График функции
0 -80 -70 -60 -50 -40 -30 -20 -10 10 0 2
Точки пересечения с осью координат X
График функции пересекает ось X при f = 0cos 2 ( 3 x ) = 0 \cos^{2}{\left(3 x \right)} = 0 cos 2 ( 3 x ) = 0 Решаем это уравнение Аналитическое решение x 1 = π 6 x_{1} = \frac{\pi}{6} x 1 = 6 π x 2 = π 2 x_{2} = \frac{\pi}{2} x 2 = 2 π Численное решение x 1 = − 67.5442421196901 x_{1} = -67.5442421196901 x 1 = − 67.5442421196901 x 2 = − 49.7418836877244 x_{2} = -49.7418836877244 x 2 = − 49.7418836877244 x 3 = − 82.2050076993983 x_{3} = -82.2050076993983 x 3 = − 82.2050076993983 x 4 = 49.7418837022546 x_{4} = 49.7418837022546 x 4 = 49.7418837022546 x 5 = 5.75958657756439 x_{5} = 5.75958657756439 x 5 = 5.75958657756439 x 6 = 75.9218225202323 x_{6} = 75.9218225202323 x 6 = 75.9218225202323 x 7 = 31.9395253611477 x_{7} = 31.9395253611477 x 7 = 31.9395253611477 x 8 = − 1.57079639846571 x_{8} = -1.57079639846571 x 8 = − 1.57079639846571 x 9 = − 36.1283154422719 x_{9} = -36.1283154422719 x 9 = − 36.1283154422719 x 10 = − 23.5619449729929 x_{10} = -23.5619449729929 x 10 = − 23.5619449729929 x 11 = − 25.6563400574942 x_{11} = -25.6563400574942 x 11 = − 25.6563400574942 x 12 = − 40.3171057809134 x_{12} = -40.3171057809134 x 12 = − 40.3171057809134 x 13 = − 43.4586982782211 x_{13} = -43.4586982782211 x 13 = − 43.4586982782211 x 14 = − 14.1371668657739 x_{14} = -14.1371668657739 x 14 = − 14.1371668657739 x 15 = 18.3259571004719 x_{15} = 18.3259571004719 x 15 = 18.3259571004719 x 16 = 34.0339203573462 x_{16} = 34.0339203573462 x 16 = 34.0339203573462 x 17 = 95.8185760099185 x_{17} = 95.8185760099185 x 17 = 95.8185760099185 x 18 = 20.4203521766952 x_{18} = 20.4203521766952 x 18 = 20.4203521766952 x 19 = − 31.9395254607427 x_{19} = -31.9395254607427 x 19 = − 31.9395254607427 x 20 = − 34.0339203815254 x_{20} = -34.0339203815254 x 20 = − 34.0339203815254 x 21 = 42.4115007524084 x_{21} = 42.4115007524084 x 21 = 42.4115007524084 x 22 = − 93.7241808498957 x_{22} = -93.7241808498957 x 22 = − 93.7241808498957 x 23 = 40.3171056798614 x_{23} = 40.3171056798614 x 23 = 40.3171056798614 x 24 = − 29.8451301070482 x_{24} = -29.8451301070482 x 24 = − 29.8451301070482 x 25 = − 80.1106125965004 x_{25} = -80.1106125965004 x 25 = − 80.1106125965004 x 26 = 73.8274274357866 x_{26} = 73.8274274357866 x 26 = 73.8274274357866 x 27 = 16.2315620580511 x_{27} = 16.2315620580511 x 27 = 16.2315620580511 x 28 = 14.1371669873783 x_{28} = 14.1371669873783 x 28 = 14.1371669873783 x 29 = 44.5058959013899 x_{29} = 44.5058959013899 x 29 = 44.5058959013899 x 30 = 29.8451302859375 x_{30} = 29.8451302859375 x 30 = 29.8451302859375 x 31 = 53.9306739409302 x_{31} = 53.9306739409302 x 31 = 53.9306739409302 x 32 = 0.523598793698386 x_{32} = 0.523598793698386 x 32 = 0.523598793698386 x 33 = − 89.5353906915371 x_{33} = -89.5353906915371 x 33 = − 89.5353906915371 x 34 = 100.007366129875 x_{34} = 100.007366129875 x 34 = 100.007366129875 x 35 = 12.0427717588149 x_{35} = 12.0427717588149 x 35 = 12.0427717588149 x 36 = − 78.0162175468244 x_{36} = -78.0162175468244 x 36 = − 78.0162175468244 x 37 = 71.7330322578684 x_{37} = 71.7330322578684 x 37 = 71.7330322578684 x 38 = − 47.6474886360566 x_{38} = -47.6474886360566 x 38 = − 47.6474886360566 x 39 = 38.2227104061931 x_{39} = 38.2227104061931 x 39 = 38.2227104061931 x 40 = − 9.94837684644357 x_{40} = -9.94837684644357 x 40 = − 9.94837684644357 x 41 = 60.2138592240082 x_{41} = 60.2138592240082 x 41 = 60.2138592240082 x 42 = − 100.007366130902 x_{42} = -100.007366130902 x 42 = − 100.007366130902 x 43 = 93.7241808065961 x_{43} = 93.7241808065961 x 43 = 93.7241808065961 x 44 = − 91.6297857922845 x_{44} = -91.6297857922845 x 44 = − 91.6297857922845 x 45 = 56.0250689512033 x_{45} = 56.0250689512033 x 45 = 56.0250689512033 x 46 = − 95.8185758690479 x_{46} = -95.8185758690479 x 46 = − 95.8185758690479 x 47 = − 96.865773562311 x_{47} = -96.865773562311 x 47 = − 96.865773562311 x 48 = − 45.5530935467806 x_{48} = -45.5530935467806 x 48 = − 45.5530935467806 x 49 = − 71.7330322689618 x_{49} = -71.7330322689618 x 49 = − 71.7330322689618 x 50 = 7.85398171030129 x_{50} = 7.85398171030129 x 50 = 7.85398171030129 x 51 = 219.387886624637 x_{51} = 219.387886624637 x 51 = 219.387886624637 x 52 = − 58.1194640191823 x_{52} = -58.1194640191823 x 52 = − 58.1194640191823 x 53 = − 96.8657733416992 x_{53} = -96.8657733416992 x 53 = − 96.8657733416992 x 54 = 27.7507351416945 x_{54} = 27.7507351416945 x 54 = 27.7507351416945 x 55 = 22.51474734437 x_{55} = 22.51474734437 x 55 = 22.51474734437 x 56 = 97.9129710990821 x_{56} = 97.9129710990821 x 56 = 97.9129710990821 x 57 = − 16.2315619891784 x_{57} = -16.2315619891784 x 57 = − 16.2315619891784 x 58 = 78.0162175417499 x_{58} = 78.0162175417499 x 58 = 78.0162175417499 x 59 = − 84.2994030191035 x_{59} = -84.2994030191035 x 59 = − 84.2994030191035 x 60 = 86.3937979052498 x_{60} = 86.3937979052498 x 60 = 86.3937979052498 x 61 = − 3.6651914786486 x_{61} = -3.6651914786486 x 61 = − 3.6651914786486 x 62 = 66.4970444630068 x_{62} = 66.4970444630068 x 62 = 66.4970444630068 x 63 = − 62.3082544993577 x_{63} = -62.3082544993577 x 63 = − 62.3082544993577 x 64 = 4.71238901629785 x_{64} = 4.71238901629785 x 64 = 4.71238901629785 x 65 = 84.2994028394378 x_{65} = 84.2994028394378 x 65 = 84.2994028394378 x 66 = − 73.8274272823298 x_{66} = -73.8274272823298 x 66 = − 73.8274272823298 x 67 = − 5.75958652424928 x_{67} = -5.75958652424928 x 67 = − 5.75958652424928 x 68 = − 53.9306740746488 x_{68} = -53.9306740746488 x 68 = − 53.9306740746488 x 69 = − 60.2138591280511 x_{69} = -60.2138591280511 x 69 = − 60.2138591280511 x 70 = − 38.2227105578705 x_{70} = -38.2227105578705 x 70 = − 38.2227105578705 x 71 = − 218.340689710019 x_{71} = -218.340689710019 x 71 = − 218.340689710019 x 72 = 82.205007807797 x_{72} = 82.205007807797 x 72 = 82.205007807797 x 73 = − 56.0250689637531 x_{73} = -56.0250689637531 x 73 = − 56.0250689637531 x 74 = 88.4881930280058 x_{74} = 88.4881930280058 x 74 = 88.4881930280058 x 75 = − 7.85398152418181 x_{75} = -7.85398152418181 x 75 = − 7.85398152418181 x 76 = 9.94837678084721 x_{76} = 9.94837678084721 x 76 = 9.94837678084721 x 77 = 51.8362788611105 x_{77} = 51.8362788611105 x 77 = 51.8362788611105 x 78 = − 51.8362786947358 x_{78} = -51.8362786947358 x 78 = − 51.8362786947358 x 79 = − 87.4409953189147 x_{79} = -87.4409953189147 x 79 = − 87.4409953189147 x 80 = − 65.4498467952353 x_{80} = -65.4498467952353 x 80 = − 65.4498467952353 x 81 = − 69.6386372143248 x_{81} = -69.6386372143248 x 81 = − 69.6386372143248 x 82 = − 21.467549748169 x_{82} = -21.467549748169 x 82 = − 21.467549748169 x 83 = 62.308254259517 x_{83} = 62.308254259517 x 83 = 62.308254259517 x 84 = − 12.0427718000216 x_{84} = -12.0427718000216 x 84 = − 12.0427718000216 x 85 = 64.4026493286099 x_{85} = 64.4026493286099 x 85 = 64.4026493286099 x 86 = − 27.7507351061626 x_{86} = -27.7507351061626 x 86 = − 27.7507351061626 x 87 = 38.2227106407841 x_{87} = 38.2227106407841 x 87 = 38.2227106407841
Точки пересечения с осью координат Y
График пересекает ось Y, когда x равняется 0:cos 2 ( 3 ⋅ 0 ) \cos^{2}{\left(3 \cdot 0 \right)} cos 2 ( 3 ⋅ 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 − 6 sin ( 3 x ) cos ( 3 x ) = 0 - 6 \sin{\left(3 x \right)} \cos{\left(3 x \right)} = 0 − 6 sin ( 3 x ) cos ( 3 x ) = 0 Решаем это уравнение x 1 = 0 x_{1} = 0 x 1 = 0 x 2 = π 6 x_{2} = \frac{\pi}{6} x 2 = 6 π x 3 = π 3 x_{3} = \frac{\pi}{3} x 3 = 3 π x 4 = π 2 x_{4} = \frac{\pi}{2} x 4 = 2 π (0, 1) pi
(--, 0)
6 pi
(--, 1)
3 pi
(--, 0)
2 Интервалы возрастания и убывания функции: x 1 = π 6 x_{1} = \frac{\pi}{6} x 1 = 6 π x 2 = π 2 x_{2} = \frac{\pi}{2} x 2 = 2 π x 2 = 0 x_{2} = 0 x 2 = 0 x 2 = π 3 x_{2} = \frac{\pi}{3} x 2 = 3 π [ π 2 , ∞ ) \left[\frac{\pi}{2}, \infty\right) [ 2 π , ∞ ) ( − ∞ , π 6 ] ∪ [ π 3 , π 2 ] \left(-\infty, \frac{\pi}{6}\right] \cup \left[\frac{\pi}{3}, \frac{\pi}{2}\right] ( − ∞ , 6 π ] ∪ [ 3 π , 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 18 ( sin 2 ( 3 x ) − cos 2 ( 3 x ) ) = 0 18 \left(\sin^{2}{\left(3 x \right)} - \cos^{2}{\left(3 x \right)}\right) = 0 18 ( sin 2 ( 3 x ) − cos 2 ( 3 x ) ) = 0 Решаем это уравнение x 1 = − π 12 x_{1} = - \frac{\pi}{12} x 1 = − 12 π x 2 = π 12 x_{2} = \frac{\pi}{12} x 2 = 12 π Интервалы выпуклости и вогнутости: ( − ∞ , − π 12 ] ∪ [ π 12 , ∞ ) \left(-\infty, - \frac{\pi}{12}\right] \cup \left[\frac{\pi}{12}, \infty\right) ( − ∞ , − 12 π ] ∪ [ 12 π , ∞ ) [ − π 12 , π 12 ] \left[- \frac{\pi}{12}, \frac{\pi}{12}\right] [ − 12 π , 12 π ]
Горизонтальные асимптоты
Горизонтальные асимптоты найдём с помощью пределов данной функции при x->+oo и x->-oolim x → − ∞ cos 2 ( 3 x ) = ⟨ 0 , 1 ⟩ \lim_{x \to -\infty} \cos^{2}{\left(3 x \right)} = \left\langle 0, 1\right\rangle x → − ∞ lim cos 2 ( 3 x ) = ⟨ 0 , 1 ⟩ Возьмём предел y = ⟨ 0 , 1 ⟩ y = \left\langle 0, 1\right\rangle y = ⟨ 0 , 1 ⟩ lim x → ∞ cos 2 ( 3 x ) = ⟨ 0 , 1 ⟩ \lim_{x \to \infty} \cos^{2}{\left(3 x \right)} = \left\langle 0, 1\right\rangle x → ∞ lim cos 2 ( 3 x ) = ⟨ 0 , 1 ⟩ Возьмём предел y = ⟨ 0 , 1 ⟩ y = \left\langle 0, 1\right\rangle y = ⟨ 0 , 1 ⟩
Наклонные асимптоты
Наклонную асимптоту можно найти, подсчитав предел функции cos(3*x)^2, делённой на x при x->+oo и x ->-oolim x → − ∞ ( cos 2 ( 3 x ) x ) = 0 \lim_{x \to -\infty}\left(\frac{\cos^{2}{\left(3 x \right)}}{x}\right) = 0 x → − ∞ lim ( x cos 2 ( 3 x ) ) = 0 Возьмём предел lim x → ∞ ( cos 2 ( 3 x ) x ) = 0 \lim_{x \to \infty}\left(\frac{\cos^{2}{\left(3 x \right)}}{x}\right) = 0 x → ∞ lim ( x cos 2 ( 3 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 ( 3 x ) = cos 2 ( 3 x ) \cos^{2}{\left(3 x \right)} = \cos^{2}{\left(3 x \right)} cos 2 ( 3 x ) = cos 2 ( 3 x ) cos 2 ( 3 x ) = − cos 2 ( 3 x ) \cos^{2}{\left(3 x \right)} = - \cos^{2}{\left(3 x \right)} cos 2 ( 3 x ) = − cos 2 ( 3 x ) 
