График функции
0 2 4 6 8 -8 -6 -4 -2 -10 10 -100 100
Точки пересечения с осью координат X
График функции пересекает ось X при f = 0cot ( x 2 − 2 ) = 0 \cot{\left (x^{2} - 2 \right )} = 0 cot ( x 2 − 2 ) = 0 Решаем это уравнение Аналитическое решение x 1 = − 1 2 2 π + 8 x_{1} = - \frac{1}{2} \sqrt{2 \pi + 8} x 1 = − 2 1 2 π + 8 x 2 = 1 2 2 π + 8 x_{2} = \frac{1}{2} \sqrt{2 \pi + 8} x 2 = 2 1 2 π + 8 Численное решение x 1 = − 27.7510252748 x_{1} = -27.7510252748 x 1 = − 27.7510252748 x 2 = 34.5576968567 x_{2} = 34.5576968567 x 2 = 34.5576968567 x 3 = − 7.75367423171 x_{3} = -7.75367423171 x 3 = − 7.75367423171 x 4 = 98.257536241 x_{4} = 98.257536241 x 4 = 98.257536241 x 5 = − 17.912890731 x_{5} = -17.912890731 x 5 = − 17.912890731 x 6 = − 51.7707227802 x_{6} = -51.7707227802 x 6 = − 51.7707227802 x 7 = 43.7812827857 x_{7} = 43.7812827857 x 7 = 43.7812827857 x 8 = 25.3861319948 x_{8} = 25.3861319948 x 8 = 25.3861319948 x 9 = 64.2288125208 x_{9} = 64.2288125208 x 9 = 64.2288125208 x 10 = − 45.7120150323 x_{10} = -45.7120150323 x 10 = − 45.7120150323 x 11 = 60.1626457248 x_{11} = 60.1626457248 x 11 = 60.1626457248 x 12 = 38.2676283593 x_{12} = 38.2676283593 x 12 = 38.2676283593 x 13 = 32.0091925663 x_{13} = 32.0091925663 x 13 = 32.0091925663 x 14 = − 53.9983998486 x_{14} = -53.9983998486 x 14 = − 53.9983998486 x 15 = 52.1937739039 x_{15} = 52.1937739039 x 15 = 52.1937739039 x 16 = 36.2871399469 x_{16} = 36.2871399469 x 16 = 36.2871399469 x 17 = − 0.655136377562 x_{17} = -0.655136377562 x 17 = − 0.655136377562 x 18 = − 84.0026565273 x_{18} = -84.0026565273 x 18 = − 84.0026565273 x 19 = − 100.110487805 x_{19} = -100.110487805 x 19 = − 100.110487805 x 20 = 54.0855984913 x_{20} = 54.0855984913 x 20 = 54.0855984913 x 21 = 4.01710927673 x_{21} = 4.01710927673 x 21 = 4.01710927673 x 22 = 70.3451322628 x_{22} = 70.3451322628 x 22 = 70.3451322628 x 23 = − 49.0920642159 x_{23} = -49.0920642159 x 23 = − 49.0920642159 x 24 = 28.2558620057 x_{24} = 28.2558620057 x 24 = 28.2558620057 x 25 = 100.626946323 x_{25} = 100.626946323 x 25 = 100.626946323 x 26 = − 19.7480807351 x_{26} = -19.7480807351 x 26 = − 19.7480807351 x 27 = 34.1001244206 x_{27} = 34.1001244206 x 27 = 34.1001244206 x 28 = − 87.5193241126 x_{28} = -87.5193241126 x 28 = − 87.5193241126 x 29 = − 65.7515071691 x_{29} = -65.7515071691 x 29 = − 65.7515071691 x 30 = − 47.9263334029 x_{30} = -47.9263334029 x 30 = − 47.9263334029 x 31 = − 93.9590578936 x_{31} = -93.9590578936 x 31 = − 93.9590578936 x 32 = − 61.8364573944 x_{32} = -61.8364573944 x 32 = − 61.8364573944 x 33 = 42.2475304712 x_{33} = 42.2475304712 x 33 = 42.2475304712 x 34 = − 64.0083277703 x_{34} = -64.0083277703 x 34 = − 64.0083277703 x 35 = − 15.7675306661 x_{35} = -15.7675306661 x 35 = − 15.7675306661 x 36 = 19.8274629081 x_{36} = 19.8274629081 x 36 = 19.8274629081 x 37 = 24.3114915739 x_{37} = 24.3114915739 x 37 = 24.3114915739 x 38 = − 24.1819237705 x_{38} = -24.1819237705 x 38 = − 24.1819237705 x 39 = − 95.829559793 x_{39} = -95.829559793 x 39 = − 95.829559793 x 40 = 22.0053282713 x_{40} = 22.0053282713 x 40 = 22.0053282713 x 41 = − 22.0053282713 x_{41} = -22.0053282713 x 41 = − 22.0053282713 x 42 = − 41.7989798807 x_{42} = -41.7989798807 x 42 = − 41.7989798807 x 43 = 96.2221531343 x_{43} = 96.2221531343 x 43 = 96.2221531343 x 44 = 56.2495600502 x_{44} = 56.2495600502 x 44 = 56.2495600502 x 45 = − 11.5054810905 x_{45} = -11.5054810905 x 45 = − 11.5054810905 x 46 = 86.0167419918 x_{46} = 86.0167419918 x 46 = 86.0167419918 x 47 = − 69.9195778925 x_{47} = -69.9195778925 x 47 = − 69.9195778925 x 48 = 39.954452464 x_{48} = 39.954452464 x 48 = 39.954452464 x 49 = − 37.7718449166 x_{49} = -37.7718449166 x 49 = − 37.7718449166 x 50 = − 98.0655103301 x_{50} = -98.0655103301 x 50 = − 98.0655103301 x 51 = 72.0003964024 x_{51} = 72.0003964024 x 51 = 72.0003964024 x 52 = − 13.7449904652 x_{52} = -13.7449904652 x 52 = − 13.7449904652 x 53 = − 1.88965508144 x_{53} = -1.88965508144 x 53 = − 1.88965508144 x 54 = 94.8409667333 x_{54} = 94.8409667333 x 54 = 94.8409667333 x 55 = 7.95368196152 x_{55} = 7.95368196152 x 55 = 7.95368196152 x 56 = − 72.0658161687 x_{56} = -72.0658161687 x 56 = − 72.0658161687 x 57 = 19.1832099541 x_{57} = 19.1832099541 x 57 = 19.1832099541 x 58 = − 67.7285351821 x_{58} = -67.7285351821 x 58 = − 67.7285351821 x 59 = 67.9138220036 x_{59} = 67.9138220036 x 59 = 67.9138220036 x 60 = − 6.17481299444 x_{60} = -6.17481299444 x 60 = − 6.17481299444 x 61 = − 92.0676174508 x_{61} = -92.0676174508 x 61 = − 92.0676174508 x 62 = − 4.01710927673 x_{62} = -4.01710927673 x 62 = − 4.01710927673 x 63 = 49.1879614025 x_{63} = 49.1879614025 x 63 = 49.1879614025 x 64 = 76.0945757872 x_{64} = 76.0945757872 x 64 = 76.0945757872 x 65 = 90.0493189457 x_{65} = 90.0493189457 x 65 = 90.0493189457 x 66 = − 91.7428794111 x_{66} = -91.7428794111 x 66 = − 91.7428794111 x 67 = 50.4180291208 x_{67} = 50.4180291208 x 67 = 50.4180291208 x 68 = 92.0505545361 x_{68} = 92.0505545361 x 68 = 92.0505545361 x 69 = − 9.73020982718 x_{69} = -9.73020982718 x 69 = − 9.73020982718 x 70 = − 72.0003964024 x_{70} = -72.0003964024 x 70 = − 72.0003964024 x 71 = 78.2521136169 x_{71} = 78.2521136169 x 71 = 78.2521136169 x 72 = − 33.7761356127 x_{72} = -33.7761356127 x 72 = − 33.7761356127 x 73 = − 26.6538767046 x_{73} = -26.6538767046 x 73 = − 26.6538767046 x 74 = 30.3465953803 x_{74} = 30.3465953803 x 74 = 30.3465953803 x 75 = 11.6412064571 x_{75} = 11.6412064571 x 75 = 11.6412064571 x 76 = 74.0015215984 x_{76} = 74.0015215984 x 76 = 74.0015215984 x 77 = − 29.8771069097 x_{77} = -29.8771069097 x 77 = − 29.8771069097 x 78 = − 44.4577437887 x_{78} = -44.4577437887 x 78 = − 44.4577437887 x 79 = 10.2030270627 x_{79} = 10.2030270627 x 79 = 10.2030270627 x 80 = − 31.7133860195 x_{80} = -31.7133860195 x 80 = − 31.7133860195 x 81 = − 34.375398782 x_{81} = -34.375398782 x 81 = − 34.375398782 x 82 = 1.88965508144 x_{82} = 1.88965508144 x 82 = 1.88965508144 x 83 = − 77.7486557387 x_{83} = -77.7486557387 x 83 = − 77.7486557387 x 84 = − 80.3517399115 x_{84} = -80.3517399115 x 84 = − 80.3517399115 x 85 = 81.5932842618 x_{85} = 81.5932842618 x 85 = 81.5932842618 x 86 = 66.2512953891 x_{86} = 66.2512953891 x 86 = 66.2512953891 x 87 = 16.2580129959 x_{87} = 16.2580129959 x 87 = 16.2580129959 x 88 = 58.3063671897 x_{88} = 58.3063671897 x 88 = 58.3063671897 x 89 = − 73.8315139873 x_{89} = -73.8315139873 x 89 = − 73.8315139873 x 90 = − 56.2216276284 x_{90} = -56.2216276284 x 90 = − 56.2216276284 x 91 = 13.9716837996 x_{91} = 13.9716837996 x 91 = 13.9716837996 x 92 = 62.2415692729 x_{92} = 62.2415692729 x 92 = 62.2415692729 x 93 = 78.9316630747 x_{93} = 78.9316630747 x 93 = 78.9316630747 x 94 = 67.8675477223 x_{94} = 67.8675477223 x 94 = 67.8675477223 x 95 = − 86.0532572782 x_{95} = -86.0532572782 x 95 = − 86.0532572782 x 96 = − 39.7968830574 x_{96} = -39.7968830574 x 96 = − 39.7968830574 x 97 = 46.1905901181 x_{97} = 46.1905901181 x 97 = 46.1905901181 x 98 = − 59.7171404665 x_{98} = -59.7171404665 x 98 = − 59.7171404665 x 99 = − 56.9434177248 x_{99} = -56.9434177248 x 99 = − 56.9434177248
Точки пересечения с осью координат Y
График пересекает ось Y, когда x равняется 0:cot ( − 2 + 0 2 ) \cot{\left (-2 + 0^{2} \right )} cot ( − 2 + 0 2 ) f ( 0 ) = − cot ( 2 ) f{\left (0 \right )} = - \cot{\left (2 \right )} f ( 0 ) = − cot ( 2 ) (0, -cot(2))
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 2 x ( − cot 2 ( x 2 − 2 ) − 1 ) = 0 2 x \left(- \cot^{2}{\left (x^{2} - 2 \right )} - 1\right) = 0 2 x ( − cot 2 ( x 2 − 2 ) − 1 ) = 0 Решаем это уравнение x 1 = 0 x_{1} = 0 x 1 = 0 (0, -cot(2)) Интервалы возрастания и убывания функции: x 1 = 0 x_{1} = 0 x 1 = 0 (-oo, 0] [0, oo)
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 2 ( 4 x 2 ( cot 2 ( x 2 − 2 ) + 1 ) cot ( x 2 − 2 ) − cot 2 ( x 2 − 2 ) − 1 ) = 0 2 \left(4 x^{2} \left(\cot^{2}{\left (x^{2} - 2 \right )} + 1\right) \cot{\left (x^{2} - 2 \right )} - \cot^{2}{\left (x^{2} - 2 \right )} - 1\right) = 0 2 ( 4 x 2 ( cot 2 ( x 2 − 2 ) + 1 ) cot ( x 2 − 2 ) − cot 2 ( x 2 − 2 ) − 1 ) = 0 Решаем это уравнение x 1 = 6.89549386648 x_{1} = 6.89549386648 x 1 = 6.89549386648 x 2 = 50.9449176976 x_{2} = 50.9449176976 x 2 = 50.9449176976 x 3 = 30.2428899501 x_{3} = 30.2428899501 x 3 = 30.2428899501 x 4 = 19.9852656914 x_{4} = 19.9852656914 x 4 = 19.9852656914 x 5 = 36.2438235091 x_{5} = 36.2438235091 x 5 = 36.2438235091 x 6 = 22.0053165405 x_{6} = 22.0053165405 x 6 = 22.0053165405 x 7 = − 9.73007413415 x_{7} = -9.73007413415 x 7 = − 9.73007413415 x 8 = 48.5774143052 x_{8} = 48.5774143052 x 8 = 48.5774143052 x 9 = − 6.17428195627 x_{9} = -6.17428195627 x 9 = − 6.17428195627 x 10 = 56.3053826227 x_{10} = 56.3053826227 x 10 = 56.3053826227 x 11 = − 59.8485150941 x_{11} = -59.8485150941 x 11 = − 59.8485150941 x 12 = − 84.0026563164 x_{12} = -84.0026563164 x 12 = − 84.0026563164 x 13 = − 69.6945578554 x_{13} = -69.6945578554 x 13 = − 69.6945578554 x 14 = − 98.0014179472 x_{14} = -98.0014179472 x 14 = − 98.0014179472 x 15 = 92.255101067 x_{15} = 92.255101067 x 15 = 92.255101067 x 16 = − 47.8935458262 x_{16} = -47.8935458262 x 16 = − 47.8935458262 x 17 = − 11.6411272211 x_{17} = -11.6411272211 x 17 = − 11.6411272211 x 18 = − 73.9378142441 x_{18} = -73.9378142441 x 18 = − 73.9378142441 x 19 = − 25.7547059133 x_{19} = -25.7547059133 x 19 = − 25.7547059133 x 20 = − 95.7475667554 x_{20} = -95.7475667554 x 20 = − 95.7475667554 x 21 = 68.0755338853 x_{21} = 68.0755338853 x 21 = 68.0755338853 x 22 = − 15.7674987786 x_{22} = -15.7674987786 x 22 = − 15.7674987786 x 23 = − 35.7639094371 x_{23} = -35.7639094371 x 23 = − 35.7639094371 x 24 = 78.2521133561 x_{24} = 78.2521133561 x 24 = 78.2521133561 x 25 = 89.9969723589 x_{25} = 89.9969723589 x 25 = 89.9969723589 x 26 = 62.2668007346 x_{26} = 62.2668007346 x 26 = 62.2668007346 x 27 = − 30.0344138966 x_{27} = -30.0344138966 x 27 = − 30.0344138966 x 28 = 64.253263652 x_{28} = 64.253263652 x 28 = 64.253263652 x 29 = − 75.7428369257 x_{29} = -75.7428369257 x 29 = − 75.7428369257 x 30 = − 31.6141648218 x_{30} = -31.6141648218 x 30 = − 31.6141648218 x 31 = − 53.9983990547 x_{31} = -53.9983990547 x 31 = − 53.9983990547 x 32 = − 7.75340605416 x_{32} = -7.75340605416 x 32 = − 7.75340605416 x 33 = − 56.1377462666 x_{33} = -56.1377462666 x 33 = − 56.1377462666 x 34 = − 18.000346536 x_{34} = -18.000346536 x 34 = − 18.000346536 x 35 = 52.2539293076 x_{35} = 52.2539293076 x 35 = 52.2539293076 x 36 = 24.24678545 x_{36} = 24.24678545 x 36 = 24.24678545 x 37 = − 81.7663643139 x_{37} = -81.7663643139 x 37 = − 81.7663643139 x 38 = 26.2380950465 x_{38} = 26.2380950465 x 38 = 26.2380950465 x 39 = − 57.9821806131 x_{39} = -57.9821806131 x 39 = − 57.9821806131 x 40 = − 89.7522854107 x_{40} = -89.7522854107 x 40 = − 89.7522854107 x 41 = 85.961939782 x_{41} = 85.961939782 x 41 = 85.961939782 x 42 = 84.0587356925 x_{42} = 84.0587356925 x 42 = 84.0587356925 x 43 = 46.2585525491 x_{43} = 46.2585525491 x 43 = 46.2585525491 x 44 = 39.9937458617 x_{44} = 39.9937458617 x 44 = 39.9937458617 x 45 = − 1.87068924915 x_{45} = -1.87068924915 x 45 = − 1.87068924915 x 46 = 81.8431713939 x_{46} = 81.8431713939 x 46 = 81.8431713939 x 47 = 75.8464587765 x_{47} = 75.8464587765 x 47 = 75.8464587765 x 48 = − 61.7856311151 x_{48} = -61.7856311151 x 48 = − 61.7856311151 x 49 = − 49.6330307618 x_{49} = -49.6330307618 x 49 = − 49.6330307618 x 50 = 89.0141890064 x_{50} = 89.0141890064 x 50 = 89.0141890064 x 51 = 16.2579839082 x_{51} = 16.2579839082 x 51 = 16.2579839082 x 52 = − 43.7453883177 x_{52} = -43.7453883177 x 52 = − 43.7453883177 x 53 = 72.0003960675 x_{53} = 72.0003960675 x 53 = 72.0003960675 x 54 = 28.2558564647 x_{54} = 28.2558564647 x 54 = 28.2558564647 x 55 = − 72.0003960675 x_{55} = -72.0003960675 x 55 = − 72.0003960675 x 56 = − 19.7480645044 x_{56} = -19.7480645044 x 56 = − 19.7480645044 x 57 = 43.2034154433 x_{57} = 43.2034154433 x 57 = 43.2034154433 x 58 = − 99.8276568681 x_{58} = -99.8276568681 x 58 = − 99.8276568681 x 59 = − 85.9984783378 x_{59} = -85.9984783378 x 59 = − 85.9984783378 x 60 = 94.2428331876 x_{60} = 94.2428331876 x 60 = 94.2428331876 x 61 = 96.2547968266 x_{61} = 96.2547968266 x 61 = 96.2547968266 x 62 = − 41.7613814794 x_{62} = -41.7613814794 x 62 = − 41.7613814794 x 63 = 32.3508781712 x_{63} = 32.3508781712 x 63 = 32.3508781712 x 64 = 13.9716379678 x_{64} = 13.9716379678 x 64 = 13.9716379678 x 65 = 58.2524608568 x_{65} = 58.2524608568 x 65 = 58.2524608568 x 66 = 38.2676261287 x_{66} = 38.2676261287 x 66 = 38.2676261287 x 67 = − 77.7486554728 x_{67} = -77.7486554728 x 67 = − 77.7486554728 x 68 = − 64.0083272937 x_{68} = -64.0083272937 x 68 = − 64.0083272937 x 69 = 7.95343351266 x_{69} = 7.95343351266 x 69 = 7.95343351266 x 70 = − 37.7302332372 x_{70} = -37.7302332372 x 70 = − 37.7302332372 x 71 = 66.2512949593 x_{71} = 66.2512949593 x 71 = 66.2512949593 x 72 = − 73.6184520687 x_{72} = -73.6184520687 x 72 = − 73.6184520687 x 73 = 42.2475288135 x_{73} = 42.2475288135 x 73 = 42.2475288135 x 74 = 80.2539351354 x_{74} = 80.2539351354 x 74 = 80.2539351354 x 75 = 33.6829917208 x_{75} = 33.6829917208 x 75 = 33.6829917208 x 76 = − 91.9993466439 x_{76} = -91.9993466439 x 76 = − 91.9993466439 x 77 = − 13.7449423281 x_{77} = -13.7449423281 x 77 = − 13.7449423281 x 78 = 10.202909374 x_{78} = 10.202909374 x 78 = 10.202909374 x 79 = − 67.7285347798 x_{79} = -67.7285347798 x 79 = − 67.7285347798 x 80 = − 24.0516490031 x_{80} = -24.0516490031 x 80 = − 24.0516490031 x 81 = 60.0319573891 x_{81} = 60.0319573891 x 81 = 60.0319573891 x 82 = 4.01517883749 x_{82} = 4.01517883749 x 82 = 4.01517883749 x 83 = 73.8102351921 x_{83} = 73.8102351921 x 83 = 73.8102351921 x 84 = − 51.7707218793 x_{84} = -51.7707218793 x 84 = − 51.7707218793 x 85 = 11.9079431485 x_{85} = 11.9079431485 x 85 = 11.9079431485 x 86 = 70.2557557202 x_{86} = 70.2557557202 x 86 = 70.2557557202 x 87 = − 27.7510194259 x_{87} = -27.7510194259 x 87 = − 27.7510194259 x 88 = 1.87068924915 x_{88} = 1.87068924915 x 88 = 1.87068924915 x 89 = 18.000346536 x_{89} = 18.000346536 x 89 = 18.000346536 x 90 = − 22.0053165405 x_{90} = -22.0053165405 x 90 = − 22.0053165405 x 91 = − 34.0078679724 x_{91} = -34.0078679724 x 91 = − 34.0078679724 x 92 = 54.0855977013 x_{92} = 54.0855977013 x 92 = 54.0855977013 x 93 = 100.282936051 x_{93} = 100.282936051 x 93 = 100.282936051 x 94 = − 65.7515067294 x_{94} = -65.7515067294 x 94 = − 65.7515067294 x 95 = − 39.7573911401 x_{95} = -39.7573911401 x 95 = − 39.7573911401 x 96 = 98.0014179472 x_{96} = 98.0014179472 x 96 = 98.0014179472 x 97 = − 93.9924875576 x_{97} = -93.9924875576 x 97 = − 93.9924875576 x 98 = − 87.7523376621 x_{98} = -87.7523376621 x 98 = − 87.7523376621 x 99 = − 45.7463636971 x_{99} = -45.7463636971 x 99 = − 45.7463636971 x 100 = − 4.01517883749 x_{100} = -4.01517883749 x 100 = − 4.01517883749 Интервалы выпуклости и вогнутости: [-1.87068924915, 1.87068924915] (-oo, -99.8276568681]
Горизонтальные асимптоты
Горизонтальные асимптоты найдём с помощью пределов данной функции при x->+oo и x->-ooTrue Возьмём предел y = lim x → − ∞ cot ( x 2 − 2 ) y = \lim_{x \to -\infty} \cot{\left (x^{2} - 2 \right )} y = x → − ∞ lim cot ( x 2 − 2 ) True Возьмём предел y = lim x → ∞ cot ( x 2 − 2 ) y = \lim_{x \to \infty} \cot{\left (x^{2} - 2 \right )} y = x → ∞ lim cot ( x 2 − 2 )
Наклонные асимптоты
Наклонную асимптоту можно найти, подсчитав предел функции cot(x^2 - 2), делённой на x при x->+oo и x ->-ooTrue Возьмём предел y = x lim x → − ∞ ( 1 x cot ( x 2 − 2 ) ) y = x \lim_{x \to -\infty}\left(\frac{1}{x} \cot{\left (x^{2} - 2 \right )}\right) y = x x → − ∞ lim ( x 1 cot ( x 2 − 2 ) ) True Возьмём предел y = x lim x → ∞ ( 1 x cot ( x 2 − 2 ) ) y = x \lim_{x \to \infty}\left(\frac{1}{x} \cot{\left (x^{2} - 2 \right )}\right) y = x x → ∞ lim ( x 1 cot ( x 2 − 2 ) )
Чётность и нечётность функции
Проверим функци чётна или нечётна с помощью соотношений f = f(-x) и f = -f(-x).cot ( x 2 − 2 ) = cot ( x 2 − 2 ) \cot{\left (x^{2} - 2 \right )} = \cot{\left (x^{2} - 2 \right )} cot ( x 2 − 2 ) = cot ( x 2 − 2 ) cot ( x 2 − 2 ) = − cot ( x 2 − 2 ) \cot{\left (x^{2} - 2 \right )} = - \cot{\left (x^{2} - 2 \right )} cot ( x 2 − 2 ) = − cot ( x 2 − 2 ) 