График функции
0 -60 -50 -40 -30 -20 -10 10 20 30 40 50 60 0 4
Точки пересечения с осью координат X
График функции пересекает ось X при f = 0cos ( x 2 ) + 1 = 0 \cos{\left(\frac{x}{2} \right)} + 1 = 0 cos ( 2 x ) + 1 = 0 Решаем это уравнение Аналитическое решение x 1 = 2 π x_{1} = 2 \pi x 1 = 2 π Численное решение x 1 = 6.28318608026041 x_{1} = 6.28318608026041 x 1 = 6.28318608026041 x 2 = − 69.1150405916551 x_{2} = -69.1150405916551 x 2 = − 69.1150405916551 x 3 = 69.115037955532 x_{3} = 69.115037955532 x 3 = 69.115037955532 x 4 = − 94.2477805989545 x_{4} = -94.2477805989545 x 4 = − 94.2477805989545 x 5 = 94.2477805288104 x_{5} = 94.2477805288104 x 5 = 94.2477805288104 x 6 = − 6.28318627375063 x_{6} = -6.28318627375063 x 6 = − 6.28318627375063 x 7 = 81.6814079898679 x_{7} = 81.6814079898679 x 7 = 81.6814079898679 x 8 = 43.9822980146158 x_{8} = 43.9822980146158 x 8 = 43.9822980146158 x 9 = − 81.68140804614 x_{9} = -81.68140804614 x 9 = − 81.68140804614 x 10 = − 18.8495543136467 x_{10} = -18.8495543136467 x 10 = − 18.8495543136467 x 11 = 56.5486672894687 x_{11} = 56.5486672894687 x 11 = 56.5486672894687 x 12 = 94.2477803922665 x_{12} = 94.2477803922665 x 12 = 94.2477803922665 x 13 = 56.5486675992574 x_{13} = 56.5486675992574 x 13 = 56.5486675992574 x 14 = 6.28318454684724 x_{14} = 6.28318454684724 x 14 = 6.28318454684724 x 15 = − 56.548667456373 x_{15} = -56.548667456373 x 15 = − 56.548667456373 x 16 = 56.5486681672153 x_{16} = 56.5486681672153 x 16 = 56.5486681672153 x 17 = 31.4159255107699 x_{17} = 31.4159255107699 x 17 = 31.4159255107699 x 18 = 69.1150368714384 x_{18} = 69.1150368714384 x 18 = 69.1150368714384 x 19 = 18.8495549415754 x_{19} = 18.8495549415754 x 19 = 18.8495549415754 x 20 = 973.893720162307 x_{20} = 973.893720162307 x 20 = 973.893720162307 x 21 = − 43.9822980632848 x_{21} = -43.9822980632848 x 21 = − 43.9822980632848 x 22 = − 56.5486683277755 x_{22} = -56.5486683277755 x 22 = − 56.5486683277755 x 23 = 31.4159275256971 x_{23} = 31.4159275256971 x 23 = 31.4159275256971 x 24 = − 6.28318512498487 x_{24} = -6.28318512498487 x 24 = − 6.28318512498487 x 25 = − 6.28318500560576 x_{25} = -6.28318500560576 x 25 = − 6.28318500560576 x 26 = − 31.4159258676764 x_{26} = -31.4159258676764 x 26 = − 31.4159258676764 x 27 = 81.6814069775521 x_{27} = 81.6814069775521 x 27 = 81.6814069775521 x 28 = 18.8495569537229 x_{28} = 18.8495569537229 x 28 = 18.8495569537229 x 29 = 94.2477788210844 x_{29} = 94.2477788210844 x 29 = 94.2477788210844 x 30 = − 56.5486677946696 x_{30} = -56.5486677946696 x 30 = − 56.5486677946696 x 31 = − 69.1150393645777 x_{31} = -69.1150393645777 x 31 = − 69.1150393645777 x 32 = − 31.4159267991785 x_{32} = -31.4159267991785 x 32 = − 31.4159267991785 x 33 = − 94.2477802857943 x_{33} = -94.2477802857943 x 33 = − 94.2477802857943 x 34 = − 43.9822979202134 x_{34} = -43.9822979202134 x 34 = − 43.9822979202134 x 35 = − 69.1150378269834 x_{35} = -69.1150378269834 x 35 = − 69.1150378269834 x 36 = 6.28318624191326 x_{36} = 6.28318624191326 x 36 = 6.28318624191326 x 37 = − 18.8495548993085 x_{37} = -18.8495548993085 x 37 = − 18.8495548993085 x 38 = − 81.6814098544517 x_{38} = -81.6814098544517 x 38 = − 81.6814098544517 x 39 = 81.6814091930517 x_{39} = 81.6814091930517 x 39 = 81.6814091930517 x 40 = − 81.6814097251289 x_{40} = -81.6814097251289 x 40 = − 81.6814097251289 x 41 = 81.681409932803 x_{41} = 81.681409932803 x 41 = 81.681409932803 x 42 = 43.982297169474 x_{42} = 43.982297169474 x 42 = 43.982297169474 x 43 = − 18.8495554800568 x_{43} = -18.8495554800568 x 43 = − 18.8495554800568 x 44 = − 6.28318630492991 x_{44} = -6.28318630492991 x 44 = − 6.28318630492991 x 45 = − 81.6814090382823 x_{45} = -81.6814090382823 x 45 = − 81.6814090382823 x 46 = − 69.115038701061 x_{46} = -69.115038701061 x 46 = − 69.115038701061 x 47 = 94.2477796093523 x_{47} = 94.2477796093523 x 47 = 94.2477796093523 x 48 = 43.9822963919738 x_{48} = 43.9822963919738 x 48 = 43.9822963919738 x 49 = 81.6814094388795 x_{49} = 81.6814094388795 x 49 = 81.6814094388795 x 50 = − 18.8495563601837 x_{50} = -18.8495563601837 x 50 = − 18.8495563601837 x 51 = − 6.28318480992034 x_{51} = -6.28318480992034 x 51 = − 6.28318480992034 x 52 = − 31.4159269573538 x_{52} = -31.4159269573538 x 52 = − 31.4159269573538 x 53 = 69.1150397628024 x_{53} = 69.1150397628024 x 53 = 69.1150397628024 x 54 = 69.1150373651011 x_{54} = 69.1150373651011 x 54 = 69.1150373651011 x 55 = 43.9822979229286 x_{55} = 43.9822979229286 x 55 = 43.9822979229286 x 56 = − 94.2477786842917 x_{56} = -94.2477786842917 x 56 = − 94.2477786842917 x 57 = 94.2477787666659 x_{57} = 94.2477787666659 x 57 = 94.2477787666659 x 58 = 31.4159259869291 x_{58} = 31.4159259869291 x 58 = 31.4159259869291 x 59 = − 69.115037346238 x_{59} = -69.115037346238 x 59 = − 69.115037346238 x 60 = − 31.4159267183799 x_{60} = -31.4159267183799 x 60 = − 31.4159267183799 x 61 = 31.415926404003 x_{61} = 31.415926404003 x 61 = 31.415926404003 x 62 = 69.115038835587 x_{62} = 69.115038835587 x 62 = 69.115038835587 x 63 = − 6.28318437561383 x_{63} = -6.28318437561383 x 63 = − 6.28318437561383 x 64 = 56.548668769019 x_{64} = 56.548668769019 x 64 = 56.548668769019 x 65 = 56.5486684458067 x_{65} = 56.5486684458067 x 65 = 56.5486684458067 x 66 = 31.4159268602638 x_{66} = 31.4159268602638 x 66 = 31.4159268602638 x 67 = − 56.5486667805662 x_{67} = -56.5486667805662 x 67 = − 56.5486667805662 x 68 = − 18.8495572113627 x_{68} = -18.8495572113627 x 68 = − 18.8495572113627 x 69 = − 31.4159274639505 x_{69} = -31.4159274639505 x 69 = − 31.4159274639505 x 70 = 18.8495564877336 x_{70} = 18.8495564877336 x 70 = 18.8495564877336 x 71 = − 18.8495569383691 x_{71} = -18.8495569383691 x 71 = − 18.8495569383691 x 72 = − 81.6814082319334 x_{72} = -81.6814082319334 x 72 = − 81.6814082319334 x 73 = 81.6814083413212 x_{73} = 81.6814083413212 x 73 = 81.6814083413212 x 74 = − 94.2477794433785 x_{74} = -94.2477794433785 x 74 = − 94.2477794433785 x 75 = − 43.9822963521153 x_{75} = -43.9822963521153 x 75 = − 43.9822963521153 x 76 = 69.1150394039813 x_{76} = 69.1150394039813 x 76 = 69.1150394039813 x 77 = 18.8495554920234 x_{77} = 18.8495554920234 x 77 = 18.8495554920234 x 78 = 43.9822962232996 x_{78} = 43.9822962232996 x 78 = 43.9822962232996 x 79 = − 420.973417050673 x_{79} = -420.973417050673 x 79 = − 420.973417050673 x 80 = 6.28318445599649 x_{80} = 6.28318445599649 x 80 = 6.28318445599649 x 81 = 56.5486668446775 x_{81} = 56.5486668446775 x 81 = 56.5486668446775 x 82 = − 43.9822971745293 x_{82} = -43.9822971745293 x 82 = − 43.9822971745293 x 83 = − 31.4159255258029 x_{83} = -31.4159255258029 x 83 = − 31.4159255258029 x 84 = − 69.1150380827831 x_{84} = -69.1150380827831 x 84 = − 69.1150380827831 x 85 = 6.28318528419795 x_{85} = 6.28318528419795 x 85 = 6.28318528419795 x 86 = − 43.982296295854 x_{86} = -43.982296295854 x 86 = − 43.982296295854 x 87 = − 56.5486687887842 x_{87} = -56.5486687887842 x 87 = − 56.5486687887842 x 88 = − 6.28318597231491 x_{88} = -6.28318597231491 x 88 = − 6.28318597231491 x 89 = 18.8495561010751 x_{89} = 18.8495561010751 x 89 = 18.8495561010751 x 90 = 18.8495556154405 x_{90} = 18.8495556154405 x 90 = 18.8495556154405 x 91 = − 94.2477790629797 x_{91} = -94.2477790629797 x 91 = − 94.2477790629797
Точки пересечения с осью координат Y
График пересекает ось Y, когда x равняется 0:cos ( 0 2 ) + 1 \cos{\left(\frac{0}{2} \right)} + 1 cos ( 2 0 ) + 1 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 2 ) 2 = 0 - \frac{\sin{\left(\frac{x}{2} \right)}}{2} = 0 − 2 sin ( 2 x ) = 0 Решаем это уравнение x 1 = 0 x_{1} = 0 x 1 = 0 x 2 = 2 π x_{2} = 2 \pi x 2 = 2 π (0, 2) (2*pi, 0) Интервалы возрастания и убывания функции: x 1 = 2 π x_{1} = 2 \pi x 1 = 2 π x 1 = 0 x_{1} = 0 x 1 = 0 ( − ∞ , 0 ] ∪ [ 2 π , ∞ ) \left(-\infty, 0\right] \cup \left[2 \pi, \infty\right) ( − ∞ , 0 ] ∪ [ 2 π , ∞ ) [ 0 , 2 π ] \left[0, 2 \pi\right] [ 0 , 2 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 2 ) 4 = 0 - \frac{\cos{\left(\frac{x}{2} \right)}}{4} = 0 − 4 cos ( 2 x ) = 0 Решаем это уравнение x 1 = π x_{1} = \pi x 1 = π x 2 = 3 π x_{2} = 3 \pi x 2 = 3 π Интервалы выпуклости и вогнутости: [ π , 3 π ] \left[\pi, 3 \pi\right] [ π , 3 π ] ( − ∞ , π ] ∪ [ 3 π , ∞ ) \left(-\infty, \pi\right] \cup \left[3 \pi, \infty\right) ( − ∞ , π ] ∪ [ 3 π , ∞ )
Горизонтальные асимптоты
Горизонтальные асимптоты найдём с помощью пределов данной функции при x->+oo и x->-oolim x → − ∞ ( cos ( x 2 ) + 1 ) = ⟨ 0 , 2 ⟩ \lim_{x \to -\infty}\left(\cos{\left(\frac{x}{2} \right)} + 1\right) = \left\langle 0, 2\right\rangle x → − ∞ lim ( cos ( 2 x ) + 1 ) = ⟨ 0 , 2 ⟩ Возьмём предел y = ⟨ 0 , 2 ⟩ y = \left\langle 0, 2\right\rangle y = ⟨ 0 , 2 ⟩ lim x → ∞ ( cos ( x 2 ) + 1 ) = ⟨ 0 , 2 ⟩ \lim_{x \to \infty}\left(\cos{\left(\frac{x}{2} \right)} + 1\right) = \left\langle 0, 2\right\rangle x → ∞ lim ( cos ( 2 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 cos(x/2) + 1, делённой на x при x->+oo и x ->-oolim x → − ∞ ( cos ( x 2 ) + 1 x ) = 0 \lim_{x \to -\infty}\left(\frac{\cos{\left(\frac{x}{2} \right)} + 1}{x}\right) = 0 x → − ∞ lim ( x cos ( 2 x ) + 1 ) = 0 Возьмём предел lim x → ∞ ( cos ( x 2 ) + 1 x ) = 0 \lim_{x \to \infty}\left(\frac{\cos{\left(\frac{x}{2} \right)} + 1}{x}\right) = 0 x → ∞ lim ( x cos ( 2 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 2 ) + 1 = cos ( x 2 ) + 1 \cos{\left(\frac{x}{2} \right)} + 1 = \cos{\left(\frac{x}{2} \right)} + 1 cos ( 2 x ) + 1 = cos ( 2 x ) + 1 cos ( x 2 ) + 1 = − cos ( x 2 ) − 1 \cos{\left(\frac{x}{2} \right)} + 1 = - \cos{\left(\frac{x}{2} \right)} - 1 cos ( 2 x ) + 1 = − cos ( 2 x ) − 1 
