График функции
0 5 -5 10 15 20 25 30 35 -10 -100 100
Точки пересечения с осью координат X
График функции пересекает ось X при f = 0tan ( 4 x ) = 0 \tan{\left(4 x \right)} = 0 tan ( 4 x ) = 0 Решаем это уравнение Аналитическое решение x 1 = 0 x_{1} = 0 x 1 = 0 Численное решение x 1 = 36.1283155162826 x_{1} = 36.1283155162826 x 1 = 36.1283155162826 x 2 = 24.3473430653209 x_{2} = 24.3473430653209 x 2 = 24.3473430653209 x 3 = 38.484510006475 x_{3} = 38.484510006475 x 3 = 38.484510006475 x 4 = 21.9911485751286 x_{4} = 21.9911485751286 x 4 = 21.9911485751286 x 5 = − 93.4623814442964 x_{5} = -93.4623814442964 x 5 = − 93.4623814442964 x 6 = 68.329640215578 x_{6} = 68.329640215578 x 6 = 68.329640215578 x 7 = − 10.2101761241668 x_{7} = -10.2101761241668 x 7 = − 10.2101761241668 x 8 = − 11.7809724509617 x_{8} = -11.7809724509617 x 8 = − 11.7809724509617 x 9 = − 54.1924732744239 x_{9} = -54.1924732744239 x 9 = − 54.1924732744239 x 10 = − 87.9645943005142 x_{10} = -87.9645943005142 x 10 = − 87.9645943005142 x 11 = 91.8915851175014 x_{11} = 91.8915851175014 x 11 = 91.8915851175014 x 12 = − 3.92699081698724 x_{12} = -3.92699081698724 x 12 = − 3.92699081698724 x 13 = − 63.6172512351933 x_{13} = -63.6172512351933 x 13 = − 63.6172512351933 x 14 = − 14.1371669411541 x_{14} = -14.1371669411541 x 14 = − 14.1371669411541 x 15 = − 18.0641577581413 x_{15} = -18.0641577581413 x 15 = − 18.0641577581413 x 16 = 16.4933614313464 x_{16} = 16.4933614313464 x 16 = 16.4933614313464 x 17 = − 43.9822971502571 x_{17} = -43.9822971502571 x 17 = − 43.9822971502571 x 18 = 42.4115008234622 x_{18} = 42.4115008234622 x 18 = 42.4115008234622 x 19 = − 5.49778714378214 x_{19} = -5.49778714378214 x 19 = − 5.49778714378214 x 20 = − 95.8185759344887 x_{20} = -95.8185759344887 x 20 = − 95.8185759344887 x 21 = − 69.9004365423729 x_{21} = -69.9004365423729 x 21 = − 69.9004365423729 x 22 = − 1.5707963267949 x_{22} = -1.5707963267949 x 22 = − 1.5707963267949 x 23 = − 33.7721210260903 x_{23} = -33.7721210260903 x 23 = − 33.7721210260903 x 24 = − 71.4712328691678 x_{24} = -71.4712328691678 x 24 = − 71.4712328691678 x 25 = 58.1194640914112 x_{25} = 58.1194640914112 x 25 = 58.1194640914112 x 26 = 6.28318530717959 x_{26} = 6.28318530717959 x 26 = 6.28318530717959 x 27 = − 47.9092879672443 x_{27} = -47.9092879672443 x 27 = − 47.9092879672443 x 28 = 14.1371669411541 x_{28} = 14.1371669411541 x 28 = 14.1371669411541 x 29 = − 49.4800842940392 x_{29} = -49.4800842940392 x 29 = − 49.4800842940392 x 30 = 72.2566310325652 x_{30} = 72.2566310325652 x 30 = 72.2566310325652 x 31 = − 58.1194640914112 x_{31} = -58.1194640914112 x 31 = − 58.1194640914112 x 32 = 3.92699081698724 x_{32} = 3.92699081698724 x 32 = 3.92699081698724 x 33 = 55.7632696012188 x_{33} = 55.7632696012188 x 33 = 55.7632696012188 x 34 = 11.7809724509617 x_{34} = 11.7809724509617 x 34 = 11.7809724509617 x 35 = 86.3937979737193 x_{35} = 86.3937979737193 x 35 = 86.3937979737193 x 36 = − 21.9911485751286 x_{36} = -21.9911485751286 x 36 = − 21.9911485751286 x 37 = − 7.85398163397448 x_{37} = -7.85398163397448 x 37 = − 7.85398163397448 x 38 = 18.0641577581413 x_{38} = 18.0641577581413 x 38 = 18.0641577581413 x 39 = − 77.7544181763474 x_{39} = -77.7544181763474 x 39 = − 77.7544181763474 x 40 = 32.2013246992954 x_{40} = 32.2013246992954 x 40 = 32.2013246992954 x 41 = 40.0553063332699 x_{41} = 40.0553063332699 x 41 = 40.0553063332699 x 42 = 10.2101761241668 x_{42} = 10.2101761241668 x 42 = 10.2101761241668 x 43 = − 25.9181393921158 x_{43} = -25.9181393921158 x 43 = − 25.9181393921158 x 44 = − 91.8915851175014 x_{44} = -91.8915851175014 x 44 = − 91.8915851175014 x 45 = 95.8185759344887 x_{45} = 95.8185759344887 x 45 = 95.8185759344887 x 46 = − 41.6261026600648 x_{46} = -41.6261026600648 x 46 = − 41.6261026600648 x 47 = − 80.1106126665397 x_{47} = -80.1106126665397 x 47 = − 80.1106126665397 x 48 = − 55.7632696012188 x_{48} = -55.7632696012188 x 48 = − 55.7632696012188 x 49 = 60.4756585816035 x_{49} = 60.4756585816035 x 49 = 60.4756585816035 x 50 = − 45.553093477052 x_{50} = -45.553093477052 x 50 = − 45.553093477052 x 51 = − 32.2013246992954 x_{51} = -32.2013246992954 x 51 = − 32.2013246992954 x 52 = − 89.5353906273091 x_{52} = -89.5353906273091 x 52 = − 89.5353906273091 x 53 = 51.8362787842316 x_{53} = 51.8362787842316 x 53 = 51.8362787842316 x 54 = 2.35619449019234 x_{54} = 2.35619449019234 x 54 = 2.35619449019234 x 55 = − 85.6083998103219 x_{55} = -85.6083998103219 x 55 = − 85.6083998103219 x 56 = 0 x_{56} = 0 x 56 = 0 x 57 = 73.8274273593601 x_{57} = 73.8274273593601 x 57 = 73.8274273593601 x 58 = 29.845130209103 x_{58} = 29.845130209103 x 58 = 29.845130209103 x 59 = 90.3207887907066 x_{59} = 90.3207887907066 x 59 = 90.3207887907066 x 60 = − 37.6991118430775 x_{60} = -37.6991118430775 x 60 = − 37.6991118430775 x 61 = 46.3384916404494 x_{61} = 46.3384916404494 x 61 = 46.3384916404494 x 62 = 65.9734457253857 x_{62} = 65.9734457253857 x 62 = 65.9734457253857 x 63 = − 27.4889357189107 x_{63} = -27.4889357189107 x 63 = − 27.4889357189107 x 64 = 82.4668071567321 x_{64} = 82.4668071567321 x 64 = 82.4668071567321 x 65 = − 40.0553063332699 x_{65} = -40.0553063332699 x 65 = − 40.0553063332699 x 66 = − 81.6814089933346 x_{66} = -81.6814089933346 x 66 = − 81.6814089933346 x 67 = 28.2743338823081 x_{67} = 28.2743338823081 x 67 = 28.2743338823081 x 68 = − 29.845130209103 x_{68} = -29.845130209103 x 68 = − 29.845130209103 x 69 = − 23.5619449019235 x_{69} = -23.5619449019235 x 69 = − 23.5619449019235 x 70 = 100.530964914873 x_{70} = 100.530964914873 x 70 = 100.530964914873 x 71 = − 67.5442420521806 x_{71} = -67.5442420521806 x 71 = − 67.5442420521806 x 72 = 84.037603483527 x_{72} = 84.037603483527 x 72 = 84.037603483527 x 73 = 62.0464549083984 x_{73} = 62.0464549083984 x 73 = 62.0464549083984 x 74 = 69.9004365423729 x_{74} = 69.9004365423729 x 74 = 69.9004365423729 x 75 = 98.174770424681 x_{75} = 98.174770424681 x 75 = 98.174770424681 x 76 = − 76.1836218495525 x_{76} = -76.1836218495525 x 76 = − 76.1836218495525 x 77 = 87.9645943005142 x_{77} = 87.9645943005142 x 77 = 87.9645943005142 x 78 = 64.4026493985908 x_{78} = 64.4026493985908 x 78 = 64.4026493985908 x 79 = 43.9822971502571 x_{79} = 43.9822971502571 x 79 = 43.9822971502571 x 80 = − 84.037603483527 x_{80} = -84.037603483527 x 80 = − 84.037603483527 x 81 = − 73.8274273593601 x_{81} = -73.8274273593601 x 81 = − 73.8274273593601 x 82 = 94.2477796076938 x_{82} = 94.2477796076938 x 82 = 94.2477796076938 x 83 = − 59.6902604182061 x_{83} = -59.6902604182061 x 83 = − 59.6902604182061 x 84 = − 51.8362787842316 x_{84} = -51.8362787842316 x 84 = − 51.8362787842316 x 85 = − 15.707963267949 x_{85} = -15.707963267949 x 85 = − 15.707963267949 x 86 = 80.1106126665397 x_{86} = 80.1106126665397 x 86 = 80.1106126665397 x 87 = 54.1924732744239 x_{87} = 54.1924732744239 x 87 = 54.1924732744239 x 88 = 25.9181393921158 x_{88} = 25.9181393921158 x 88 = 25.9181393921158 x 89 = 7.85398163397448 x_{89} = 7.85398163397448 x 89 = 7.85398163397448 x 90 = 78.5398163397448 x_{90} = 78.5398163397448 x 90 = 78.5398163397448 x 91 = 76.1836218495525 x_{91} = 76.1836218495525 x 91 = 76.1836218495525 x 92 = − 98.174770424681 x_{92} = -98.174770424681 x 92 = − 98.174770424681 x 93 = 33.7721210260903 x_{93} = 33.7721210260903 x 93 = 33.7721210260903 x 94 = − 65.9734457253857 x_{94} = -65.9734457253857 x 94 = − 65.9734457253857 x 95 = 50.2654824574367 x_{95} = 50.2654824574367 x 95 = 50.2654824574367 x 96 = 20.4203522483337 x_{96} = 20.4203522483337 x 96 = 20.4203522483337 x 97 = − 36.1283155162826 x_{97} = -36.1283155162826 x 97 = − 36.1283155162826 x 98 = − 99.7455667514759 x_{98} = -99.7455667514759 x 98 = − 99.7455667514759 x 99 = − 19.6349540849362 x_{99} = -19.6349540849362 x 99 = − 19.6349540849362 x 100 = − 62.0464549083984 x_{100} = -62.0464549083984 x 100 = − 62.0464549083984 x 101 = 47.9092879672443 x_{101} = 47.9092879672443 x 101 = 47.9092879672443
Точки пересечения с осью координат Y
График пересекает ось Y, когда x равняется 0:tan ( 4 ⋅ 0 ) \tan{\left(4 \cdot 0 \right)} tan ( 4 ⋅ 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 4 tan 2 ( 4 x ) + 4 = 0 4 \tan^{2}{\left(4 x \right)} + 4 = 0 4 tan 2 ( 4 x ) + 4 = 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 32 ( tan 2 ( 4 x ) + 1 ) tan ( 4 x ) = 0 32 \left(\tan^{2}{\left(4 x \right)} + 1\right) \tan{\left(4 x \right)} = 0 32 ( tan 2 ( 4 x ) + 1 ) tan ( 4 x ) = 0 Решаем это уравнение x 1 = 0 x_{1} = 0 x 1 = 0 Интервалы выпуклости и вогнутости: [ 0 , ∞ ) \left[0, \infty\right) [ 0 , ∞ ) ( − ∞ , 0 ] \left(-\infty, 0\right] ( − ∞ , 0 ]
Горизонтальные асимптоты
Горизонтальные асимптоты найдём с помощью пределов данной функции при x->+oo и x->-oolim x → − ∞ tan ( 4 x ) = ⟨ − ∞ , ∞ ⟩ \lim_{x \to -\infty} \tan{\left(4 x \right)} = \left\langle -\infty, \infty\right\rangle x → − ∞ lim tan ( 4 x ) = ⟨ − ∞ , ∞ ⟩ Возьмём предел y = ⟨ − ∞ , ∞ ⟩ y = \left\langle -\infty, \infty\right\rangle y = ⟨ − ∞ , ∞ ⟩ lim x → ∞ tan ( 4 x ) = ⟨ − ∞ , ∞ ⟩ \lim_{x \to \infty} \tan{\left(4 x \right)} = \left\langle -\infty, \infty\right\rangle x → ∞ lim tan ( 4 x ) = ⟨ − ∞ , ∞ ⟩ Возьмём предел y = ⟨ − ∞ , ∞ ⟩ y = \left\langle -\infty, \infty\right\rangle y = ⟨ − ∞ , ∞ ⟩
Наклонные асимптоты
Наклонную асимптоту можно найти, подсчитав предел функции tan(4*x), делённой на x при x->+oo и x ->-oolim x → − ∞ ( tan ( 4 x ) x ) = lim x → − ∞ ( tan ( 4 x ) x ) \lim_{x \to -\infty}\left(\frac{\tan{\left(4 x \right)}}{x}\right) = \lim_{x \to -\infty}\left(\frac{\tan{\left(4 x \right)}}{x}\right) x → − ∞ lim ( x tan ( 4 x ) ) = x → − ∞ lim ( x tan ( 4 x ) ) Возьмём предел y = x lim x → − ∞ ( tan ( 4 x ) x ) y = x \lim_{x \to -\infty}\left(\frac{\tan{\left(4 x \right)}}{x}\right) y = x x → − ∞ lim ( x tan ( 4 x ) ) lim x → ∞ ( tan ( 4 x ) x ) = lim x → ∞ ( tan ( 4 x ) x ) \lim_{x \to \infty}\left(\frac{\tan{\left(4 x \right)}}{x}\right) = \lim_{x \to \infty}\left(\frac{\tan{\left(4 x \right)}}{x}\right) x → ∞ lim ( x tan ( 4 x ) ) = x → ∞ lim ( x tan ( 4 x ) ) Возьмём предел y = x lim x → ∞ ( tan ( 4 x ) x ) y = x \lim_{x \to \infty}\left(\frac{\tan{\left(4 x \right)}}{x}\right) y = x x → ∞ lim ( x tan ( 4 x ) )
Чётность и нечётность функции
Проверим функци чётна или нечётна с помощью соотношений f = f(-x) и f = -f(-x).tan ( 4 x ) = − tan ( 4 x ) \tan{\left(4 x \right)} = - \tan{\left(4 x \right)} tan ( 4 x ) = − tan ( 4 x ) tan ( 4 x ) = tan ( 4 x ) \tan{\left(4 x \right)} = \tan{\left(4 x \right)} tan ( 4 x ) = tan ( 4 x ) 
