График функции
0 -40 -30 -20 -10 10 20 30 40 50 60 70 80 2 -2
Точки пересечения с осью координат X
График функции пересекает ось X при f = 0cos ( 3 x ) = 0 \cos{\left(3 x \right)} = 0 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 = − 45.553093477052 x_{1} = -45.553093477052 x 1 = − 45.553093477052 x 2 = − 19.3731546971371 x_{2} = -19.3731546971371 x 2 = − 19.3731546971371 x 3 = 86.3937979737193 x_{3} = 86.3937979737193 x 3 = 86.3937979737193 x 4 = 40.317105721069 x_{4} = 40.317105721069 x 4 = 40.317105721069 x 5 = 5.75958653158129 x_{5} = 5.75958653158129 x 5 = 5.75958653158129 x 6 = 91.6297857297023 x_{6} = 91.6297857297023 x 6 = 91.6297857297023 x 7 = 15.1843644923507 x_{7} = 15.1843644923507 x 7 = 15.1843644923507 x 8 = − 3.66519142918809 x_{8} = -3.66519142918809 x 8 = − 3.66519142918809 x 9 = 1.5707963267949 x_{9} = 1.5707963267949 x 9 = 1.5707963267949 x 10 = 36.1283155162826 x_{10} = 36.1283155162826 x 10 = 36.1283155162826 x 11 = 4.71238898038469 x_{11} = 4.71238898038469 x 11 = 4.71238898038469 x 12 = − 7.85398163397448 x_{12} = -7.85398163397448 x 12 = − 7.85398163397448 x 13 = 22.5147473507269 x_{13} = 22.5147473507269 x 13 = 22.5147473507269 x 14 = 95.8185759344887 x_{14} = 95.8185759344887 x 14 = 95.8185759344887 x 15 = 88.4881930761125 x_{15} = 88.4881930761125 x 15 = 88.4881930761125 x 16 = − 73.8274273593601 x_{16} = -73.8274273593601 x 16 = − 73.8274273593601 x 17 = − 23.5619449019235 x_{17} = -23.5619449019235 x 17 = − 23.5619449019235 x 18 = 84.2994028713261 x_{18} = 84.2994028713261 x 18 = 84.2994028713261 x 19 = − 93.7241808320955 x_{19} = -93.7241808320955 x 19 = − 93.7241808320955 x 20 = 82.2050077689329 x_{20} = 82.2050077689329 x 20 = 82.2050077689329 x 21 = 0.523598775598299 x_{21} = 0.523598775598299 x 21 = 0.523598775598299 x 22 = − 36.1283155162826 x_{22} = -36.1283155162826 x 22 = − 36.1283155162826 x 23 = − 58.1194640914112 x_{23} = -58.1194640914112 x 23 = − 58.1194640914112 x 24 = 42.4115008234622 x_{24} = 42.4115008234622 x 24 = 42.4115008234622 x 25 = − 53.9306738866248 x_{25} = -53.9306738866248 x 25 = − 53.9306738866248 x 26 = − 34.0339204138894 x_{26} = -34.0339204138894 x 26 = − 34.0339204138894 x 27 = 7.85398163397448 x_{27} = 7.85398163397448 x 27 = 7.85398163397448 x 28 = − 14.1371669411541 x_{28} = -14.1371669411541 x 28 = − 14.1371669411541 x 29 = − 61.261056745001 x_{29} = -61.261056745001 x 29 = − 61.261056745001 x 30 = 20.4203522483337 x_{30} = 20.4203522483337 x 30 = 20.4203522483337 x 31 = − 12.0427718387609 x_{31} = -12.0427718387609 x 31 = − 12.0427718387609 x 32 = − 21.4675497995303 x_{32} = -21.4675497995303 x 32 = − 21.4675497995303 x 33 = 26.7035375555132 x_{33} = 26.7035375555132 x 33 = 26.7035375555132 x 34 = − 56.025068989018 x_{34} = -56.025068989018 x 34 = − 56.025068989018 x 35 = − 27.7507351067098 x_{35} = -27.7507351067098 x 35 = − 27.7507351067098 x 36 = − 51.8362787842316 x_{36} = -51.8362787842316 x 36 = − 51.8362787842316 x 37 = − 31.9395253114962 x_{37} = -31.9395253114962 x 37 = − 31.9395253114962 x 38 = 53.9306738866248 x_{38} = 53.9306738866248 x 38 = 53.9306738866248 x 39 = 62.3082542961976 x_{39} = 62.3082542961976 x 39 = 62.3082542961976 x 40 = − 60.2138591938044 x_{40} = -60.2138591938044 x 40 = − 60.2138591938044 x 41 = 18.3259571459405 x_{41} = 18.3259571459405 x 41 = 18.3259571459405 x 42 = 27.7507351067098 x_{42} = 27.7507351067098 x 42 = 27.7507351067098 x 43 = 14.1371669411541 x_{43} = 14.1371669411541 x 43 = 14.1371669411541 x 44 = − 78.0162175641465 x_{44} = -78.0162175641465 x 44 = − 78.0162175641465 x 45 = − 16.2315620435473 x_{45} = -16.2315620435473 x 45 = − 16.2315620435473 x 46 = 41.3643032722656 x_{46} = 41.3643032722656 x 46 = 41.3643032722656 x 47 = − 98.9601685880785 x_{47} = -98.9601685880785 x 47 = − 98.9601685880785 x 48 = 44.5058959258554 x_{48} = 44.5058959258554 x 48 = 44.5058959258554 x 49 = − 43.4586983746588 x_{49} = -43.4586983746588 x 49 = − 43.4586983746588 x 50 = − 69.6386371545737 x_{50} = -69.6386371545737 x 50 = − 69.6386371545737 x 51 = − 67.5442420521806 x_{51} = -67.5442420521806 x 51 = − 67.5442420521806 x 52 = − 80.1106126665397 x_{52} = -80.1106126665397 x 52 = − 80.1106126665397 x 53 = 34.0339204138894 x_{53} = 34.0339204138894 x 53 = 34.0339204138894 x 54 = − 1.5707963267949 x_{54} = -1.5707963267949 x 54 = − 1.5707963267949 x 55 = − 71.733032256967 x_{55} = -71.733032256967 x 55 = − 71.733032256967 x 56 = 80.1106126665397 x_{56} = 80.1106126665397 x 56 = 80.1106126665397 x 57 = 78.0162175641465 x_{57} = 78.0162175641465 x 57 = 78.0162175641465 x 58 = 66.497044500984 x_{58} = 66.497044500984 x 58 = 66.497044500984 x 59 = 64.4026493985908 x_{59} = 64.4026493985908 x 59 = 64.4026493985908 x 60 = − 91.6297857297023 x_{60} = -91.6297857297023 x 60 = − 91.6297857297023 x 61 = − 25.6563400043166 x_{61} = -25.6563400043166 x 61 = − 25.6563400043166 x 62 = 1676.03968069015 x_{62} = 1676.03968069015 x 62 = 1676.03968069015 x 63 = 58.1194640914112 x_{63} = 58.1194640914112 x 63 = 58.1194640914112 x 64 = − 97.9129710368819 x_{64} = -97.9129710368819 x 64 = − 97.9129710368819 x 65 = − 82.2050077689329 x_{65} = -82.2050077689329 x 65 = − 82.2050077689329 x 66 = − 62.3082542961976 x_{66} = -62.3082542961976 x 66 = − 62.3082542961976 x 67 = − 65.4498469497874 x_{67} = -65.4498469497874 x 67 = − 65.4498469497874 x 68 = 60.2138591938044 x_{68} = 60.2138591938044 x 68 = 60.2138591938044 x 69 = 49.7418836818384 x_{69} = 49.7418836818384 x 69 = 49.7418836818384 x 70 = 100.007366139275 x_{70} = 100.007366139275 x 70 = 100.007366139275 x 71 = − 47.6474885794452 x_{71} = -47.6474885794452 x 71 = − 47.6474885794452 x 72 = 97.9129710368819 x_{72} = 97.9129710368819 x 72 = 97.9129710368819 x 73 = − 5.75958653158129 x_{73} = -5.75958653158129 x 73 = − 5.75958653158129 x 74 = 68.5914396033772 x_{74} = 68.5914396033772 x 74 = 68.5914396033772 x 75 = 56.025068989018 x_{75} = 56.025068989018 x 75 = 56.025068989018 x 76 = − 100.007366139275 x_{76} = -100.007366139275 x 76 = − 100.007366139275 x 77 = 73.8274273593601 x_{77} = 73.8274273593601 x 77 = 73.8274273593601 x 78 = 9.94837673636768 x_{78} = 9.94837673636768 x 78 = 9.94837673636768 x 79 = − 95.8185759344887 x_{79} = -95.8185759344887 x 79 = − 95.8185759344887 x 80 = 31.9395253114962 x_{80} = 31.9395253114962 x 80 = 31.9395253114962 x 81 = 75.9218224617533 x_{81} = 75.9218224617533 x 81 = 75.9218224617533 x 82 = − 89.5353906273091 x_{82} = -89.5353906273091 x 82 = − 89.5353906273091 x 83 = 71.733032256967 x_{83} = 71.733032256967 x 83 = 71.733032256967 x 84 = − 49.7418836818384 x_{84} = -49.7418836818384 x 84 = − 49.7418836818384 x 85 = 12.0427718387609 x_{85} = 12.0427718387609 x 85 = 12.0427718387609 x 86 = 29.845130209103 x_{86} = 29.845130209103 x 86 = 29.845130209103 x 87 = − 38.2227106186758 x_{87} = -38.2227106186758 x 87 = − 38.2227106186758 x 88 = 38.2227106186758 x_{88} = 38.2227106186758 x 88 = 38.2227106186758 x 89 = 87.4409955249159 x_{89} = 87.4409955249159 x 89 = 87.4409955249159 x 90 = 93.7241808320955 x_{90} = 93.7241808320955 x 90 = 93.7241808320955 x 91 = − 75.9218224617533 x_{91} = -75.9218224617533 x 91 = − 75.9218224617533 x 92 = − 29.845130209103 x_{92} = -29.845130209103 x 92 = − 29.845130209103 x 93 = − 84.2994028713261 x_{93} = -84.2994028713261 x 93 = − 84.2994028713261 x 94 = 16.2315620435473 x_{94} = 16.2315620435473 x 94 = 16.2315620435473 x 95 = − 9.94837673636768 x_{95} = -9.94837673636768 x 95 = − 9.94837673636768 x 96 = 51.8362787842316 x_{96} = 51.8362787842316 x 96 = 51.8362787842316
Точки пересечения с осью координат Y
График пересекает ось Y, когда x равняется 0:cos ( 3 ⋅ 0 ) \cos{\left(3 \cdot 0 \right)} cos ( 3 ⋅ 0 ) f ( 0 ) = 1 f{\left(0 \right)} = 1 f ( 0 ) = 1 (0, 1)
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 − 3 sin ( 3 x ) = 0 - 3 \sin{\left(3 x \right)} = 0 − 3 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, 1) pi
(--, -1)
3 Интервалы возрастания и убывания функции: x 1 = π 3 x_{1} = \frac{\pi}{3} x 1 = 3 π x 1 = 0 x_{1} = 0 x 1 = 0 ( − ∞ , 0 ] ∪ [ π 3 , ∞ ) \left(-\infty, 0\right] \cup \left[\frac{\pi}{3}, \infty\right) ( − ∞ , 0 ] ∪ [ 3 π , ∞ ) [ 0 , π 3 ] \left[0, \frac{\pi}{3}\right] [ 0 , 3 π ]
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 − 9 cos ( 3 x ) = 0 - 9 \cos{\left(3 x \right)} = 0 − 9 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: [ π 6 , π 2 ] \left[\frac{\pi}{6}, \frac{\pi}{2}\right] [ 6 π , 2 π ] ( − ∞ , π 6 ] ∪ [ π 2 , ∞ ) \left(-\infty, \frac{\pi}{6}\right] \cup \left[\frac{\pi}{2}, \infty\right) ( − ∞ , 6 π ] ∪ [ 2 π , ∞ )
Горизонтальные асимптоты
Горизонтальные асимптоты найдём с помощью пределов данной функции при x->+oo и x->-oolim x → − ∞ cos ( 3 x ) = ⟨ − 1 , 1 ⟩ \lim_{x \to -\infty} \cos{\left(3 x \right)} = \left\langle -1, 1\right\rangle x → − ∞ lim cos ( 3 x ) = ⟨ − 1 , 1 ⟩ Возьмём предел y = ⟨ − 1 , 1 ⟩ y = \left\langle -1, 1\right\rangle y = ⟨ − 1 , 1 ⟩ lim x → ∞ cos ( 3 x ) = ⟨ − 1 , 1 ⟩ \lim_{x \to \infty} \cos{\left(3 x \right)} = \left\langle -1, 1\right\rangle x → ∞ lim cos ( 3 x ) = ⟨ − 1 , 1 ⟩ Возьмём предел y = ⟨ − 1 , 1 ⟩ y = \left\langle -1, 1\right\rangle y = ⟨ − 1 , 1 ⟩
Наклонные асимптоты
Наклонную асимптоту можно найти, подсчитав предел функции cos(3*x), делённой на x при x->+oo и x ->-oolim x → − ∞ ( cos ( 3 x ) x ) = 0 \lim_{x \to -\infty}\left(\frac{\cos{\left(3 x \right)}}{x}\right) = 0 x → − ∞ lim ( x cos ( 3 x ) ) = 0 Возьмём предел lim x → ∞ ( cos ( 3 x ) x ) = 0 \lim_{x \to \infty}\left(\frac{\cos{\left(3 x \right)}}{x}\right) = 0 x → ∞ lim ( x 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).cos ( 3 x ) = cos ( 3 x ) \cos{\left(3 x \right)} = \cos{\left(3 x \right)} cos ( 3 x ) = cos ( 3 x ) cos ( 3 x ) = − cos ( 3 x ) \cos{\left(3 x \right)} = - \cos{\left(3 x \right)} cos ( 3 x ) = − cos ( 3 x ) 
