График функции
0 -80 -70 -60 -50 -40 -30 -20 -10 10 0 4
Точки пересечения с осью координат X
График функции пересекает ось X при f = 01 − cos ( 3 x ) = 0 1 - \cos{\left(3 x \right)} = 0 1 − cos ( 3 x ) = 0 Решаем это уравнение Аналитическое решение x 1 = 0 x_{1} = 0 x 1 = 0 x 2 = 2 π 3 x_{2} = \frac{2 \pi}{3} x 2 = 3 2 π Численное решение x 1 = − 4.18879008400713 x_{1} = -4.18879008400713 x 1 = − 4.18879008400713 x 2 = − 16.755160961199 x_{2} = -16.755160961199 x 2 = − 16.755160961199 x 3 = − 85.8701992330716 x_{3} = -85.8701992330716 x 3 = − 85.8701992330716 x 4 = 41.8879021332297 x_{4} = 41.8879021332297 x 4 = 41.8879021332297 x 5 = − 8.37758024222806 x_{5} = -8.37758024222806 x 5 = − 8.37758024222806 x 6 = 48.1710874363538 x_{6} = 48.1710874363538 x 6 = 48.1710874363538 x 7 = 35.6047169091797 x_{7} = 35.6047169091797 x 7 = 35.6047169091797 x 8 = − 23.0383458748855 x_{8} = -23.0383458748855 x 8 = − 23.0383458748855 x 9 = − 92.1533843995496 x_{9} = -92.1533843995496 x 9 = − 92.1533843995496 x 10 = − 35.6047172724316 x_{10} = -35.6047172724316 x 10 = − 35.6047172724316 x 11 = 161.26842260871 x_{11} = 161.26842260871 x 11 = 161.26842260871 x 12 = − 77.4926189173344 x_{12} = -77.4926189173344 x 12 = − 77.4926189173344 x 13 = 14.6607655553299 x_{13} = 14.6607655553299 x 13 = 14.6607655553299 x 14 = − 69.115038422181 x_{14} = -69.115038422181 x 14 = − 69.115038422181 x 15 = − 90.0589893499645 x_{15} = -90.0589893499645 x 15 = − 90.0589893499645 x 16 = − 79.5870139829524 x_{16} = -79.5870139829524 x 16 = − 79.5870139829524 x 17 = − 441575121.501282 x_{17} = -441575121.501282 x 17 = − 441575121.501282 x 18 = − 43.9822971746469 x_{18} = -43.9822971746469 x 18 = − 43.9822971746469 x 19 = − 81.6814090376256 x_{19} = -81.6814090376256 x 19 = − 81.6814090376256 x 20 = 33.5103217994214 x_{20} = 33.5103217994214 x 20 = 33.5103217994214 x 21 = − 71.2094336231345 x_{21} = -71.2094336231345 x 21 = − 71.2094336231345 x 22 = − 100.530964767287 x_{22} = -100.530964767287 x 22 = − 100.530964767287 x 23 = − 14.6607657359666 x_{23} = -14.6607657359666 x 23 = − 14.6607657359666 x 24 = − 56.548667625889 x_{24} = -56.548667625889 x 24 = − 56.548667625889 x 25 = 8.37758033410453 x_{25} = 8.37758033410453 x 25 = 8.37758033410453 x 26 = − 60.7374583801782 x_{26} = -60.7374583801782 x 26 = − 60.7374583801782 x 27 = 58.6430627071515 x_{27} = 58.6430627071515 x 27 = 58.6430627071515 x 28 = 79.5870140598991 x_{28} = 79.5870140598991 x 28 = 79.5870140598991 x 29 = 85.8701992926441 x_{29} = 85.8701992926441 x 29 = 85.8701992926441 x 30 = 92.1533846197794 x_{30} = 92.1533846197794 x 30 = 92.1533846197794 x 31 = − 37.6991118769598 x_{31} = -37.6991118769598 x 31 = − 37.6991118769598 x 32 = 96.3421746527392 x_{32} = 96.3421746527392 x 32 = 96.3421746527392 x 33 = − 48.1710872415186 x_{33} = -48.1710872415186 x 33 = − 48.1710872415186 x 34 = 87.9645943355056 x_{34} = 87.9645943355056 x 34 = 87.9645943355056 x 35 = 60.7374578111775 x_{35} = 60.7374578111775 x 35 = 60.7374578111775 x 36 = − 98.4365696440843 x_{36} = -98.4365696440843 x 36 = − 98.4365696440843 x 37 = 20.943951173337 x_{37} = 20.943951173337 x 37 = 20.943951173337 x 38 = − 50.2654823108466 x_{38} = -50.2654823108466 x 38 = − 50.2654823108466 x 39 = 75.3982237804204 x_{39} = 75.3982237804204 x 39 = 75.3982237804204 x 40 = − 33.5103217610131 x_{40} = -33.5103217610131 x 40 = − 33.5103217610131 x 41 = − 29.3215315957413 x_{41} = -29.3215315957413 x 41 = − 29.3215315957413 x 42 = 77.4926189449124 x_{42} = 77.4926189449124 x 42 = 77.4926189449124 x 43 = 52.3598774932141 x_{43} = 52.3598774932141 x 43 = 52.3598774932141 x 44 = − 12.5663704869723 x_{44} = -12.5663704869723 x 44 = − 12.5663704869723 x 45 = 31.4159266487811 x_{45} = 31.4159266487811 x 45 = 31.4159266487811 x 46 = 98.4365697102782 x_{46} = 98.4365697102782 x 46 = 98.4365697102782 x 47 = − 27.2271364801679 x_{47} = -27.2271364801679 x 47 = − 27.2271364801679 x 48 = − 25.1327413022544 x_{48} = -25.1327413022544 x 48 = − 25.1327413022544 x 49 = − 87.964594359041 x_{49} = -87.964594359041 x 49 = − 87.964594359041 x 50 = 56.5486676233132 x_{50} = 56.5486676233132 x 50 = 56.5486676233132 x 51 = − 83.7758040750549 x_{51} = -83.7758040750549 x 51 = − 83.7758040750549 x 52 = 37.6991119985219 x_{52} = 37.6991119985219 x 52 = 37.6991119985219 x 53 = 4.18879026156543 x_{53} = 4.18879026156543 x 53 = 4.18879026156543 x 54 = 18.8495558248099 x_{54} = 18.8495558248099 x 54 = 18.8495558248099 x 55 = − 96.3421745469196 x_{55} = -96.3421745469196 x 55 = − 96.3421745469196 x 56 = 12.5663704684363 x_{56} = 12.5663704684363 x 56 = 12.5663704684363 x 57 = 833.569250843322 x_{57} = 833.569250843322 x 57 = 833.569250843322 x 58 = 62.8318529572331 x_{58} = 62.8318529572331 x 58 = 62.8318529572331 x 59 = 0 x_{59} = 0 x 59 = 0 x 60 = − 52.3598773942745 x_{60} = -52.3598773942745 x 60 = − 52.3598773942745 x 61 = − 2.09439502690814 x_{61} = -2.09439502690814 x 61 = − 2.09439502690814 x 62 = − 67.0206429069494 x_{62} = -67.0206429069494 x 62 = − 67.0206429069494 x 63 = − 35.604716824613 x_{63} = -35.604716824613 x 63 = − 35.604716824613 x 64 = − 39.7935069082666 x_{64} = -39.7935069082666 x 64 = − 39.7935069082666 x 65 = − 54.4542724952377 x_{65} = -54.4542724952377 x 65 = − 54.4542724952377 x 66 = 54.4542725528033 x_{66} = 54.4542725528033 x 66 = 54.4542725528033 x 67 = − 41.8879015909008 x_{67} = -41.8879015909008 x 67 = − 41.8879015909008 x 68 = − 46.0766921879046 x_{68} = -46.0766921879046 x 68 = − 46.0766921879046 x 69 = 43.9822971693627 x_{69} = 43.9822971693627 x 69 = 43.9822971693627 x 70 = − 20.9439509761532 x_{70} = -20.9439509761532 x 70 = − 20.9439509761532 x 71 = 64.9262482454786 x_{71} = 64.9262482454786 x 71 = 64.9262482454786 x 72 = 50.2654824463613 x_{72} = 50.2654824463613 x 72 = 50.2654824463613 x 73 = 81.6814091524463 x_{73} = 81.6814091524463 x 73 = 81.6814091524463 x 74 = 73.3038284231128 x_{74} = 73.3038284231128 x 74 = 73.3038284231128 x 75 = − 69.1150389029232 x_{75} = -69.1150389029232 x 75 = − 69.1150389029232 x 76 = 46.076691966571 x_{76} = 46.076691966571 x 76 = 46.076691966571 x 77 = 6.28318528432449 x_{77} = 6.28318528432449 x 77 = 6.28318528432449 x 78 = − 31.4159266859544 x_{78} = -31.4159266859544 x 78 = − 31.4159266859544 x 79 = − 94.2477794662643 x_{79} = -94.2477794662643 x 79 = − 94.2477794662643 x 80 = − 75.3982238398266 x_{80} = -75.3982238398266 x 80 = − 75.3982238398266 x 81 = − 58.6430628407171 x_{81} = -58.6430628407171 x 81 = − 58.6430628407171 x 82 = 98.4365698045197 x_{82} = 98.4365698045197 x 82 = 98.4365698045197 x 83 = 100.530964778649 x_{83} = 100.530964778649 x 83 = 100.530964778649 x 84 = 16.7551606647126 x_{84} = 16.7551606647126 x 84 = 16.7551606647126 x 85 = 94.2477796093527 x_{85} = 94.2477796093527 x 85 = 94.2477796093527 x 86 = 90.0589892608483 x_{86} = 90.0589892608483 x 86 = 90.0589892608483 x 87 = 37.6991113434848 x_{87} = 37.6991113434848 x 87 = 37.6991113434848 x 88 = 10.4719753957034 x_{88} = 10.4719753957034 x 88 = 10.4719753957034 x 89 = 39.7935070730078 x_{89} = 39.7935070730078 x 89 = 39.7935070730078 x 90 = − 10.4719753473561 x_{90} = -10.4719753473561 x 90 = − 10.4719753473561 x 91 = 83.7758042295396 x_{91} = 83.7758042295396 x 91 = 83.7758042295396 x 92 = 29.3215313536198 x_{92} = 29.3215313536198 x 92 = 29.3215313536198 x 93 = − 73.3038287459741 x_{93} = -73.3038287459741 x 93 = − 73.3038287459741 x 94 = − 6.28318515591421 x_{94} = -6.28318515591421 x 94 = − 6.28318515591421
Точки пересечения с осью координат Y
График пересекает ось Y, когда x равняется 0:1 − cos ( 3 ⋅ 0 ) 1 - \cos{\left(3 \cdot 0 \right)} 1 − cos ( 3 ⋅ 0 ) f ( 0 ) = 0 f{\left(0 \right)} = 0 f ( 0 ) = 0 (0, 0)
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 3 sin ( 3 x ) = 0 3 \sin{\left(3 x \right)} = 0 3 sin ( 3 x ) = 0 Решаем это уравнение x 1 = 0 x_{1} = 0 x 1 = 0 x 2 = π 3 x_{2} = \frac{\pi}{3} x 2 = 3 π (0, 0) pi
(--, 2)
3 Интервалы возрастания и убывания функции: x 1 = 0 x_{1} = 0 x 1 = 0 x 1 = π 3 x_{1} = \frac{\pi}{3} x 1 = 3 π [ 0 , π 3 ] \left[0, \frac{\pi}{3}\right] [ 0 , 3 π ] ( − ∞ , 0 ] ∪ [ π 3 , ∞ ) \left(-\infty, 0\right] \cup \left[\frac{\pi}{3}, \infty\right) ( − ∞ , 0 ] ∪ [ 3 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 9 cos ( 3 x ) = 0 9 \cos{\left(3 x \right)} = 0 9 cos ( 3 x ) = 0 Решаем это уравнение 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 Интервалы выпуклости и вогнутости: ( − ∞ , π 6 ] ∪ [ π 2 , ∞ ) \left(-\infty, \frac{\pi}{6}\right] \cup \left[\frac{\pi}{2}, \infty\right) ( − ∞ , 6 π ] ∪ [ 2 π , ∞ ) [ π 6 , π 2 ] \left[\frac{\pi}{6}, \frac{\pi}{2}\right] [ 6 π , 2 π ]
Горизонтальные асимптоты
Горизонтальные асимптоты найдём с помощью пределов данной функции при x->+oo и x->-oolim x → − ∞ ( 1 − cos ( 3 x ) ) = ⟨ 0 , 2 ⟩ \lim_{x \to -\infty}\left(1 - \cos{\left(3 x \right)}\right) = \left\langle 0, 2\right\rangle x → − ∞ lim ( 1 − cos ( 3 x ) ) = ⟨ 0 , 2 ⟩ Возьмём предел y = ⟨ 0 , 2 ⟩ y = \left\langle 0, 2\right\rangle y = ⟨ 0 , 2 ⟩ lim x → ∞ ( 1 − cos ( 3 x ) ) = ⟨ 0 , 2 ⟩ \lim_{x \to \infty}\left(1 - \cos{\left(3 x \right)}\right) = \left\langle 0, 2\right\rangle x → ∞ lim ( 1 − cos ( 3 x ) ) = ⟨ 0 , 2 ⟩ Возьмём предел y = ⟨ 0 , 2 ⟩ y = \left\langle 0, 2\right\rangle y = ⟨ 0 , 2 ⟩
Наклонные асимптоты
Наклонную асимптоту можно найти, подсчитав предел функции 1 - cos(3*x), делённой на x при x->+oo и x ->-oolim x → − ∞ ( 1 − cos ( 3 x ) x ) = 0 \lim_{x \to -\infty}\left(\frac{1 - \cos{\left(3 x \right)}}{x}\right) = 0 x → − ∞ lim ( x 1 − cos ( 3 x ) ) = 0 Возьмём предел lim x → ∞ ( 1 − cos ( 3 x ) x ) = 0 \lim_{x \to \infty}\left(\frac{1 - \cos{\left(3 x \right)}}{x}\right) = 0 x → ∞ lim ( x 1 − cos ( 3 x ) ) = 0 Возьмём предел
Чётность и нечётность функции
Проверим функци чётна или нечётна с помощью соотношений f = f(-x) и f = -f(-x).1 − cos ( 3 x ) = 1 − cos ( 3 x ) 1 - \cos{\left(3 x \right)} = 1 - \cos{\left(3 x \right)} 1 − cos ( 3 x ) = 1 − cos ( 3 x ) 1 − cos ( 3 x ) = cos ( 3 x ) − 1 1 - \cos{\left(3 x \right)} = \cos{\left(3 x \right)} - 1 1 − cos ( 3 x ) = cos ( 3 x ) − 1 
