График функции
0 -2000 -1500 -1000 -500 500 1000 1500 2000 2 -2
Точки пересечения с осью координат X
График функции пересекает ось X при f = 0cos ( 10 x ) = 0 \cos{\left (10 x \right )} = 0 cos ( 10 x ) = 0 Решаем это уравнение Аналитическое решение x 1 = π 20 x_{1} = \frac{\pi}{20} x 1 = 20 π x 2 = 3 π 20 x_{2} = \frac{3 \pi}{20} x 2 = 20 3 π Численное решение x 1 = − 95.9756555672 x_{1} = -95.9756555672 x 1 = − 95.9756555672 x 2 = − 77.7544181763 x_{2} = -77.7544181763 x 2 = − 77.7544181763 x 3 = − 45.081854579 x_{3} = -45.081854579 x 3 = − 45.081854579 x 4 = − 35.9712358836 x_{4} = -35.9712358836 x 4 = − 35.9712358836 x 5 = − 5.81194640914 x_{5} = -5.81194640914 x 5 = − 5.81194640914 x 6 = 60.1614993162 x_{6} = 60.1614993162 x 6 = 60.1614993162 x 7 = − 68.0154809502 x_{7} = -68.0154809502 x 7 = − 68.0154809502 x 8 = − 31.8871654339 x_{8} = -31.8871654339 x 8 = − 31.8871654339 x 9 = − 13.9800873085 x_{9} = -13.9800873085 x 9 = − 13.9800873085 x 10 = 70.2145958077 x_{10} = 70.2145958077 x 10 = 70.2145958077 x 11 = 44.1393767829 x_{11} = 44.1393767829 x 11 = 44.1393767829 x 12 = − 47.9092879672 x_{12} = -47.9092879672 x 12 = − 47.9092879672 x 13 = − 97.8606111593 x_{13} = -97.8606111593 x 13 = − 97.8606111593 x 14 = 82.1526478914 x_{14} = 82.1526478914 x 14 = 82.1526478914 x 15 = − 41.9402619254 x_{15} = -41.9402619254 x 15 = − 41.9402619254 x 16 = − 73.984506992 x_{16} = -73.984506992 x 16 = − 73.984506992 x 17 = 8.01106126665 x_{17} = 8.01106126665 x 17 = 8.01106126665 x 18 = − 22.7765467385 x_{18} = -22.7765467385 x 18 = − 22.7765467385 x 19 = − 25.9181393921 x_{19} = -25.9181393921 x 19 = − 25.9181393921 x 20 = 100.059726017 x_{20} = 100.059726017 x 20 = 100.059726017 x 21 = 42.2544211908 x_{21} = 42.2544211908 x 21 = 42.2544211908 x 22 = 90.0066295253 x_{22} = 90.0066295253 x 22 = 90.0066295253 x 23 = − 9.89601685881 x_{23} = -9.89601685881 x 23 = − 9.89601685881 x 24 = 84.0376034835 x_{24} = 84.0376034835 x 24 = 84.0376034835 x 25 = 54.1924732744 x_{25} = 54.1924732744 x 25 = 54.1924732744 x 26 = 48.2234472326 x_{26} = 48.2234472326 x 26 = 48.2234472326 x 27 = − 43.8252175176 x_{27} = -43.8252175176 x 27 = − 43.8252175176 x 28 = 38.1703507411 x_{28} = 38.1703507411 x 28 = 38.1703507411 x 29 = − 33.7721210261 x_{29} = -33.7721210261 x 29 = − 33.7721210261 x 30 = − 30.0022098418 x_{30} = -30.0022098418 x 30 = − 30.0022098418 x 31 = 10.2101761242 x_{31} = 10.2101761242 x 31 = 10.2101761242 x 32 = − 17.7499984928 x_{32} = -17.7499984928 x 32 = − 17.7499984928 x 33 = 62.0464549084 x_{33} = 62.0464549084 x 33 = 62.0464549084 x 34 = − 19.9491133503 x_{34} = -19.9491133503 x 34 = − 19.9491133503 x 35 = − 15.8650429006 x_{35} = -15.8650429006 x 35 = − 15.8650429006 x 36 = − 65.8163660927 x_{36} = -65.8163660927 x 36 = − 65.8163660927 x 37 = 76.1836218496 x_{37} = 76.1836218496 x 37 = 76.1836218496 x 38 = − 21.8340689424 x_{38} = -21.8340689424 x 38 = − 21.8340689424 x 39 = 37.227872945 x_{39} = 37.227872945 x 39 = 37.227872945 x 40 = 2.04203522483 x_{40} = 2.04203522483 x 40 = 2.04203522483 x 41 = − 55.7632696012 x_{41} = -55.7632696012 x 41 = − 55.7632696012 x 42 = 62.6747734391 x_{42} = 62.6747734391 x 42 = 62.6747734391 x 43 = 4.24115008235 x_{43} = 4.24115008235 x 43 = 4.24115008235 x 44 = 51.9933584169 x_{44} = 51.9933584169 x 44 = 51.9933584169 x 45 = 80.2676922992 x_{45} = 80.2676922992 x 45 = 80.2676922992 x 46 = 40.0553063333 x_{46} = 40.0553063333 x 46 = 40.0553063333 x 47 = 73.984506992 x_{47} = 73.984506992 x 47 = 73.984506992 x 48 = − 87.8075146678 x_{48} = -87.8075146678 x 48 = − 87.8075146678 x 49 = 92.2057443829 x_{49} = 92.2057443829 x 49 = 92.2057443829 x 50 = 30.0022098418 x_{50} = 30.0022098418 x 50 = 30.0022098418 x 51 = − 39.7411470679 x_{51} = -39.7411470679 x 51 = − 39.7411470679 x 52 = − 71.7853921345 x_{52} = -71.7853921345 x 52 = − 71.7853921345 x 53 = − 91.8915851175 x_{53} = -91.8915851175 x 53 = − 91.8915851175 x 54 = 88.1216739332 x_{54} = 88.1216739332 x 54 = 88.1216739332 x 55 = − 51.9933584169 x_{55} = -51.9933584169 x 55 = − 51.9933584169 x 56 = − 75.8694625842 x_{56} = -75.8694625842 x 56 = − 75.8694625842 x 57 = 16.179202166 x_{57} = 16.179202166 x 57 = 16.179202166 x 58 = − 11.780972451 x_{58} = -11.780972451 x 58 = − 11.780972451 x 59 = − 27.8030949843 x_{59} = -27.8030949843 x 59 = − 27.8030949843 x 60 = 6.1261056745 x_{60} = 6.1261056745 x 60 = 6.1261056745 x 61 = 32.2013246993 x_{61} = 32.2013246993 x 61 = 32.2013246993 x 62 = − 61.732295643 x_{62} = -61.732295643 x 62 = − 61.732295643 x 63 = 18.0641577581 x_{63} = 18.0641577581 x 63 = 18.0641577581 x 64 = 86.236718341 x_{64} = 86.236718341 x 64 = 86.236718341 x 65 = − 99.7455667515 x_{65} = -99.7455667515 x 65 = − 99.7455667515 x 66 = − 69.9004365424 x_{66} = -69.9004365424 x 66 = − 69.9004365424 x 67 = 28.1172542496 x_{67} = 28.1172542496 x 67 = 28.1172542496 x 68 = 20.2632726157 x_{68} = 20.2632726157 x 68 = 20.2632726157 x 69 = 78.0685774417 x_{69} = 78.0685774417 x 69 = 78.0685774417 x 70 = − 63.9314105006 x_{70} = -63.9314105006 x 70 = − 63.9314105006 x 71 = 72.0995513999 x_{71} = 72.0995513999 x 71 = 72.0995513999 x 72 = 98.1747704247 x_{72} = 98.1747704247 x 72 = 98.1747704247 x 73 = − 59.8473400509 x_{73} = -59.8473400509 x 73 = − 59.8473400509 x 74 = 12.0951317163 x_{74} = 12.0951317163 x 74 = 12.0951317163 x 75 = − 53.8783140091 x_{75} = -53.8783140091 x 75 = − 53.8783140091 x 76 = 0.157079632679 x_{76} = 0.157079632679 x 76 = 0.157079632679 x 77 = 68.0154809502 x_{77} = 68.0154809502 x 77 = 68.0154809502 x 78 = − 79.9535330339 x_{78} = -79.9535330339 x 78 = − 79.9535330339 x 79 = − 49.7942435594 x_{79} = -49.7942435594 x 79 = − 49.7942435594 x 80 = 46.0243323751 x_{80} = 46.0243323751 x 80 = 46.0243323751 x 81 = − 57.9623844587 x_{81} = -57.9623844587 x 81 = − 57.9623844587 x 82 = 22.1482282078 x_{82} = 22.1482282078 x 82 = 22.1482282078 x 83 = − 81.838488626 x_{83} = -81.838488626 x 83 = − 81.838488626 x 84 = − 90.0066295253 x_{84} = -90.0066295253 x 84 = − 90.0066295253 x 85 = − 1.09955742876 x_{85} = -1.09955742876 x 85 = − 1.09955742876 x 86 = − 93.7765407097 x_{86} = -93.7765407097 x 86 = − 93.7765407097 x 87 = 56.0774288666 x_{87} = 56.0774288666 x 87 = 56.0774288666 x 88 = 66.1305253581 x_{88} = 66.1305253581 x 88 = 66.1305253581 x 89 = 24.0331838 x_{89} = 24.0331838 x 89 = 24.0331838 x 90 = 64.2455697659 x_{90} = 64.2455697659 x 90 = 64.2455697659 x 91 = − 3.92699081699 x_{91} = -3.92699081699 x 91 = − 3.92699081699 x 92 = − 85.9225590757 x_{92} = -85.9225590757 x 92 = − 85.9225590757 x 93 = − 8.01106126665 x_{93} = -8.01106126665 x 93 = − 8.01106126665 x 94 = − 89.69247026 x_{94} = -89.69247026 x 94 = − 89.69247026 x 95 = 13.9800873085 x_{95} = 13.9800873085 x 95 = 13.9800873085 x 96 = 34.0862802914 x_{96} = 34.0862802914 x 96 = 34.0862802914 x 97 = − 37.8561914758 x_{97} = -37.8561914758 x 97 = − 37.8561914758 x 98 = 26.2322986575 x_{98} = 26.2322986575 x 98 = 26.2322986575 x 99 = 94.090699975 x_{99} = 94.090699975 x 99 = 94.090699975 x 100 = 50.1084028248 x_{100} = 50.1084028248 x 100 = 50.1084028248
Точки пересечения с осью координат Y
График пересекает ось Y, когда x равняется 0:cos ( 0 ⋅ 10 ) \cos{\left (0 \cdot 10 \right )} cos ( 0 ⋅ 10 ) 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 − 10 sin ( 10 x ) = 0 - 10 \sin{\left (10 x \right )} = 0 − 10 sin ( 10 x ) = 0 Решаем это уравнение x 1 = 0 x_{1} = 0 x 1 = 0 x 2 = π 10 x_{2} = \frac{\pi}{10} x 2 = 10 π (0, 1) pi
(--, -1)
10 Интервалы возрастания и убывания функции: x 2 = π 10 x_{2} = \frac{\pi}{10} x 2 = 10 π x 2 = 0 x_{2} = 0 x 2 = 0 (-oo, 0] U [pi/10, oo) [0, pi/10]
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 − 100 cos ( 10 x ) = 0 - 100 \cos{\left (10 x \right )} = 0 − 100 cos ( 10 x ) = 0 Решаем это уравнение x 1 = π 20 x_{1} = \frac{\pi}{20} x 1 = 20 π x 2 = 3 π 20 x_{2} = \frac{3 \pi}{20} x 2 = 20 3 π Интервалы выпуклости и вогнутости: [pi/20, 3*pi/20] (-oo, pi/20] U [3*pi/20, oo)
Горизонтальные асимптоты
Горизонтальные асимптоты найдём с помощью пределов данной функции при x->+oo и x->-oolim x → − ∞ cos ( 10 x ) = ⟨ − 1 , 1 ⟩ \lim_{x \to -\infty} \cos{\left (10 x \right )} = \langle -1, 1\rangle x → − ∞ lim cos ( 10 x ) = ⟨ − 1 , 1 ⟩ Возьмём предел y = ⟨ − 1 , 1 ⟩ y = \langle -1, 1\rangle y = ⟨ − 1 , 1 ⟩ lim x → ∞ cos ( 10 x ) = ⟨ − 1 , 1 ⟩ \lim_{x \to \infty} \cos{\left (10 x \right )} = \langle -1, 1\rangle x → ∞ lim cos ( 10 x ) = ⟨ − 1 , 1 ⟩ Возьмём предел y = ⟨ − 1 , 1 ⟩ y = \langle -1, 1\rangle y = ⟨ − 1 , 1 ⟩
Наклонные асимптоты
Наклонную асимптоту можно найти, подсчитав предел функции cos(10*x), делённой на x при x->+oo и x ->-oolim x → − ∞ ( 1 x cos ( 10 x ) ) = 0 \lim_{x \to -\infty}\left(\frac{1}{x} \cos{\left (10 x \right )}\right) = 0 x → − ∞ lim ( x 1 cos ( 10 x ) ) = 0 Возьмём предел lim x → ∞ ( 1 x cos ( 10 x ) ) = 0 \lim_{x \to \infty}\left(\frac{1}{x} \cos{\left (10 x \right )}\right) = 0 x → ∞ lim ( x 1 cos ( 10 x ) ) = 0 Возьмём предел
Чётность и нечётность функции
Проверим функци чётна или нечётна с помощью соотношений f = f(-x) и f = -f(-x).cos ( 10 x ) = cos ( 10 x ) \cos{\left (10 x \right )} = \cos{\left (10 x \right )} cos ( 10 x ) = cos ( 10 x ) cos ( 10 x ) = − cos ( 10 x ) \cos{\left (10 x \right )} = - \cos{\left (10 x \right )} cos ( 10 x ) = − cos ( 10 x ) 