График функции
0 -10 10 20 30 40 50 60 70 80 -1000 1000
Точки пересечения с осью координат X
График функции пересекает ось X при f = 0tan ( 2 x + π 3 ) = 0 \tan{\left(2 x + \frac{\pi}{3} \right)} = 0 tan ( 2 x + 3 π ) = 0 Решаем это уравнение Аналитическое решение x 1 = − π 6 x_{1} = - \frac{\pi}{6} x 1 = − 6 π Численное решение x 1 = 82.7286065445312 x_{1} = 82.7286065445312 x 1 = 82.7286065445312 x 2 = − 16.2315620435473 x_{2} = -16.2315620435473 x 2 = − 16.2315620435473 x 3 = 71.733032256967 x_{3} = 71.733032256967 x 3 = 71.733032256967 x 4 = − 55.5014702134197 x_{4} = -55.5014702134197 x 4 = − 55.5014702134197 x 5 = − 38.2227106186758 x_{5} = -38.2227106186758 x 5 = − 38.2227106186758 x 6 = − 31.9395253114962 x_{6} = -31.9395253114962 x 6 = − 31.9395253114962 x 7 = 87.4409955249159 x_{7} = 87.4409955249159 x 7 = 87.4409955249159 x 8 = 38.7463093942741 x_{8} = 38.7463093942741 x 8 = 38.7463093942741 x 9 = − 13.0899693899575 x_{9} = -13.0899693899575 x 9 = − 13.0899693899575 x 10 = 27.7507351067098 x_{10} = 27.7507351067098 x 10 = 27.7507351067098 x 11 = 84.2994028713261 x_{11} = 84.2994028713261 x 11 = 84.2994028713261 x 12 = 90.5825881785057 x_{12} = 90.5825881785057 x 12 = 90.5825881785057 x 13 = 13.6135681655558 x_{13} = 13.6135681655558 x 13 = 13.6135681655558 x 14 = − 99.4837673636768 x_{14} = -99.4837673636768 x 14 = − 99.4837673636768 x 15 = − 91.6297857297023 x_{15} = -91.6297857297023 x 15 = − 91.6297857297023 x 16 = − 96.342174710087 x_{16} = -96.342174710087 x 16 = − 96.342174710087 x 17 = 26.1799387799149 x_{17} = 26.1799387799149 x 17 = 26.1799387799149 x 18 = − 33.5103216382911 x_{18} = -33.5103216382911 x 18 = − 33.5103216382911 x 19 = − 25.6563400043166 x_{19} = -25.6563400043166 x 19 = − 25.6563400043166 x 20 = − 5.23598775598299 x_{20} = -5.23598775598299 x 20 = − 5.23598775598299 x 21 = 68.5914396033772 x_{21} = 68.5914396033772 x 21 = 68.5914396033772 x 22 = 48.1710873550435 x_{22} = 48.1710873550435 x 22 = 48.1710873550435 x 23 = 74.8746249105567 x_{23} = 74.8746249105567 x 23 = 74.8746249105567 x 24 = − 90.0589894029074 x_{24} = -90.0589894029074 x 24 = − 90.0589894029074 x 25 = − 30.3687289847013 x_{25} = -30.3687289847013 x 25 = − 30.3687289847013 x 26 = − 75.9218224617533 x_{26} = -75.9218224617533 x 26 = − 75.9218224617533 x 27 = − 47.6474885794452 x_{27} = -47.6474885794452 x 27 = − 47.6474885794452 x 28 = − 44.5058959258554 x_{28} = -44.5058959258554 x 28 = − 44.5058959258554 x 29 = − 71.2094334813686 x_{29} = -71.2094334813686 x 29 = − 71.2094334813686 x 30 = 34.0339204138894 x_{30} = 34.0339204138894 x 30 = 34.0339204138894 x 31 = − 88.4881930761125 x_{31} = -88.4881930761125 x 31 = − 88.4881930761125 x 32 = 43.4586983746588 x_{32} = 43.4586983746588 x 32 = 43.4586983746588 x 33 = − 52.3598775598299 x_{33} = -52.3598775598299 x 33 = − 52.3598775598299 x 34 = 30.8923277602996 x_{34} = 30.8923277602996 x 34 = 30.8923277602996 x 35 = 10.471975511966 x_{35} = 10.471975511966 x 35 = 10.471975511966 x 36 = 62.3082542961976 x_{36} = 62.3082542961976 x 36 = 62.3082542961976 x 37 = 35.6047167406843 x_{37} = 35.6047167406843 x 37 = 35.6047167406843 x 38 = − 3.66519142918809 x_{38} = -3.66519142918809 x 38 = − 3.66519142918809 x 39 = − 57.0722665402146 x_{39} = -57.0722665402146 x 39 = − 57.0722665402146 x 40 = 96.8657734856853 x_{40} = 96.8657734856853 x 40 = 96.8657734856853 x 41 = 54.4542726622231 x_{41} = 54.4542726622231 x 41 = 54.4542726622231 x 42 = − 22.5147473507269 x_{42} = -22.5147473507269 x 42 = − 22.5147473507269 x 43 = − 0.523598775598299 x_{43} = -0.523598775598299 x 43 = − 0.523598775598299 x 44 = − 63.3554518473942 x_{44} = -63.3554518473942 x 44 = − 63.3554518473942 x 45 = 12.0427718387609 x_{45} = 12.0427718387609 x 45 = 12.0427718387609 x 46 = − 79.0634151153431 x_{46} = -79.0634151153431 x 46 = − 79.0634151153431 x 47 = − 49.2182849062401 x_{47} = -49.2182849062401 x 47 = − 49.2182849062401 x 48 = 98.4365698124802 x_{48} = 98.4365698124802 x 48 = 98.4365698124802 x 49 = − 39.7935069454707 x_{49} = -39.7935069454707 x 49 = − 39.7935069454707 x 50 = − 74.3510261349584 x_{50} = -74.3510261349584 x 50 = − 74.3510261349584 x 51 = 93.7241808320955 x_{51} = 93.7241808320955 x 51 = 93.7241808320955 x 52 = − 27.2271363311115 x_{52} = -27.2271363311115 x 52 = − 27.2271363311115 x 53 = − 97.9129710368819 x_{53} = -97.9129710368819 x 53 = − 97.9129710368819 x 54 = − 24.0855436775217 x_{54} = -24.0855436775217 x 54 = − 24.0855436775217 x 55 = 16.7551608191456 x_{55} = 16.7551608191456 x 55 = 16.7551608191456 x 56 = 2.61799387799149 x_{56} = 2.61799387799149 x 56 = 2.61799387799149 x 57 = 8.90117918517108 x_{57} = 8.90117918517108 x 57 = 8.90117918517108 x 58 = − 85.3466004225227 x_{58} = -85.3466004225227 x 58 = − 85.3466004225227 x 59 = 63.8790506229925 x_{59} = 63.8790506229925 x 59 = 63.8790506229925 x 60 = − 60.2138591938044 x_{60} = -60.2138591938044 x 60 = − 60.2138591938044 x 61 = 100.007366139275 x_{61} = 100.007366139275 x 61 = 100.007366139275 x 62 = − 8.37758040957278 x_{62} = -8.37758040957278 x 62 = − 8.37758040957278 x 63 = 19.8967534727354 x_{63} = 19.8967534727354 x 63 = 19.8967534727354 x 64 = 85.870199198121 x_{64} = 85.870199198121 x 64 = 85.870199198121 x 65 = − 61.7846555205993 x_{65} = -61.7846555205993 x 65 = − 61.7846555205993 x 66 = 24.60914245312 x_{66} = 24.60914245312 x 66 = 24.60914245312 x 67 = 60.7374579694027 x_{67} = 60.7374579694027 x 67 = 60.7374579694027 x 68 = 41.8879020478639 x_{68} = 41.8879020478639 x 68 = 41.8879020478639 x 69 = − 53.9306738866248 x_{69} = -53.9306738866248 x 69 = − 53.9306738866248 x 70 = − 66.497044500984 x_{70} = -66.497044500984 x 70 = − 66.497044500984 x 71 = 21.4675497995303 x_{71} = 21.4675497995303 x 71 = 21.4675497995303 x 72 = − 46.0766922526503 x_{72} = -46.0766922526503 x 72 = − 46.0766922526503 x 73 = 52.8834763354282 x_{73} = 52.8834763354282 x 73 = 52.8834763354282 x 74 = − 82.2050077689329 x_{74} = -82.2050077689329 x 74 = − 82.2050077689329 x 75 = 56.025068989018 x_{75} = 56.025068989018 x 75 = 56.025068989018 x 76 = 5.75958653158129 x_{76} = 5.75958653158129 x 76 = 5.75958653158129 x 77 = 78.0162175641465 x_{77} = 78.0162175641465 x 77 = 78.0162175641465 x 78 = − 77.4926187885482 x_{78} = -77.4926187885482 x 78 = − 77.4926187885482 x 79 = 70.162235930172 x_{79} = 70.162235930172 x 79 = 70.162235930172 x 80 = − 2.0943951023932 x_{80} = -2.0943951023932 x 80 = − 2.0943951023932 x 81 = 4.18879020478639 x_{81} = 4.18879020478639 x 81 = 4.18879020478639 x 82 = − 35.081117965086 x_{82} = -35.081117965086 x 82 = − 35.081117965086 x 83 = 46.6002910282486 x_{83} = 46.6002910282486 x 83 = 46.6002910282486 x 84 = 76.4454212373516 x_{84} = 76.4454212373516 x 84 = 76.4454212373516 x 85 = − 41.3643032722656 x_{85} = -41.3643032722656 x 85 = − 41.3643032722656 x 86 = 40.317105721069 x_{86} = 40.317105721069 x 86 = 40.317105721069 x 87 = − 69.6386371545737 x_{87} = -69.6386371545737 x 87 = − 69.6386371545737 x 88 = 65.4498469497874 x_{88} = 65.4498469497874 x 88 = 65.4498469497874 x 89 = − 11.5191730631626 x_{89} = -11.5191730631626 x 89 = − 11.5191730631626 x 90 = 32.4631240870945 x_{90} = 32.4631240870945 x 90 = 32.4631240870945 x 91 = − 17.8023583703422 x_{91} = -17.8023583703422 x 91 = − 17.8023583703422 x 92 = 18.3259571459405 x_{92} = 18.3259571459405 x 92 = 18.3259571459405 x 93 = 57.5958653158129 x_{93} = 57.5958653158129 x 93 = 57.5958653158129 x 94 = − 68.0678408277789 x_{94} = -68.0678408277789 x 94 = − 68.0678408277789 x 95 = 49.7418836818384 x_{95} = 49.7418836818384 x 95 = 49.7418836818384 x 96 = − 19.3731546971371 x_{96} = -19.3731546971371 x 96 = − 19.3731546971371 x 97 = − 9.94837673636768 x_{97} = -9.94837673636768 x 97 = − 9.94837673636768 x 98 = − 93.2005820564972 x_{98} = -93.2005820564972 x 98 = − 93.2005820564972 x 99 = 92.1533845053006 x_{99} = 92.1533845053006 x 99 = 92.1533845053006 x 100 = − 83.7758040957278 x_{100} = -83.7758040957278 x 100 = − 83.7758040957278 x 101 = 79.5870138909414 x_{101} = 79.5870138909414 x 101 = 79.5870138909414
Точки пересечения с осью координат Y
График пересекает ось Y, когда x равняется 0:tan ( 2 ⋅ 0 + π 3 ) \tan{\left(2 \cdot 0 + \frac{\pi}{3} \right)} tan ( 2 ⋅ 0 + 3 π ) f ( 0 ) = 3 f{\left(0 \right)} = \sqrt{3} f ( 0 ) = 3 (0, sqrt(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 tan 2 ( 2 x + π 3 ) + 2 = 0 2 \tan^{2}{\left(2 x + \frac{\pi}{3} \right)} + 2 = 0 2 tan 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 3 ) + 1 ) tan ( 2 x + π 3 ) = 0 8 \left(\tan^{2}{\left(2 x + \frac{\pi}{3} \right)} + 1\right) \tan{\left(2 x + \frac{\pi}{3} \right)} = 0 8 ( tan 2 ( 2 x + 3 π ) + 1 ) tan ( 2 x + 3 π ) = 0 Решаем это уравнение x 1 = − π 6 x_{1} = - \frac{\pi}{6} x 1 = − 6 π Интервалы выпуклости и вогнутости: [ − π 6 , ∞ ) \left[- \frac{\pi}{6}, \infty\right) [ − 6 π , ∞ ) ( − ∞ , − π 6 ] \left(-\infty, - \frac{\pi}{6}\right] ( − ∞ , − 6 π ]
Горизонтальные асимптоты
Горизонтальные асимптоты найдём с помощью пределов данной функции при x->+oo и x->-oolim x → − ∞ tan ( 2 x + π 3 ) = ⟨ − ∞ , ∞ ⟩ \lim_{x \to -\infty} \tan{\left(2 x + \frac{\pi}{3} \right)} = \left\langle -\infty, \infty\right\rangle x → − ∞ lim tan ( 2 x + 3 π ) = ⟨ − ∞ , ∞ ⟩ Возьмём предел y = ⟨ − ∞ , ∞ ⟩ y = \left\langle -\infty, \infty\right\rangle y = ⟨ − ∞ , ∞ ⟩ lim x → ∞ tan ( 2 x + π 3 ) = ⟨ − ∞ , ∞ ⟩ \lim_{x \to \infty} \tan{\left(2 x + \frac{\pi}{3} \right)} = \left\langle -\infty, \infty\right\rangle x → ∞ lim tan ( 2 x + 3 π ) = ⟨ − ∞ , ∞ ⟩ Возьмём предел y = ⟨ − ∞ , ∞ ⟩ y = \left\langle -\infty, \infty\right\rangle y = ⟨ − ∞ , ∞ ⟩
Наклонные асимптоты
Наклонную асимптоту можно найти, подсчитав предел функции tan(2*x + pi/3), делённой на x при x->+oo и x ->-oolim x → − ∞ ( tan ( 2 x + π 3 ) x ) = lim x → − ∞ ( tan ( 2 x + π 3 ) x ) \lim_{x \to -\infty}\left(\frac{\tan{\left(2 x + \frac{\pi}{3} \right)}}{x}\right) = \lim_{x \to -\infty}\left(\frac{\tan{\left(2 x + \frac{\pi}{3} \right)}}{x}\right) x → − ∞ lim ( x tan ( 2 x + 3 π ) ) = x → − ∞ lim ( x tan ( 2 x + 3 π ) ) Возьмём предел y = x lim x → − ∞ ( tan ( 2 x + π 3 ) x ) y = x \lim_{x \to -\infty}\left(\frac{\tan{\left(2 x + \frac{\pi}{3} \right)}}{x}\right) y = x x → − ∞ lim ( x tan ( 2 x + 3 π ) ) lim x → ∞ ( tan ( 2 x + π 3 ) x ) = lim x → ∞ ( tan ( 2 x + π 3 ) x ) \lim_{x \to \infty}\left(\frac{\tan{\left(2 x + \frac{\pi}{3} \right)}}{x}\right) = \lim_{x \to \infty}\left(\frac{\tan{\left(2 x + \frac{\pi}{3} \right)}}{x}\right) x → ∞ lim ( x tan ( 2 x + 3 π ) ) = x → ∞ lim ( x tan ( 2 x + 3 π ) ) Возьмём предел y = x lim x → ∞ ( tan ( 2 x + π 3 ) x ) y = x \lim_{x \to \infty}\left(\frac{\tan{\left(2 x + \frac{\pi}{3} \right)}}{x}\right) y = x x → ∞ lim ( x tan ( 2 x + 3 π ) )
Чётность и нечётность функции
Проверим функци чётна или нечётна с помощью соотношений f = f(-x) и f = -f(-x).tan ( 2 x + π 3 ) = cot ( 2 x + π 6 ) \tan{\left(2 x + \frac{\pi}{3} \right)} = \cot{\left(2 x + \frac{\pi}{6} \right)} tan ( 2 x + 3 π ) = cot ( 2 x + 6 π ) tan ( 2 x + π 3 ) = − cot ( 2 x + π 6 ) \tan{\left(2 x + \frac{\pi}{3} \right)} = - \cot{\left(2 x + \frac{\pi}{6} \right)} tan ( 2 x + 3 π ) = − cot ( 2 x + 6 π ) 
