График функции
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Точки пересечения с осью координат X
График функции пересекает ось X при f = 0cot ( 2 x − π 3 ) = 0 \cot{\left(2 x - \frac{\pi}{3} \right)} = 0 cot ( 2 x − 3 π ) = 0 Решаем это уравнение Аналитическое решение x 1 = − π 12 x_{1} = - \frac{\pi}{12} x 1 = − 12 π Численное решение x 1 = 65.7116463375865 x_{1} = 65.7116463375865 x 1 = 65.7116463375865 x 2 = 28.012534494509 x_{2} = 28.012534494509 x 2 = 28.012534494509 x 3 = − 72.5184304203644 x_{3} = -72.5184304203644 x 3 = − 72.5184304203644 x 4 = − 39.5317075576716 x_{4} = -39.5317075576716 x 4 = − 39.5317075576716 x 5 = − 23.8237442897226 x_{5} = -23.8237442897226 x 5 = − 23.8237442897226 x 6 = 57.857664703612 x_{6} = 57.857664703612 x 6 = 57.857664703612 x 7 = 26.4417381677141 x_{7} = 26.4417381677141 x 7 = 26.4417381677141 x 8 = 81.4196096055355 x_{8} = 81.4196096055355 x 8 = 81.4196096055355 x 9 = − 66.2352451131848 x_{9} = -66.2352451131848 x 9 = − 66.2352451131848 x 10 = − 15.9697626557481 x_{10} = -15.9697626557481 x 10 = − 15.9697626557481 x 11 = 43.720497762458 x_{11} = 43.720497762458 x 11 = 43.720497762458 x 12 = − 94.5095789954929 x_{12} = -94.5095789954929 x 12 = − 94.5095789954929 x 13 = − 41.1025038844665 x_{13} = -41.1025038844665 x 13 = − 41.1025038844665 x 14 = − 59.9520598060052 x_{14} = -59.9520598060052 x 14 = − 59.9520598060052 x 15 = 48.4328867428426 x_{15} = 48.4328867428426 x 15 = 48.4328867428426 x 16 = − 19.1113553093379 x_{16} = -19.1113553093379 x 16 = − 19.1113553093379 x 17 = 6.02138591938044 x_{17} = 6.02138591938044 x 17 = 6.02138591938044 x 18 = 54.7160720500222 x_{18} = 54.7160720500222 x 18 = 54.7160720500222 x 19 = − 44.2440965380563 x_{19} = -44.2440965380563 x 19 = − 44.2440965380563 x 20 = 71.9948316447661 x_{20} = 71.9948316447661 x 20 = 71.9948316447661 x 21 = − 81.9432083811338 x_{21} = -81.9432083811338 x 21 = − 81.9432083811338 x 22 = − 28.5361332701073 x_{22} = -28.5361332701073 x 22 = − 28.5361332701073 x 23 = 90.8443875663049 x_{23} = 90.8443875663049 x 23 = 90.8443875663049 x 24 = 40.5789051088682 x_{24} = 40.5789051088682 x 24 = 40.5789051088682 x 25 = 37.4373124552784 x_{25} = 37.4373124552784 x 25 = 37.4373124552784 x 26 = 92.4151838930998 x_{26} = 92.4151838930998 x 26 = 92.4151838930998 x 27 = 62.5700536839967 x_{27} = 62.5700536839967 x 27 = 62.5700536839967 x 28 = 10.7337748997651 x_{28} = 10.7337748997651 x 28 = 10.7337748997651 x 29 = 4.45058959258554 x_{29} = 4.45058959258554 x 29 = 4.45058959258554 x 30 = 12.30457122656 x_{30} = 12.30457122656 x 30 = 12.30457122656 x 31 = − 97.6511716490827 x_{31} = -97.6511716490827 x 31 = − 97.6511716490827 x 32 = − 31.6777259236971 x_{32} = -31.6777259236971 x 32 = − 31.6777259236971 x 33 = − 3.40339204138894 x_{33} = -3.40339204138894 x 33 = − 3.40339204138894 x 34 = 95.5567765466895 x_{34} = 95.5567765466895 x 34 = 95.5567765466895 x 35 = − 52.0980781720307 x_{35} = -52.0980781720307 x 35 = − 52.0980781720307 x 36 = − 33.248522250492 x_{36} = -33.248522250492 x 36 = − 33.248522250492 x 37 = 86.1319985859202 x_{37} = 86.1319985859202 x 37 = 86.1319985859202 x 38 = 42.1497014356631 x_{38} = 42.1497014356631 x 38 = 42.1497014356631 x 39 = − 6.54498469497874 x_{39} = -6.54498469497874 x 39 = − 6.54498469497874 x 40 = − 11.2573736753634 x_{40} = -11.2573736753634 x 40 = − 11.2573736753634 x 41 = − 83.5140047079287 x_{41} = -83.5140047079287 x 41 = − 83.5140047079287 x 42 = 46.8620904160477 x_{42} = 46.8620904160477 x 42 = 46.8620904160477 x 43 = 13.8753675533549 x_{43} = 13.8753675533549 x 43 = 13.8753675533549 x 44 = − 91.3679863419031 x_{44} = -91.3679863419031 x 44 = − 91.3679863419031 x 45 = − 96.0803753222878 x_{45} = -96.0803753222878 x 45 = − 96.0803753222878 x 46 = 93.9859802198946 x_{46} = 93.9859802198946 x 46 = 93.9859802198946 x 47 = 100.269165527074 x_{47} = 100.269165527074 x 47 = 100.269165527074 x 48 = − 17.540558982543 x_{48} = -17.540558982543 x 48 = − 17.540558982543 x 49 = − 74.0892267471593 x_{49} = -74.0892267471593 x 49 = − 74.0892267471593 x 50 = − 14.3989663289532 x_{50} = -14.3989663289532 x 50 = − 14.3989663289532 x 51 = − 1.83259571459405 x_{51} = -1.83259571459405 x 51 = − 1.83259571459405 x 52 = 68.8532389911763 x_{52} = 68.8532389911763 x 52 = 68.8532389911763 x 53 = 15.4461638801498 x_{53} = 15.4461638801498 x 53 = 15.4461638801498 x 54 = 35.8665161284835 x_{54} = 35.8665161284835 x 54 = 35.8665161284835 x 55 = 32.7249234748937 x_{55} = 32.7249234748937 x 55 = 32.7249234748937 x 56 = − 63.093652459595 x_{56} = -63.093652459595 x 56 = − 63.093652459595 x 57 = − 30.1069295969022 x_{57} = -30.1069295969022 x 57 = − 30.1069295969022 x 58 = − 47.3856891916461 x_{58} = -47.3856891916461 x 58 = − 47.3856891916461 x 59 = 76.7072206251508 x_{59} = 76.7072206251508 x 59 = 76.7072206251508 x 60 = − 50.5272818452358 x_{60} = -50.5272818452358 x 60 = − 50.5272818452358 x 61 = 64.1408500107916 x_{61} = 64.1408500107916 x 61 = 64.1408500107916 x 62 = − 53.6688744988256 x_{62} = -53.6688744988256 x 62 = − 53.6688744988256 x 63 = 78.2780169519457 x_{63} = 78.2780169519457 x 63 = 78.2780169519457 x 64 = 56.2868683768171 x_{64} = 56.2868683768171 x 64 = 56.2868683768171 x 65 = − 85.0848010347236 x_{65} = -85.0848010347236 x 65 = − 85.0848010347236 x 66 = 73.565627971561 x_{66} = 73.565627971561 x 66 = 73.565627971561 x 67 = − 22.2529479629277 x_{67} = -22.2529479629277 x 67 = − 22.2529479629277 x 68 = − 0.261799387799149 x_{68} = -0.261799387799149 x 68 = − 0.261799387799149 x 69 = 34.2957198016886 x_{69} = 34.2957198016886 x 69 = 34.2957198016886 x 70 = 29.5833308213039 x_{70} = 29.5833308213039 x 70 = 29.5833308213039 x 71 = 79.8488132787406 x_{71} = 79.8488132787406 x 71 = 79.8488132787406 x 72 = 50.0036830696375 x_{72} = 50.0036830696375 x 72 = 50.0036830696375 x 73 = 70.4240353179712 x_{73} = 70.4240353179712 x 73 = 70.4240353179712 x 74 = 24.8709418409192 x_{74} = 24.8709418409192 x 74 = 24.8709418409192 x 75 = 59.4284610304069 x_{75} = 59.4284610304069 x 75 = 59.4284610304069 x 76 = − 55.2396708256205 x_{76} = -55.2396708256205 x 76 = − 55.2396708256205 x 77 = 98.6983692002793 x_{77} = 98.6983692002793 x 77 = 98.6983692002793 x 78 = − 77.2308194007491 x_{78} = -77.2308194007491 x 78 = − 77.2308194007491 x 79 = − 58.3812634792103 x_{79} = -58.3812634792103 x 79 = − 58.3812634792103 x 80 = 7.59218224617533 x_{80} = 7.59218224617533 x 80 = 7.59218224617533 x 81 = 84.5612022591253 x_{81} = 84.5612022591253 x 81 = 84.5612022591253 x 82 = 51.5744793964324 x_{82} = 51.5744793964324 x 82 = 51.5744793964324 x 83 = − 9.68657734856853 x_{83} = -9.68657734856853 x 83 = − 9.68657734856853 x 84 = 20.1585528605345 x_{84} = 20.1585528605345 x 84 = 20.1585528605345 x 85 = − 89.7971900151083 x_{85} = -89.7971900151083 x 85 = − 89.7971900151083 x 86 = − 37.9609112308767 x_{86} = -37.9609112308767 x 86 = − 37.9609112308767 x 87 = − 75.6600230739542 x_{87} = -75.6600230739542 x 87 = − 75.6600230739542 x 88 = − 99.2219679758776 x_{88} = -99.2219679758776 x 88 = − 99.2219679758776 x 89 = − 8.11578102177363 x_{89} = -8.11578102177363 x 89 = − 8.11578102177363 x 90 = − 36.3901149040818 x_{90} = -36.3901149040818 x 90 = − 36.3901149040818 x 91 = − 45.8148928648512 x_{91} = -45.8148928648512 x 91 = − 45.8148928648512 x 92 = − 88.2263936883134 x_{92} = -88.2263936883134 x 92 = − 88.2263936883134 x 93 = − 80.3724120543389 x_{93} = -80.3724120543389 x 93 = − 80.3724120543389 x 94 = − 69.3768377667746 x_{94} = -69.3768377667746 x 94 = − 69.3768377667746 x 95 = − 61.5228561328001 x_{95} = -61.5228561328001 x 95 = − 61.5228561328001 x 96 = 18.5877565337396 x_{96} = 18.5877565337396 x 96 = 18.5877565337396 x 97 = − 67.8060414399797 x_{97} = -67.8060414399797 x 97 = − 67.8060414399797 x 98 = 21.7293491873294 x_{98} = 21.7293491873294 x 98 = 21.7293491873294 x 99 = 87.7027949127151 x_{99} = 87.7027949127151 x 99 = 87.7027949127151 x 100 = 2.87979326579064 x_{100} = 2.87979326579064 x 100 = 2.87979326579064 x 101 = − 25.3945406165175 x_{101} = -25.3945406165175 x 101 = − 25.3945406165175
Точки пересечения с осью координат Y
График пересекает ось Y, когда x равняется 0:cot ( − π 3 + 2 ⋅ 0 ) \cot{\left(- \frac{\pi}{3} + 2 \cdot 0 \right)} cot ( − 3 π + 2 ⋅ 0 ) f ( 0 ) = − 3 3 f{\left(0 \right)} = - \frac{\sqrt{3}}{3} f ( 0 ) = − 3 3 (0, -sqrt(3)/3)
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 − 2 cot 2 ( 2 x − π 3 ) − 2 = 0 - 2 \cot^{2}{\left(2 x - \frac{\pi}{3} \right)} - 2 = 0 − 2 cot 2 ( 2 x − 3 π ) − 2 = 0 Решаем это уравнение
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 − 8 ( tan 2 ( 2 x + π 6 ) + 1 ) tan ( 2 x + π 6 ) = 0 - 8 \left(\tan^{2}{\left(2 x + \frac{\pi}{6} \right)} + 1\right) \tan{\left(2 x + \frac{\pi}{6} \right)} = 0 − 8 ( tan 2 ( 2 x + 6 π ) + 1 ) tan ( 2 x + 6 π ) = 0 Решаем это уравнение x 1 = − π 12 x_{1} = - \frac{\pi}{12} x 1 = − 12 π Интервалы выпуклости и вогнутости: ( − ∞ , − π 12 ] \left(-\infty, - \frac{\pi}{12}\right] ( − ∞ , − 12 π ] [ − π 12 , ∞ ) \left[- \frac{\pi}{12}, \infty\right) [ − 12 π , ∞ )
Горизонтальные асимптоты
Горизонтальные асимптоты найдём с помощью пределов данной функции при x->+oo и x->-oolim x → − ∞ cot ( 2 x − π 3 ) = ⟨ − ∞ , ∞ ⟩ \lim_{x \to -\infty} \cot{\left(2 x - \frac{\pi}{3} \right)} = \left\langle -\infty, \infty\right\rangle x → − ∞ lim cot ( 2 x − 3 π ) = ⟨ − ∞ , ∞ ⟩ Возьмём предел y = ⟨ − ∞ , ∞ ⟩ y = \left\langle -\infty, \infty\right\rangle y = ⟨ − ∞ , ∞ ⟩ lim x → ∞ cot ( 2 x − π 3 ) = ⟨ − ∞ , ∞ ⟩ \lim_{x \to \infty} \cot{\left(2 x - \frac{\pi}{3} \right)} = \left\langle -\infty, \infty\right\rangle x → ∞ lim cot ( 2 x − 3 π ) = ⟨ − ∞ , ∞ ⟩ Возьмём предел y = ⟨ − ∞ , ∞ ⟩ y = \left\langle -\infty, \infty\right\rangle y = ⟨ − ∞ , ∞ ⟩
Наклонные асимптоты
Наклонную асимптоту можно найти, подсчитав предел функции cot(2*x - pi/3), делённой на x при x->+oo и x ->-oolim x → − ∞ ( cot ( 2 x − π 3 ) x ) = lim x → − ∞ ( cot ( 2 x − π 3 ) x ) \lim_{x \to -\infty}\left(\frac{\cot{\left(2 x - \frac{\pi}{3} \right)}}{x}\right) = \lim_{x \to -\infty}\left(\frac{\cot{\left(2 x - \frac{\pi}{3} \right)}}{x}\right) x → − ∞ lim ( x cot ( 2 x − 3 π ) ) = x → − ∞ lim ( x cot ( 2 x − 3 π ) ) Возьмём предел y = x lim x → − ∞ ( cot ( 2 x − π 3 ) x ) y = x \lim_{x \to -\infty}\left(\frac{\cot{\left(2 x - \frac{\pi}{3} \right)}}{x}\right) y = x x → − ∞ lim ( x cot ( 2 x − 3 π ) ) lim x → ∞ ( cot ( 2 x − π 3 ) x ) = lim x → ∞ ( cot ( 2 x − π 3 ) x ) \lim_{x \to \infty}\left(\frac{\cot{\left(2 x - \frac{\pi}{3} \right)}}{x}\right) = \lim_{x \to \infty}\left(\frac{\cot{\left(2 x - \frac{\pi}{3} \right)}}{x}\right) x → ∞ lim ( x cot ( 2 x − 3 π ) ) = x → ∞ lim ( x cot ( 2 x − 3 π ) ) Возьмём предел y = x lim x → ∞ ( cot ( 2 x − π 3 ) x ) y = x \lim_{x \to \infty}\left(\frac{\cot{\left(2 x - \frac{\pi}{3} \right)}}{x}\right) y = x x → ∞ lim ( x cot ( 2 x − 3 π ) )
Чётность и нечётность функции
Проверим функци чётна или нечётна с помощью соотношений f = f(-x) и f = -f(-x).cot ( 2 x − π 3 ) = − cot ( 2 x + π 3 ) \cot{\left(2 x - \frac{\pi}{3} \right)} = - \cot{\left(2 x + \frac{\pi}{3} \right)} cot ( 2 x − 3 π ) = − cot ( 2 x + 3 π ) cot ( 2 x − π 3 ) = cot ( 2 x + π 3 ) \cot{\left(2 x - \frac{\pi}{3} \right)} = \cot{\left(2 x + \frac{\pi}{3} \right)} cot ( 2 x − 3 π ) = cot ( 2 x + 3 π ) 
