График функции
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Точки пересечения с осью координат X
График функции пересекает ось X при f = 0cos ( x ) + 1 = 0 \cos{\left(x \right)} + 1 = 0 cos ( x ) + 1 = 0 Решаем это уравнение Аналитическое решение x 1 = π x_{1} = \pi x 1 = π Численное решение x 1 = − 28.2743340989896 x_{1} = -28.2743340989896 x 1 = − 28.2743340989896 x 2 = 91.1061865667532 x_{2} = 91.1061865667532 x 2 = 91.1061865667532 x 3 = 9.42477826738203 x_{3} = 9.42477826738203 x 3 = 9.42477826738203 x 4 = 59.6902599104079 x_{4} = 59.6902599104079 x 4 = 59.6902599104079 x 5 = − 53.407075294995 x_{5} = -53.407075294995 x 5 = − 53.407075294995 x 6 = 53.4070746418597 x_{6} = 53.4070746418597 x 6 = 53.4070746418597 x 7 = 53.4070766553897 x_{7} = 53.4070766553897 x 7 = 53.4070766553897 x 8 = 53.407075424589 x_{8} = 53.407075424589 x 8 = 53.407075424589 x 9 = 91.1061873718352 x_{9} = 91.1061873718352 x 9 = 91.1061873718352 x 10 = 84.8230013636028 x_{10} = 84.8230013636028 x 10 = 84.8230013636028 x 11 = 78.5398161804942 x_{11} = 78.5398161804942 x 11 = 78.5398161804942 x 12 = 78.5398152766482 x_{12} = 78.5398152766482 x 12 = 78.5398152766482 x 13 = 59.6902606104322 x_{13} = 59.6902606104322 x 13 = 59.6902606104322 x 14 = − 9.4247781365785 x_{14} = -9.4247781365785 x 14 = − 9.4247781365785 x 15 = 15.7079627593774 x_{15} = 15.7079627593774 x 15 = 15.7079627593774 x 16 = 65.9734460390947 x_{16} = 65.9734460390947 x 16 = 65.9734460390947 x 17 = − 3.14159295109225 x_{17} = -3.14159295109225 x 17 = − 3.14159295109225 x 18 = − 97.3893716284562 x_{18} = -97.3893716284562 x 18 = − 97.3893716284562 x 19 = 78.5398166181283 x_{19} = 78.5398166181283 x 19 = 78.5398166181283 x 20 = − 34.5575196658297 x_{20} = -34.5575196658297 x 20 = − 34.5575196658297 x 21 = 65.9734452390837 x_{21} = 65.9734452390837 x 21 = 65.9734452390837 x 22 = 21.9911480932338 x_{22} = 21.9911480932338 x 22 = 21.9911480932338 x 23 = 78.5398168562347 x_{23} = 78.5398168562347 x 23 = 78.5398168562347 x 24 = 15.7079634518075 x_{24} = 15.7079634518075 x 24 = 15.7079634518075 x 25 = − 47.1238901083229 x_{25} = -47.1238901083229 x 25 = − 47.1238901083229 x 26 = − 59.6902604578012 x_{26} = -59.6902604578012 x 26 = − 59.6902604578012 x 27 = 65.9734457529812 x_{27} = 65.9734457529812 x 27 = 65.9734457529812 x 28 = 47.123889410773 x_{28} = 47.123889410773 x 28 = 47.123889410773 x 29 = 40.8407045792514 x_{29} = 40.8407045792514 x 29 = 40.8407045792514 x 30 = − 21.9911490521325 x_{30} = -21.9911490521325 x 30 = − 21.9911490521325 x 31 = − 59.6902606928653 x_{31} = -59.6902606928653 x 31 = − 59.6902606928653 x 32 = 21.9911485852059 x_{32} = 21.9911485852059 x 32 = 21.9911485852059 x 33 = − 65.9734461969855 x_{33} = -65.9734461969855 x 33 = − 65.9734461969855 x 34 = 40.8407045848602 x_{34} = 40.8407045848602 x 34 = 40.8407045848602 x 35 = − 15.707962774825 x_{35} = -15.707962774825 x 35 = − 15.707962774825 x 36 = − 91.1061864815274 x_{36} = -91.1061864815274 x 36 = − 91.1061864815274 x 37 = 34.5575195449229 x_{37} = 34.5575195449229 x 37 = 34.5575195449229 x 38 = − 28.2743343914215 x_{38} = -28.2743343914215 x 38 = − 28.2743343914215 x 39 = − 72.2566315419804 x_{39} = -72.2566315419804 x 39 = − 72.2566315419804 x 40 = 59.6902600526626 x_{40} = 59.6902600526626 x 40 = 59.6902600526626 x 41 = − 15.7079632965989 x_{41} = -15.7079632965989 x 41 = − 15.7079632965989 x 42 = 97.3893717959212 x_{42} = 97.3893717959212 x 42 = 97.3893717959212 x 43 = − 65.9734449870253 x_{43} = -65.9734449870253 x 43 = − 65.9734449870253 x 44 = − 84.8230012511693 x_{44} = -84.8230012511693 x 44 = − 84.8230012511693 x 45 = − 72.2566311847166 x_{45} = -72.2566311847166 x 45 = − 72.2566311847166 x 46 = − 53.4070745963886 x_{46} = -53.4070745963886 x 46 = − 53.4070745963886 x 47 = 9.42477748794163 x_{47} = 9.42477748794163 x 47 = 9.42477748794163 x 48 = − 59.6902599212271 x_{48} = -59.6902599212271 x 48 = − 59.6902599212271 x 49 = 28.2743335663982 x_{49} = 28.2743335663982 x 49 = 28.2743335663982 x 50 = 3.1415922548952 x_{50} = 3.1415922548952 x 50 = 3.1415922548952 x 51 = 72.2566310277176 x_{51} = 72.2566310277176 x 51 = 72.2566310277176 x 52 = − 84.8230020565447 x_{52} = -84.8230020565447 x 52 = − 84.8230020565447 x 53 = − 34.5575188899093 x_{53} = -34.5575188899093 x 53 = − 34.5575188899093 x 54 = − 15.7079635641079 x_{54} = -15.7079635641079 x 54 = − 15.7079635641079 x 55 = − 78.5398160472843 x_{55} = -78.5398160472843 x 55 = − 78.5398160472843 x 56 = − 65.9734457649277 x_{56} = -65.9734457649277 x 56 = − 65.9734457649277 x 57 = 3.14159306054457 x_{57} = 3.14159306054457 x 57 = 3.14159306054457 x 58 = 34.5575197055812 x_{58} = 34.5575197055812 x 58 = 34.5575197055812 x 59 = − 53.4070745786761 x_{59} = -53.4070745786761 x 59 = − 53.4070745786761 x 60 = 47.1238902162437 x_{60} = 47.1238902162437 x 60 = 47.1238902162437 x 61 = 28.2743343711514 x_{61} = 28.2743343711514 x 61 = 28.2743343711514 x 62 = − 97.3893724533348 x_{62} = -97.3893724533348 x 62 = − 97.3893724533348 x 63 = − 47.1238893275319 x_{63} = -47.1238893275319 x 63 = − 47.1238893275319 x 64 = − 78.5398168194507 x_{64} = -78.5398168194507 x 64 = − 78.5398168194507 x 65 = 72.2566315166773 x_{65} = 72.2566315166773 x 65 = 72.2566315166773 x 66 = 40.8407042062167 x_{66} = 40.8407042062167 x 66 = 40.8407042062167 x 67 = 15.707963957033 x_{67} = 15.707963957033 x 67 = 15.707963957033 x 68 = − 21.991148226056 x_{68} = -21.991148226056 x 68 = − 21.991148226056 x 69 = − 3.14159217367683 x_{69} = -3.14159217367683 x 69 = − 3.14159217367683 x 70 = − 9.42477752082051 x_{70} = -9.42477752082051 x 70 = − 9.42477752082051 x 71 = − 21.9911485864417 x_{71} = -21.9911485864417 x 71 = − 21.9911485864417 x 72 = 28.2743338651796 x_{72} = 28.2743338651796 x 72 = 28.2743338651796 x 73 = 21.9911489072506 x_{73} = 21.9911489072506 x 73 = 21.9911489072506 x 74 = − 97.3893717476911 x_{74} = -97.3893717476911 x 74 = − 97.3893717476911 x 75 = 84.8230021335997 x_{75} = 84.8230021335997 x 75 = 84.8230021335997 x 76 = − 72.2566308657983 x_{76} = -72.2566308657983 x 76 = − 72.2566308657983 x 77 = 72.2566306985 x_{77} = 72.2566306985 x 77 = 72.2566306985 x 78 = 15.7079629803241 x_{78} = 15.7079629803241 x 78 = 15.7079629803241 x 79 = − 40.8407049290801 x_{79} = -40.8407049290801 x 79 = − 40.8407049290801 x 80 = − 9.42477744529557 x_{80} = -9.42477744529557 x 80 = − 9.42477744529557 x 81 = − 91.106187265474 x_{81} = -91.106187265474 x 81 = − 91.106187265474 x 82 = 78.5398149750205 x_{82} = 78.5398149750205 x 82 = 78.5398149750205 x 83 = − 1127.83176318906 x_{83} = -1127.83176318906 x 83 = − 1127.83176318906 x 84 = − 40.8407040952604 x_{84} = -40.8407040952604 x 84 = − 40.8407040952604 x 85 = 34.5575190219169 x_{85} = 34.5575190219169 x 85 = 34.5575190219169 x 86 = − 28.2743337069329 x_{86} = -28.2743337069329 x 86 = − 28.2743337069329 x 87 = 97.389372581711 x_{87} = 97.389372581711 x 87 = 97.389372581711 x 88 = − 65.9734453607004 x_{88} = -65.9734453607004 x 88 = − 65.9734453607004 x 89 = − 40.8407049008781 x_{89} = -40.8407049008781 x 89 = − 40.8407049008781 x 90 = 40.8407049800347 x_{90} = 40.8407049800347 x 90 = 40.8407049800347
Точки пересечения с осью координат Y
График пересекает ось Y, когда x равняется 0:1 + cos ( 0 ) 1 + \cos{\left(0 \right)} 1 + cos ( 0 ) 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 ) = 0 - \sin{\left(x \right)} = 0 − sin ( x ) = 0 Решаем это уравнение x 1 = 0 x_{1} = 0 x 1 = 0 x 2 = π x_{2} = \pi x 2 = π (0, 2) (pi, 0) Интервалы возрастания и убывания функции: x 1 = π x_{1} = \pi x 1 = π x 1 = 0 x_{1} = 0 x 1 = 0 ( − ∞ , 0 ] ∪ [ π , ∞ ) \left(-\infty, 0\right] \cup \left[\pi, \infty\right) ( − ∞ , 0 ] ∪ [ π , ∞ ) [ 0 , π ] \left[0, \pi\right] [ 0 , 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 − cos ( x ) = 0 - \cos{\left(x \right)} = 0 − cos ( x ) = 0 Решаем это уравнение x 1 = π 2 x_{1} = \frac{\pi}{2} x 1 = 2 π x 2 = 3 π 2 x_{2} = \frac{3 \pi}{2} x 2 = 2 3 π Интервалы выпуклости и вогнутости: [ π 2 , 3 π 2 ] \left[\frac{\pi}{2}, \frac{3 \pi}{2}\right] [ 2 π , 2 3 π ] ( − ∞ , π 2 ] ∪ [ 3 π 2 , ∞ ) \left(-\infty, \frac{\pi}{2}\right] \cup \left[\frac{3 \pi}{2}, \infty\right) ( − ∞ , 2 π ] ∪ [ 2 3 π , ∞ )
Горизонтальные асимптоты
Горизонтальные асимптоты найдём с помощью пределов данной функции при x->+oo и x->-oolim x → − ∞ ( cos ( x ) + 1 ) = ⟨ 0 , 2 ⟩ \lim_{x \to -\infty}\left(\cos{\left(x \right)} + 1\right) = \left\langle 0, 2\right\rangle x → − ∞ lim ( cos ( x ) + 1 ) = ⟨ 0 , 2 ⟩ Возьмём предел y = ⟨ 0 , 2 ⟩ y = \left\langle 0, 2\right\rangle y = ⟨ 0 , 2 ⟩ lim x → ∞ ( cos ( x ) + 1 ) = ⟨ 0 , 2 ⟩ \lim_{x \to \infty}\left(\cos{\left(x \right)} + 1\right) = \left\langle 0, 2\right\rangle x → ∞ lim ( cos ( x ) + 1 ) = ⟨ 0 , 2 ⟩ Возьмём предел y = ⟨ 0 , 2 ⟩ y = \left\langle 0, 2\right\rangle y = ⟨ 0 , 2 ⟩
Наклонные асимптоты
Наклонную асимптоту можно найти, подсчитав предел функции 1 + cos(x), делённой на x при x->+oo и x ->-oolim x → − ∞ ( cos ( x ) + 1 x ) = 0 \lim_{x \to -\infty}\left(\frac{\cos{\left(x \right)} + 1}{x}\right) = 0 x → − ∞ lim ( x cos ( x ) + 1 ) = 0 Возьмём предел lim x → ∞ ( cos ( x ) + 1 x ) = 0 \lim_{x \to \infty}\left(\frac{\cos{\left(x \right)} + 1}{x}\right) = 0 x → ∞ lim ( x cos ( x ) + 1 ) = 0 Возьмём предел
Чётность и нечётность функции
Проверим функци чётна или нечётна с помощью соотношений f = f(-x) и f = -f(-x).cos ( x ) + 1 = cos ( x ) + 1 \cos{\left(x \right)} + 1 = \cos{\left(x \right)} + 1 cos ( x ) + 1 = cos ( x ) + 1 cos ( x ) + 1 = − cos ( x ) − 1 \cos{\left(x \right)} + 1 = - \cos{\left(x \right)} - 1 cos ( x ) + 1 = − cos ( x ) − 1 
