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 ( 7 x ) = 0 \cos{\left (7 x \right )} = 0 cos ( 7 x ) = 0 Решаем это уравнение Аналитическое решение x 1 = π 14 x_{1} = \frac{\pi}{14} x 1 = 14 π x 2 = 3 π 14 x_{2} = \frac{3 \pi}{14} x 2 = 14 3 π Численное решение x 1 = − 59.9146598935 x_{1} = -59.9146598935 x 1 = − 59.9146598935 x 2 = 30.2939291596 x_{2} = 30.2939291596 x 2 = 30.2939291596 x 3 = − 95.8185759345 x_{3} = -95.8185759345 x 3 = − 95.8185759345 x 4 = 78.3154168645 x_{4} = 78.3154168645 x 4 = 78.3154168645 x 5 = 62.158654646 x_{5} = 62.158654646 x 5 = 62.158654646 x 6 = − 11.8931721886 x_{6} = -11.8931721886 x 6 = − 11.8931721886 x 7 = 96.267374885 x_{7} = 96.267374885 x 7 = 96.267374885 x 8 = 10.0979763865 x_{8} = 10.0979763865 x 8 = 10.0979763865 x 9 = − 81.9058084686 x_{9} = -81.9058084686 x 9 = − 81.9058084686 x 10 = − 15.9323627432 x_{10} = -15.9323627432 x 10 = − 15.9323627432 x 11 = − 94.0233801324 x_{11} = -94.0233801324 x 11 = − 94.0233801324 x 12 = 56.7730672399 x_{12} = 56.7730672399 x 12 = 56.7730672399 x 13 = − 46.0018924276 x_{13} = -46.0018924276 x 13 = − 46.0018924276 x 14 = 36.1283155163 x_{14} = 36.1283155163 x 14 = 36.1283155163 x 15 = − 72.0322315573 x_{15} = -72.0322315573 x 15 = − 72.0322315573 x 16 = 80.1106126665 x_{16} = 80.1106126665 x 16 = 80.1106126665 x 17 = − 67.9930410027 x_{17} = -67.9930410027 x 17 = − 67.9930410027 x 18 = − 61.7098556955 x_{18} = -61.7098556955 x 18 = − 61.7098556955 x 19 = 14.1371669412 x_{19} = 14.1371669412 x 19 = 14.1371669412 x 20 = 85.9449990232 x_{20} = 85.9449990232 x 20 = 85.9449990232 x 21 = 70.2370357553 x_{21} = 70.2370357553 x 21 = 70.2370357553 x 22 = − 63.9538504481 x_{22} = -63.9538504481 x 22 = − 63.9538504481 x 23 = − 80.5594116171 x_{23} = -80.5594116171 x 23 = − 80.5594116171 x 24 = − 51.8362787842 x_{24} = -51.8362787842 x 24 = − 51.8362787842 x 25 = 92.2281843304 x_{25} = 92.2281843304 x 25 = 92.2281843304 x 26 = − 7.85398163397 x_{26} = -7.85398163397 x 26 = − 7.85398163397 x 27 = − 83.7010042706 x_{27} = -83.7010042706 x 27 = − 83.7010042706 x 28 = 22.2155480504 x_{28} = 22.2155480504 x 28 = 22.2155480504 x 29 = 66.1978452006 x_{29} = 66.1978452006 x 29 = 66.1978452006 x 30 = 94.0233801324 x_{30} = 94.0233801324 x 30 = 94.0233801324 x 31 = 76.0714221119 x_{31} = 76.0714221119 x 31 = 76.0714221119 x 32 = − 25.8059396545 x_{32} = -25.8059396545 x 32 = − 25.8059396545 x 33 = − 2.01959527731 x_{33} = -2.01959527731 x 33 = − 2.01959527731 x 34 = 32.0891249617 x_{34} = 32.0891249617 x 34 = 32.0891249617 x 35 = 50.0410829822 x_{35} = 50.0410829822 x 35 = 50.0410829822 x 36 = 19.9715532978 x_{36} = 19.9715532978 x 36 = 19.9715532978 x 37 = − 28.0499344071 x_{37} = -28.0499344071 x 37 = − 28.0499344071 x 38 = − 13.6883679906 x_{38} = -13.6883679906 x 38 = − 13.6883679906 x 39 = 98.0625706871 x_{39} = 98.0625706871 x 39 = 98.0625706871 x 40 = − 55.8754693388 x_{40} = -55.8754693388 x 40 = − 55.8754693388 x 41 = 72.0322315573 x_{41} = 72.0322315573 x 41 = 72.0322315573 x 42 = − 37.9235113183 x_{42} = -37.9235113183 x 42 = − 37.9235113183 x 43 = − 17.7275585453 x_{43} = -17.7275585453 x 43 = − 17.7275585453 x 44 = − 85.9449990232 x_{44} = -85.9449990232 x 44 = − 85.9449990232 x 45 = 4.26359002987 x_{45} = 4.26359002987 x 45 = 4.26359002987 x 46 = − 91.7793853799 x_{46} = -91.7793853799 x 46 = − 91.7793853799 x 47 = − 33.8843207637 x_{47} = -33.8843207637 x 47 = − 33.8843207637 x 48 = 0.224399475256 x_{48} = 0.224399475256 x 48 = 0.224399475256 x 49 = 52.2850777347 x_{49} = 52.2850777347 x 49 = 52.2850777347 x 50 = − 99.8577664891 x_{50} = -99.8577664891 x 50 = − 99.8577664891 x 51 = − 39.7187071204 x_{51} = -39.7187071204 x 51 = − 39.7187071204 x 52 = − 19.9715532978 x_{52} = -19.9715532978 x 52 = − 19.9715532978 x 53 = 54.0802735368 x_{53} = 54.0802735368 x 53 = 54.0802735368 x 54 = 67.9930410027 x_{54} = 67.9930410027 x 54 = 67.9930410027 x 55 = 26.254738605 x_{55} = 26.254738605 x 55 = 26.254738605 x 56 = 37.4747123678 x_{56} = 37.4747123678 x 56 = 37.4747123678 x 57 = − 47.7970882296 x_{57} = -47.7970882296 x 57 = − 47.7970882296 x 58 = − 32.5379239122 x_{58} = -32.5379239122 x 58 = − 32.5379239122 x 59 = − 87.7401948253 x_{59} = -87.7401948253 x 59 = − 87.7401948253 x 60 = − 35.6795165658 x_{60} = -35.6795165658 x 60 = − 35.6795165658 x 61 = 40.1675060709 x_{61} = 40.1675060709 x 61 = 40.1675060709 x 62 = − 3.81479107936 x_{62} = -3.81479107936 x 62 = − 3.81479107936 x 63 = − 77.866617914 x_{63} = -77.866617914 x 63 = − 77.866617914 x 64 = 89.9841895778 x_{64} = 89.9841895778 x 64 = 89.9841895778 x 65 = 3.81479107936 x_{65} = 3.81479107936 x 65 = 3.81479107936 x 66 = − 73.8274273594 x_{66} = -73.8274273594 x 66 = − 73.8274273594 x 67 = − 98.0625706871 x_{67} = -98.0625706871 x 67 = − 98.0625706871 x 68 = − 43.757897675 x_{68} = -43.757897675 x 68 = − 43.757897675 x 69 = 41.9627018729 x_{69} = 41.9627018729 x 69 = 41.9627018729 x 70 = − 76.0714221119 x_{70} = -76.0714221119 x 70 = − 76.0714221119 x 71 = 84.1498032212 x_{71} = 84.1498032212 x 71 = 84.1498032212 x 72 = 24.0107438524 x_{72} = 24.0107438524 x 72 = 24.0107438524 x 73 = − 29.8451302091 x_{73} = -29.8451302091 x 73 = − 29.8451302091 x 74 = − 24.0107438524 x_{74} = -24.0107438524 x 74 = − 24.0107438524 x 75 = 2.01959527731 x_{75} = 2.01959527731 x 75 = 2.01959527731 x 76 = − 65.7490462501 x_{76} = -65.7490462501 x 76 = − 65.7490462501 x 77 = 74.2762263099 x_{77} = 74.2762263099 x 77 = 74.2762263099 x 78 = − 69.7882368047 x_{78} = -69.7882368047 x 78 = − 69.7882368047 x 79 = 48.2458871801 x_{79} = 48.2458871801 x 79 = 48.2458871801 x 80 = − 89.9841895778 x_{80} = -89.9841895778 x 80 = − 89.9841895778 x 81 = 58.1194640914 x_{81} = 58.1194640914 x 81 = 58.1194640914 x 82 = 15.9323627432 x_{82} = 15.9323627432 x 82 = 15.9323627432 x 83 = 100.30656544 x_{83} = 100.30656544 x 83 = 100.30656544 x 84 = 46.0018924276 x_{84} = 46.0018924276 x 84 = 46.0018924276 x 85 = 63.9538504481 x_{85} = 63.9538504481 x 85 = 63.9538504481 x 86 = 8.30278058449 x_{86} = 8.30278058449 x 86 = 8.30278058449 x 87 = 28.0499344071 x_{87} = 28.0499344071 x 87 = 28.0499344071 x 88 = − 21.7667490999 x_{88} = -21.7667490999 x 88 = − 21.7667490999 x 89 = − 50.0410829822 x_{89} = -50.0410829822 x 89 = − 50.0410829822 x 90 = − 41.9627018729 x_{90} = -41.9627018729 x 90 = − 41.9627018729 x 91 = 18.1763574958 x_{91} = 18.1763574958 x 91 = 18.1763574958 x 92 = 44.2066966255 x_{92} = 44.2066966255 x 92 = 44.2066966255 x 93 = 6.05878583192 x_{93} = 6.05878583192 x 93 = 6.05878583192 x 94 = − 6.05878583192 x_{94} = -6.05878583192 x 94 = − 6.05878583192 x 95 = 88.1889937758 x_{95} = 88.1889937758 x 95 = 88.1889937758
Точки пересечения с осью координат Y
График пересекает ось Y, когда x равняется 0:cos ( 0 ⋅ 7 ) \cos{\left (0 \cdot 7 \right )} cos ( 0 ⋅ 7 ) 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 − 7 sin ( 7 x ) = 0 - 7 \sin{\left (7 x \right )} = 0 − 7 sin ( 7 x ) = 0 Решаем это уравнение x 1 = 0 x_{1} = 0 x 1 = 0 x 2 = π 7 x_{2} = \frac{\pi}{7} x 2 = 7 π (0, 1) pi
(--, -1)
7 Интервалы возрастания и убывания функции: x 2 = π 7 x_{2} = \frac{\pi}{7} x 2 = 7 π x 2 = 0 x_{2} = 0 x 2 = 0 (-oo, 0] U [pi/7, oo) [0, pi/7]
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 − 49 cos ( 7 x ) = 0 - 49 \cos{\left (7 x \right )} = 0 − 49 cos ( 7 x ) = 0 Решаем это уравнение x 1 = π 14 x_{1} = \frac{\pi}{14} x 1 = 14 π x 2 = 3 π 14 x_{2} = \frac{3 \pi}{14} x 2 = 14 3 π Интервалы выпуклости и вогнутости: [pi/14, 3*pi/14] (-oo, pi/14] U [3*pi/14, oo)
Горизонтальные асимптоты
Горизонтальные асимптоты найдём с помощью пределов данной функции при x->+oo и x->-oolim x → − ∞ cos ( 7 x ) = ⟨ − 1 , 1 ⟩ \lim_{x \to -\infty} \cos{\left (7 x \right )} = \langle -1, 1\rangle x → − ∞ lim cos ( 7 x ) = ⟨ − 1 , 1 ⟩ Возьмём предел y = ⟨ − 1 , 1 ⟩ y = \langle -1, 1\rangle y = ⟨ − 1 , 1 ⟩ lim x → ∞ cos ( 7 x ) = ⟨ − 1 , 1 ⟩ \lim_{x \to \infty} \cos{\left (7 x \right )} = \langle -1, 1\rangle x → ∞ lim cos ( 7 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(7*x), делённой на x при x->+oo и x ->-oolim x → − ∞ ( 1 x cos ( 7 x ) ) = 0 \lim_{x \to -\infty}\left(\frac{1}{x} \cos{\left (7 x \right )}\right) = 0 x → − ∞ lim ( x 1 cos ( 7 x ) ) = 0 Возьмём предел lim x → ∞ ( 1 x cos ( 7 x ) ) = 0 \lim_{x \to \infty}\left(\frac{1}{x} \cos{\left (7 x \right )}\right) = 0 x → ∞ lim ( x 1 cos ( 7 x ) ) = 0 Возьмём предел
Чётность и нечётность функции
Проверим функци чётна или нечётна с помощью соотношений f = f(-x) и f = -f(-x).cos ( 7 x ) = cos ( 7 x ) \cos{\left (7 x \right )} = \cos{\left (7 x \right )} cos ( 7 x ) = cos ( 7 x ) cos ( 7 x ) = − cos ( 7 x ) \cos{\left (7 x \right )} = - \cos{\left (7 x \right )} cos ( 7 x ) = − cos ( 7 x ) 