График функции
0 -3500 -3000 -2500 -2000 -1500 -1000 -500 500 1000 1500 2000 2500 3000 3500 0.25 -0.25
Область определения функции
Точки, в которых функция точно неопределена:x 1 = 0 x_{1} = 0 x 1 = 0
Точки пересечения с осью координат X
График функции пересекает ось X при f = 01 x ( cos ( x ) − 1 ) = 0 \frac{1}{x} \left(\cos{\left (x \right )} - 1\right) = 0 x 1 ( cos ( x ) − 1 ) = 0 Решаем это уравнение Аналитическое решение x 1 = 2 π x_{1} = 2 \pi x 1 = 2 π Численное решение x 1 = 12.5663704427 x_{1} = 12.5663704427 x 1 = 12.5663704427 x 2 = 62.8318535568 x_{2} = 62.8318535568 x 2 = 62.8318535568 x 3 = 56.5486682809 x_{3} = 56.5486682809 x 3 = 56.5486682809 x 4 = 25.1327408328 x_{4} = 25.1327408328 x 4 = 25.1327408328 x 5 = 56.5486668532 x_{5} = 56.5486668532 x 5 = 56.5486668532 x 6 = 87.9645938122 x_{6} = 87.9645938122 x 6 = 87.9645938122 x 7 = − 6.28318581605 x_{7} = -6.28318581605 x 7 = − 6.28318581605 x 8 = 25.1327416384 x_{8} = 25.1327416384 x 8 = 25.1327416384 x 9 = − 37.699111348 x_{9} = -37.699111348 x 9 = − 37.699111348 x 10 = − 94.2477794453 x_{10} = -94.2477794453 x 10 = − 94.2477794453 x 11 = 50.265482944 x_{11} = 50.265482944 x 11 = 50.265482944 x 12 = 94.2477792651 x_{12} = 94.2477792651 x 12 = 94.2477792651 x 13 = 69.1150387941 x_{13} = 69.1150387941 x 13 = 69.1150387941 x 14 = − 43.9822967932 x_{14} = -43.9822967932 x 14 = − 43.9822967932 x 15 = 31.415926846 x_{15} = 31.415926846 x 15 = 31.415926846 x 16 = − 69.1150379045 x_{16} = -69.1150379045 x 16 = − 69.1150379045 x 17 = 87.9645946044 x_{17} = 87.9645946044 x 17 = 87.9645946044 x 18 = 31.4159260649 x_{18} = 31.4159260649 x 18 = 31.4159260649 x 19 = 50.2654821323 x_{19} = 50.2654821323 x 19 = 50.2654821323 x 20 = 6.28318579822 x_{20} = 6.28318579822 x 20 = 6.28318579822 x 21 = 56.5486680806 x_{21} = 56.5486680806 x 21 = 56.5486680806 x 22 = 12.5663710111 x_{22} = 12.5663710111 x 22 = 12.5663710111 x 23 = − 18.849556323 x_{23} = -18.849556323 x 23 = − 18.849556323 x 24 = − 81.6814090382 x_{24} = -81.6814090382 x 24 = − 81.6814090382 x 25 = − 6.2831855585 x_{25} = -6.2831855585 x 25 = − 6.2831855585 x 26 = 12.5663711302 x_{26} = 12.5663711302 x 26 = 12.5663711302 x 27 = 75.3982232189 x_{27} = 75.3982232189 x 27 = 75.3982232189 x 28 = − 87.9645947692 x_{28} = -87.9645947692 x 28 = − 87.9645947692 x 29 = − 4.79511606159 ⋅ 1 0 − 7 x_{29} = -4.79511606159 \cdot 10^{-7} x 29 = − 4.79511606159 ⋅ 1 0 − 7 x 30 = 6.28318626748 x_{30} = 6.28318626748 x 30 = 6.28318626748 x 31 = − 37.6991121287 x_{31} = -37.6991121287 x 31 = − 37.6991121287 x 32 = 37.6991113349 x_{32} = 37.6991113349 x 32 = 37.6991113349 x 33 = − 87.9645939285 x_{33} = -87.9645939285 x 33 = − 87.9645939285 x 34 = 81.6814085861 x_{34} = 81.6814085861 x 34 = 81.6814085861 x 35 = − 75.3982238742 x_{35} = -75.3982238742 x 35 = − 75.3982238742 x 36 = − 62.8318534787 x_{36} = -62.8318534787 x 36 = − 62.8318534787 x 37 = − 50.2654829667 x_{37} = -50.2654829667 x 37 = − 50.2654829667 x 38 = 94.2477800893 x_{38} = 94.2477800893 x 38 = 94.2477800893 x 39 = 56.5486676012 x_{39} = 56.5486676012 x 39 = 56.5486676012 x 40 = 43.9822974734 x_{40} = 43.9822974734 x 40 = 43.9822974734 x 41 = − 37.6991118773 x_{41} = -37.6991118773 x 41 = − 37.6991118773 x 42 = − 12.5663703113 x_{42} = -12.5663703113 x 42 = − 12.5663703113 x 43 = − 18.8495552124 x_{43} = -18.8495552124 x 43 = − 18.8495552124 x 44 = 0 x_{44} = 0 x 44 = 0 x 45 = 100.53096476 x_{45} = 100.53096476 x 45 = 100.53096476 x 46 = − 81.6814092565 x_{46} = -81.6814092565 x 46 = − 81.6814092565 x 47 = − 69.1150386869 x_{47} = -69.1150386869 x 47 = − 69.1150386869 x 48 = 6.28318528421 x_{48} = 6.28318528421 x 48 = 6.28318528421 x 49 = 3.40772025643 ⋅ 1 0 − 7 x_{49} = 3.40772025643 \cdot 10^{-7} x 49 = 3.40772025643 ⋅ 1 0 − 7 x 50 = − 75.3982231046 x_{50} = -75.3982231046 x 50 = − 75.3982231046 x 51 = − 62.8318526732 x_{51} = -62.8318526732 x 51 = − 62.8318526732 x 52 = − 81.6814084946 x_{52} = -81.6814084946 x 52 = − 81.6814084946 x 53 = − 81.6814075578 x_{53} = -81.6814075578 x 53 = − 81.6814075578 x 54 = 37.6991120311 x_{54} = 37.6991120311 x 54 = 37.6991120311 x 55 = 37.6991115174 x_{55} = 37.6991115174 x 55 = 37.6991115174 x 56 = − 75.398223172 x_{56} = -75.398223172 x 56 = − 75.398223172 x 57 = 75.3982240032 x_{57} = 75.3982240032 x 57 = 75.3982240032 x 58 = − 31.4159260508 x_{58} = -31.4159260508 x 58 = − 31.4159260508 x 59 = 62.8318527849 x_{59} = 62.8318527849 x 59 = 62.8318527849 x 60 = − 50.2654826411 x_{60} = -50.2654826411 x 60 = − 50.2654826411 x 61 = − 50.2654822863 x_{61} = -50.2654822863 x 61 = − 50.2654822863 x 62 = − 18.8495555173 x_{62} = -18.8495555173 x 62 = − 18.8495555173 x 63 = − 31.4159260208 x_{63} = -31.4159260208 x 63 = − 31.4159260208 x 64 = − 25.1327407506 x_{64} = -25.1327407506 x 64 = − 25.1327407506 x 65 = 6.28318500094 x_{65} = 6.28318500094 x 65 = 6.28318500094 x 66 = 69.1150379888 x_{66} = 69.1150379888 x 66 = 69.1150379888 x 67 = − 12.566371089 x_{67} = -12.566371089 x 67 = − 12.566371089 x 68 = − 56.5486674686 x_{68} = -56.5486674686 x 68 = − 56.5486674686 x 69 = − 43.9822971745 x_{69} = -43.9822971745 x 69 = − 43.9822971745 x 70 = 94.2477796094 x_{70} = 94.2477796094 x 70 = 94.2477796094 x 71 = − 100.530964626 x_{71} = -100.530964626 x 71 = − 100.530964626 x 72 = − 56.5486682427 x_{72} = -56.5486682427 x 72 = − 56.5486682427 x 73 = 50.2654824463 x_{73} = 50.2654824463 x 73 = 50.2654824463 x 74 = 43.9822966661 x_{74} = 43.9822966661 x 74 = 43.9822966661 x 75 = − 31.4159267158 x_{75} = -31.4159267158 x 75 = − 31.4159267158 x 76 = − 94.2477797298 x_{76} = -94.2477797298 x 76 = − 94.2477797298 x 77 = − 25.1327415297 x_{77} = -25.1327415297 x 77 = − 25.1327415297 x 78 = 81.6814091897 x_{78} = 81.6814091897 x 78 = 81.6814091897 x 79 = 100.530965157 x_{79} = 100.530965157 x 79 = 100.530965157 x 80 = − 6.28318512755 x_{80} = -6.28318512755 x 80 = − 6.28318512755 x 81 = − 94.2477801172 x_{81} = -94.2477801172 x 81 = − 94.2477801172 x 82 = 43.9822971695 x_{82} = 43.9822971695 x 82 = 43.9822971695 x 83 = − 87.9645943586 x_{83} = -87.9645943586 x 83 = − 87.9645943586 x 84 = − 43.9822976246 x_{84} = -43.9822976246 x 84 = − 43.9822976246 x 85 = 87.9645943359 x_{85} = 87.9645943359 x 85 = 87.9645943359 x 86 = 18.8495564032 x_{86} = 18.8495564032 x 86 = 18.8495564032 x 87 = 18.8495556276 x_{87} = 18.8495556276 x 87 = 18.8495556276 x 88 = 81.681408486 x_{88} = 81.681408486 x 88 = 81.681408486
Точки пересечения с осью координат Y
График пересекает ось Y, когда x равняется 0:1 0 ( − 1 + cos ( 0 ) ) \frac{1}{0} \left(-1 + \cos{\left (0 \right )}\right) 0 1 ( − 1 + cos ( 0 ) ) f ( 0 ) = N a N f{\left (0 \right )} = \mathrm{NaN} f ( 0 ) = NaN
Горизонтальные асимптоты
Горизонтальные асимптоты найдём с помощью пределов данной функции при x->+oo и x->-oolim x → − ∞ ( 1 x ( cos ( x ) − 1 ) ) = 0 \lim_{x \to -\infty}\left(\frac{1}{x} \left(\cos{\left (x \right )} - 1\right)\right) = 0 x → − ∞ lim ( x 1 ( cos ( x ) − 1 ) ) = 0 Возьмём предел y = 0 y = 0 y = 0 lim x → ∞ ( 1 x ( cos ( x ) − 1 ) ) = 0 \lim_{x \to \infty}\left(\frac{1}{x} \left(\cos{\left (x \right )} - 1\right)\right) = 0 x → ∞ lim ( x 1 ( cos ( x ) − 1 ) ) = 0 Возьмём предел y = 0 y = 0 y = 0
Наклонные асимптоты
Наклонную асимптоту можно найти, подсчитав предел функции (cos(x) - 1)/x, делённой на x при x->+oo и x ->-oolim x → − ∞ ( 1 x 2 ( cos ( x ) − 1 ) ) = 0 \lim_{x \to -\infty}\left(\frac{1}{x^{2}} \left(\cos{\left (x \right )} - 1\right)\right) = 0 x → − ∞ lim ( x 2 1 ( cos ( x ) − 1 ) ) = 0 Возьмём предел lim x → ∞ ( 1 x 2 ( cos ( x ) − 1 ) ) = 0 \lim_{x \to \infty}\left(\frac{1}{x^{2}} \left(\cos{\left (x \right )} - 1\right)\right) = 0 x → ∞ lim ( x 2 1 ( 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).1 x ( cos ( x ) − 1 ) = − 1 x ( cos ( x ) − 1 ) \frac{1}{x} \left(\cos{\left (x \right )} - 1\right) = - \frac{1}{x} \left(\cos{\left (x \right )} - 1\right) x 1 ( cos ( x ) − 1 ) = − x 1 ( cos ( x ) − 1 ) 1 x ( cos ( x ) − 1 ) = − 1 x ( − cos ( x ) + 1 ) \frac{1}{x} \left(\cos{\left (x \right )} - 1\right) = - \frac{1}{x} \left(- \cos{\left (x \right )} + 1\right) x 1 ( cos ( x ) − 1 ) = − x 1 ( − cos ( x ) + 1 ) 