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 = 0sin ( x ) + cos ( x ) + cos ( 3 x ) + sin ( 3 x ) = 0 \sin{\left (x \right )} + \cos{\left (x \right )} + \cos{\left (3 x \right )} + \sin{\left (3 x \right )} = 0 sin ( x ) + cos ( x ) + cos ( 3 x ) + sin ( 3 x ) = 0 Решаем это уравнение Аналитическое решение x 1 = − π 2 x_{1} = - \frac{\pi}{2} x 1 = − 2 π x 2 = π 2 x_{2} = \frac{\pi}{2} x 2 = 2 π x 3 = − i log ( − 1 2 − 2 + 2 − i 2 2 + 2 ) x_{3} = - i \log{\left (- \frac{1}{2} \sqrt{- \sqrt{2} + 2} - \frac{i}{2} \sqrt{\sqrt{2} + 2} \right )} x 3 = − i log ( − 2 1 − 2 + 2 − 2 i 2 + 2 ) x 4 = − i log ( 1 2 − 2 + 2 + i 2 2 + 2 ) x_{4} = - i \log{\left (\frac{1}{2} \sqrt{- \sqrt{2} + 2} + \frac{i}{2} \sqrt{\sqrt{2} + 2} \right )} x 4 = − i log ( 2 1 − 2 + 2 + 2 i 2 + 2 ) x 5 = − i log ( − 1 2 2 + 2 + i 2 − 2 + 2 ) x_{5} = - i \log{\left (- \frac{1}{2} \sqrt{\sqrt{2} + 2} + \frac{i}{2} \sqrt{- \sqrt{2} + 2} \right )} x 5 = − i log ( − 2 1 2 + 2 + 2 i − 2 + 2 ) x 6 = − i log ( 1 2 2 + 2 − i 2 − 2 + 2 ) x_{6} = - i \log{\left (\frac{1}{2} \sqrt{\sqrt{2} + 2} - \frac{i}{2} \sqrt{- \sqrt{2} + 2} \right )} x 6 = − i log ( 2 1 2 + 2 − 2 i − 2 + 2 ) Численное решение x 1 = 93.855080526 x_{1} = 93.855080526 x 1 = 93.855080526 x 2 = − 95.8185759345 x_{2} = -95.8185759345 x 2 = − 95.8185759345 x 3 = 65.5807466437 x_{3} = 65.5807466437 x 3 = 65.5807466437 x 4 = − 47.5165888855 x_{4} = -47.5165888855 x 4 = − 47.5165888855 x 5 = − 61.6537558267 x_{5} = -61.6537558267 x 5 = − 61.6537558267 x 6 = − 91.4988860358 x_{6} = -91.4988860358 x 6 = − 91.4988860358 x 7 = 1.1780972451 x_{7} = 1.1780972451 x 7 = 1.1780972451 x 8 = − 82.074108075 x_{8} = -82.074108075 x 8 = − 82.074108075 x 9 = 100.138265833 x_{9} = 100.138265833 x 9 = 100.138265833 x 10 = − 42.8041999052 x_{10} = -42.8041999052 x 10 = − 42.8041999052 x 11 = 36.1283155163 x_{11} = 36.1283155163 x 11 = 36.1283155163 x 12 = − 55.3705705195 x_{12} = -55.3705705195 x 12 = − 55.3705705195 x 13 = 27.8816348006 x_{13} = 27.8816348006 x 13 = 27.8816348006 x 14 = 80.1106126665 x_{14} = 80.1106126665 x 14 = 80.1106126665 x 15 = 26.3108384738 x_{15} = 26.3108384738 x 15 = 26.3108384738 x 16 = 71.8639319509 x_{16} = 71.8639319509 x 16 = 71.8639319509 x 17 = 42.0188017418 x_{17} = 42.0188017418 x 17 = 42.0188017418 x 18 = 51.8362787842 x_{18} = 51.8362787842 x 18 = 51.8362787842 x 19 = − 44.374996232 x_{19} = -44.374996232 x 19 = − 44.374996232 x 20 = 87.5718952188 x_{20} = 87.5718952188 x 20 = 87.5718952188 x 21 = 4.31968989869 x_{21} = 4.31968989869 x 21 = 4.31968989869 x 22 = 14.1371669412 x_{22} = 14.1371669412 x 22 = 14.1371669412 x 23 = − 25.5254403104 x_{23} = -25.5254403104 x 23 = − 25.5254403104 x 24 = 76.5763209313 x_{24} = 76.5763209313 x 24 = 76.5763209313 x 25 = − 3.53429173529 x_{25} = -3.53429173529 x 25 = − 3.53429173529 x 26 = − 51.8362787842 x_{26} = -51.8362787842 x 26 = − 51.8362787842 x 27 = − 45.9457925588 x_{27} = -45.9457925588 x 27 = − 45.9457925588 x 28 = − 7.85398163397 x_{28} = -7.85398163397 x 28 = − 7.85398163397 x 29 = 56.1559686829 x_{29} = 56.1559686829 x 29 = 56.1559686829 x 30 = 29.8451302091 x_{30} = 29.8451302091 x 30 = 29.8451302091 x 31 = 42.4115008235 x_{31} = 42.4115008235 x 31 = 42.4115008235 x 32 = 32.594023781 x_{32} = 32.594023781 x 32 = 32.594023781 x 33 = − 75.7909227679 x_{33} = -75.7909227679 x 33 = − 75.7909227679 x 34 = − 86.7864970554 x_{34} = -86.7864970554 x 34 = − 86.7864970554 x 35 = − 80.1106126665 x_{35} = -80.1106126665 x 35 = − 80.1106126665 x 36 = − 58.1194640914 x_{36} = -58.1194640914 x 36 = − 58.1194640914 x 37 = − 31.8086256176 x_{37} = -31.8086256176 x 37 = − 31.8086256176 x 38 = − 83.6449044018 x_{38} = -83.6449044018 x 38 = − 83.6449044018 x 39 = 21.5984494934 x_{39} = 21.5984494934 x 39 = 21.5984494934 x 40 = − 53.7997741927 x_{40} = -53.7997741927 x 40 = − 53.7997741927 x 41 = − 71.0785337875 x_{41} = -71.0785337875 x 41 = − 71.0785337875 x 42 = 34.1648201078 x_{42} = 34.1648201078 x 42 = 34.1648201078 x 43 = 5.89048622548 x_{43} = 5.89048622548 x 43 = 5.89048622548 x 44 = 26.7035375555 x_{44} = 26.7035375555 x 44 = 26.7035375555 x 45 = − 97.782071343 x_{45} = -97.782071343 x 45 = − 97.782071343 x 46 = 98.5674695064 x_{46} = 98.5674695064 x 46 = 98.5674695064 x 47 = − 88.3572933822 x_{47} = -88.3572933822 x 47 = − 88.3572933822 x 48 = − 16.1006623496 x_{48} = -16.1006623496 x 48 = − 16.1006623496 x 49 = − 9.81747704247 x_{49} = -9.81747704247 x 49 = − 9.81747704247 x 50 = 49.8727833757 x_{50} = 49.8727833757 x 50 = 49.8727833757 x 51 = 70.2931356241 x_{51} = 70.2931356241 x 51 = 70.2931356241 x 52 = − 1.57079632679 x_{52} = -1.57079632679 x 52 = − 1.57079632679 x 53 = 92.2842841992 x_{53} = 92.2842841992 x 53 = 92.2842841992 x 54 = − 33.3794219444 x_{54} = -33.3794219444 x 54 = − 33.3794219444 x 55 = − 67.9369411339 x_{55} = -67.9369411339 x 55 = − 67.9369411339 x 56 = − 99.3528676698 x_{56} = -99.3528676698 x 56 = − 99.3528676698 x 57 = 40.448005415 x_{57} = 40.448005415 x 57 = 40.448005415 x 58 = − 89.928089709 x_{58} = -89.928089709 x 58 = − 89.928089709 x 59 = − 64.7953484803 x_{59} = -64.7953484803 x 59 = − 64.7953484803 x 60 = 78.147117258 x_{60} = 78.147117258 x 60 = 78.147117258 x 61 = − 1.96349540849 x_{61} = -1.96349540849 x 61 = − 1.96349540849 x 62 = − 77.3617190946 x_{62} = -77.3617190946 x 62 = − 77.3617190946 x 63 = − 14.1371669412 x_{63} = -14.1371669412 x 63 = − 14.1371669412 x 64 = 7.85398163397 x_{64} = 7.85398163397 x 64 = 7.85398163397 x 65 = − 73.8274273594 x_{65} = -73.8274273594 x 65 = − 73.8274273594 x 66 = 86.001098892 x_{66} = 86.001098892 x 66 = 86.001098892 x 67 = − 0.392699081699 x_{67} = -0.392699081699 x 67 = − 0.392699081699 x 68 = 84.4303025652 x_{68} = 84.4303025652 x 68 = 84.4303025652 x 69 = − 38.0918109248 x_{69} = -38.0918109248 x 69 = − 38.0918109248 x 70 = − 11.3882733693 x_{70} = -11.3882733693 x 70 = − 11.3882733693 x 71 = − 1198.91029643 x_{71} = -1198.91029643 x 71 = − 1198.91029643 x 72 = − 29.8451302091 x_{72} = -29.8451302091 x 72 = − 29.8451302091 x 73 = − 69.5077374607 x_{73} = -69.5077374607 x 73 = − 69.5077374607 x 74 = − 23.9546439836 x_{74} = -23.9546439836 x 74 = − 23.9546439836 x 75 = − 60.0829594999 x_{75} = -60.0829594999 x 75 = − 60.0829594999 x 76 = 43.5895980686 x_{76} = 43.5895980686 x 76 = 43.5895980686 x 77 = − 39.6626072516 x_{77} = -39.6626072516 x 77 = − 39.6626072516 x 78 = 58.1194640914 x_{78} = 58.1194640914 x 78 = 58.1194640914 x 79 = 95.8185759345 x_{79} = 95.8185759345 x 79 = 95.8185759345 x 80 = − 36.1283155163 x_{80} = -36.1283155163 x 80 = − 36.1283155163 x 81 = − 20.81305133 x_{81} = -20.81305133 x 81 = − 20.81305133 x 82 = 73.8274273594 x_{82} = 73.8274273594 x 82 = 73.8274273594 x 83 = − 66.3661448071 x_{83} = -66.3661448071 x 83 = − 66.3661448071 x 84 = 10.6028752059 x_{84} = 10.6028752059 x 84 = 10.6028752059 x 85 = 23.1692458202 x_{85} = 23.1692458202 x 85 = 23.1692458202 x 86 = 48.3019870489 x_{86} = 48.3019870489 x 86 = 48.3019870489 x 87 = 54.5851723561 x_{87} = 54.5851723561 x 87 = 54.5851723561 x 88 = 20.0276531666 x_{88} = 20.0276531666 x 88 = 20.0276531666 x 89 = − 17.6714586764 x_{89} = -17.6714586764 x 89 = − 17.6714586764 x 90 = − 22.3838476568 x_{90} = -22.3838476568 x 90 = − 22.3838476568 x 91 = 64.0099503169 x_{91} = 64.0099503169 x 91 = 64.0099503169 x 92 = 62.4391539901 x_{92} = 62.4391539901 x 92 = 62.4391539901 x 93 = 12.1736715327 x_{93} = 12.1736715327 x 93 = 12.1736715327 x 94 = 18.4568568398 x_{94} = 18.4568568398 x 94 = 18.4568568398
Точки пересечения с осью координат Y
График пересекает ось Y, когда x равняется 0:sin ( 0 ⋅ 3 ) + sin ( 0 ) + cos ( 0 ) + cos ( 0 ⋅ 3 ) \sin{\left (0 \cdot 3 \right )} + \sin{\left (0 \right )} + \cos{\left (0 \right )} + \cos{\left (0 \cdot 3 \right )} sin ( 0 ⋅ 3 ) + sin ( 0 ) + cos ( 0 ) + cos ( 0 ⋅ 3 ) f ( 0 ) = 2 f{\left (0 \right )} = 2 f ( 0 ) = 2 (0, 2)
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 − sin ( x ) − 3 sin ( 3 x ) + cos ( x ) + 3 cos ( 3 x ) = 0 - \sin{\left (x \right )} - 3 \sin{\left (3 x \right )} + \cos{\left (x \right )} + 3 \cos{\left (3 x \right )} = 0 − sin ( x ) − 3 sin ( 3 x ) + cos ( x ) + 3 cos ( 3 x ) = 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 − sin ( x ) + 9 sin ( 3 x ) + cos ( x ) + 9 cos ( 3 x ) = 0 - \sin{\left (x \right )} + 9 \sin{\left (3 x \right )} + \cos{\left (x \right )} + 9 \cos{\left (3 x \right )} = 0 − sin ( x ) + 9 sin ( 3 x ) + cos ( x ) + 9 cos ( 3 x ) = 0 Решаем это уравнение
Горизонтальные асимптоты
Горизонтальные асимптоты найдём с помощью пределов данной функции при x->+oo и x->-oolim x → − ∞ ( sin ( x ) + cos ( x ) + cos ( 3 x ) + sin ( 3 x ) ) = ⟨ − 4 , 4 ⟩ \lim_{x \to -\infty}\left(\sin{\left (x \right )} + \cos{\left (x \right )} + \cos{\left (3 x \right )} + \sin{\left (3 x \right )}\right) = \langle -4, 4\rangle x → − ∞ lim ( sin ( x ) + cos ( x ) + cos ( 3 x ) + sin ( 3 x ) ) = ⟨ − 4 , 4 ⟩ Возьмём предел y = ⟨ − 4 , 4 ⟩ y = \langle -4, 4\rangle y = ⟨ − 4 , 4 ⟩ lim x → ∞ ( sin ( x ) + cos ( x ) + cos ( 3 x ) + sin ( 3 x ) ) = ⟨ − 4 , 4 ⟩ \lim_{x \to \infty}\left(\sin{\left (x \right )} + \cos{\left (x \right )} + \cos{\left (3 x \right )} + \sin{\left (3 x \right )}\right) = \langle -4, 4\rangle x → ∞ lim ( sin ( x ) + cos ( x ) + cos ( 3 x ) + sin ( 3 x ) ) = ⟨ − 4 , 4 ⟩ Возьмём предел y = ⟨ − 4 , 4 ⟩ y = \langle -4, 4\rangle y = ⟨ − 4 , 4 ⟩
Наклонные асимптоты
Наклонную асимптоту можно найти, подсчитав предел функции cos(x) + sin(x) + cos(3*x) + sin(3*x), делённой на x при x->+oo и x ->-oolim x → − ∞ ( 1 x ( sin ( x ) + cos ( x ) + cos ( 3 x ) + sin ( 3 x ) ) ) = 0 \lim_{x \to -\infty}\left(\frac{1}{x} \left(\sin{\left (x \right )} + \cos{\left (x \right )} + \cos{\left (3 x \right )} + \sin{\left (3 x \right )}\right)\right) = 0 x → − ∞ lim ( x 1 ( sin ( x ) + cos ( x ) + cos ( 3 x ) + sin ( 3 x ) ) ) = 0 Возьмём предел lim x → ∞ ( 1 x ( sin ( x ) + cos ( x ) + cos ( 3 x ) + sin ( 3 x ) ) ) = 0 \lim_{x \to \infty}\left(\frac{1}{x} \left(\sin{\left (x \right )} + \cos{\left (x \right )} + \cos{\left (3 x \right )} + \sin{\left (3 x \right )}\right)\right) = 0 x → ∞ lim ( x 1 ( sin ( x ) + cos ( x ) + cos ( 3 x ) + sin ( 3 x ) ) ) = 0 Возьмём предел
Чётность и нечётность функции
Проверим функци чётна или нечётна с помощью соотношений f = f(-x) и f = -f(-x).sin ( x ) + cos ( x ) + cos ( 3 x ) + sin ( 3 x ) = − sin ( x ) − sin ( 3 x ) + cos ( x ) + cos ( 3 x ) \sin{\left (x \right )} + \cos{\left (x \right )} + \cos{\left (3 x \right )} + \sin{\left (3 x \right )} = - \sin{\left (x \right )} - \sin{\left (3 x \right )} + \cos{\left (x \right )} + \cos{\left (3 x \right )} sin ( x ) + cos ( x ) + cos ( 3 x ) + sin ( 3 x ) = − sin ( x ) − sin ( 3 x ) + cos ( x ) + cos ( 3 x ) sin ( x ) + cos ( x ) + cos ( 3 x ) + sin ( 3 x ) = − − 1 sin ( x ) − − sin ( 3 x ) − cos ( x ) − cos ( 3 x ) \sin{\left (x \right )} + \cos{\left (x \right )} + \cos{\left (3 x \right )} + \sin{\left (3 x \right )} = - -1 \sin{\left (x \right )} - - \sin{\left (3 x \right )} - \cos{\left (x \right )} - \cos{\left (3 x \right )} sin ( x ) + cos ( x ) + cos ( 3 x ) + sin ( 3 x ) = − − 1 sin ( x ) − − sin ( 3 x ) − cos ( x ) − cos ( 3 x ) 