График функции
0 -6000 -5000 -4000 -3000 -2000 -1000 1000 2000 3000 4000 5000 6000 5 -5
Точки пересечения с осью координат X
График функции пересекает ось X при f = 0sin ( x ) + cos ( 2 x ) = 0 \sin{\left (x \right )} + \cos{\left (2 x \right )} = 0 sin ( x ) + cos ( 2 x ) = 0 Решаем это уравнение Аналитическое решение x 1 = π 2 x_{1} = \frac{\pi}{2} x 1 = 2 π x 2 = − i log ( − 3 2 − i 2 ) x_{2} = - i \log{\left (- \frac{\sqrt{3}}{2} - \frac{i}{2} \right )} x 2 = − i log ( − 2 3 − 2 i ) x 3 = − i log ( 3 2 − i 2 ) x_{3} = - i \log{\left (\frac{\sqrt{3}}{2} - \frac{i}{2} \right )} x 3 = − i log ( 2 3 − 2 i ) Численное решение x 1 = − 10.995574825 x_{1} = -10.995574825 x 1 = − 10.995574825 x 2 = 64.4026493092 x_{2} = 64.4026493092 x 2 = 64.4026493092 x 3 = − 67.5442421665 x_{3} = -67.5442421665 x 3 = − 67.5442421665 x 4 = − 73.8274272801 x_{4} = -73.8274272801 x 4 = − 73.8274272801 x 5 = 41.3643032723 x_{5} = 41.3643032723 x 5 = 41.3643032723 x 6 = − 69.6386371546 x_{6} = -69.6386371546 x 6 = − 69.6386371546 x 7 = − 21.4675497995 x_{7} = -21.4675497995 x 7 = − 21.4675497995 x 8 = − 2.61799387799 x_{8} = -2.61799387799 x 8 = − 2.61799387799 x 9 = − 71.733032257 x_{9} = -71.733032257 x 9 = − 71.733032257 x 10 = 47.6474885794 x_{10} = 47.6474885794 x 10 = 47.6474885794 x 11 = − 44.5058959259 x_{11} = -44.5058959259 x 11 = − 44.5058959259 x 12 = 1.57079653523 x_{12} = 1.57079653523 x 12 = 1.57079653523 x 13 = 76.9690198128 x_{13} = 76.9690198128 x 13 = 76.9690198128 x 14 = 51.8362788989 x_{14} = 51.8362788989 x 14 = 51.8362788989 x 15 = − 0.523598775598 x_{15} = -0.523598775598 x 15 = − 0.523598775598 x 16 = 18.3259571459 x_{16} = 18.3259571459 x 16 = 18.3259571459 x 17 = − 13.08996939 x_{17} = -13.08996939 x 17 = − 13.08996939 x 18 = − 36.1283154199 x_{18} = -36.1283154199 x 18 = − 36.1283154199 x 19 = 26.7035373553 x_{19} = 26.7035373553 x 19 = 26.7035373553 x 20 = − 59.1666616426 x_{20} = -59.1666616426 x 20 = − 59.1666616426 x 21 = − 31.9395253115 x_{21} = -31.9395253115 x 21 = − 31.9395253115 x 22 = − 25.6563400043 x_{22} = -25.6563400043 x 22 = − 25.6563400043 x 23 = − 75.9218224618 x_{23} = -75.9218224618 x 23 = − 75.9218224618 x 24 = 95.8185760577 x_{24} = 95.8185760577 x 24 = 95.8185760577 x 25 = 7.85398173973 x_{25} = 7.85398173973 x 25 = 7.85398173973 x 26 = 70.6858345098 x_{26} = 70.6858345098 x 26 = 70.6858345098 x 27 = 5.75958653158 x_{27} = 5.75958653158 x 27 = 5.75958653158 x 28 = − 27.7507351067 x_{28} = -27.7507351067 x 28 = − 27.7507351067 x 29 = − 4.71238877564 x_{29} = -4.71238877564 x 29 = − 4.71238877564 x 30 = 72.7802298082 x_{30} = 72.7802298082 x 30 = 72.7802298082 x 31 = − 40.3171057211 x_{31} = -40.3171057211 x 31 = − 40.3171057211 x 32 = 62.3082542962 x_{32} = 62.3082542962 x 32 = 62.3082542962 x 33 = 22.5147473507 x_{33} = 22.5147473507 x 33 = 22.5147473507 x 34 = 49.7418836818 x_{34} = 49.7418836818 x 34 = 49.7418836818 x 35 = 58.119464472 x_{35} = 58.119464472 x 35 = 58.119464472 x 36 = − 23.5619450082 x_{36} = -23.5619450082 x 36 = − 23.5619450082 x 37 = 45.5530936891 x_{37} = 45.5530936891 x 37 = 45.5530936891 x 38 = − 38.2227106187 x_{38} = -38.2227106187 x 38 = − 38.2227106187 x 39 = − 80.11061258 x_{39} = -80.11061258 x 39 = − 80.11061258 x 40 = 32.98672267 x_{40} = 32.98672267 x 40 = 32.98672267 x 41 = 66.497044501 x_{41} = 66.497044501 x 41 = 66.497044501 x 42 = 97.9129710369 x_{42} = 97.9129710369 x 42 = 97.9129710369 x 43 = − 61.2610569526 x_{43} = -61.2610569526 x 43 = − 61.2610569526 x 44 = 14.1371671029 x_{44} = 14.1371671029 x 44 = 14.1371671029 x 45 = 12.0427718388 x_{45} = 12.0427718388 x 45 = 12.0427718388 x 46 = − 48.6946859199 x_{46} = -48.6946859199 x 46 = − 48.6946859199 x 47 = − 42.411500619 x_{47} = -42.411500619 x 47 = − 42.411500619 x 48 = − 34.0339204139 x_{48} = -34.0339204139 x 48 = − 34.0339204139 x 49 = 89.5353908427 x_{49} = 89.5353908427 x 49 = 89.5353908427 x 50 = − 105.243352994 x_{50} = -105.243352994 x 50 = − 105.243352994 x 51 = − 92.6769830654 x_{51} = -92.6769830654 x 51 = − 92.6769830654 x 52 = 16.2315620435 x_{52} = 16.2315620435 x 52 = 16.2315620435 x 53 = 83.2522055084 x_{53} = 83.2522055084 x 53 = 83.2522055084 x 54 = 100.007366139 x_{54} = 100.007366139 x 54 = 100.007366139 x 55 = − 17.2787597988 x_{55} = -17.2787597988 x 55 = − 17.2787597988 x 56 = − 29.8451300967 x_{56} = -29.8451300967 x 56 = − 29.8451300967 x 57 = 9.94837673637 x_{57} = 9.94837673637 x 57 = 9.94837673637 x 58 = − 90.5825881785 x_{58} = -90.5825881785 x 58 = − 90.5825881785 x 59 = − 46.6002910282 x_{59} = -46.6002910282 x 59 = − 46.6002910282 x 60 = − 63.3554518474 x_{60} = -63.3554518474 x 60 = − 63.3554518474 x 61 = − 78.0162175641 x_{61} = -78.0162175641 x 61 = − 78.0162175641 x 62 = 68.5914396034 x_{62} = 68.5914396034 x 62 = 68.5914396034 x 63 = − 54.9778716097 x_{63} = -54.9778716097 x 63 = − 54.9778716097 x 64 = − 65.4498469498 x_{64} = -65.4498469498 x 64 = − 65.4498469498 x 65 = 56.025068989 x_{65} = 56.025068989 x 65 = 56.025068989 x 66 = − 82.2050077689 x_{66} = -82.2050077689 x 66 = − 82.2050077689 x 67 = − 57.0722665402 x_{67} = -57.0722665402 x 67 = − 57.0722665402 x 68 = 85.3466004225 x_{68} = 85.3466004225 x 68 = 85.3466004225 x 69 = 30.8923277603 x_{69} = 30.8923277603 x 69 = 30.8923277603 x 70 = − 10.9955744709 x_{70} = -10.9955744709 x 70 = − 10.9955744709 x 71 = − 54.9778719401 x_{71} = -54.9778719401 x 71 = − 54.9778719401 x 72 = 24.6091424531 x_{72} = 24.6091424531 x 72 = 24.6091424531 x 73 = − 98.96016896 x_{73} = -98.96016896 x 73 = − 98.96016896 x 74 = 91.6297857297 x_{74} = 91.6297857297 x 74 = 91.6297857297 x 75 = 74.8746249106 x_{75} = 74.8746249106 x 75 = 74.8746249106 x 76 = − 88.4881930761 x_{76} = -88.4881930761 x 76 = − 88.4881930761 x 77 = − 98.9601687457 x_{77} = -98.9601687457 x 77 = − 98.9601687457 x 78 = − 84.2994028713 x_{78} = -84.2994028713 x 78 = − 84.2994028713 x 79 = 39.2699083672 x_{79} = 39.2699083672 x 79 = 39.2699083672 x 80 = 93.7241808321 x_{80} = 93.7241808321 x 80 = 93.7241808321 x 81 = 20.4203521504 x_{81} = 20.4203521504 x 81 = 20.4203521504 x 82 = 629.88932728 x_{82} = 629.88932728 x 82 = 629.88932728 x 83 = 83.2522056281 x_{83} = 83.2522056281 x 83 = 83.2522056281 x 84 = 53.9306738866 x_{84} = 53.9306738866 x 84 = 53.9306738866 x 85 = − 142.942465507 x_{85} = -142.942465507 x 85 = − 142.942465507 x 86 = 3.66519142919 x_{86} = 3.66519142919 x 86 = 3.66519142919 x 87 = − 86.3937977737 x_{87} = -86.3937977737 x 87 = − 86.3937977737 x 88 = 60.2138591938 x_{88} = 60.2138591938 x 88 = 60.2138591938 x 89 = − 19.3731546971 x_{89} = -19.3731546971 x 89 = − 19.3731546971
Точки пересечения с осью координат Y
График пересекает ось Y, когда x равняется 0:sin ( 0 ) + cos ( 0 ⋅ 2 ) \sin{\left (0 \right )} + \cos{\left (0 \cdot 2 \right )} sin ( 0 ) + cos ( 0 ⋅ 2 ) f ( 0 ) = 1 f{\left (0 \right )} = 1 f ( 0 ) = 1 (0, 1)
Горизонтальные асимптоты
Горизонтальные асимптоты найдём с помощью пределов данной функции при x->+oo и x->-oolim x → − ∞ ( sin ( x ) + cos ( 2 x ) ) = ⟨ − 2 , 2 ⟩ \lim_{x \to -\infty}\left(\sin{\left (x \right )} + \cos{\left (2 x \right )}\right) = \langle -2, 2\rangle x → − ∞ lim ( sin ( x ) + cos ( 2 x ) ) = ⟨ − 2 , 2 ⟩ Возьмём предел y = ⟨ − 2 , 2 ⟩ y = \langle -2, 2\rangle y = ⟨ − 2 , 2 ⟩ lim x → ∞ ( sin ( x ) + cos ( 2 x ) ) = ⟨ − 2 , 2 ⟩ \lim_{x \to \infty}\left(\sin{\left (x \right )} + \cos{\left (2 x \right )}\right) = \langle -2, 2\rangle x → ∞ lim ( sin ( x ) + cos ( 2 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 cos(2*x) + sin(x), делённой на x при x->+oo и x ->-oolim x → − ∞ ( 1 x ( sin ( x ) + cos ( 2 x ) ) ) = 0 \lim_{x \to -\infty}\left(\frac{1}{x} \left(\sin{\left (x \right )} + \cos{\left (2 x \right )}\right)\right) = 0 x → − ∞ lim ( x 1 ( sin ( x ) + cos ( 2 x ) ) ) = 0 Возьмём предел lim x → ∞ ( 1 x ( sin ( x ) + cos ( 2 x ) ) ) = 0 \lim_{x \to \infty}\left(\frac{1}{x} \left(\sin{\left (x \right )} + \cos{\left (2 x \right )}\right)\right) = 0 x → ∞ lim ( x 1 ( sin ( x ) + cos ( 2 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 ( 2 x ) = − sin ( x ) + cos ( 2 x ) \sin{\left (x \right )} + \cos{\left (2 x \right )} = - \sin{\left (x \right )} + \cos{\left (2 x \right )} sin ( x ) + cos ( 2 x ) = − sin ( x ) + cos ( 2 x ) sin ( x ) + cos ( 2 x ) = − − 1 sin ( x ) − cos ( 2 x ) \sin{\left (x \right )} + \cos{\left (2 x \right )} = - -1 \sin{\left (x \right )} - \cos{\left (2 x \right )} sin ( x ) + cos ( 2 x ) = − − 1 sin ( x ) − cos ( 2 x ) 