График функции
0 5 -30 -25 -20 -15 -10 -5 10 15 2 -2
Точки пересечения с осью координат X
График функции пересекает ось X при f = 0sin ( 5 x ) = 0 \sin{\left(5 x \right)} = 0 sin ( 5 x ) = 0 Решаем это уравнение Аналитическое решение x 1 = 0 x_{1} = 0 x 1 = 0 x 2 = π 5 x_{2} = \frac{\pi}{5} x 2 = 5 π Численное решение x 1 = 16.3362817986669 x_{1} = 16.3362817986669 x 1 = 16.3362817986669 x 2 = 5.65486677646163 x_{2} = 5.65486677646163 x 2 = 5.65486677646163 x 3 = − 32.0442450666159 x_{3} = -32.0442450666159 x 3 = − 32.0442450666159 x 4 = 60.318578948924 x_{4} = 60.318578948924 x 4 = 60.318578948924 x 5 = 84.1946831162065 x_{5} = 84.1946831162065 x 5 = 84.1946831162065 x 6 = 11.3097335529233 x_{6} = 11.3097335529233 x 6 = 11.3097335529233 x 7 = − 91.734505484822 x_{7} = -91.734505484822 x 7 = − 91.734505484822 x 8 = 50.2654824574367 x_{8} = 50.2654824574367 x 8 = 50.2654824574367 x 9 = 42.0973415581032 x_{9} = 42.0973415581032 x 9 = 42.0973415581032 x 10 = − 47.7522083345649 x_{10} = -47.7522083345649 x 10 = − 47.7522083345649 x 11 = 45.867252742411 x_{11} = 45.867252742411 x 11 = 45.867252742411 x 12 = − 79.7964534011807 x_{12} = -79.7964534011807 x 12 = − 79.7964534011807 x 13 = − 35.8141562509236 x_{13} = -35.8141562509236 x 13 = − 35.8141562509236 x 14 = 92.3628240155399 x_{14} = 92.3628240155399 x 14 = 92.3628240155399 x 15 = 10.0530964914873 x_{15} = 10.0530964914873 x 15 = 10.0530964914873 x 16 = − 57.8053048260522 x_{16} = -57.8053048260522 x 16 = − 57.8053048260522 x 17 = − 99.9026463841554 x_{17} = -99.9026463841554 x 17 = − 99.9026463841554 x 18 = 30.159289474462 x_{18} = 30.159289474462 x 18 = 30.159289474462 x 19 = 38.3274303737955 x_{19} = 38.3274303737955 x 19 = 38.3274303737955 x 20 = 33.9292006587698 x_{20} = 33.9292006587698 x 20 = 33.9292006587698 x 21 = − 65.9734457253857 x_{21} = -65.9734457253857 x 21 = − 65.9734457253857 x 22 = − 93.6194610769758 x_{22} = -93.6194610769758 x 22 = − 93.6194610769758 x 23 = 48.3805268652828 x_{23} = 48.3805268652828 x 23 = 48.3805268652828 x 24 = 87.9645943005142 x_{24} = 87.9645943005142 x 24 = 87.9645943005142 x 25 = − 5.65486677646163 x_{25} = -5.65486677646163 x 25 = − 5.65486677646163 x 26 = − 37.6991118430775 x_{26} = -37.6991118430775 x 26 = − 37.6991118430775 x 27 = 55.9203492338983 x_{27} = 55.9203492338983 x 27 = 55.9203492338983 x 28 = − 21.9911485751286 x_{28} = -21.9911485751286 x 28 = − 21.9911485751286 x 29 = 96.1327351998477 x_{29} = 96.1327351998477 x 29 = 96.1327351998477 x 30 = 64.0884901332318 x_{30} = 64.0884901332318 x 30 = 64.0884901332318 x 31 = − 49.6371639267187 x_{31} = -49.6371639267187 x 31 = − 49.6371639267187 x 32 = − 71.6283125018473 x_{32} = -71.6283125018473 x 32 = − 71.6283125018473 x 33 = − 67.8584013175395 x_{33} = -67.8584013175395 x 33 = − 67.8584013175395 x 34 = 62.2035345410779 x_{34} = 62.2035345410779 x 34 = 62.2035345410779 x 35 = − 86.0796387083603 x_{35} = -86.0796387083603 x 35 = − 86.0796387083603 x 36 = 8.16814089933346 x_{36} = 8.16814089933346 x 36 = 8.16814089933346 x 37 = − 96.1327351998477 x_{37} = -96.1327351998477 x 37 = − 96.1327351998477 x 38 = 28.2743338823081 x_{38} = 28.2743338823081 x 38 = 28.2743338823081 x 39 = − 76.026542216873 x_{39} = -76.026542216873 x 39 = − 76.026542216873 x 40 = − 52.7787565803085 x_{40} = -52.7787565803085 x 40 = − 52.7787565803085 x 41 = − 55.9203492338983 x_{41} = -55.9203492338983 x 41 = − 55.9203492338983 x 42 = − 15.707963267949 x_{42} = -15.707963267949 x 42 = − 15.707963267949 x 43 = 67.8584013175395 x_{43} = 67.8584013175395 x 43 = 67.8584013175395 x 44 = 26.3893782901543 x_{44} = 26.3893782901543 x 44 = 26.3893782901543 x 45 = − 98.0176907920015 x_{45} = -98.0176907920015 x 45 = − 98.0176907920015 x 46 = 40.2123859659494 x_{46} = 40.2123859659494 x 46 = 40.2123859659494 x 47 = 65.9734457253857 x_{47} = 65.9734457253857 x 47 = 65.9734457253857 x 48 = − 27.6460153515902 x_{48} = -27.6460153515902 x 48 = − 27.6460153515902 x 49 = 89.2212313619501 x_{49} = 89.2212313619501 x 49 = 89.2212313619501 x 50 = − 54.0353936417444 x_{50} = -54.0353936417444 x 50 = − 54.0353936417444 x 51 = − 11.9380520836412 x_{51} = -11.9380520836412 x 51 = − 11.9380520836412 x 52 = 18.2212373908208 x_{52} = 18.2212373908208 x 52 = 18.2212373908208 x 53 = 98.0176907920015 x_{53} = 98.0176907920015 x 53 = 98.0176907920015 x 54 = − 89.8495498926681 x_{54} = -89.8495498926681 x 54 = − 89.8495498926681 x 55 = − 20.1061929829747 x_{55} = -20.1061929829747 x 55 = − 20.1061929829747 x 56 = 1.88495559215388 x_{56} = 1.88495559215388 x 56 = 1.88495559215388 x 57 = − 77.9114978090269 x_{57} = -77.9114978090269 x 57 = − 77.9114978090269 x 58 = 74.1415866247191 x_{58} = 74.1415866247191 x 58 = 74.1415866247191 x 59 = 72.2566310325652 x_{59} = 72.2566310325652 x 59 = 72.2566310325652 x 60 = 23.8761041672824 x_{60} = 23.8761041672824 x 60 = 23.8761041672824 x 61 = 0 x_{61} = 0 x 61 = 0 x 62 = 76.026542216873 x_{62} = 76.026542216873 x 62 = 76.026542216873 x 63 = − 74.1415866247191 x_{63} = -74.1415866247191 x 63 = − 74.1415866247191 x 64 = 43.9822971502571 x_{64} = 43.9822971502571 x 64 = 43.9822971502571 x 65 = 21.9911485751286 x_{65} = 21.9911485751286 x 65 = 21.9911485751286 x 66 = 70.3716754404114 x_{66} = 70.3716754404114 x 66 = 70.3716754404114 x 67 = 54.0353936417444 x_{67} = 54.0353936417444 x 67 = 54.0353936417444 x 68 = − 43.9822971502571 x_{68} = -43.9822971502571 x 68 = − 43.9822971502571 x 69 = 189.752196276824 x_{69} = 189.752196276824 x 69 = 189.752196276824 x 70 = − 25.7610597594363 x_{70} = -25.7610597594363 x 70 = − 25.7610597594363 x 71 = 99.9026463841554 x_{71} = 99.9026463841554 x 71 = 99.9026463841554 x 72 = − 33.9292006587698 x_{72} = -33.9292006587698 x 72 = − 33.9292006587698 x 73 = 6.28318530717959 x_{73} = 6.28318530717959 x 73 = 6.28318530717959 x 74 = 52.7787565803085 x_{74} = 52.7787565803085 x 74 = 52.7787565803085 x 75 = − 69.7433569096934 x_{75} = -69.7433569096934 x 75 = − 69.7433569096934 x 76 = 11.9380520836412 x_{76} = 11.9380520836412 x 76 = 11.9380520836412 x 77 = 32.0442450666159 x_{77} = 32.0442450666159 x 77 = 32.0442450666159 x 78 = − 42.0973415581032 x_{78} = -42.0973415581032 x 78 = − 42.0973415581032 x 79 = 101.159283445591 x_{79} = 101.159283445591 x 79 = 101.159283445591 x 80 = − 59.6902604182061 x_{80} = -59.6902604182061 x 80 = − 59.6902604182061 x 81 = − 3.76991118430775 x_{81} = -3.76991118430775 x 81 = − 3.76991118430775 x 82 = 20.1061929829747 x_{82} = 20.1061929829747 x 82 = 20.1061929829747 x 83 = 77.9114978090269 x_{83} = 77.9114978090269 x 83 = 77.9114978090269 x 84 = − 64.0884901332318 x_{84} = -64.0884901332318 x 84 = − 64.0884901332318 x 85 = − 45.867252742411 x_{85} = -45.867252742411 x 85 = − 45.867252742411 x 86 = 82.3097275240526 x_{86} = 82.3097275240526 x 86 = 82.3097275240526 x 87 = 52.1504380495906 x_{87} = 52.1504380495906 x 87 = 52.1504380495906 x 88 = − 1.88495559215388 x_{88} = -1.88495559215388 x 88 = − 1.88495559215388 x 89 = − 13.8230076757951 x_{89} = -13.8230076757951 x 89 = − 13.8230076757951 x 90 = − 10.0530964914873 x_{90} = -10.0530964914873 x 90 = − 10.0530964914873 x 91 = − 87.9645943005142 x_{91} = -87.9645943005142 x 91 = − 87.9645943005142 x 92 = 86.0796387083603 x_{92} = 86.0796387083603 x 92 = 86.0796387083603 x 93 = 5.02654824574367 x_{93} = 5.02654824574367 x 93 = 5.02654824574367 x 94 = 94.2477796076938 x_{94} = 94.2477796076938 x 94 = 94.2477796076938 x 95 = − 81.6814089933346 x_{95} = -81.6814089933346 x 95 = − 81.6814089933346 x 96 = − 23.8761041672824 x_{96} = -23.8761041672824 x 96 = − 23.8761041672824
Точки пересечения с осью координат Y
График пересекает ось Y, когда x равняется 0:sin ( 5 ⋅ 0 ) \sin{\left(5 \cdot 0 \right)} sin ( 5 ⋅ 0 ) f ( 0 ) = 0 f{\left(0 \right)} = 0 f ( 0 ) = 0 (0, 0)
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 cos ( 5 x ) = 0 5 \cos{\left(5 x \right)} = 0 5 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 pi
(--, 1)
10 3*pi
(----, -1)
10 Интервалы возрастания и убывания функции: x 1 = 3 π 10 x_{1} = \frac{3 \pi}{10} x 1 = 10 3 π x 1 = π 10 x_{1} = \frac{\pi}{10} x 1 = 10 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 , ∞ ) [ π 10 , 3 π 10 ] \left[\frac{\pi}{10}, \frac{3 \pi}{10}\right] [ 10 π , 10 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 − 25 sin ( 5 x ) = 0 - 25 \sin{\left(5 x \right)} = 0 − 25 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 Интервалы выпуклости и вогнутости: ( − ∞ , 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 ]
Горизонтальные асимптоты
Горизонтальные асимптоты найдём с помощью пределов данной функции при x->+oo и x->-oolim x → − ∞ sin ( 5 x ) = ⟨ − 1 , 1 ⟩ \lim_{x \to -\infty} \sin{\left(5 x \right)} = \left\langle -1, 1\right\rangle x → − ∞ lim sin ( 5 x ) = ⟨ − 1 , 1 ⟩ Возьмём предел y = ⟨ − 1 , 1 ⟩ y = \left\langle -1, 1\right\rangle y = ⟨ − 1 , 1 ⟩ lim x → ∞ sin ( 5 x ) = ⟨ − 1 , 1 ⟩ \lim_{x \to \infty} \sin{\left(5 x \right)} = \left\langle -1, 1\right\rangle x → ∞ lim sin ( 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 sin(5*x), делённой на x при x->+oo и x ->-oolim x → − ∞ ( sin ( 5 x ) x ) = 0 \lim_{x \to -\infty}\left(\frac{\sin{\left(5 x \right)}}{x}\right) = 0 x → − ∞ lim ( x sin ( 5 x ) ) = 0 Возьмём предел lim x → ∞ ( sin ( 5 x ) x ) = 0 \lim_{x \to \infty}\left(\frac{\sin{\left(5 x \right)}}{x}\right) = 0 x → ∞ lim ( x sin ( 5 x ) ) = 0 Возьмём предел
Чётность и нечётность функции
Проверим функци чётна или нечётна с помощью соотношений f = f(-x) и f = -f(-x).sin ( 5 x ) = − sin ( 5 x ) \sin{\left(5 x \right)} = - \sin{\left(5 x \right)} sin ( 5 x ) = − sin ( 5 x ) sin ( 5 x ) = sin ( 5 x ) \sin{\left(5 x \right)} = \sin{\left(5 x \right)} sin ( 5 x ) = sin ( 5 x ) 
