График функции
0 -30 -20 -10 10 20 30 40 50 60 2 -2
Точки пересечения с осью координат X
График функции пересекает ось X при f = 0cos ( 6 x ) = 0 \cos{\left(6 x \right)} = 0 cos ( 6 x ) = 0 Решаем это уравнение Аналитическое решение x 1 = π 12 x_{1} = \frac{\pi}{12} x 1 = 12 π x 2 = π 4 x_{2} = \frac{\pi}{4} x 2 = 4 π Численное решение x 1 = − 30.6305283725005 x_{1} = -30.6305283725005 x 1 = − 30.6305283725005 x 2 = − 35.8665161284835 x_{2} = -35.8665161284835 x 2 = − 35.8665161284835 x 3 = 69.9004365423729 x_{3} = 69.9004365423729 x 3 = 69.9004365423729 x 4 = − 43.720497762458 x_{4} = -43.720497762458 x 4 = − 43.720497762458 x 5 = 2.87979326579064 x_{5} = 2.87979326579064 x 5 = 2.87979326579064 x 6 = 78.2780169519457 x_{6} = 78.2780169519457 x 6 = 78.2780169519457 x 7 = − 99.7455667514759 x_{7} = -99.7455667514759 x 7 = − 99.7455667514759 x 8 = − 84.037603483527 x_{8} = -84.037603483527 x 8 = − 84.037603483527 x 9 = − 79.8488132787406 x_{9} = -79.8488132787406 x 9 = − 79.8488132787406 x 10 = − 89.7971900151083 x_{10} = -89.7971900151083 x 10 = − 89.7971900151083 x 11 = − 9.68657734856853 x_{11} = -9.68657734856853 x 11 = − 9.68657734856853 x 12 = − 40.5789051088682 x_{12} = -40.5789051088682 x 12 = − 40.5789051088682 x 13 = 64.1408500107916 x_{13} = 64.1408500107916 x 13 = 64.1408500107916 x 14 = 34.2957198016886 x_{14} = 34.2957198016886 x 14 = 34.2957198016886 x 15 = 6.02138591938044 x_{15} = 6.02138591938044 x 15 = 6.02138591938044 x 16 = 56.2868683768171 x_{16} = 56.2868683768171 x 16 = 56.2868683768171 x 17 = 76.1836218495525 x_{17} = 76.1836218495525 x 17 = 76.1836218495525 x 18 = 74.0892267471593 x_{18} = 74.0892267471593 x 18 = 74.0892267471593 x 19 = − 3.92699081698724 x_{19} = -3.92699081698724 x 19 = − 3.92699081698724 x 20 = − 47.9092879672443 x_{20} = -47.9092879672443 x 20 = − 47.9092879672443 x 21 = 12.30457122656 x_{21} = 12.30457122656 x 21 = 12.30457122656 x 22 = 68.329640215578 x_{22} = 68.329640215578 x 22 = 68.329640215578 x 23 = − 40.0553063332699 x_{23} = -40.0553063332699 x 23 = − 40.0553063332699 x 24 = 81.9432083811338 x_{24} = 81.9432083811338 x 24 = 81.9432083811338 x 25 = 80.3724120543389 x_{25} = 80.3724120543389 x 25 = 80.3724120543389 x 26 = − 33.7721210260903 x_{26} = -33.7721210260903 x 26 = − 33.7721210260903 x 27 = 25.9181393921158 x_{27} = 25.9181393921158 x 27 = 25.9181393921158 x 28 = − 50.0036830696375 x_{28} = -50.0036830696375 x 28 = − 50.0036830696375 x 29 = 62.0464549083984 x_{29} = 62.0464549083984 x 29 = 62.0464549083984 x 30 = − 15.9697626557481 x_{30} = -15.9697626557481 x 30 = − 15.9697626557481 x 31 = 0.261799387799149 x_{31} = 0.261799387799149 x 31 = 0.261799387799149 x 32 = − 74.6128255227576 x_{32} = -74.6128255227576 x 32 = − 74.6128255227576 x 33 = − 53.6688744988256 x_{33} = -53.6688744988256 x 33 = − 53.6688744988256 x 34 = 59.9520598060052 x_{34} = 59.9520598060052 x 34 = 59.9520598060052 x 35 = − 77.7544181763474 x_{35} = -77.7544181763474 x 35 = − 77.7544181763474 x 36 = 32.7249234748937 x_{36} = 32.7249234748937 x 36 = 32.7249234748937 x 37 = − 67.8060414399797 x_{37} = -67.8060414399797 x 37 = − 67.8060414399797 x 38 = − 45.8148928648512 x_{38} = -45.8148928648512 x 38 = − 45.8148928648512 x 39 = 91.3679863419031 x_{39} = 91.3679863419031 x 39 = 91.3679863419031 x 40 = 52.0980781720307 x_{40} = 52.0980781720307 x 40 = 52.0980781720307 x 41 = 32.2013246992954 x_{41} = 32.2013246992954 x 41 = 32.2013246992954 x 42 = 84.037603483527 x_{42} = 84.037603483527 x 42 = 84.037603483527 x 43 = − 74.0892267471593 x_{43} = -74.0892267471593 x 43 = − 74.0892267471593 x 44 = 86.1319985859202 x_{44} = 86.1319985859202 x 44 = 86.1319985859202 x 45 = 22.2529479629277 x_{45} = 22.2529479629277 x 45 = 22.2529479629277 x 46 = − 69.9004365423729 x_{46} = -69.9004365423729 x 46 = − 69.9004365423729 x 47 = 40.0553063332699 x_{47} = 40.0553063332699 x 47 = 40.0553063332699 x 48 = 90.3207887907066 x_{48} = 90.3207887907066 x 48 = 90.3207887907066 x 49 = − 18.0641577581413 x_{49} = -18.0641577581413 x 49 = − 18.0641577581413 x 50 = − 55.7632696012188 x_{50} = -55.7632696012188 x 50 = − 55.7632696012188 x 51 = − 87.7027949127151 x_{51} = -87.7027949127151 x 51 = − 87.7027949127151 x 52 = 50.0036830696375 x_{52} = 50.0036830696375 x 52 = 50.0036830696375 x 53 = − 96.0803753222878 x_{53} = -96.0803753222878 x 53 = − 96.0803753222878 x 54 = − 91.8915851175014 x_{54} = -91.8915851175014 x 54 = − 91.8915851175014 x 55 = − 32.2013246992954 x_{55} = -32.2013246992954 x 55 = − 32.2013246992954 x 56 = 37.9609112308767 x_{56} = 37.9609112308767 x 56 = 37.9609112308767 x 57 = 66.2352451131848 x_{57} = 66.2352451131848 x 57 = 66.2352451131848 x 58 = − 93.9859802198946 x_{58} = -93.9859802198946 x 58 = − 93.9859802198946 x 59 = 44.2440965380563 x_{59} = 44.2440965380563 x 59 = 44.2440965380563 x 60 = 10.2101761241668 x_{60} = 10.2101761241668 x 60 = 10.2101761241668 x 61 = − 25.9181393921158 x_{61} = -25.9181393921158 x 61 = − 25.9181393921158 x 62 = 71.9948316447661 x_{62} = 71.9948316447661 x 62 = 71.9948316447661 x 63 = 3.92699081698724 x_{63} = 3.92699081698724 x 63 = 3.92699081698724 x 64 = − 6.02138591938044 x_{64} = -6.02138591938044 x 64 = − 6.02138591938044 x 65 = − 75.6600230739542 x_{65} = -75.6600230739542 x 65 = − 75.6600230739542 x 66 = − 31.6777259236971 x_{66} = -31.6777259236971 x 66 = − 31.6777259236971 x 67 = 54.1924732744239 x_{67} = 54.1924732744239 x 67 = 54.1924732744239 x 68 = − 13.8753675533549 x_{68} = -13.8753675533549 x 68 = − 13.8753675533549 x 69 = − 23.8237442897226 x_{69} = -23.8237442897226 x 69 = − 23.8237442897226 x 70 = − 71.9948316447661 x_{70} = -71.9948316447661 x 70 = − 71.9948316447661 x 71 = 93.9859802198946 x_{71} = 93.9859802198946 x 71 = 93.9859802198946 x 72 = 96.0803753222878 x_{72} = 96.0803753222878 x 72 = 96.0803753222878 x 73 = 18.0641577581413 x_{73} = 18.0641577581413 x 73 = 18.0641577581413 x 74 = − 11.7809724509617 x_{74} = -11.7809724509617 x 74 = − 11.7809724509617 x 75 = − 59.9520598060052 x_{75} = -59.9520598060052 x 75 = − 59.9520598060052 x 76 = − 21.7293491873294 x_{76} = -21.7293491873294 x 76 = − 21.7293491873294 x 77 = − 28.012534494509 x_{77} = -28.012534494509 x 77 = − 28.012534494509 x 78 = − 81.9432083811338 x_{78} = -81.9432083811338 x 78 = − 81.9432083811338 x 79 = − 65.7116463375865 x_{79} = -65.7116463375865 x 79 = − 65.7116463375865 x 80 = 98.174770424681 x_{80} = 98.174770424681 x 80 = 98.174770424681 x 81 = − 37.9609112308767 x_{81} = -37.9609112308767 x 81 = − 37.9609112308767 x 82 = − 1.83259571459405 x_{82} = -1.83259571459405 x 82 = − 1.83259571459405 x 83 = 47.9092879672443 x_{83} = 47.9092879672443 x 83 = 47.9092879672443 x 84 = 89.7971900151083 x_{84} = 89.7971900151083 x 84 = 89.7971900151083 x 85 = − 52.0980781720307 x_{85} = -52.0980781720307 x 85 = − 52.0980781720307 x 86 = 20.1585528605345 x_{86} = 20.1585528605345 x 86 = 20.1585528605345 x 87 = 100.269165527074 x_{87} = 100.269165527074 x 87 = 100.269165527074 x 88 = 15.9697626557481 x_{88} = 15.9697626557481 x 88 = 15.9697626557481 x 89 = 28.012534494509 x_{89} = 28.012534494509 x 89 = 28.012534494509 x 90 = − 62.0464549083984 x_{90} = -62.0464549083984 x 90 = − 62.0464549083984 x 91 = 24.3473430653209 x_{91} = 24.3473430653209 x 91 = 24.3473430653209 x 92 = 88.2263936883134 x_{92} = 88.2263936883134 x 92 = 88.2263936883134 x 93 = − 61.5228561328001 x_{93} = -61.5228561328001 x 93 = − 61.5228561328001 x 94 = − 57.857664703612 x_{94} = -57.857664703612 x 94 = − 57.857664703612 x 95 = − 97.6511716490827 x_{95} = -97.6511716490827 x 95 = − 97.6511716490827 x 96 = 42.1497014356631 x_{96} = 42.1497014356631 x 96 = 42.1497014356631 x 97 = 8.11578102177363 x_{97} = 8.11578102177363 x 97 = 8.11578102177363 x 98 = 30.1069295969022 x_{98} = 30.1069295969022 x 98 = 30.1069295969022 x 99 = 46.3384916404494 x_{99} = 46.3384916404494 x 99 = 46.3384916404494
Точки пересечения с осью координат Y
График пересекает ось Y, когда x равняется 0:cos ( 6 ⋅ 0 ) \cos{\left(6 \cdot 0 \right)} cos ( 6 ⋅ 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 ( 6 x ) = 0 - 6 \sin{\left(6 x \right)} = 0 − 6 sin ( 6 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 (0, 1) pi
(--, -1)
6 Интервалы возрастания и убывания функции: x 1 = π 6 x_{1} = \frac{\pi}{6} x 1 = 6 π x 1 = 0 x_{1} = 0 x 1 = 0 ( − ∞ , 0 ] ∪ [ π 6 , ∞ ) \left(-\infty, 0\right] \cup \left[\frac{\pi}{6}, \infty\right) ( − ∞ , 0 ] ∪ [ 6 π , ∞ ) [ 0 , π 6 ] \left[0, \frac{\pi}{6}\right] [ 0 , 6 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 − 36 cos ( 6 x ) = 0 - 36 \cos{\left(6 x \right)} = 0 − 36 cos ( 6 x ) = 0 Решаем это уравнение x 1 = π 12 x_{1} = \frac{\pi}{12} x 1 = 12 π x 2 = π 4 x_{2} = \frac{\pi}{4} x 2 = 4 π Интервалы выпуклости и вогнутости: [ π 12 , π 4 ] \left[\frac{\pi}{12}, \frac{\pi}{4}\right] [ 12 π , 4 π ] ( − ∞ , π 12 ] ∪ [ π 4 , ∞ ) \left(-\infty, \frac{\pi}{12}\right] \cup \left[\frac{\pi}{4}, \infty\right) ( − ∞ , 12 π ] ∪ [ 4 π , ∞ )
Горизонтальные асимптоты
Горизонтальные асимптоты найдём с помощью пределов данной функции при x->+oo и x->-oolim x → − ∞ cos ( 6 x ) = ⟨ − 1 , 1 ⟩ \lim_{x \to -\infty} \cos{\left(6 x \right)} = \left\langle -1, 1\right\rangle x → − ∞ lim cos ( 6 x ) = ⟨ − 1 , 1 ⟩ Возьмём предел y = ⟨ − 1 , 1 ⟩ y = \left\langle -1, 1\right\rangle y = ⟨ − 1 , 1 ⟩ lim x → ∞ cos ( 6 x ) = ⟨ − 1 , 1 ⟩ \lim_{x \to \infty} \cos{\left(6 x \right)} = \left\langle -1, 1\right\rangle x → ∞ lim cos ( 6 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(6*x), делённой на x при x->+oo и x ->-oolim x → − ∞ ( cos ( 6 x ) x ) = 0 \lim_{x \to -\infty}\left(\frac{\cos{\left(6 x \right)}}{x}\right) = 0 x → − ∞ lim ( x cos ( 6 x ) ) = 0 Возьмём предел lim x → ∞ ( cos ( 6 x ) x ) = 0 \lim_{x \to \infty}\left(\frac{\cos{\left(6 x \right)}}{x}\right) = 0 x → ∞ lim ( x cos ( 6 x ) ) = 0 Возьмём предел
Чётность и нечётность функции
Проверим функци чётна или нечётна с помощью соотношений f = f(-x) и f = -f(-x).cos ( 6 x ) = cos ( 6 x ) \cos{\left(6 x \right)} = \cos{\left(6 x \right)} cos ( 6 x ) = cos ( 6 x ) cos ( 6 x ) = − cos ( 6 x ) \cos{\left(6 x \right)} = - \cos{\left(6 x \right)} cos ( 6 x ) = − cos ( 6 x ) 
