График функции
0 5 -5 10 15 20 25 30 35 40 45 50 55 60 65 -10 2 -2
Точки пересечения с осью координат X
График функции пересекает ось X при f = 0cos ( 5 x ) = 0 \cos{\left(5 x \right)} = 0 cos ( 5 x ) = 0 Решаем это уравнение Аналитическое решение x 1 = π 10 x_{1} = \frac{\pi}{10} x 1 = 10 π x 2 = 3 π 10 x_{2} = \frac{3 \pi}{10} x 2 = 10 3 π Численное решение x 1 = 17.9070781254618 x_{1} = 17.9070781254618 x 1 = 17.9070781254618 x 2 = 65.0309679293087 x_{2} = 65.0309679293087 x 2 = 65.0309679293087 x 3 = 36.1283155162826 x_{3} = 36.1283155162826 x 3 = 36.1283155162826 x 4 = − 65.6592864600267 x_{4} = -65.6592864600267 x 4 = − 65.6592864600267 x 5 = − 76.9690200129499 x_{5} = -76.9690200129499 x 5 = − 76.9690200129499 x 6 = 61.8893752757189 x_{6} = 61.8893752757189 x 6 = 61.8893752757189 x 7 = 48.0663675999238 x_{7} = 48.0663675999238 x 7 = 48.0663675999238 x 8 = − 71.9424717672063 x_{8} = -71.9424717672063 x 8 = − 71.9424717672063 x 9 = 46.18141200777 x_{9} = 46.18141200777 x 9 = 46.18141200777 x 10 = 98.3318500573605 x_{10} = 98.3318500573605 x 10 = 98.3318500573605 x 11 = 27.3318560862312 x_{11} = 27.3318560862312 x 11 = 27.3318560862312 x 12 = − 14.1371669411541 x_{12} = -14.1371669411541 x 12 = − 14.1371669411541 x 13 = − 16.0221225333079 x_{13} = -16.0221225333079 x 13 = − 16.0221225333079 x 14 = − 53.0929158456675 x_{14} = -53.0929158456675 x 14 = − 53.0929158456675 x 15 = 60.0044196835651 x_{15} = 60.0044196835651 x 15 = 60.0044196835651 x 16 = − 9.73893722612836 x_{16} = -9.73893722612836 x 16 = − 9.73893722612836 x 17 = 95.1902574037707 x_{17} = 95.1902574037707 x 17 = 95.1902574037707 x 18 = 56.2345084992573 x_{18} = 56.2345084992573 x 18 = 56.2345084992573 x 19 = − 95.8185759344887 x_{19} = -95.8185759344887 x 19 = − 95.8185759344887 x 20 = 32.3584043319749 x_{20} = 32.3584043319749 x 20 = 32.3584043319749 x 21 = 58.1194640914112 x_{21} = 58.1194640914112 x 21 = 58.1194640914112 x 22 = 10.3672557568463 x_{22} = 10.3672557568463 x 22 = 10.3672557568463 x 23 = 22.3053078404875 x_{23} = 22.3053078404875 x 23 = 22.3053078404875 x 24 = 14.1371669411541 x_{24} = 14.1371669411541 x 24 = 14.1371669411541 x 25 = − 61.8893752757189 x_{25} = -61.8893752757189 x 25 = − 61.8893752757189 x 26 = − 58.1194640914112 x_{26} = -58.1194640914112 x 26 = − 58.1194640914112 x 27 = − 33.6150413934108 x_{27} = -33.6150413934108 x 27 = − 33.6150413934108 x 28 = − 39.8982267005904 x_{28} = -39.8982267005904 x 28 = − 39.8982267005904 x 29 = − 7.85398163397448 x_{29} = -7.85398163397448 x 29 = − 7.85398163397448 x 30 = 66.2876049907446 x_{30} = 66.2876049907446 x 30 = 66.2876049907446 x 31 = 92.0486647501809 x_{31} = 92.0486647501809 x 31 = 92.0486647501809 x 32 = 71.9424717672063 x_{32} = 71.9424717672063 x 32 = 71.9424717672063 x 33 = 81.9955682586936 x_{33} = 81.9955682586936 x 33 = 81.9955682586936 x 34 = − 53.7212343763855 x_{34} = -53.7212343763855 x 34 = − 53.7212343763855 x 35 = − 48.0663675999238 x_{35} = -48.0663675999238 x 35 = − 48.0663675999238 x 36 = 90.1637091580271 x_{36} = 90.1637091580271 x 36 = 90.1637091580271 x 37 = − 5.96902604182061 x_{37} = -5.96902604182061 x 37 = − 5.96902604182061 x 38 = − 63.7743308678728 x_{38} = -63.7743308678728 x 38 = − 63.7743308678728 x 39 = − 80.1106126665397 x_{39} = -80.1106126665397 x 39 = − 80.1106126665397 x 40 = 88.2787535658732 x_{40} = 88.2787535658732 x 40 = 88.2787535658732 x 41 = 16.0221225333079 x_{41} = 16.0221225333079 x 41 = 16.0221225333079 x 42 = 4.08407044966673 x_{42} = 4.08407044966673 x 42 = 4.08407044966673 x 43 = − 27.9601746169492 x_{43} = -27.9601746169492 x 43 = − 27.9601746169492 x 44 = 49.9513231920777 x_{44} = 49.9513231920777 x 44 = 49.9513231920777 x 45 = − 38.0132711084365 x_{45} = -38.0132711084365 x 45 = − 38.0132711084365 x 46 = − 75.712382951514 x_{46} = -75.712382951514 x 46 = − 75.712382951514 x 47 = − 11.6238928182822 x_{47} = -11.6238928182822 x 47 = − 11.6238928182822 x 48 = − 19.7920337176157 x_{48} = -19.7920337176157 x 48 = − 19.7920337176157 x 49 = − 69.4291976443344 x_{49} = -69.4291976443344 x 49 = − 69.4291976443344 x 50 = − 21.6769893097696 x_{50} = -21.6769893097696 x 50 = − 21.6769893097696 x 51 = 12.2522113490002 x_{51} = 12.2522113490002 x 51 = 12.2522113490002 x 52 = 93.9336203423348 x_{52} = 93.9336203423348 x 52 = 93.9336203423348 x 53 = 2.19911485751286 x_{53} = 2.19911485751286 x 53 = 2.19911485751286 x 54 = 38.0132711084365 x_{54} = 38.0132711084365 x 54 = 38.0132711084365 x 55 = 76.340701482232 x_{55} = 76.340701482232 x 55 = 76.340701482232 x 56 = 70.0575161750524 x_{56} = 70.0575161750524 x 56 = 70.0575161750524 x 57 = − 83.8805238508475 x_{57} = -83.8805238508475 x 57 = − 83.8805238508475 x 58 = 68.1725605828985 x_{58} = 68.1725605828985 x 58 = 68.1725605828985 x 59 = − 92.0486647501809 x_{59} = -92.0486647501809 x 59 = − 92.0486647501809 x 60 = 7.22566310325652 x_{60} = 7.22566310325652 x 60 = 7.22566310325652 x 61 = − 49.9513231920777 x_{61} = -49.9513231920777 x 61 = − 49.9513231920777 x 62 = − 17.9070781254618 x_{62} = -17.9070781254618 x 62 = − 17.9070781254618 x 63 = − 87.6504350351552 x_{63} = -87.6504350351552 x 63 = − 87.6504350351552 x 64 = 44.2964564156161 x_{64} = 44.2964564156161 x 64 = 44.2964564156161 x 65 = − 29.845130209103 x_{65} = -29.845130209103 x 65 = − 29.845130209103 x 66 = 88.9070720965912 x_{66} = 88.9070720965912 x 66 = 88.9070720965912 x 67 = 5.96902604182061 x_{67} = 5.96902604182061 x 67 = 5.96902604182061 x 68 = − 70.0575161750524 x_{68} = -70.0575161750524 x 68 = − 70.0575161750524 x 69 = − 31.7300858012569 x_{69} = -31.7300858012569 x 69 = − 31.7300858012569 x 70 = 27.9601746169492 x_{70} = 27.9601746169492 x 70 = 27.9601746169492 x 71 = − 81.9955682586936 x_{71} = -81.9955682586936 x 71 = − 81.9955682586936 x 72 = 100.216805649514 x_{72} = 100.216805649514 x 72 = 100.216805649514 x 73 = − 60.0044196835651 x_{73} = -60.0044196835651 x 73 = − 60.0044196835651 x 74 = 72.5707902979242 x_{74} = 72.5707902979242 x 74 = 72.5707902979242 x 75 = 26.0752190247953 x_{75} = 26.0752190247953 x 75 = 26.0752190247953 x 76 = − 85.7654794430014 x_{76} = -85.7654794430014 x 76 = − 85.7654794430014 x 77 = − 26.0752190247953 x_{77} = -26.0752190247953 x 77 = − 26.0752190247953 x 78 = − 41.7831822927443 x_{78} = -41.7831822927443 x 78 = − 41.7831822927443 x 79 = − 73.8274273593601 x_{79} = -73.8274273593601 x 79 = − 73.8274273593601 x 80 = 0.314159265358979 x_{80} = 0.314159265358979 x 80 = 0.314159265358979 x 81 = − 4.08407044966673 x_{81} = -4.08407044966673 x 81 = − 4.08407044966673 x 82 = − 102.730079772386 x_{82} = -102.730079772386 x 82 = − 102.730079772386 x 83 = − 51.8362787842316 x_{83} = -51.8362787842316 x 83 = − 51.8362787842316 x 84 = 80.1106126665397 x_{84} = 80.1106126665397 x 84 = 80.1106126665397 x 85 = 21.0486707790516 x_{85} = 21.0486707790516 x 85 = 21.0486707790516 x 86 = − 93.9336203423348 x_{86} = -93.9336203423348 x 86 = − 93.9336203423348 x 87 = 39.8982267005904 x_{87} = 39.8982267005904 x 87 = 39.8982267005904 x 88 = 54.3495529071034 x_{88} = 54.3495529071034 x 88 = 54.3495529071034 x 89 = 78.2256570743859 x_{89} = 78.2256570743859 x 89 = 78.2256570743859 x 90 = − 36.1283155162826 x_{90} = -36.1283155162826 x 90 = − 36.1283155162826 x 91 = 83.8805238508475 x_{91} = 83.8805238508475 x 91 = 83.8805238508475 x 92 = − 97.7035315266426 x_{92} = -97.7035315266426 x 92 = − 97.7035315266426 x 93 = 34.2433599241287 x_{93} = 34.2433599241287 x 93 = 34.2433599241287 x 94 = − 103.358398303104 x_{94} = -103.358398303104 x 94 = − 103.358398303104 x 95 = 24.1902634326414 x_{95} = 24.1902634326414 x 95 = 24.1902634326414 x 96 = − 43.6681378848981 x_{96} = -43.6681378848981 x 96 = − 43.6681378848981
Точки пересечения с осью координат Y
График пересекает ось Y, когда x равняется 0:cos ( 5 ⋅ 0 ) \cos{\left(5 \cdot 0 \right)} cos ( 5 ⋅ 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 − 5 sin ( 5 x ) = 0 - 5 \sin{\left(5 x \right)} = 0 − 5 sin ( 5 x ) = 0 Решаем это уравнение x 1 = 0 x_{1} = 0 x 1 = 0 x 2 = π 5 x_{2} = \frac{\pi}{5} x 2 = 5 π (0, 1) pi
(--, -1)
5 Интервалы возрастания и убывания функции: x 1 = π 5 x_{1} = \frac{\pi}{5} x 1 = 5 π x 1 = 0 x_{1} = 0 x 1 = 0 ( − ∞ , 0 ] ∪ [ π 5 , ∞ ) \left(-\infty, 0\right] \cup \left[\frac{\pi}{5}, \infty\right) ( − ∞ , 0 ] ∪ [ 5 π , ∞ ) [ 0 , π 5 ] \left[0, \frac{\pi}{5}\right] [ 0 , 5 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 − 25 cos ( 5 x ) = 0 - 25 \cos{\left(5 x \right)} = 0 − 25 cos ( 5 x ) = 0 Решаем это уравнение x 1 = π 10 x_{1} = \frac{\pi}{10} x 1 = 10 π x 2 = 3 π 10 x_{2} = \frac{3 \pi}{10} x 2 = 10 3 π Интервалы выпуклости и вогнутости: [ π 10 , 3 π 10 ] \left[\frac{\pi}{10}, \frac{3 \pi}{10}\right] [ 10 π , 10 3 π ] ( − ∞ , π 10 ] ∪ [ 3 π 10 , ∞ ) \left(-\infty, \frac{\pi}{10}\right] \cup \left[\frac{3 \pi}{10}, \infty\right) ( − ∞ , 10 π ] ∪ [ 10 3 π , ∞ )
Горизонтальные асимптоты
Горизонтальные асимптоты найдём с помощью пределов данной функции при x->+oo и x->-oolim x → − ∞ cos ( 5 x ) = ⟨ − 1 , 1 ⟩ \lim_{x \to -\infty} \cos{\left(5 x \right)} = \left\langle -1, 1\right\rangle x → − ∞ lim cos ( 5 x ) = ⟨ − 1 , 1 ⟩ Возьмём предел y = ⟨ − 1 , 1 ⟩ y = \left\langle -1, 1\right\rangle y = ⟨ − 1 , 1 ⟩ lim x → ∞ cos ( 5 x ) = ⟨ − 1 , 1 ⟩ \lim_{x \to \infty} \cos{\left(5 x \right)} = \left\langle -1, 1\right\rangle x → ∞ lim cos ( 5 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(5*x), делённой на x при x->+oo и x ->-oolim x → − ∞ ( cos ( 5 x ) x ) = 0 \lim_{x \to -\infty}\left(\frac{\cos{\left(5 x \right)}}{x}\right) = 0 x → − ∞ lim ( x cos ( 5 x ) ) = 0 Возьмём предел lim x → ∞ ( cos ( 5 x ) x ) = 0 \lim_{x \to \infty}\left(\frac{\cos{\left(5 x \right)}}{x}\right) = 0 x → ∞ lim ( x cos ( 5 x ) ) = 0 Возьмём предел
Чётность и нечётность функции
Проверим функци чётна или нечётна с помощью соотношений f = f(-x) и f = -f(-x).cos ( 5 x ) = cos ( 5 x ) \cos{\left(5 x \right)} = \cos{\left(5 x \right)} cos ( 5 x ) = cos ( 5 x ) cos ( 5 x ) = − cos ( 5 x ) \cos{\left(5 x \right)} = - \cos{\left(5 x \right)} cos ( 5 x ) = − cos ( 5 x ) 
