График функции
0 -25000 -20000 -15000 -10000 -5000 5000 10000 15000 20000 25000 5 -5
Точки пересечения с осью координат X
График функции пересекает ось X при f = 02 cos ( 3 x ) = 0 2 \cos{\left (3 x \right )} = 0 2 cos ( 3 x ) = 0 Решаем это уравнение Аналитическое решение x 1 = π 6 x_{1} = \frac{\pi}{6} x 1 = 6 π x 2 = π 2 x_{2} = \frac{\pi}{2} x 2 = 2 π Численное решение x 1 = − 3.66519142919 x_{1} = -3.66519142919 x 1 = − 3.66519142919 x 2 = − 100.007366139 x_{2} = -100.007366139 x 2 = − 100.007366139 x 3 = − 95.8185759345 x_{3} = -95.8185759345 x 3 = − 95.8185759345 x 4 = 71.733032257 x_{4} = 71.733032257 x 4 = 71.733032257 x 5 = 1676.03968069 x_{5} = 1676.03968069 x 5 = 1676.03968069 x 6 = 41.3643032723 x_{6} = 41.3643032723 x 6 = 41.3643032723 x 7 = 88.4881930761 x_{7} = 88.4881930761 x 7 = 88.4881930761 x 8 = − 69.6386371546 x_{8} = -69.6386371546 x 8 = − 69.6386371546 x 9 = − 51.8362787842 x_{9} = -51.8362787842 x 9 = − 51.8362787842 x 10 = − 21.4675497995 x_{10} = -21.4675497995 x 10 = − 21.4675497995 x 11 = − 60.2138591938 x_{11} = -60.2138591938 x 11 = − 60.2138591938 x 12 = − 43.4586983747 x_{12} = -43.4586983747 x 12 = − 43.4586983747 x 13 = 36.1283155163 x_{13} = 36.1283155163 x 13 = 36.1283155163 x 14 = − 71.733032257 x_{14} = -71.733032257 x 14 = − 71.733032257 x 15 = 80.1106126665 x_{15} = 80.1106126665 x 15 = 80.1106126665 x 16 = 34.0339204139 x_{16} = 34.0339204139 x 16 = 34.0339204139 x 17 = 51.8362787842 x_{17} = 51.8362787842 x 17 = 51.8362787842 x 18 = − 12.0427718388 x_{18} = -12.0427718388 x 18 = − 12.0427718388 x 19 = 14.1371669412 x_{19} = 14.1371669412 x 19 = 14.1371669412 x 20 = 31.9395253115 x_{20} = 31.9395253115 x 20 = 31.9395253115 x 21 = − 5.75958653158 x_{21} = -5.75958653158 x 21 = − 5.75958653158 x 22 = − 36.1283155163 x_{22} = -36.1283155163 x 22 = − 36.1283155163 x 23 = − 47.6474885794 x_{23} = -47.6474885794 x 23 = − 47.6474885794 x 24 = 0.523598775598 x_{24} = 0.523598775598 x 24 = 0.523598775598 x 25 = − 7.85398163397 x_{25} = -7.85398163397 x 25 = − 7.85398163397 x 26 = 18.3259571459 x_{26} = 18.3259571459 x 26 = 18.3259571459 x 27 = − 23.5619449019 x_{27} = -23.5619449019 x 27 = − 23.5619449019 x 28 = 29.8451302091 x_{28} = 29.8451302091 x 28 = 29.8451302091 x 29 = 42.4115008235 x_{29} = 42.4115008235 x 29 = 42.4115008235 x 30 = − 31.9395253115 x_{30} = -31.9395253115 x 30 = − 31.9395253115 x 31 = − 25.6563400043 x_{31} = -25.6563400043 x 31 = − 25.6563400043 x 32 = − 75.9218224618 x_{32} = -75.9218224618 x 32 = − 75.9218224618 x 33 = − 62.3082542962 x_{33} = -62.3082542962 x 33 = − 62.3082542962 x 34 = − 27.7507351067 x_{34} = -27.7507351067 x 34 = − 27.7507351067 x 35 = 62.3082542962 x_{35} = 62.3082542962 x 35 = 62.3082542962 x 36 = 22.5147473507 x_{36} = 22.5147473507 x 36 = 22.5147473507 x 37 = − 80.1106126665 x_{37} = -80.1106126665 x 37 = − 80.1106126665 x 38 = − 58.1194640914 x_{38} = -58.1194640914 x 38 = − 58.1194640914 x 39 = 49.7418836818 x_{39} = 49.7418836818 x 39 = 49.7418836818 x 40 = 5.75958653158 x_{40} = 5.75958653158 x 40 = 5.75958653158 x 41 = 64.4026493986 x_{41} = 64.4026493986 x 41 = 64.4026493986 x 42 = 26.7035375555 x_{42} = 26.7035375555 x 42 = 26.7035375555 x 43 = 86.3937979737 x_{43} = 86.3937979737 x 43 = 86.3937979737 x 44 = 66.497044501 x_{44} = 66.497044501 x 44 = 66.497044501 x 45 = 97.9129710369 x_{45} = 97.9129710369 x 45 = 97.9129710369 x 46 = 75.9218224618 x_{46} = 75.9218224618 x 46 = 75.9218224618 x 47 = − 16.2315620435 x_{47} = -16.2315620435 x 47 = − 16.2315620435 x 48 = 12.0427718388 x_{48} = 12.0427718388 x 48 = 12.0427718388 x 49 = − 67.5442420522 x_{49} = -67.5442420522 x 49 = − 67.5442420522 x 50 = − 34.0339204139 x_{50} = -34.0339204139 x 50 = − 34.0339204139 x 51 = − 97.9129710369 x_{51} = -97.9129710369 x 51 = − 97.9129710369 x 52 = 78.0162175641 x_{52} = 78.0162175641 x 52 = 78.0162175641 x 53 = − 1.57079632679 x_{53} = -1.57079632679 x 53 = − 1.57079632679 x 54 = − 91.6297857297 x_{54} = -91.6297857297 x 54 = − 91.6297857297 x 55 = 16.2315620435 x_{55} = 16.2315620435 x 55 = 16.2315620435 x 56 = 100.007366139 x_{56} = 100.007366139 x 56 = 100.007366139 x 57 = 84.2994028713 x_{57} = 84.2994028713 x 57 = 84.2994028713 x 58 = − 49.7418836818 x_{58} = -49.7418836818 x 58 = − 49.7418836818 x 59 = − 93.7241808321 x_{59} = -93.7241808321 x 59 = − 93.7241808321 x 60 = 9.94837673637 x_{60} = 9.94837673637 x 60 = 9.94837673637 x 61 = − 45.5530934771 x_{61} = -45.5530934771 x 61 = − 45.5530934771 x 62 = 4.71238898038 x_{62} = 4.71238898038 x 62 = 4.71238898038 x 63 = 7.85398163397 x_{63} = 7.85398163397 x 63 = 7.85398163397 x 64 = 68.5914396034 x_{64} = 68.5914396034 x 64 = 68.5914396034 x 65 = − 73.8274273594 x_{65} = -73.8274273594 x 65 = − 73.8274273594 x 66 = − 14.1371669412 x_{66} = -14.1371669412 x 66 = − 14.1371669412 x 67 = − 65.4498469498 x_{67} = -65.4498469498 x 67 = − 65.4498469498 x 68 = − 98.9601685881 x_{68} = -98.9601685881 x 68 = − 98.9601685881 x 69 = − 56.025068989 x_{69} = -56.025068989 x 69 = − 56.025068989 x 70 = 56.025068989 x_{70} = 56.025068989 x 70 = 56.025068989 x 71 = − 82.2050077689 x_{71} = -82.2050077689 x 71 = − 82.2050077689 x 72 = − 61.261056745 x_{72} = -61.261056745 x 72 = − 61.261056745 x 73 = 91.6297857297 x_{73} = 91.6297857297 x 73 = 91.6297857297 x 74 = 82.2050077689 x_{74} = 82.2050077689 x 74 = 82.2050077689 x 75 = 20.4203522483 x_{75} = 20.4203522483 x 75 = 20.4203522483 x 76 = − 29.8451302091 x_{76} = -29.8451302091 x 76 = − 29.8451302091 x 77 = 44.5058959259 x_{77} = 44.5058959259 x 77 = 44.5058959259 x 78 = − 78.0162175641 x_{78} = -78.0162175641 x 78 = − 78.0162175641 x 79 = − 38.2227106187 x_{79} = -38.2227106187 x 79 = − 38.2227106187 x 80 = − 9.94837673637 x_{80} = -9.94837673637 x 80 = − 9.94837673637 x 81 = − 89.5353906273 x_{81} = -89.5353906273 x 81 = − 89.5353906273 x 82 = 38.2227106187 x_{82} = 38.2227106187 x 82 = 38.2227106187 x 83 = 58.1194640914 x_{83} = 58.1194640914 x 83 = 58.1194640914 x 84 = 95.8185759345 x_{84} = 95.8185759345 x 84 = 95.8185759345 x 85 = 40.3171057211 x_{85} = 40.3171057211 x 85 = 40.3171057211 x 86 = 73.8274273594 x_{86} = 73.8274273594 x 86 = 73.8274273594 x 87 = − 84.2994028713 x_{87} = -84.2994028713 x 87 = − 84.2994028713 x 88 = − 53.9306738866 x_{88} = -53.9306738866 x 88 = − 53.9306738866 x 89 = 93.7241808321 x_{89} = 93.7241808321 x 89 = 93.7241808321 x 90 = 15.1843644924 x_{90} = 15.1843644924 x 90 = 15.1843644924 x 91 = 53.9306738866 x_{91} = 53.9306738866 x 91 = 53.9306738866 x 92 = 87.4409955249 x_{92} = 87.4409955249 x 92 = 87.4409955249 x 93 = 27.7507351067 x_{93} = 27.7507351067 x 93 = 27.7507351067 x 94 = 60.2138591938 x_{94} = 60.2138591938 x 94 = 60.2138591938 x 95 = − 19.3731546971 x_{95} = -19.3731546971 x 95 = − 19.3731546971 x 96 = 1.57079632679 x_{96} = 1.57079632679 x 96 = 1.57079632679
Точки пересечения с осью координат Y
График пересекает ось Y, когда x равняется 0:2 cos ( 0 ⋅ 3 ) 2 \cos{\left (0 \cdot 3 \right )} 2 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 − 6 sin ( 3 x ) = 0 - 6 \sin{\left (3 x \right )} = 0 − 6 sin ( 3 x ) = 0 Решаем это уравнение x 1 = 0 x_{1} = 0 x 1 = 0 x 2 = π 3 x_{2} = \frac{\pi}{3} x 2 = 3 π (0, 2) pi
(--, -2)
3 Интервалы возрастания и убывания функции: x 2 = π 3 x_{2} = \frac{\pi}{3} x 2 = 3 π x 2 = 0 x_{2} = 0 x 2 = 0 (-oo, 0] U [pi/3, oo) [0, pi/3]
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 − 18 cos ( 3 x ) = 0 - 18 \cos{\left (3 x \right )} = 0 − 18 cos ( 3 x ) = 0 Решаем это уравнение x 1 = π 6 x_{1} = \frac{\pi}{6} x 1 = 6 π x 2 = π 2 x_{2} = \frac{\pi}{2} x 2 = 2 π Интервалы выпуклости и вогнутости: [pi/6, pi/2] (-oo, pi/6] U [pi/2, oo)
Горизонтальные асимптоты
Горизонтальные асимптоты найдём с помощью пределов данной функции при x->+oo и x->-oolim x → − ∞ ( 2 cos ( 3 x ) ) = ⟨ − 2 , 2 ⟩ \lim_{x \to -\infty}\left(2 \cos{\left (3 x \right )}\right) = \langle -2, 2\rangle x → − ∞ lim ( 2 cos ( 3 x ) ) = ⟨ − 2 , 2 ⟩ Возьмём предел y = ⟨ − 2 , 2 ⟩ y = \langle -2, 2\rangle y = ⟨ − 2 , 2 ⟩ lim x → ∞ ( 2 cos ( 3 x ) ) = ⟨ − 2 , 2 ⟩ \lim_{x \to \infty}\left(2 \cos{\left (3 x \right )}\right) = \langle -2, 2\rangle x → ∞ lim ( 2 cos ( 3 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*cos(3*x), делённой на x при x->+oo и x ->-oolim x → − ∞ ( 2 x cos ( 3 x ) ) = 0 \lim_{x \to -\infty}\left(\frac{2}{x} \cos{\left (3 x \right )}\right) = 0 x → − ∞ lim ( x 2 cos ( 3 x ) ) = 0 Возьмём предел lim x → ∞ ( 2 x cos ( 3 x ) ) = 0 \lim_{x \to \infty}\left(\frac{2}{x} \cos{\left (3 x \right )}\right) = 0 x → ∞ lim ( x 2 cos ( 3 x ) ) = 0 Возьмём предел
Чётность и нечётность функции
Проверим функци чётна или нечётна с помощью соотношений f = f(-x) и f = -f(-x).2 cos ( 3 x ) = 2 cos ( 3 x ) 2 \cos{\left (3 x \right )} = 2 \cos{\left (3 x \right )} 2 cos ( 3 x ) = 2 cos ( 3 x ) 2 cos ( 3 x ) = − 2 cos ( 3 x ) 2 \cos{\left (3 x \right )} = - 2 \cos{\left (3 x \right )} 2 cos ( 3 x ) = − 2 cos ( 3 x ) 