График функции
0 5 -50 -45 -40 -35 -30 -25 -20 -15 -10 -5 10 2 -2
Точки пересечения с осью координат X
График функции пересекает ось X при f = 0sin ( 4 x ) = 0 \sin{\left(4 x \right)} = 0 sin ( 4 x ) = 0 Решаем это уравнение Аналитическое решение x 1 = 0 x_{1} = 0 x 1 = 0 x 2 = π 4 x_{2} = \frac{\pi}{4} x 2 = 4 π Численное решение x 1 = − 47.9092879672443 x_{1} = -47.9092879672443 x 1 = − 47.9092879672443 x 2 = − 45.553093477052 x_{2} = -45.553093477052 x 2 = − 45.553093477052 x 3 = − 54.1924732744239 x_{3} = -54.1924732744239 x 3 = − 54.1924732744239 x 4 = − 99.7455667514759 x_{4} = -99.7455667514759 x 4 = − 99.7455667514759 x 5 = − 11.7809724509617 x_{5} = -11.7809724509617 x 5 = − 11.7809724509617 x 6 = − 85.6083998103219 x_{6} = -85.6083998103219 x 6 = − 85.6083998103219 x 7 = 58.1194640914112 x_{7} = 58.1194640914112 x 7 = 58.1194640914112 x 8 = 29.845130209103 x_{8} = 29.845130209103 x 8 = 29.845130209103 x 9 = − 40.0553063332699 x_{9} = -40.0553063332699 x 9 = − 40.0553063332699 x 10 = 21.9911485751286 x_{10} = 21.9911485751286 x 10 = 21.9911485751286 x 11 = 40.0553063332699 x_{11} = 40.0553063332699 x 11 = 40.0553063332699 x 12 = − 25.9181393921158 x_{12} = -25.9181393921158 x 12 = − 25.9181393921158 x 13 = − 21.9911485751286 x_{13} = -21.9911485751286 x 13 = − 21.9911485751286 x 14 = − 84.037603483527 x_{14} = -84.037603483527 x 14 = − 84.037603483527 x 15 = − 37.6991118430775 x_{15} = -37.6991118430775 x 15 = − 37.6991118430775 x 16 = 50.2654824574367 x_{16} = 50.2654824574367 x 16 = 50.2654824574367 x 17 = − 3.92699081698724 x_{17} = -3.92699081698724 x 17 = − 3.92699081698724 x 18 = − 58.1194640914112 x_{18} = -58.1194640914112 x 18 = − 58.1194640914112 x 19 = 51.8362787842316 x_{19} = 51.8362787842316 x 19 = 51.8362787842316 x 20 = 76.1836218495525 x_{20} = 76.1836218495525 x 20 = 76.1836218495525 x 21 = 181.426975744811 x_{21} = 181.426975744811 x 21 = 181.426975744811 x 22 = − 109.170344712245 x_{22} = -109.170344712245 x 22 = − 109.170344712245 x 23 = 84.037603483527 x_{23} = 84.037603483527 x 23 = 84.037603483527 x 24 = 72.2566310325652 x_{24} = 72.2566310325652 x 24 = 72.2566310325652 x 25 = − 63.6172512351933 x_{25} = -63.6172512351933 x 25 = − 63.6172512351933 x 26 = 0 x_{26} = 0 x 26 = 0 x 27 = 68.329640215578 x_{27} = 68.329640215578 x 27 = 68.329640215578 x 28 = − 10.9955742875643 x_{28} = -10.9955742875643 x 28 = − 10.9955742875643 x 29 = 62.0464549083984 x_{29} = 62.0464549083984 x 29 = 62.0464549083984 x 30 = − 32.2013246992954 x_{30} = -32.2013246992954 x 30 = − 32.2013246992954 x 31 = 83.2522053201295 x_{31} = 83.2522053201295 x 31 = 83.2522053201295 x 32 = 73.8274273593601 x_{32} = 73.8274273593601 x 32 = 73.8274273593601 x 33 = − 18.0641577581413 x_{33} = -18.0641577581413 x 33 = − 18.0641577581413 x 34 = 94.2477796076938 x_{34} = 94.2477796076938 x 34 = 94.2477796076938 x 35 = 20.4203522483337 x_{35} = 20.4203522483337 x 35 = 20.4203522483337 x 36 = 14.1371669411541 x_{36} = 14.1371669411541 x 36 = 14.1371669411541 x 37 = 54.1924732744239 x_{37} = 54.1924732744239 x 37 = 54.1924732744239 x 38 = − 59.6902604182061 x_{38} = -59.6902604182061 x 38 = − 59.6902604182061 x 39 = − 64.4026493985908 x_{39} = -64.4026493985908 x 39 = − 64.4026493985908 x 40 = 87.9645943005142 x_{40} = 87.9645943005142 x 40 = 87.9645943005142 x 41 = 98.174770424681 x_{41} = 98.174770424681 x 41 = 98.174770424681 x 42 = − 95.8185759344887 x_{42} = -95.8185759344887 x 42 = − 95.8185759344887 x 43 = 65.9734457253857 x_{43} = 65.9734457253857 x 43 = 65.9734457253857 x 44 = − 23.5619449019235 x_{44} = -23.5619449019235 x 44 = − 23.5619449019235 x 45 = 54.9778714378214 x_{45} = 54.9778714378214 x 45 = 54.9778714378214 x 46 = 95.8185759344887 x_{46} = 95.8185759344887 x 46 = 95.8185759344887 x 47 = 90.3207887907066 x_{47} = 90.3207887907066 x 47 = 90.3207887907066 x 48 = 25.9181393921158 x_{48} = 25.9181393921158 x 48 = 25.9181393921158 x 49 = − 29.845130209103 x_{49} = -29.845130209103 x 49 = − 29.845130209103 x 50 = 64.4026493985908 x_{50} = 64.4026493985908 x 50 = 64.4026493985908 x 51 = 42.4115008234622 x_{51} = 42.4115008234622 x 51 = 42.4115008234622 x 52 = 7.85398163397448 x_{52} = 7.85398163397448 x 52 = 7.85398163397448 x 53 = − 62.0464549083984 x_{53} = -62.0464549083984 x 53 = − 62.0464549083984 x 54 = 18.0641577581413 x_{54} = 18.0641577581413 x 54 = 18.0641577581413 x 55 = 91.8915851175014 x_{55} = 91.8915851175014 x 55 = 91.8915851175014 x 56 = 86.3937979737193 x_{56} = 86.3937979737193 x 56 = 86.3937979737193 x 57 = 24.3473430653209 x_{57} = 24.3473430653209 x 57 = 24.3473430653209 x 58 = 3.92699081698724 x_{58} = 3.92699081698724 x 58 = 3.92699081698724 x 59 = − 81.6814089933346 x_{59} = -81.6814089933346 x 59 = − 81.6814089933346 x 60 = 10.9955742875643 x_{60} = 10.9955742875643 x 60 = 10.9955742875643 x 61 = − 36.1283155162826 x_{61} = -36.1283155162826 x 61 = − 36.1283155162826 x 62 = − 14.1371669411541 x_{62} = -14.1371669411541 x 62 = − 14.1371669411541 x 63 = − 77.7544181763474 x_{63} = -77.7544181763474 x 63 = − 77.7544181763474 x 64 = − 43.9822971502571 x_{64} = -43.9822971502571 x 64 = − 43.9822971502571 x 65 = − 80.1106126665397 x_{65} = -80.1106126665397 x 65 = − 80.1106126665397 x 66 = − 51.8362787842316 x_{66} = -51.8362787842316 x 66 = − 51.8362787842316 x 67 = 43.9822971502571 x_{67} = 43.9822971502571 x 67 = 43.9822971502571 x 68 = 47.9092879672443 x_{68} = 47.9092879672443 x 68 = 47.9092879672443 x 69 = − 88.7499924639117 x_{69} = -88.7499924639117 x 69 = − 88.7499924639117 x 70 = − 87.9645943005142 x_{70} = -87.9645943005142 x 70 = − 87.9645943005142 x 71 = − 33.7721210260903 x_{71} = -33.7721210260903 x 71 = − 33.7721210260903 x 72 = − 1.5707963267949 x_{72} = -1.5707963267949 x 72 = − 1.5707963267949 x 73 = 28.2743338823081 x_{73} = 28.2743338823081 x 73 = 28.2743338823081 x 74 = − 91.8915851175014 x_{74} = -91.8915851175014 x 74 = − 91.8915851175014 x 75 = 80.1106126665397 x_{75} = 80.1106126665397 x 75 = 80.1106126665397 x 76 = − 55.7632696012188 x_{76} = -55.7632696012188 x 76 = − 55.7632696012188 x 77 = 36.1283155162826 x_{77} = 36.1283155162826 x 77 = 36.1283155162826 x 78 = 17.2787595947439 x_{78} = 17.2787595947439 x 78 = 17.2787595947439 x 79 = − 19.6349540849362 x_{79} = -19.6349540849362 x 79 = − 19.6349540849362 x 80 = − 98.174770424681 x_{80} = -98.174770424681 x 80 = − 98.174770424681 x 81 = − 76.1836218495525 x_{81} = -76.1836218495525 x 81 = − 76.1836218495525 x 82 = − 73.8274273593601 x_{82} = -73.8274273593601 x 82 = − 73.8274273593601 x 83 = 69.9004365423729 x_{83} = 69.9004365423729 x 83 = 69.9004365423729 x 84 = 2.35619449019234 x_{84} = 2.35619449019234 x 84 = 2.35619449019234 x 85 = − 41.6261026600648 x_{85} = -41.6261026600648 x 85 = − 41.6261026600648 x 86 = 46.3384916404494 x_{86} = 46.3384916404494 x 86 = 46.3384916404494 x 87 = − 15.707963267949 x_{87} = -15.707963267949 x 87 = − 15.707963267949 x 88 = − 69.1150383789755 x_{88} = -69.1150383789755 x 88 = − 69.1150383789755 x 89 = − 69.9004365423729 x_{89} = -69.9004365423729 x 89 = − 69.9004365423729 x 90 = 6.28318530717959 x_{90} = 6.28318530717959 x 90 = 6.28318530717959 x 91 = 32.2013246992954 x_{91} = 32.2013246992954 x 91 = 32.2013246992954 x 92 = − 7.85398163397448 x_{92} = -7.85398163397448 x 92 = − 7.85398163397448 x 93 = 10.2101761241668 x_{93} = 10.2101761241668 x 93 = 10.2101761241668 x 94 = − 65.9734457253857 x_{94} = -65.9734457253857 x 94 = − 65.9734457253857
Точки пересечения с осью координат Y
График пересекает ось Y, когда x равняется 0:sin ( 4 ⋅ 0 ) \sin{\left(4 \cdot 0 \right)} sin ( 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 cos ( 4 x ) = 0 4 \cos{\left(4 x \right)} = 0 4 cos ( 4 x ) = 0 Решаем это уравнение x 1 = π 8 x_{1} = \frac{\pi}{8} x 1 = 8 π x 2 = 3 π 8 x_{2} = \frac{3 \pi}{8} x 2 = 8 3 π pi
(--, 1)
8 3*pi
(----, -1)
8 Интервалы возрастания и убывания функции: x 1 = 3 π 8 x_{1} = \frac{3 \pi}{8} x 1 = 8 3 π x 1 = π 8 x_{1} = \frac{\pi}{8} x 1 = 8 π ( − ∞ , π 8 ] ∪ [ 3 π 8 , ∞ ) \left(-\infty, \frac{\pi}{8}\right] \cup \left[\frac{3 \pi}{8}, \infty\right) ( − ∞ , 8 π ] ∪ [ 8 3 π , ∞ ) [ π 8 , 3 π 8 ] \left[\frac{\pi}{8}, \frac{3 \pi}{8}\right] [ 8 π , 8 3 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 − 16 sin ( 4 x ) = 0 - 16 \sin{\left(4 x \right)} = 0 − 16 sin ( 4 x ) = 0 Решаем это уравнение x 1 = 0 x_{1} = 0 x 1 = 0 x 2 = π 4 x_{2} = \frac{\pi}{4} x 2 = 4 π Интервалы выпуклости и вогнутости: ( − ∞ , 0 ] ∪ [ π 4 , ∞ ) \left(-\infty, 0\right] \cup \left[\frac{\pi}{4}, \infty\right) ( − ∞ , 0 ] ∪ [ 4 π , ∞ ) [ 0 , π 4 ] \left[0, \frac{\pi}{4}\right] [ 0 , 4 π ]
Горизонтальные асимптоты
Горизонтальные асимптоты найдём с помощью пределов данной функции при x->+oo и x->-oolim x → − ∞ sin ( 4 x ) = ⟨ − 1 , 1 ⟩ \lim_{x \to -\infty} \sin{\left(4 x \right)} = \left\langle -1, 1\right\rangle x → − ∞ lim sin ( 4 x ) = ⟨ − 1 , 1 ⟩ Возьмём предел y = ⟨ − 1 , 1 ⟩ y = \left\langle -1, 1\right\rangle y = ⟨ − 1 , 1 ⟩ lim x → ∞ sin ( 4 x ) = ⟨ − 1 , 1 ⟩ \lim_{x \to \infty} \sin{\left(4 x \right)} = \left\langle -1, 1\right\rangle x → ∞ lim sin ( 4 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 sin(4*x), делённой на x при x->+oo и x ->-oolim x → − ∞ ( sin ( 4 x ) x ) = 0 \lim_{x \to -\infty}\left(\frac{\sin{\left(4 x \right)}}{x}\right) = 0 x → − ∞ lim ( x sin ( 4 x ) ) = 0 Возьмём предел lim x → ∞ ( sin ( 4 x ) x ) = 0 \lim_{x \to \infty}\left(\frac{\sin{\left(4 x \right)}}{x}\right) = 0 x → ∞ lim ( x sin ( 4 x ) ) = 0 Возьмём предел
Чётность и нечётность функции
Проверим функци чётна или нечётна с помощью соотношений f = f(-x) и f = -f(-x).sin ( 4 x ) = − sin ( 4 x ) \sin{\left(4 x \right)} = - \sin{\left(4 x \right)} sin ( 4 x ) = − sin ( 4 x ) sin ( 4 x ) = sin ( 4 x ) \sin{\left(4 x \right)} = \sin{\left(4 x \right)} sin ( 4 x ) = sin ( 4 x ) 
