График функции
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Точки пересечения с осью координат X
График функции пересекает ось X при f = 0cos 2 ( x 2 ) = 0 \cos^{2}{\left (\frac{x}{2} \right )} = 0 cos 2 ( 2 x ) = 0 Решаем это уравнение Аналитическое решение x 1 = π x_{1} = \pi x 1 = π x 2 = 3 π x_{2} = 3 \pi x 2 = 3 π Численное решение x 1 = 91.1061873718 x_{1} = 91.1061873718 x 1 = 91.1061873718 x 2 = − 53.4070745787 x_{2} = -53.4070745787 x 2 = − 53.4070745787 x 3 = 84.8230021336 x_{3} = 84.8230021336 x 3 = 84.8230021336 x 4 = − 97.3893716285 x_{4} = -97.3893716285 x 4 = − 97.3893716285 x 5 = 72.2566306985 x_{5} = 72.2566306985 x 5 = 72.2566306985 x 6 = 59.6902599104 x_{6} = 59.6902599104 x 6 = 59.6902599104 x 7 = 47.1238894108 x_{7} = 47.1238894108 x 7 = 47.1238894108 x 8 = − 21.9911485864 x_{8} = -21.9911485864 x 8 = − 21.9911485864 x 9 = − 34.5575196658 x_{9} = -34.5575196658 x 9 = − 34.5575196658 x 10 = − 72.2566308658 x_{10} = -72.2566308658 x 10 = − 72.2566308658 x 11 = − 59.6902599212 x_{11} = -59.6902599212 x 11 = − 59.6902599212 x 12 = 84.8230013636 x_{12} = 84.8230013636 x 12 = 84.8230013636 x 13 = 34.5575190219 x_{13} = 34.5575190219 x 13 = 34.5575190219 x 14 = 15.7079629803 x_{14} = 15.7079629803 x 14 = 15.7079629803 x 15 = 65.9734452391 x_{15} = 65.9734452391 x 15 = 65.9734452391 x 16 = 59.6902606104 x_{16} = 59.6902606104 x 16 = 59.6902606104 x 17 = − 34.5575188899 x_{17} = -34.5575188899 x 17 = − 34.5575188899 x 18 = − 47.1238893275 x_{18} = -47.1238893275 x 18 = − 47.1238893275 x 19 = − 97.3893717477 x_{19} = -97.3893717477 x 19 = − 97.3893717477 x 20 = 72.2566315167 x_{20} = 72.2566315167 x 20 = 72.2566315167 x 21 = − 78.5398160473 x_{21} = -78.5398160473 x 21 = − 78.5398160473 x 22 = − 59.6902606929 x_{22} = -59.6902606929 x 22 = − 59.6902606929 x 23 = 15.707963957 x_{23} = 15.707963957 x 23 = 15.707963957 x 24 = − 3.14159217368 x_{24} = -3.14159217368 x 24 = − 3.14159217368 x 25 = − 15.7079632966 x_{25} = -15.7079632966 x 25 = − 15.7079632966 x 26 = − 91.1061872655 x_{26} = -91.1061872655 x 26 = − 91.1061872655 x 27 = − 15.7079627748 x_{27} = -15.7079627748 x 27 = − 15.7079627748 x 28 = − 72.256631542 x_{28} = -72.256631542 x 28 = − 72.256631542 x 29 = 53.4070766554 x_{29} = 53.4070766554 x 29 = 53.4070766554 x 30 = − 47.1238901083 x_{30} = -47.1238901083 x 30 = − 47.1238901083 x 31 = 34.5575197056 x_{31} = 34.5575197056 x 31 = 34.5575197056 x 32 = − 65.973446197 x_{32} = -65.973446197 x 32 = − 65.973446197 x 33 = − 65.9734457649 x_{33} = -65.9734457649 x 33 = − 65.9734457649 x 34 = 97.3893725817 x_{34} = 97.3893725817 x 34 = 97.3893725817 x 35 = 40.8407045793 x_{35} = 40.8407045793 x 35 = 40.8407045793 x 36 = 78.5398166181 x_{36} = 78.5398166181 x 36 = 78.5398166181 x 37 = − 1127.83176319 x_{37} = -1127.83176319 x 37 = − 1127.83176319 x 38 = 15.7079627594 x_{38} = 15.7079627594 x 38 = 15.7079627594 x 39 = − 40.8407049009 x_{39} = -40.8407049009 x 39 = − 40.8407049009 x 40 = − 72.2566311847 x_{40} = -72.2566311847 x 40 = − 72.2566311847 x 41 = 9.42477826738 x_{41} = 9.42477826738 x 41 = 9.42477826738 x 42 = 15.7079634518 x_{42} = 15.7079634518 x 42 = 15.7079634518 x 43 = 72.2566310277 x_{43} = 72.2566310277 x 43 = 72.2566310277 x 44 = − 65.973444987 x_{44} = -65.973444987 x 44 = − 65.973444987 x 45 = 65.9734460391 x_{45} = 65.9734460391 x 45 = 65.9734460391 x 46 = − 9.42477752082 x_{46} = -9.42477752082 x 46 = − 9.42477752082 x 47 = − 28.2743343914 x_{47} = -28.2743343914 x 47 = − 28.2743343914 x 48 = − 21.9911490521 x_{48} = -21.9911490521 x 48 = − 21.9911490521 x 49 = − 91.1061864815 x_{49} = -91.1061864815 x 49 = − 91.1061864815 x 50 = − 84.8230012512 x_{50} = -84.8230012512 x 50 = − 84.8230012512 x 51 = − 9.42477813658 x_{51} = -9.42477813658 x 51 = − 9.42477813658 x 52 = − 59.6902604578 x_{52} = -59.6902604578 x 52 = − 59.6902604578 x 53 = − 84.8230020565 x_{53} = -84.8230020565 x 53 = − 84.8230020565 x 54 = 97.3893717959 x_{54} = 97.3893717959 x 54 = 97.3893717959 x 55 = − 28.274334099 x_{55} = -28.274334099 x 55 = − 28.274334099 x 56 = − 65.9734453607 x_{56} = -65.9734453607 x 56 = − 65.9734453607 x 57 = − 21.9911482261 x_{57} = -21.9911482261 x 57 = − 21.9911482261 x 58 = − 53.4070745964 x_{58} = -53.4070745964 x 58 = − 53.4070745964 x 59 = 28.2743335664 x_{59} = 28.2743335664 x 59 = 28.2743335664 x 60 = − 15.7079635641 x_{60} = -15.7079635641 x 60 = − 15.7079635641 x 61 = 3.14159306054 x_{61} = 3.14159306054 x 61 = 3.14159306054 x 62 = − 9.4247774453 x_{62} = -9.4247774453 x 62 = − 9.4247774453 x 63 = 65.973445753 x_{63} = 65.973445753 x 63 = 65.973445753 x 64 = − 78.5398168195 x_{64} = -78.5398168195 x 64 = − 78.5398168195 x 65 = 21.9911489073 x_{65} = 21.9911489073 x 65 = 21.9911489073 x 66 = 91.1061865668 x_{66} = 91.1061865668 x 66 = 91.1061865668 x 67 = 3.1415922549 x_{67} = 3.1415922549 x 67 = 3.1415922549 x 68 = 78.5398152766 x_{68} = 78.5398152766 x 68 = 78.5398152766 x 69 = − 28.2743337069 x_{69} = -28.2743337069 x 69 = − 28.2743337069 x 70 = − 3.14159295109 x_{70} = -3.14159295109 x 70 = − 3.14159295109 x 71 = 78.5398161805 x_{71} = 78.5398161805 x 71 = 78.5398161805 x 72 = 40.84070498 x_{72} = 40.84070498 x 72 = 40.84070498 x 73 = 78.5398168562 x_{73} = 78.5398168562 x 73 = 78.5398168562 x 74 = 53.4070754246 x_{74} = 53.4070754246 x 74 = 53.4070754246 x 75 = 28.2743343712 x_{75} = 28.2743343712 x 75 = 28.2743343712 x 76 = − 53.407075295 x_{76} = -53.407075295 x 76 = − 53.407075295 x 77 = 40.8407042062 x_{77} = 40.8407042062 x 77 = 40.8407042062 x 78 = − 40.8407040953 x_{78} = -40.8407040953 x 78 = − 40.8407040953 x 79 = 34.5575195449 x_{79} = 34.5575195449 x 79 = 34.5575195449 x 80 = − 97.3893724533 x_{80} = -97.3893724533 x 80 = − 97.3893724533 x 81 = 78.539814975 x_{81} = 78.539814975 x 81 = 78.539814975 x 82 = 9.42477748794 x_{82} = 9.42477748794 x 82 = 9.42477748794 x 83 = 21.9911480932 x_{83} = 21.9911480932 x 83 = 21.9911480932 x 84 = 40.8407045849 x_{84} = 40.8407045849 x 84 = 40.8407045849 x 85 = 21.9911485852 x_{85} = 21.9911485852 x 85 = 21.9911485852 x 86 = 47.1238902162 x_{86} = 47.1238902162 x 86 = 47.1238902162 x 87 = − 40.8407049291 x_{87} = -40.8407049291 x 87 = − 40.8407049291 x 88 = 59.6902600527 x_{88} = 59.6902600527 x 88 = 59.6902600527 x 89 = 53.4070746419 x_{89} = 53.4070746419 x 89 = 53.4070746419 x 90 = 28.2743338652 x_{90} = 28.2743338652 x 90 = 28.2743338652
Точки пересечения с осью координат Y
График пересекает ось Y, когда x равняется 0:cos 2 ( 0 2 ) \cos^{2}{\left (\frac{0}{2} \right )} cos 2 ( 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 − sin ( x 2 ) cos ( x 2 ) = 0 - \sin{\left (\frac{x}{2} \right )} \cos{\left (\frac{x}{2} \right )} = 0 − sin ( 2 x ) cos ( 2 x ) = 0 Решаем это уравнение x 1 = 0 x_{1} = 0 x 1 = 0 x 2 = π x_{2} = \pi x 2 = π x 3 = 2 π x_{3} = 2 \pi x 3 = 2 π x 4 = 3 π x_{4} = 3 \pi x 4 = 3 π (0, 1) (pi, 0) (2*pi, 1) (3*pi, 0) Интервалы возрастания и убывания функции: x 4 = π x_{4} = \pi x 4 = π x 4 = 3 π x_{4} = 3 \pi x 4 = 3 π x 4 = 0 x_{4} = 0 x 4 = 0 x 4 = 2 π x_{4} = 2 \pi x 4 = 2 π [3*pi, oo) (-oo, pi] U [2*pi, 3*pi]
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 1 2 ( sin 2 ( x 2 ) − cos 2 ( x 2 ) ) = 0 \frac{1}{2} \left(\sin^{2}{\left (\frac{x}{2} \right )} - \cos^{2}{\left (\frac{x}{2} \right )}\right) = 0 2 1 ( sin 2 ( 2 x ) − cos 2 ( 2 x ) ) = 0 Решаем это уравнение x 1 = − 3 π 2 x_{1} = - \frac{3 \pi}{2} x 1 = − 2 3 π x 2 = − π 2 x_{2} = - \frac{\pi}{2} x 2 = − 2 π x 3 = π 2 x_{3} = \frac{\pi}{2} x 3 = 2 π x 4 = 3 π 2 x_{4} = \frac{3 \pi}{2} x 4 = 2 3 π Интервалы выпуклости и вогнутости: [-3*pi/2, -pi/2] U [pi/2, oo) (-oo, -3*pi/2]
Горизонтальные асимптоты
Горизонтальные асимптоты найдём с помощью пределов данной функции при x->+oo и x->-oolim x → − ∞ cos 2 ( x 2 ) = ⟨ 0 , 1 ⟩ \lim_{x \to -\infty} \cos^{2}{\left (\frac{x}{2} \right )} = \langle 0, 1\rangle x → − ∞ lim cos 2 ( 2 x ) = ⟨ 0 , 1 ⟩ Возьмём предел y = ⟨ 0 , 1 ⟩ y = \langle 0, 1\rangle y = ⟨ 0 , 1 ⟩ lim x → ∞ cos 2 ( x 2 ) = ⟨ 0 , 1 ⟩ \lim_{x \to \infty} \cos^{2}{\left (\frac{x}{2} \right )} = \langle 0, 1\rangle x → ∞ lim cos 2 ( 2 x ) = ⟨ 0 , 1 ⟩ Возьмём предел y = ⟨ 0 , 1 ⟩ y = \langle 0, 1\rangle y = ⟨ 0 , 1 ⟩
Наклонные асимптоты
Наклонную асимптоту можно найти, подсчитав предел функции cos(x/2)^2, делённой на x при x->+oo и x ->-oolim x → − ∞ ( 1 x cos 2 ( x 2 ) ) = 0 \lim_{x \to -\infty}\left(\frac{1}{x} \cos^{2}{\left (\frac{x}{2} \right )}\right) = 0 x → − ∞ lim ( x 1 cos 2 ( 2 x ) ) = 0 Возьмём предел lim x → ∞ ( 1 x cos 2 ( x 2 ) ) = 0 \lim_{x \to \infty}\left(\frac{1}{x} \cos^{2}{\left (\frac{x}{2} \right )}\right) = 0 x → ∞ lim ( x 1 cos 2 ( 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 2 ) = cos 2 ( x 2 ) \cos^{2}{\left (\frac{x}{2} \right )} = \cos^{2}{\left (\frac{x}{2} \right )} cos 2 ( 2 x ) = cos 2 ( 2 x ) cos 2 ( x 2 ) = − cos 2 ( x 2 ) \cos^{2}{\left (\frac{x}{2} \right )} = - \cos^{2}{\left (\frac{x}{2} \right )} cos 2 ( 2 x ) = − cos 2 ( 2 x ) 