График функции
0 -20000 -15000 -10000 -5000 5000 10000 15000 20000 2 -2
Точки пересечения с осью координат X
График функции пересекает ось X при f = 0sin ( 7 x ) = 0 \sin{\left (7 x \right )} = 0 sin ( 7 x ) = 0 Решаем это уравнение Аналитическое решение x 1 = 0 x_{1} = 0 x 1 = 0 x 2 = π 7 x_{2} = \frac{\pi}{7} x 2 = 7 π Численное решение x 1 = 74.0518268346 x_{1} = 74.0518268346 x 1 = 74.0518268346 x 2 = 30.0695296844 x_{2} = 30.0695296844 x 2 = 30.0695296844 x 3 = 12.1175716638 x_{3} = 12.1175716638 x 3 = 12.1175716638 x 4 = − 71.8078320821 x_{4} = -71.8078320821 x 4 = − 71.8078320821 x 5 = − 79.8862131913 x_{5} = -79.8862131913 x 5 = − 79.8862131913 x 6 = 98.2869701623 x_{6} = 98.2869701623 x 6 = 98.2869701623 x 7 = − 27.8255349318 x_{7} = -27.8255349318 x 7 = − 27.8255349318 x 8 = − 83.9254037459 x_{8} = -83.9254037459 x 8 = − 83.9254037459 x 9 = − 48.0214877049 x_{9} = -48.0214877049 x 9 = − 48.0214877049 x 10 = 26.0303391297 x_{10} = 26.0303391297 x 10 = 26.0303391297 x 11 = 52.0606782595 x_{11} = 52.0606782595 x 11 = 52.0606782595 x 12 = − 45.7774929523 x_{12} = -45.7774929523 x 12 = − 45.7774929523 x 13 = − 96.0429754097 x_{13} = -96.0429754097 x 13 = − 96.0429754097 x 14 = 43.9822971503 x_{14} = 43.9822971503 x 14 = 43.9822971503 x 15 = − 39.9431065956 x_{15} = -39.9431065956 x 15 = − 39.9431065956 x 16 = 61.9342551708 x_{16} = 61.9342551708 x 16 = 61.9342551708 x 17 = − 87.9645943005 x_{17} = -87.9645943005 x 17 = − 87.9645943005 x 18 = 24.2351433277 x_{18} = 24.2351433277 x 18 = 24.2351433277 x 19 = − 4.03919055462 x_{19} = -4.03919055462 x 19 = − 4.03919055462 x 20 = 54.3046730121 x_{20} = 54.3046730121 x 20 = 54.3046730121 x 21 = 87.9645943005 x_{21} = 87.9645943005 x 21 = 87.9645943005 x 22 = − 13.9127674659 x_{22} = -13.9127674659 x 22 = − 13.9127674659 x 23 = − 89.7597901026 x_{23} = -89.7597901026 x 23 = − 89.7597901026 x 24 = 90.2085890531 x_{24} = 90.2085890531 x 24 = 90.2085890531 x 25 = − 32.3135244369 x_{25} = -32.3135244369 x 25 = − 32.3135244369 x 26 = 8.07838110923 x_{26} = 8.07838110923 x 26 = 8.07838110923 x 27 = 56.0998688141 x_{27} = 56.0998688141 x 27 = 56.0998688141 x 28 = − 17.9519580205 x_{28} = -17.9519580205 x 28 = − 17.9519580205 x 29 = − 30.0695296844 x_{29} = -30.0695296844 x 29 = − 30.0695296844 x 30 = − 8.97597901026 x_{30} = -8.97597901026 x 30 = − 8.97597901026 x 31 = 48.0214877049 x_{31} = 48.0214877049 x 31 = 48.0214877049 x 32 = − 21.9911485751 x_{32} = -21.9911485751 x 32 = − 21.9911485751 x 33 = − 37.6991118431 x_{33} = -37.6991118431 x 33 = − 37.6991118431 x 34 = 63.2806520223 x_{34} = 63.2806520223 x 34 = 63.2806520223 x 35 = 21.9911485751 x_{35} = 21.9911485751 x 35 = 21.9911485751 x 36 = 34.108720239 x_{36} = 34.108720239 x 36 = 34.108720239 x 37 = − 92.0037848551 x_{37} = -92.0037848551 x 37 = − 92.0037848551 x 38 = 0 x_{38} = 0 x 38 = 0 x 39 = − 70.01263628 x_{39} = -70.01263628 x 39 = − 70.01263628 x 40 = 68.2174404779 x_{40} = 68.2174404779 x 40 = 68.2174404779 x 41 = 50.2654824574 x_{41} = 50.2654824574 x 41 = 50.2654824574 x 42 = − 59.6902604182 x_{42} = -59.6902604182 x 42 = − 59.6902604182 x 43 = 28.2743338823 x_{43} = 28.2743338823 x 43 = 28.2743338823 x 44 = − 43.9822971503 x_{44} = -43.9822971503 x 44 = − 43.9822971503 x 45 = 76.2958215872 x_{45} = 76.2958215872 x 45 = 76.2958215872 x 46 = − 81.6814089933 x_{46} = -81.6814089933 x 46 = − 81.6814089933 x 47 = − 23.7863443772 x_{47} = -23.7863443772 x 47 = − 23.7863443772 x 48 = 82.1302079438 x_{48} = 82.1302079438 x 48 = 82.1302079438 x 49 = − 75.8470226367 x_{49} = -75.8470226367 x 49 = − 75.8470226367 x 50 = 72.2566310326 x_{50} = 72.2566310326 x 50 = 72.2566310326 x 51 = − 65.9734457254 x_{51} = -65.9734457254 x 51 = − 65.9734457254 x 52 = − 72.7054299831 x_{52} = -72.7054299831 x 52 = − 72.7054299831 x 53 = − 5.83438635667 x_{53} = -5.83438635667 x 53 = − 5.83438635667 x 54 = 60.1390593687 x_{54} = 60.1390593687 x 54 = 60.1390593687 x 55 = 32.3135244369 x_{55} = 32.3135244369 x 55 = 32.3135244369 x 56 = 64.1782499233 x_{56} = 64.1782499233 x 56 = 64.1782499233 x 57 = 17.9519580205 x_{57} = 17.9519580205 x 57 = 17.9519580205 x 58 = 38.1479107936 x_{58} = 38.1479107936 x 58 = 38.1479107936 x 59 = 92.0037848551 x_{59} = 92.0037848551 x 59 = 92.0037848551 x 60 = − 101.877361766 x_{60} = -101.877361766 x 60 = − 101.877361766 x 61 = − 53.8558740615 x_{61} = -53.8558740615 x 61 = − 53.8558740615 x 62 = − 67.7686415274 x_{62} = -67.7686415274 x 62 = − 67.7686415274 x 63 = − 1.79519580205 x_{63} = -1.79519580205 x 63 = − 1.79519580205 x 64 = 42.1871013482 x_{64} = 42.1871013482 x 64 = 42.1871013482 x 65 = 94.2477796077 x_{65} = 94.2477796077 x 65 = 94.2477796077 x 66 = 20.1959527731 x_{66} = 20.1959527731 x 66 = 20.1959527731 x 67 = − 35.903916041 x_{67} = -35.903916041 x 67 = − 35.903916041 x 68 = − 74.0518268346 x_{68} = -74.0518268346 x 68 = − 74.0518268346 x 69 = 46.2262919028 x_{69} = 46.2262919028 x 69 = 46.2262919028 x 70 = − 31.8647254864 x_{70} = -31.8647254864 x 70 = − 31.8647254864 x 71 = − 85.720599548 x_{71} = -85.720599548 x 71 = − 85.720599548 x 72 = 16.1567622185 x_{72} = 16.1567622185 x 72 = 16.1567622185 x 73 = 100.082165964 x_{73} = 100.082165964 x 73 = 100.082165964 x 74 = − 49.8166835069 x_{74} = -49.8166835069 x 74 = − 49.8166835069 x 75 = − 15.7079632679 x_{75} = -15.7079632679 x 75 = − 15.7079632679 x 76 = − 57.8950646162 x_{76} = -57.8950646162 x 76 = − 57.8950646162 x 77 = 83.4766047954 x_{77} = 83.4766047954 x 77 = 83.4766047954 x 78 = 65.9734457254 x_{78} = 65.9734457254 x 78 = 65.9734457254 x 79 = − 19.7471538226 x_{79} = -19.7471538226 x 79 = − 19.7471538226 x 80 = 70.01263628 x_{80} = 70.01263628 x 80 = 70.01263628 x 81 = 2.24399475256 x_{81} = 2.24399475256 x 81 = 2.24399475256 x 82 = − 63.7294509728 x_{82} = -63.7294509728 x 82 = − 63.7294509728 x 83 = − 97.8381712118 x_{83} = -97.8381712118 x 83 = − 97.8381712118 x 84 = − 9.87357691128 x_{84} = -9.87357691128 x 84 = − 9.87357691128 x 85 = 4.03919055462 x_{85} = 4.03919055462 x 85 = 4.03919055462 x 86 = − 61.9342551708 x_{86} = -61.9342551708 x 86 = − 61.9342551708 x 87 = 6.28318530718 x_{87} = 6.28318530718 x 87 = 6.28318530718 x 88 = 96.0429754097 x_{88} = 96.0429754097 x 88 = 96.0429754097 x 89 = − 93.7989806572 x_{89} = -93.7989806572 x 89 = − 93.7989806572 x 90 = − 41.7383023977 x_{90} = -41.7383023977 x 90 = − 41.7383023977 x 91 = − 26.0303391297 x_{91} = -26.0303391297 x 91 = − 26.0303391297 x 92 = 78.0910173892 x_{92} = 78.0910173892 x 92 = 78.0910173892 x 93 = 86.1693984985 x_{93} = 86.1693984985 x 93 = 86.1693984985 x 94 = 39.9431065956 x_{94} = 39.9431065956 x 94 = 39.9431065956 x 95 = − 52.0606782595 x_{95} = -52.0606782595 x 95 = − 52.0606782595
Точки пересечения с осью координат Y
График пересекает ось Y, когда x равняется 0:sin ( 0 ⋅ 7 ) \sin{\left (0 \cdot 7 \right )} sin ( 0 ⋅ 7 ) 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 7 cos ( 7 x ) = 0 7 \cos{\left (7 x \right )} = 0 7 cos ( 7 x ) = 0 Решаем это уравнение x 1 = π 14 x_{1} = \frac{\pi}{14} x 1 = 14 π x 2 = 3 π 14 x_{2} = \frac{3 \pi}{14} x 2 = 14 3 π pi
(--, 1)
14 3*pi
(----, -1)
14 Интервалы возрастания и убывания функции: x 2 = 3 π 14 x_{2} = \frac{3 \pi}{14} x 2 = 14 3 π x 2 = π 14 x_{2} = \frac{\pi}{14} x 2 = 14 π (-oo, pi/14] U [3*pi/14, oo) [pi/14, 3*pi/14]
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 − 49 sin ( 7 x ) = 0 - 49 \sin{\left (7 x \right )} = 0 − 49 sin ( 7 x ) = 0 Решаем это уравнение x 1 = 0 x_{1} = 0 x 1 = 0 x 2 = π 7 x_{2} = \frac{\pi}{7} x 2 = 7 π Интервалы выпуклости и вогнутости: (-oo, 0] U [pi/7, oo) [0, pi/7]
Горизонтальные асимптоты
Горизонтальные асимптоты найдём с помощью пределов данной функции при x->+oo и x->-oolim x → − ∞ sin ( 7 x ) = ⟨ − 1 , 1 ⟩ \lim_{x \to -\infty} \sin{\left (7 x \right )} = \langle -1, 1\rangle x → − ∞ lim sin ( 7 x ) = ⟨ − 1 , 1 ⟩ Возьмём предел y = ⟨ − 1 , 1 ⟩ y = \langle -1, 1\rangle y = ⟨ − 1 , 1 ⟩ lim x → ∞ sin ( 7 x ) = ⟨ − 1 , 1 ⟩ \lim_{x \to \infty} \sin{\left (7 x \right )} = \langle -1, 1\rangle x → ∞ lim sin ( 7 x ) = ⟨ − 1 , 1 ⟩ Возьмём предел y = ⟨ − 1 , 1 ⟩ y = \langle -1, 1\rangle y = ⟨ − 1 , 1 ⟩
Наклонные асимптоты
Наклонную асимптоту можно найти, подсчитав предел функции sin(7*x), делённой на x при x->+oo и x ->-oolim x → − ∞ ( 1 x sin ( 7 x ) ) = 0 \lim_{x \to -\infty}\left(\frac{1}{x} \sin{\left (7 x \right )}\right) = 0 x → − ∞ lim ( x 1 sin ( 7 x ) ) = 0 Возьмём предел lim x → ∞ ( 1 x sin ( 7 x ) ) = 0 \lim_{x \to \infty}\left(\frac{1}{x} \sin{\left (7 x \right )}\right) = 0 x → ∞ lim ( x 1 sin ( 7 x ) ) = 0 Возьмём предел
Чётность и нечётность функции
Проверим функци чётна или нечётна с помощью соотношений f = f(-x) и f = -f(-x).sin ( 7 x ) = − sin ( 7 x ) \sin{\left (7 x \right )} = - \sin{\left (7 x \right )} sin ( 7 x ) = − sin ( 7 x ) sin ( 7 x ) = − − 1 sin ( 7 x ) \sin{\left (7 x \right )} = - -1 \sin{\left (7 x \right )} sin ( 7 x ) = − − 1 sin ( 7 x ) 