График функции
0 2 4 6 8 -8 -6 -4 -2 -10 10 -100 100
Точки пересечения с осью координат X
График функции пересекает ось X при f = 0tan ( x + π 4 ) = 0 \tan{\left (x + \frac{\pi}{4} \right )} = 0 tan ( x + 4 π ) = 0 Решаем это уравнение Аналитическое решение x 1 = − π 4 x_{1} = - \frac{\pi}{4} x 1 = − 4 π Численное решение x 1 = 99.7455667515 x_{1} = 99.7455667515 x 1 = 99.7455667515 x 2 = 21.2057504117 x_{2} = 21.2057504117 x 2 = 21.2057504117 x 3 = 90.3207887907 x_{3} = 90.3207887907 x 3 = 90.3207887907 x 4 = − 69.9004365424 x_{4} = -69.9004365424 x 4 = − 69.9004365424 x 5 = − 38.4845100065 x_{5} = -38.4845100065 x 5 = − 38.4845100065 x 6 = − 73.042029196 x_{6} = -73.042029196 x 6 = − 73.042029196 x 7 = 96.6039740979 x_{7} = 96.6039740979 x 7 = 96.6039740979 x 8 = 62.0464549084 x_{8} = 62.0464549084 x 8 = 62.0464549084 x 9 = 74.6128255228 x_{9} = 74.6128255228 x 9 = 74.6128255228 x 10 = − 63.6172512352 x_{10} = -63.6172512352 x 10 = − 63.6172512352 x 11 = 77.7544181763 x_{11} = 77.7544181763 x 11 = 77.7544181763 x 12 = − 76.1836218496 x_{12} = -76.1836218496 x 12 = − 76.1836218496 x 13 = − 22.7765467385 x_{13} = -22.7765467385 x 13 = − 22.7765467385 x 14 = 18.0641577581 x_{14} = 18.0641577581 x 14 = 18.0641577581 x 15 = 33.7721210261 x_{15} = 33.7721210261 x 15 = 33.7721210261 x 16 = 49.480084294 x_{16} = 49.480084294 x 16 = 49.480084294 x 17 = 36.9137136797 x_{17} = 36.9137136797 x 17 = 36.9137136797 x 18 = 58.9048622548 x_{18} = 58.9048622548 x 18 = 58.9048622548 x 19 = − 13.3517687778 x_{19} = -13.3517687778 x 19 = − 13.3517687778 x 20 = − 88.7499924639 x_{20} = -88.7499924639 x 20 = − 88.7499924639 x 21 = − 16.4933614313 x_{21} = -16.4933614313 x 21 = − 16.4933614313 x 22 = − 85.6083998103 x_{22} = -85.6083998103 x 22 = − 85.6083998103 x 23 = − 82.4668071567 x_{23} = -82.4668071567 x 23 = − 82.4668071567 x 24 = − 47.9092879672 x_{24} = -47.9092879672 x 24 = − 47.9092879672 x 25 = 71.4712328692 x_{25} = 71.4712328692 x 25 = 71.4712328692 x 26 = − 10.2101761242 x_{26} = -10.2101761242 x 26 = − 10.2101761242 x 27 = 80.8960108299 x_{27} = 80.8960108299 x 27 = 80.8960108299 x 28 = − 0.785398163397 x_{28} = -0.785398163397 x 28 = − 0.785398163397 x 29 = 24.3473430653 x_{29} = 24.3473430653 x 29 = 24.3473430653 x 30 = − 54.1924732744 x_{30} = -54.1924732744 x 30 = − 54.1924732744 x 31 = 93.4623814443 x_{31} = 93.4623814443 x 31 = 93.4623814443 x 32 = − 29.0597320457 x_{32} = -29.0597320457 x 32 = − 29.0597320457 x 33 = 43.1968989869 x_{33} = 43.1968989869 x 33 = 43.1968989869 x 34 = 14.9225651046 x_{34} = 14.9225651046 x 34 = 14.9225651046 x 35 = − 98.1747704247 x_{35} = -98.1747704247 x 35 = − 98.1747704247 x 36 = − 95.0331777711 x_{36} = -95.0331777711 x 36 = − 95.0331777711 x 37 = 40.0553063333 x_{37} = 40.0553063333 x 37 = 40.0553063333 x 38 = 65.188047562 x_{38} = 65.188047562 x 38 = 65.188047562 x 39 = − 66.7588438888 x_{39} = -66.7588438888 x 39 = − 66.7588438888 x 40 = 11.780972451 x_{40} = 11.780972451 x 40 = 11.780972451 x 41 = − 101.316363078 x_{41} = -101.316363078 x 41 = − 101.316363078 x 42 = 68.3296402156 x_{42} = 68.3296402156 x 42 = 68.3296402156 x 43 = − 25.9181393921 x_{43} = -25.9181393921 x 43 = − 25.9181393921 x 44 = 52.6216769476 x_{44} = 52.6216769476 x 44 = 52.6216769476 x 45 = − 19.6349540849 x_{45} = -19.6349540849 x 45 = − 19.6349540849 x 46 = − 32.2013246993 x_{46} = -32.2013246993 x 46 = − 32.2013246993 x 47 = 5.49778714378 x_{47} = 5.49778714378 x 47 = 5.49778714378 x 48 = − 79.3252145031 x_{48} = -79.3252145031 x 48 = − 79.3252145031 x 49 = − 3.92699081699 x_{49} = -3.92699081699 x 49 = − 3.92699081699 x 50 = − 44.7676953137 x_{50} = -44.7676953137 x 50 = − 44.7676953137 x 51 = 30.6305283725 x_{51} = 30.6305283725 x 51 = 30.6305283725 x 52 = − 60.4756585816 x_{52} = -60.4756585816 x 52 = − 60.4756585816 x 53 = 87.1791961371 x_{53} = 87.1791961371 x 53 = 87.1791961371 x 54 = − 57.334065928 x_{54} = -57.334065928 x 54 = − 57.334065928 x 55 = − 35.3429173529 x_{55} = -35.3429173529 x 55 = − 35.3429173529 x 56 = − 41.6261026601 x_{56} = -41.6261026601 x 56 = − 41.6261026601 x 57 = 55.7632696012 x_{57} = 55.7632696012 x 57 = 55.7632696012 x 58 = 84.0376034835 x_{58} = 84.0376034835 x 58 = 84.0376034835 x 59 = 46.3384916404 x_{59} = 46.3384916404 x 59 = 46.3384916404 x 60 = − 51.0508806208 x_{60} = -51.0508806208 x 60 = − 51.0508806208 x 61 = 27.4889357189 x_{61} = 27.4889357189 x 61 = 27.4889357189 x 62 = 2.35619449019 x_{62} = 2.35619449019 x 62 = 2.35619449019 x 63 = 8.63937979737 x_{63} = 8.63937979737 x 63 = 8.63937979737 x 64 = − 7.06858347058 x_{64} = -7.06858347058 x 64 = − 7.06858347058 x 65 = − 91.8915851175 x_{65} = -91.8915851175 x 65 = − 91.8915851175
Точки пересечения с осью координат Y
График пересекает ось Y, когда x равняется 0:tan ( π 4 ) \tan{\left (\frac{\pi}{4} \right )} tan ( 4 π ) 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 tan 2 ( x + π 4 ) + 1 = 0 \tan^{2}{\left (x + \frac{\pi}{4} \right )} + 1 = 0 tan 2 ( x + 4 π ) + 1 = 0 Решаем это уравнение
Горизонтальные асимптоты
Горизонтальные асимптоты найдём с помощью пределов данной функции при x->+oo и x->-oolim x → − ∞ tan ( x + π 4 ) = ⟨ − ∞ , ∞ ⟩ \lim_{x \to -\infty} \tan{\left (x + \frac{\pi}{4} \right )} = \langle -\infty, \infty\rangle x → − ∞ lim tan ( x + 4 π ) = ⟨ − ∞ , ∞ ⟩ Возьмём предел y = ⟨ − ∞ , ∞ ⟩ y = \langle -\infty, \infty\rangle y = ⟨ − ∞ , ∞ ⟩ lim x → ∞ tan ( x + π 4 ) = ⟨ − ∞ , ∞ ⟩ \lim_{x \to \infty} \tan{\left (x + \frac{\pi}{4} \right )} = \langle -\infty, \infty\rangle x → ∞ lim tan ( x + 4 π ) = ⟨ − ∞ , ∞ ⟩ Возьмём предел y = ⟨ − ∞ , ∞ ⟩ y = \langle -\infty, \infty\rangle y = ⟨ − ∞ , ∞ ⟩
Наклонные асимптоты
Наклонную асимптоту можно найти, подсчитав предел функции tan(x + pi/4), делённой на x при x->+oo и x ->-ooTrue Возьмём предел y = x lim x → − ∞ ( 1 x tan ( x + π 4 ) ) y = x \lim_{x \to -\infty}\left(\frac{1}{x} \tan{\left (x + \frac{\pi}{4} \right )}\right) y = x x → − ∞ lim ( x 1 tan ( x + 4 π ) ) True Возьмём предел y = x lim x → ∞ ( 1 x tan ( x + π 4 ) ) y = x \lim_{x \to \infty}\left(\frac{1}{x} \tan{\left (x + \frac{\pi}{4} \right )}\right) y = x x → ∞ lim ( x 1 tan ( x + 4 π ) )
Чётность и нечётность функции
Проверим функци чётна или нечётна с помощью соотношений f = f(-x) и f = -f(-x).tan ( x + π 4 ) = cot ( x + π 4 ) \tan{\left (x + \frac{\pi}{4} \right )} = \cot{\left (x + \frac{\pi}{4} \right )} tan ( x + 4 π ) = cot ( x + 4 π ) tan ( x + π 4 ) = − cot ( x + π 4 ) \tan{\left (x + \frac{\pi}{4} \right )} = - \cot{\left (x + \frac{\pi}{4} \right )} tan ( x + 4 π ) = − cot ( x + 4 π ) 