График функции
0 -2000 -1500 -1000 -500 500 1000 1500 2000 -50 50
Точки пересечения с осью координат X
График функции пересекает ось X при f = 015 sin ( 10 x ) = 0 15 \sin{\left (10 x \right )} = 0 15 sin ( 10 x ) = 0 Решаем это уравнение Аналитическое решение x 1 = 0 x_{1} = 0 x 1 = 0 x 2 = π 10 x_{2} = \frac{\pi}{10} x 2 = 10 π Численное решение x 1 = − 83.8805238508 x_{1} = -83.8805238508 x 1 = − 83.8805238508 x 2 = 74.1415866247 x_{2} = 74.1415866247 x 2 = 74.1415866247 x 3 = − 95.8185759345 x_{3} = -95.8185759345 x 3 = − 95.8185759345 x 4 = − 61.8893752757 x_{4} = -61.8893752757 x 4 = − 61.8893752757 x 5 = 98.017690792 x_{5} = 98.017690792 x 5 = 98.017690792 x 6 = 10.0530964915 x_{6} = 10.0530964915 x 6 = 10.0530964915 x 7 = − 27.9601746169 x_{7} = -27.9601746169 x 7 = − 27.9601746169 x 8 = 78.2256570744 x_{8} = 78.2256570744 x 8 = 78.2256570744 x 9 = − 57.8053048261 x_{9} = -57.8053048261 x 9 = − 57.8053048261 x 10 = 36.1283155163 x_{10} = 36.1283155163 x 10 = 36.1283155163 x 11 = − 33.9292006588 x_{11} = -33.9292006588 x 11 = − 33.9292006588 x 12 = 80.1106126665 x_{12} = 80.1106126665 x 12 = 80.1106126665 x 13 = 88.5929128312 x_{13} = 88.5929128312 x 13 = 88.5929128312 x 14 = 16.0221225333 x_{14} = 16.0221225333 x 14 = 16.0221225333 x 15 = 14.1371669412 x_{15} = 14.1371669412 x 15 = 14.1371669412 x 16 = 54.0353936417 x_{16} = 54.0353936417 x 16 = 54.0353936417 x 17 = 84.1946831162 x_{17} = 84.1946831162 x 17 = 84.1946831162 x 18 = 100.21680565 x_{18} = 100.21680565 x 18 = 100.21680565 x 19 = − 51.8362787842 x_{19} = -51.8362787842 x 19 = − 51.8362787842 x 20 = − 25.7610597594 x_{20} = -25.7610597594 x 20 = − 25.7610597594 x 21 = 81.9955682587 x_{21} = 81.9955682587 x 21 = 81.9955682587 x 22 = 12.252211349 x_{22} = 12.252211349 x 22 = 12.252211349 x 23 = − 7.85398163397 x_{23} = -7.85398163397 x 23 = − 7.85398163397 x 24 = 70.0575161751 x_{24} = 70.0575161751 x 24 = 70.0575161751 x 25 = 43.9822971503 x_{25} = 43.9822971503 x 25 = 43.9822971503 x 26 = 48.0663675999 x_{26} = 48.0663675999 x 26 = 48.0663675999 x 27 = − 35.8141562509 x_{27} = -35.8141562509 x 27 = − 35.8141562509 x 28 = 18.2212373908 x_{28} = 18.2212373908 x 28 = 18.2212373908 x 29 = 90.163709158 x_{29} = 90.163709158 x 29 = 90.163709158 x 30 = 30.1592894745 x_{30} = 30.1592894745 x 30 = 30.1592894745 x 31 = − 47.7522083346 x_{31} = -47.7522083346 x 31 = − 47.7522083346 x 32 = − 11.9380520836 x_{32} = -11.9380520836 x 32 = − 11.9380520836 x 33 = 20.106192983 x_{33} = 20.106192983 x 33 = 20.106192983 x 34 = − 79.7964534012 x_{34} = -79.7964534012 x 34 = − 79.7964534012 x 35 = − 45.8672527424 x_{35} = -45.8672527424 x 35 = − 45.8672527424 x 36 = − 39.8982267006 x_{36} = -39.8982267006 x 36 = − 39.8982267006 x 37 = − 13.8230076758 x_{37} = -13.8230076758 x 37 = − 13.8230076758 x 38 = 40.2123859659 x_{38} = 40.2123859659 x 38 = 40.2123859659 x 39 = − 17.9070781255 x_{39} = -17.9070781255 x 39 = − 17.9070781255 x 40 = 64.0884901332 x_{40} = 64.0884901332 x 40 = 64.0884901332 x 41 = 2.19911485751 x_{41} = 2.19911485751 x 41 = 2.19911485751 x 42 = − 89.8495498927 x_{42} = -89.8495498927 x 42 = − 89.8495498927 x 43 = − 71.9424717672 x_{43} = -71.9424717672 x 43 = − 71.9424717672 x 44 = 8.16814089933 x_{44} = 8.16814089933 x 44 = 8.16814089933 x 45 = − 9.73893722613 x_{45} = -9.73893722613 x 45 = − 9.73893722613 x 46 = − 21.9911485751 x_{46} = -21.9911485751 x 46 = − 21.9911485751 x 47 = 86.0796387084 x_{47} = 86.0796387084 x 47 = 86.0796387084 x 48 = 21.9911485751 x_{48} = 21.9911485751 x 48 = 21.9911485751 x 49 = 0 x_{49} = 0 x 49 = 0 x 50 = − 16.0221225333 x_{50} = -16.0221225333 x 50 = − 16.0221225333 x 51 = − 69.7433569097 x_{51} = -69.7433569097 x 51 = − 69.7433569097 x 52 = 52.1504380496 x_{52} = 52.1504380496 x 52 = 52.1504380496 x 53 = − 60.0044196836 x_{53} = -60.0044196836 x 53 = − 60.0044196836 x 54 = − 91.7345054848 x_{54} = -91.7345054848 x 54 = − 91.7345054848 x 55 = 92.0486647502 x_{55} = 92.0486647502 x 55 = 92.0486647502 x 56 = 24.1902634326 x_{56} = 24.1902634326 x 56 = 24.1902634326 x 57 = 4.08407044967 x_{57} = 4.08407044967 x 57 = 4.08407044967 x 58 = 50.2654824574 x_{58} = 50.2654824574 x 58 = 50.2654824574 x 59 = − 53.407075111 x_{59} = -53.407075111 x 59 = − 53.407075111 x 60 = − 43.9822971503 x_{60} = -43.9822971503 x 60 = − 43.9822971503 x 61 = − 63.7743308679 x_{61} = -63.7743308679 x 61 = − 63.7743308679 x 62 = − 19.7920337176 x_{62} = -19.7920337176 x 62 = − 19.7920337176 x 63 = 46.1814120078 x_{63} = 46.1814120078 x 63 = 46.1814120078 x 64 = 72.2566310326 x_{64} = 72.2566310326 x 64 = 72.2566310326 x 65 = − 65.9734457254 x_{65} = -65.9734457254 x 65 = − 65.9734457254 x 66 = − 73.8274273594 x_{66} = -73.8274273594 x 66 = − 73.8274273594 x 67 = − 1.88495559215 x_{67} = -1.88495559215 x 67 = − 1.88495559215 x 68 = 42.0973415581 x_{68} = 42.0973415581 x 68 = 42.0973415581 x 69 = 34.2433599241 x_{69} = 34.2433599241 x 69 = 34.2433599241 x 70 = − 31.7300858013 x_{70} = -31.7300858013 x 70 = − 31.7300858013 x 71 = 60.0044196836 x_{71} = 60.0044196836 x 71 = 60.0044196836 x 72 = − 49.9513231921 x_{72} = -49.9513231921 x 72 = − 49.9513231921 x 73 = − 55.9203492339 x_{73} = -55.9203492339 x 73 = − 55.9203492339 x 74 = 94.2477796077 x_{74} = 94.2477796077 x 74 = 94.2477796077 x 75 = 56.2345084993 x_{75} = 56.2345084993 x 75 = 56.2345084993 x 76 = − 3.76991118431 x_{76} = -3.76991118431 x 76 = − 3.76991118431 x 77 = − 98.017690792 x_{77} = -98.017690792 x 77 = − 98.017690792 x 78 = − 29.8451302091 x_{78} = -29.8451302091 x 78 = − 29.8451302091 x 79 = 32.0442450666 x_{79} = 32.0442450666 x 79 = 32.0442450666 x 80 = − 5.96902604182 x_{80} = -5.96902604182 x 80 = − 5.96902604182 x 81 = − 93.9336203423 x_{81} = -93.9336203423 x 81 = − 93.9336203423 x 82 = − 23.8761041673 x_{82} = -23.8761041673 x 82 = − 23.8761041673 x 83 = − 81.9955682587 x_{83} = -81.9955682587 x 83 = − 81.9955682587 x 84 = − 38.0132711084 x_{84} = -38.0132711084 x 84 = − 38.0132711084 x 85 = 62.2035345411 x_{85} = 62.2035345411 x 85 = 62.2035345411 x 86 = 58.1194640914 x_{86} = 58.1194640914 x 86 = 58.1194640914 x 87 = − 67.8584013175 x_{87} = -67.8584013175 x 87 = − 67.8584013175 x 88 = − 136.0309619 x_{88} = -136.0309619 x 88 = − 136.0309619 x 89 = − 41.7831822927 x_{89} = -41.7831822927 x 89 = − 41.7831822927 x 90 = 26.0752190248 x_{90} = 26.0752190248 x 90 = 26.0752190248 x 91 = − 87.9645943005 x_{91} = -87.9645943005 x 91 = − 87.9645943005 x 92 = 96.1327351998 x_{92} = 96.1327351998 x 92 = 96.1327351998 x 93 = 68.1725605829 x_{93} = 68.1725605829 x 93 = 68.1725605829 x 94 = − 85.765479443 x_{94} = -85.765479443 x 94 = − 85.765479443 x 95 = − 99.9026463842 x_{95} = -99.9026463842 x 95 = − 99.9026463842 x 96 = 38.0132711084 x_{96} = 38.0132711084 x 96 = 38.0132711084 x 97 = 76.0265422169 x_{97} = 76.0265422169 x 97 = 76.0265422169 x 98 = − 77.911497809 x_{98} = -77.911497809 x 98 = − 77.911497809 x 99 = 37.6991118431 x_{99} = 37.6991118431 x 99 = 37.6991118431
Точки пересечения с осью координат Y
График пересекает ось Y, когда x равняется 0:15 sin ( 0 ⋅ 10 ) 15 \sin{\left (0 \cdot 10 \right )} 15 sin ( 0 ⋅ 10 ) 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 150 cos ( 10 x ) = 0 150 \cos{\left (10 x \right )} = 0 150 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 pi
(--, 15)
20 3*pi
(----, -15)
20 Интервалы возрастания и убывания функции: x 2 = 3 π 20 x_{2} = \frac{3 \pi}{20} x 2 = 20 3 π x 2 = π 20 x_{2} = \frac{\pi}{20} x 2 = 20 π (-oo, pi/20] U [3*pi/20, oo) [pi/20, 3*pi/20]
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 − 1500 sin ( 10 x ) = 0 - 1500 \sin{\left (10 x \right )} = 0 − 1500 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 Интервалы выпуклости и вогнутости: (-oo, 0] U [pi/10, oo) [0, pi/10]
Горизонтальные асимптоты
Горизонтальные асимптоты найдём с помощью пределов данной функции при x->+oo и x->-oolim x → − ∞ ( 15 sin ( 10 x ) ) = ⟨ − 15 , 15 ⟩ \lim_{x \to -\infty}\left(15 \sin{\left (10 x \right )}\right) = \langle -15, 15\rangle x → − ∞ lim ( 15 sin ( 10 x ) ) = ⟨ − 15 , 15 ⟩ Возьмём предел y = ⟨ − 15 , 15 ⟩ y = \langle -15, 15\rangle y = ⟨ − 15 , 15 ⟩ lim x → ∞ ( 15 sin ( 10 x ) ) = ⟨ − 15 , 15 ⟩ \lim_{x \to \infty}\left(15 \sin{\left (10 x \right )}\right) = \langle -15, 15\rangle x → ∞ lim ( 15 sin ( 10 x ) ) = ⟨ − 15 , 15 ⟩ Возьмём предел y = ⟨ − 15 , 15 ⟩ y = \langle -15, 15\rangle y = ⟨ − 15 , 15 ⟩
Наклонные асимптоты
Наклонную асимптоту можно найти, подсчитав предел функции sin(10*x)*15, делённой на x при x->+oo и x ->-oolim x → − ∞ ( 15 x sin ( 10 x ) ) = 0 \lim_{x \to -\infty}\left(\frac{15}{x} \sin{\left (10 x \right )}\right) = 0 x → − ∞ lim ( x 15 sin ( 10 x ) ) = 0 Возьмём предел lim x → ∞ ( 15 x sin ( 10 x ) ) = 0 \lim_{x \to \infty}\left(\frac{15}{x} \sin{\left (10 x \right )}\right) = 0 x → ∞ lim ( x 15 sin ( 10 x ) ) = 0 Возьмём предел
Чётность и нечётность функции
Проверим функци чётна или нечётна с помощью соотношений f = f(-x) и f = -f(-x).15 sin ( 10 x ) = − 15 sin ( 10 x ) 15 \sin{\left (10 x \right )} = - 15 \sin{\left (10 x \right )} 15 sin ( 10 x ) = − 15 sin ( 10 x ) 15 sin ( 10 x ) = − − 1 ⋅ 15 sin ( 10 x ) 15 \sin{\left (10 x \right )} = - -1 \cdot 15 \sin{\left (10 x \right )} 15 sin ( 10 x ) = − − 1 ⋅ 15 sin ( 10 x ) 