График функции
0 -2000 -1500 -1000 -500 500 1000 1500 2000 5 -5
Точки пересечения с осью координат X
График функции пересекает ось X при f = 02 sin ( 4 x ) = 0 2 \sin{\left (4 x \right )} = 0 2 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 = − 77.7544181763 x_{1} = -77.7544181763 x 1 = − 77.7544181763 x 2 = − 95.8185759345 x_{2} = -95.8185759345 x 2 = − 95.8185759345 x 3 = 90.3207887907 x_{3} = 90.3207887907 x 3 = 90.3207887907 x 4 = 36.1283155163 x_{4} = 36.1283155163 x 4 = 36.1283155163 x 5 = − 47.9092879672 x_{5} = -47.9092879672 x 5 = − 47.9092879672 x 6 = 80.1106126665 x_{6} = 80.1106126665 x 6 = 80.1106126665 x 7 = 10.9955742876 x_{7} = 10.9955742876 x 7 = 10.9955742876 x 8 = 51.8362787842 x_{8} = 51.8362787842 x 8 = 51.8362787842 x 9 = 14.1371669412 x_{9} = 14.1371669412 x 9 = 14.1371669412 x 10 = 69.9004365424 x_{10} = 69.9004365424 x 10 = 69.9004365424 x 11 = − 25.9181393921 x_{11} = -25.9181393921 x 11 = − 25.9181393921 x 12 = − 51.8362787842 x_{12} = -51.8362787842 x 12 = − 51.8362787842 x 13 = − 7.85398163397 x_{13} = -7.85398163397 x 13 = − 7.85398163397 x 14 = 84.0376034835 x_{14} = 84.0376034835 x 14 = 84.0376034835 x 15 = 54.1924732744 x_{15} = 54.1924732744 x 15 = 54.1924732744 x 16 = − 23.5619449019 x_{16} = -23.5619449019 x 16 = − 23.5619449019 x 17 = 29.8451302091 x_{17} = 29.8451302091 x 17 = 29.8451302091 x 18 = 43.9822971503 x_{18} = 43.9822971503 x 18 = 43.9822971503 x 19 = 94.2477796077 x_{19} = 94.2477796077 x 19 = 94.2477796077 x 20 = − 33.7721210261 x_{20} = -33.7721210261 x 20 = − 33.7721210261 x 21 = 10.2101761242 x_{21} = 10.2101761242 x 21 = 10.2101761242 x 22 = − 98.1747704247 x_{22} = -98.1747704247 x 22 = − 98.1747704247 x 23 = 62.0464549084 x_{23} = 62.0464549084 x 23 = 62.0464549084 x 24 = 76.1836218496 x_{24} = 76.1836218496 x 24 = 76.1836218496 x 25 = − 80.1106126665 x_{25} = -80.1106126665 x 25 = − 80.1106126665 x 26 = − 58.1194640914 x_{26} = -58.1194640914 x 26 = − 58.1194640914 x 27 = 87.9645943005 x_{27} = 87.9645943005 x 27 = 87.9645943005 x 28 = − 55.7632696012 x_{28} = -55.7632696012 x 28 = − 55.7632696012 x 29 = − 54.1924732744 x_{29} = -54.1924732744 x 29 = − 54.1924732744 x 30 = 64.4026493986 x_{30} = 64.4026493986 x 30 = 64.4026493986 x 31 = 40.0553063333 x_{31} = 40.0553063333 x 31 = 40.0553063333 x 32 = − 32.2013246993 x_{32} = -32.2013246993 x 32 = − 32.2013246993 x 33 = 42.4115008235 x_{33} = 42.4115008235 x 33 = 42.4115008235 x 34 = − 18.0641577581 x_{34} = -18.0641577581 x 34 = − 18.0641577581 x 35 = − 62.0464549084 x_{35} = -62.0464549084 x 35 = − 62.0464549084 x 36 = − 21.9911485751 x_{36} = -21.9911485751 x 36 = − 21.9911485751 x 37 = − 37.6991118431 x_{37} = -37.6991118431 x 37 = − 37.6991118431 x 38 = 46.3384916404 x_{38} = 46.3384916404 x 38 = 46.3384916404 x 39 = 21.9911485751 x_{39} = 21.9911485751 x 39 = 21.9911485751 x 40 = − 11.780972451 x_{40} = -11.780972451 x 40 = − 11.780972451 x 41 = − 1.57079632679 x_{41} = -1.57079632679 x 41 = − 1.57079632679 x 42 = − 29.8451302091 x_{42} = -29.8451302091 x 42 = − 29.8451302091 x 43 = 0 x_{43} = 0 x 43 = 0 x 44 = − 14.1371669412 x_{44} = -14.1371669412 x 44 = − 14.1371669412 x 45 = 86.3937979737 x_{45} = 86.3937979737 x 45 = 86.3937979737 x 46 = 32.2013246993 x_{46} = 32.2013246993 x 46 = 32.2013246993 x 47 = − 63.6172512352 x_{47} = -63.6172512352 x 47 = − 63.6172512352 x 48 = 50.2654824574 x_{48} = 50.2654824574 x 48 = 50.2654824574 x 49 = − 76.1836218496 x_{49} = -76.1836218496 x 49 = − 76.1836218496 x 50 = 18.0641577581 x_{50} = 18.0641577581 x 50 = 18.0641577581 x 51 = − 59.6902604182 x_{51} = -59.6902604182 x 51 = − 59.6902604182 x 52 = 28.2743338823 x_{52} = 28.2743338823 x 52 = 28.2743338823 x 53 = − 43.9822971503 x_{53} = -43.9822971503 x 53 = − 43.9822971503 x 54 = − 99.7455667515 x_{54} = -99.7455667515 x 54 = − 99.7455667515 x 55 = − 81.6814089933 x_{55} = -81.6814089933 x 55 = − 81.6814089933 x 56 = − 69.9004365424 x_{56} = -69.9004365424 x 56 = − 69.9004365424 x 57 = 3.92699081699 x_{57} = 3.92699081699 x 57 = 3.92699081699 x 58 = 72.2566310326 x_{58} = 72.2566310326 x 58 = 72.2566310326 x 59 = 7.85398163397 x_{59} = 7.85398163397 x 59 = 7.85398163397 x 60 = − 65.9734457254 x_{60} = -65.9734457254 x 60 = − 65.9734457254 x 61 = − 73.8274273594 x_{61} = -73.8274273594 x 61 = − 73.8274273594 x 62 = − 45.5530934771 x_{62} = -45.5530934771 x 62 = − 45.5530934771 x 63 = 98.1747704247 x_{63} = 98.1747704247 x 63 = 98.1747704247 x 64 = − 19.6349540849 x_{64} = -19.6349540849 x 64 = − 19.6349540849 x 65 = − 109.170344712 x_{65} = -109.170344712 x 65 = − 109.170344712 x 66 = 24.3473430653 x_{66} = 24.3473430653 x 66 = 24.3473430653 x 67 = − 84.0376034835 x_{67} = -84.0376034835 x 67 = − 84.0376034835 x 68 = 2.35619449019 x_{68} = 2.35619449019 x 68 = 2.35619449019 x 69 = − 41.6261026601 x_{69} = -41.6261026601 x 69 = − 41.6261026601 x 70 = − 91.8915851175 x_{70} = -91.8915851175 x 70 = − 91.8915851175 x 71 = 20.4203522483 x_{71} = 20.4203522483 x 71 = 20.4203522483 x 72 = − 15.7079632679 x_{72} = -15.7079632679 x 72 = − 15.7079632679 x 73 = 65.9734457254 x_{73} = 65.9734457254 x 73 = 65.9734457254 x 74 = 25.9181393921 x_{74} = 25.9181393921 x 74 = 25.9181393921 x 75 = 17.2787595947 x_{75} = 17.2787595947 x 75 = 17.2787595947 x 76 = 181.426975745 x_{76} = 181.426975745 x 76 = 181.426975745 x 77 = 58.1194640914 x_{77} = 58.1194640914 x 77 = 58.1194640914 x 78 = 95.8185759345 x_{78} = 95.8185759345 x 78 = 95.8185759345 x 79 = − 36.1283155163 x_{79} = -36.1283155163 x 79 = − 36.1283155163 x 80 = 73.8274273594 x_{80} = 73.8274273594 x 80 = 73.8274273594 x 81 = − 40.0553063333 x_{81} = -40.0553063333 x 81 = − 40.0553063333 x 82 = − 85.6083998103 x_{82} = -85.6083998103 x 82 = − 85.6083998103 x 83 = − 69.115038379 x_{83} = -69.115038379 x 83 = − 69.115038379 x 84 = 6.28318530718 x_{84} = 6.28318530718 x 84 = 6.28318530718 x 85 = − 87.9645943005 x_{85} = -87.9645943005 x 85 = − 87.9645943005 x 86 = 83.2522053201 x_{86} = 83.2522053201 x 86 = 83.2522053201 x 87 = 47.9092879672 x_{87} = 47.9092879672 x 87 = 47.9092879672 x 88 = − 3.92699081699 x_{88} = -3.92699081699 x 88 = − 3.92699081699 x 89 = 68.3296402156 x_{89} = 68.3296402156 x 89 = 68.3296402156 x 90 = − 64.4026493986 x_{90} = -64.4026493986 x 90 = − 64.4026493986 x 91 = 54.9778714378 x_{91} = 54.9778714378 x 91 = 54.9778714378 x 92 = − 88.7499924639 x_{92} = -88.7499924639 x 92 = − 88.7499924639 x 93 = 91.8915851175 x_{93} = 91.8915851175 x 93 = 91.8915851175 x 94 = − 10.9955742876 x_{94} = -10.9955742876 x 94 = − 10.9955742876
Точки пересечения с осью координат Y
График пересекает ось Y, когда x равняется 0:2 sin ( 0 ⋅ 4 ) 2 \sin{\left (0 \cdot 4 \right )} 2 sin ( 0 ⋅ 4 ) 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 8 cos ( 4 x ) = 0 8 \cos{\left (4 x \right )} = 0 8 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
(--, 2)
8 3*pi
(----, -2)
8 Интервалы возрастания и убывания функции: x 2 = 3 π 8 x_{2} = \frac{3 \pi}{8} x 2 = 8 3 π x 2 = π 8 x_{2} = \frac{\pi}{8} x 2 = 8 π (-oo, pi/8] U [3*pi/8, oo) [pi/8, 3*pi/8]
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 sin ( 4 x ) = 0 - 32 \sin{\left (4 x \right )} = 0 − 32 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: (-oo, 0] U [pi/4, oo) [0, pi/4]
Горизонтальные асимптоты
Горизонтальные асимптоты найдём с помощью пределов данной функции при x->+oo и x->-oolim x → − ∞ ( 2 sin ( 4 x ) ) = ⟨ − 2 , 2 ⟩ \lim_{x \to -\infty}\left(2 \sin{\left (4 x \right )}\right) = \langle -2, 2\rangle x → − ∞ lim ( 2 sin ( 4 x ) ) = ⟨ − 2 , 2 ⟩ Возьмём предел y = ⟨ − 2 , 2 ⟩ y = \langle -2, 2\rangle y = ⟨ − 2 , 2 ⟩ lim x → ∞ ( 2 sin ( 4 x ) ) = ⟨ − 2 , 2 ⟩ \lim_{x \to \infty}\left(2 \sin{\left (4 x \right )}\right) = \langle -2, 2\rangle x → ∞ lim ( 2 sin ( 4 x ) ) = ⟨ − 2 , 2 ⟩ Возьмём предел y = ⟨ − 2 , 2 ⟩ y = \langle -2, 2\rangle y = ⟨ − 2 , 2 ⟩
Наклонные асимптоты
Наклонную асимптоту можно найти, подсчитав предел функции 2*sin(4*x), делённой на x при x->+oo и x ->-oolim x → − ∞ ( 2 x sin ( 4 x ) ) = 0 \lim_{x \to -\infty}\left(\frac{2}{x} \sin{\left (4 x \right )}\right) = 0 x → − ∞ lim ( x 2 sin ( 4 x ) ) = 0 Возьмём предел lim x → ∞ ( 2 x sin ( 4 x ) ) = 0 \lim_{x \to \infty}\left(\frac{2}{x} \sin{\left (4 x \right )}\right) = 0 x → ∞ lim ( x 2 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).2 sin ( 4 x ) = − 2 sin ( 4 x ) 2 \sin{\left (4 x \right )} = - 2 \sin{\left (4 x \right )} 2 sin ( 4 x ) = − 2 sin ( 4 x ) 2 sin ( 4 x ) = − − 1 ⋅ 2 sin ( 4 x ) 2 \sin{\left (4 x \right )} = - -1 \cdot 2 \sin{\left (4 x \right )} 2 sin ( 4 x ) = − − 1 ⋅ 2 sin ( 4 x ) 