 2 x cos 2 ( x ) ) = 0 \lim_{x \to -\infty}\left(\frac{2}{x} \cos^{2}{\left (x \right )}\right) = 0 x → − ∞ lim ( x 2 cos 2 ( x ) ) = 0 Возьмём предел lim x → ∞ ( 2 x cos 2 ( x ) ) = 0 \lim_{x \to \infty}\left(\frac{2}{x} \cos^{2}{\left (x \right )}\right) = 0 x → ∞ lim ( x 2 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).2 cos 2 ( x ) = 2 cos 2 ( x ) 2 \cos^{2}{\left (x \right )} = 2 \cos^{2}{\left (x \right )} 2 cos 2 ( x ) = 2 cos 2 ( x ) 2 cos 2 ( x ) = − 2 cos 2 ( x ) 2 \cos^{2}{\left (x \right )} = - 2 \cos^{2}{\left (x \right )} 2 cos 2 ( x ) = − 2 cos 2 ( x ) 