График функции
0 2 4 6 8 -8 -6 -4 -2 -10 10 -2000 1000
Точки пересечения с осью координат X
График функции пересекает ось X при f = 0tan ( x 2 + 1 ) = 0 \tan{\left (x^{2} + 1 \right )} = 0 tan ( x 2 + 1 ) = 0 Решаем это уравнение Численное решение x 1 = 45.6963796666 x_{1} = 45.6963796666 x 1 = 45.6963796666 x 2 = − 91.7522114808 x_{2} = -91.7522114808 x 2 = − 91.7522114808 x 3 = 56.0409908334 x_{3} = 56.0409908334 x 3 = 56.0409908334 x 4 = 20.3393763492 x_{4} = 20.3393763492 x 4 = 20.3393763492 x 5 = − 55.6754160208 x_{5} = -55.6754160208 x 5 = − 55.6754160208 x 6 = − 97.4798326687 x_{6} = -97.4798326687 x 6 = − 97.4798326687 x 7 = − 14.0328306901 x_{7} = -14.0328306901 x 7 = − 14.0328306901 x 8 = − 12.2391358916 x_{8} = -12.2391358916 x 8 = − 12.2391358916 x 9 = 39.021435609 x_{9} = 39.021435609 x 9 = 39.021435609 x 10 = 96.1330618823 x_{10} = 96.1330618823 x 10 = 96.1330618823 x 11 = − 8.25318352995 x_{11} = -8.25318352995 x 11 = − 8.25318352995 x 12 = 18.221109222 x_{12} = 18.221109222 x 12 = 18.221109222 x 13 = 63.9480541922 x_{13} = 63.9480541922 x 13 = 63.9480541922 x 14 = 48.4655682406 x_{14} = 48.4655682406 x 14 = 48.4655682406 x 15 = 31.4420369232 x_{15} = 31.4420369232 x 15 = 31.4420369232 x 16 = 51.2690262347 x_{16} = 51.2690262347 x 16 = 51.2690262347 x 17 = 26.2707134237 x_{17} = 26.2707134237 x 17 = 26.2707134237 x 18 = 60.3332900272 x_{18} = 60.3332900272 x 18 = 60.3332900272 x 19 = 38.2489500061 x_{19} = 38.2489500061 x 19 = 38.2489500061 x 20 = 8.25318352995 x_{20} = 8.25318352995 x 20 = 8.25318352995 x 21 = 76.2501672247 x_{21} = 76.2501672247 x 21 = 76.2501672247 x 22 = − 57.6165205028 x_{22} = -57.6165205028 x 22 = − 57.6165205028 x 23 = − 27.7252628505 x_{23} = -27.7252628505 x 23 = − 27.7252628505 x 24 = − 39.8183913545 x_{24} = -39.8183913545 x 24 = − 39.8183913545 x 25 = − 89.7793225031 x_{25} = -89.7793225031 x 25 = − 89.7793225031 x 26 = 100.401049922 x_{26} = 100.401049922 x 26 = 100.401049922 x 27 = − 35.8317093552 x_{27} = -35.8317093552 x 27 = − 35.8317093552 x 28 = − 79.7541542712 x_{28} = -79.7541542712 x 28 = − 79.7541542712 x 29 = − 42.0815667083 x_{29} = -42.0815667083 x 29 = − 42.0815667083 x 30 = − 89.9715747795 x_{30} = -89.9715747795 x 30 = − 89.9715747795 x 31 = 30.1151170824 x_{31} = 30.1151170824 x 31 = 30.1151170824 x 32 = 92.2473549679 x_{32} = 92.2473549679 x 32 = 92.2473549679 x 33 = − 3.83509625276 x_{33} = -3.83509625276 x 33 = − 3.83509625276 x 34 = 98.2502632286 x_{34} = 98.2502632286 x 34 = 98.2502632286 x 35 = − 68.0419544942 x_{35} = -68.0419544942 x 35 = − 68.0419544942 x 36 = 40.2499914193 x_{36} = 40.2499914193 x 36 = 40.2499914193 x 37 = 89.9017124291 x_{37} = 89.9017124291 x 37 = 89.9017124291 x 38 = 2.29851806762 x_{38} = 2.29851806762 x 38 = 2.29851806762 x 39 = 84.1809540603 x_{39} = 84.1809540603 x 39 = 84.1809540603 x 40 = 82.4844814961 x_{40} = 82.4844814961 x 40 = 82.4844814961 x 41 = − 51.8781736776 x_{41} = -51.8781736776 x 41 = − 51.8781736776 x 42 = − 1.46341814038 x_{42} = -1.46341814038 x 42 = − 1.46341814038 x 43 = 68.2494095599 x_{43} = 68.2494095599 x 43 = 68.2494095599 x 44 = − 71.7500547682 x_{44} = -71.7500547682 x 44 = − 71.7500547682 x 45 = 86.2273147619 x_{45} = 86.2273147619 x 45 = 86.2273147619 x 46 = − 35.7878443422 x_{46} = -35.7878443422 x 46 = − 35.7878443422 x 47 = − 21.7573089028 x_{47} = -21.7573089028 x 47 = − 21.7573089028 x 48 = 54.2464037187 x_{48} = 54.2464037187 x 48 = 54.2464037187 x 49 = − 29.7477583746 x_{49} = -29.7477583746 x 49 = − 29.7477583746 x 50 = − 46.749850127 x_{50} = -46.749850127 x 50 = − 46.749850127 x 51 = − 69.7068420968 x_{51} = -69.7068420968 x 51 = − 69.7068420968 x 52 = − 78.0016985072 x_{52} = -78.0016985072 x 52 = − 78.0016985072 x 53 = 34.1252234349 x_{53} = 34.1252234349 x 53 = 34.1252234349 x 54 = 80.2450306241 x_{54} = 80.2450306241 x 54 = 80.2450306241 x 55 = − 99.9463087783 x_{55} = -99.9463087783 x 55 = − 99.9463087783 x 56 = 3.83509625276 x_{56} = 3.83509625276 x 56 = 3.83509625276 x 57 = − 85.7340469674 x_{57} = -85.7340469674 x 57 = − 85.7340469674 x 58 = − 82.7506615371 x_{58} = -82.7506615371 x 58 = − 82.7506615371 x 59 = 66.0029469837 x_{59} = 66.0029469837 x 59 = 66.0029469837 x 60 = 90.2504833824 x_{60} = 90.2504833824 x 60 = 90.2504833824 x 61 = 89.4638367238 x_{61} = 89.4638367238 x 61 = 89.4638367238 x 62 = − 23.7589283775 x_{62} = -23.7589283775 x 62 = − 23.7589283775 x 63 = 69.5489229717 x_{63} = 69.5489229717 x 63 = 69.5489229717 x 64 = 57.7798666558 x_{64} = 57.7798666558 x 64 = 57.7798666558 x 65 = − 42.8216079765 x_{65} = -42.8216079765 x 65 = − 42.8216079765 x 66 = − 93.7338500868 x_{66} = -93.7338500868 x 66 = − 93.7338500868 x 67 = − 62.7579252971 x_{67} = -62.7579252971 x 67 = − 62.7579252971 x 68 = − 61.7486314136 x_{68} = -61.7486314136 x 68 = − 61.7486314136 x 69 = − 19.949492901 x_{69} = -19.949492901 x 69 = − 19.949492901 x 70 = − 47.7472121333 x_{70} = -47.7472121333 x 70 = − 47.7472121333 x 71 = − 15.9207099383 x_{71} = -15.9207099383 x 71 = − 15.9207099383 x 72 = − 9.97652068182 x_{72} = -9.97652068182 x 72 = − 9.97652068182 x 73 = 52.5400793919 x_{73} = 52.5400793919 x 73 = 52.5400793919 x 74 = − 49.8082141452 x_{74} = -49.8082141452 x 74 = − 49.8082141452 x 75 = − 73.8643786902 x_{75} = -73.8643786902 x 75 = − 73.8643786902 x 76 = − 30.8367082231 x_{76} = -30.8367082231 x 76 = − 30.8367082231 x 77 = 35.9192786763 x_{77} = 35.9192786763 x 77 = 35.9192786763 x 78 = − 97.8176426972 x_{78} = -97.8176426972 x 78 = − 97.8176426972 x 79 = − 25.7269518192 x_{79} = -25.7269518192 x 79 = − 25.7269518192 x 80 = 12.6182893188 x_{80} = 12.6182893188 x 80 = 12.6182893188 x 81 = 61.9264453205 x_{81} = 61.9264453205 x 81 = 61.9264453205 x 82 = 14.2549472985 x_{82} = 14.2549472985 x 82 = 14.2549472985 x 83 = − 17.7848491141 x_{83} = -17.7848491141 x 83 = − 17.7848491141 x 84 = 15.6219149588 x_{84} = 15.6219149588 x 84 = 15.6219149588 x 85 = 46.4464654523 x_{85} = 46.4464654523 x 85 = 46.4464654523 x 86 = − 33.8014752031 x_{86} = -33.8014752031 x 86 = − 33.8014752031 x 87 = − 83.7506799702 x_{87} = -83.7506799702 x 87 = − 83.7506799702 x 88 = 72.2518303593 x_{88} = 72.2518303593 x 88 = 72.2518303593 x 89 = 23.9564491444 x_{89} = 23.9564491444 x 89 = 23.9564491444 x 90 = 42.637801848 x_{90} = 42.637801848 x 90 = 42.637801848 x 91 = − 53.7518859325 x_{91} = -53.7518859325 x 91 = − 53.7518859325 x 92 = 22.2569458657 x_{92} = 22.2569458657 x 92 = 22.2569458657 x 93 = 28.063138742 x_{93} = 28.063138742 x 93 = 28.063138742 x 94 = − 59.7840483723 x_{94} = -59.7840483723 x 94 = − 59.7840483723 x 95 = − 65.8122805356 x_{95} = -65.8122805356 x 95 = − 65.8122805356 x 96 = − 76.0232233111 x_{96} = -76.0232233111 x 96 = − 76.0232233111 x 97 = 74.330757279 x_{97} = 74.330757279 x 97 = 74.330757279 x 98 = 4.22487347994 x_{98} = 4.22487347994 x 98 = 4.22487347994
Точки пересечения с осью координат Y
График пересекает ось Y, когда x равняется 0:tan ( 0 2 + 1 ) \tan{\left (0^{2} + 1 \right )} tan ( 0 2 + 1 ) f ( 0 ) = tan ( 1 ) f{\left (0 \right )} = \tan{\left (1 \right )} f ( 0 ) = tan ( 1 ) (0, tan(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 2 x ( tan 2 ( x 2 + 1 ) + 1 ) = 0 2 x \left(\tan^{2}{\left (x^{2} + 1 \right )} + 1\right) = 0 2 x ( tan 2 ( x 2 + 1 ) + 1 ) = 0 Решаем это уравнение x 1 = 0 x_{1} = 0 x 1 = 0 (0, tan(1)) Интервалы возрастания и убывания функции: x 1 = 0 x_{1} = 0 x 1 = 0 [0, oo) (-oo, 0]
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 ( tan 2 ( x 2 + 1 ) + 1 ) tan ( x 2 + 1 ) + tan 2 ( x 2 + 1 ) + 1 ) = 0 2 \left(4 x^{2} \left(\tan^{2}{\left (x^{2} + 1 \right )} + 1\right) \tan{\left (x^{2} + 1 \right )} + \tan^{2}{\left (x^{2} + 1 \right )} + 1\right) = 0 2 ( 4 x 2 ( tan 2 ( x 2 + 1 ) + 1 ) tan ( x 2 + 1 ) + tan 2 ( x 2 + 1 ) + 1 ) = 0 Решаем это уравнение x 1 = 98.2502630968 x_{1} = 98.2502630968 x 1 = 98.2502630968 x 2 = 24.02191947 x_{2} = 24.02191947 x 2 = 24.02191947 x 3 = 14.254904145 x_{3} = 14.254904145 x 3 = 14.254904145 x 4 = 4.22321440164 x_{4} = 4.22321440164 x 4 = 4.22321440164 x 5 = − 5.51431819183 x_{5} = -5.51431819183 x 5 = − 5.51431819183 x 6 = − 90.041382753 x_{6} = -90.041382753 x 6 = − 90.041382753 x 7 = − 39.7789207925 x_{7} = -39.7789207925 x 7 = − 39.7789207925 x 8 = − 49.9656487946 x_{8} = -49.9656487946 x 8 = − 49.9656487946 x 9 = − 43.5129893029 x_{9} = -43.5129893029 x 9 = − 43.5129893029 x 10 = − 1.42068899323 x_{10} = -1.42068899323 x 10 = − 1.42068899323 x 11 = − 63.7512418093 x_{11} = -63.7512418093 x 11 = − 63.7512418093 x 12 = − 79.7541540248 x_{12} = -79.7541540248 x 12 = − 79.7541540248 x 13 = 36.0066323488 x_{13} = 36.0066323488 x 13 = 36.0066323488 x 14 = − 55.7599914513 x_{14} = -55.7599914513 x 14 = − 55.7599914513 x 15 = 56.0690125356 x_{15} = 56.0690125356 x 15 = 56.0690125356 x 16 = 51.1463252333 x_{16} = 51.1463252333 x 16 = 51.1463252333 x 17 = − 17.7848268933 x_{17} = -17.7848268933 x 17 = − 17.7848268933 x 18 = − 28.0071034683 x_{18} = -28.0071034683 x 18 = − 28.0071034683 x 19 = − 91.752211319 x_{19} = -91.752211319 x 19 = − 91.752211319 x 20 = − 65.9553316097 x_{20} = -65.9553316097 x 20 = − 65.9553316097 x 21 = 42.0068438429 x_{21} = 42.0068438429 x 21 = 42.0068438429 x 22 = − 21.7572967663 x_{22} = -21.7572967663 x 22 = − 21.7572967663 x 23 = 80.2450303822 x_{23} = 80.2450303822 x 23 = 80.2450303822 x 24 = − 71.7500544297 x_{24} = -71.7500544297 x 24 = − 71.7500544297 x 25 = − 9.97639479349 x_{25} = -9.97639479349 x 25 = − 9.97639479349 x 26 = − 15.9206789624 x_{26} = -15.9206789624 x 26 = − 15.9206789624 x 27 = 32.2314586653 x_{27} = 32.2314586653 x 27 = 32.2314586653 x 28 = 60.1507661451 x_{28} = 60.1507661451 x 28 = 60.1507661451 x 29 = − 53.7518851276 x_{29} = -53.7518851276 x 29 = − 53.7518851276 x 30 = − 46.0047098719 x_{30} = -46.0047098719 x 30 = − 46.0047098719 x 31 = − 31.5915528796 x_{31} = -31.5915528796 x 31 = − 31.5915528796 x 32 = 2.2881150211 x_{32} = 2.2881150211 x 32 = 2.2881150211 x 33 = − 36.3107246539 x_{33} = -36.3107246539 x 33 = − 36.3107246539 x 34 = 46.4464642048 x_{34} = 46.4464642048 x 34 = 46.4464642048 x 35 = 66.0029465489 x_{35} = 66.0029465489 x 35 = 66.0029465489 x 36 = − 78.0016982438 x_{36} = -78.0016982438 x 36 = − 78.0016982438 x 37 = 8.25296116168 x_{37} = 8.25296116168 x 37 = 8.25296116168 x 38 = − 97.7694554486 x_{38} = -97.7694554486 x 38 = − 97.7694554486 x 39 = 86.2455295658 x_{39} = 86.2455295658 x 39 = 86.2455295658 x 40 = − 73.86437838 x_{40} = -73.86437838 x 40 = − 73.86437838 x 41 = − 57.8613661613 x_{41} = -57.8613661613 x 41 = − 57.8613661613 x 42 = 84.0128481674 x_{42} = 84.0128481674 x 42 = 84.0128481674 x 43 = − 69.9318218733 x_{43} = -69.9318218733 x 43 = − 69.9318218733 x 44 = 22.2569345283 x_{44} = 22.2569345283 x 44 = 22.2569345283 x 45 = 76.2501669427 x_{45} = 76.2501669427 x 45 = 76.2501669427 x 46 = 100.307134565 x_{46} = 100.307134565 x 46 = 100.307134565 x 47 = − 96.0022542366 x_{47} = -96.0022542366 x 47 = − 96.0022542366 x 48 = 12.2390677106 x_{48} = 12.2390677106 x 48 = 12.2390677106 x 49 = − 61.7486308827 x_{49} = -61.7486308827 x 49 = − 61.7486308827 x 50 = − 59.836573726 x_{50} = -59.836573726 x 50 = − 59.836573726 x 51 = − 37.7529189511 x_{51} = -37.7529189511 x 51 = − 37.7529189511 x 52 = 82.3510686162 x_{52} = 82.3510686162 x 52 = 82.3510686162 x 53 = 90.2504832124 x_{53} = 90.2504832124 x 53 = 90.2504832124 x 54 = 10.2864856749 x_{54} = 10.2864856749 x 54 = 10.2864856749 x 55 = 92.2473548087 x_{55} = 92.2473548087 x 55 = 92.2473548087 x 56 = 72.2518300279 x_{56} = 72.2518300279 x 56 = 72.2518300279 x 57 = − 85.7523665449 x_{57} = -85.7523665449 x 57 = − 85.7523665449 x 58 = 28.0071034683 x_{58} = 28.0071034683 x 58 = 28.0071034683 x 59 = 44.4769968117 x_{59} = 44.4769968117 x 59 = 44.4769968117 x 60 = − 47.747210985 x_{60} = -47.747210985 x 60 = − 47.747210985 x 61 = − 25.7269444784 x_{61} = -25.7269444784 x 61 = − 25.7269444784 x 62 = − 87.7441938904 x_{62} = -87.7441938904 x 62 = − 87.7441938904 x 63 = 38.2489477723 x_{63} = 38.2489477723 x 63 = 38.2489477723 x 64 = 16.0190394017 x_{64} = 16.0190394017 x 64 = 16.0190394017 x 65 = 57.9969452443 x_{65} = 57.9969452443 x 65 = 57.9969452443 x 66 = − 51.726558434 x_{66} = -51.726558434 x 66 = − 51.726558434 x 67 = 20.2619848159 x_{67} = 20.2619848159 x 67 = 20.2619848159 x 68 = − 3.83287719973 x_{68} = -3.83287719973 x 68 = − 3.83287719973 x 69 = 40.2499895023 x_{69} = 40.2499895023 x 69 = 40.2499895023 x 70 = 34.1252202894 x_{70} = 34.1252202894 x 70 = 34.1252202894 x 71 = − 41.8194567767 x_{71} = -41.8194567767 x 71 = − 41.8194567767 x 72 = 4.91145373377 x_{72} = 4.91145373377 x 72 = 4.91145373377 x 73 = 96.2473724756 x_{73} = 96.2473724756 x 73 = 96.2473724756 x 74 = 74.2461787683 x_{74} = 74.2461787683 x 74 = 74.2461787683 x 75 = 68.2724208176 x_{75} = 68.2724208176 x 75 = 68.2724208176 x 76 = 70.2008463108 x_{76} = 70.2008463108 x 76 = 70.2008463108 x 77 = − 68.0188644767 x_{77} = -68.0188644767 x 77 = − 68.0188644767 x 78 = − 23.7589190572 x_{78} = -23.7589190572 x 78 = − 23.7589190572 x 79 = 87.5291062988 x_{79} = 87.5291062988 x 79 = 87.5291062988 x 80 = − 14.0327854548 x_{80} = -14.0327854548 x 80 = − 14.0327854548 x 81 = 77.6585980651 x_{81} = 77.6585980651 x 81 = 77.6585980651 x 82 = 93.1117376869 x_{82} = 93.1117376869 x 82 = 93.1117376869 x 83 = − 83.7506797574 x_{83} = -83.7506797574 x 83 = − 83.7506797574 x 84 = − 99.9934467421 x_{84} = -99.9934467421 x 84 = − 99.9934467421 x 85 = 61.8503014279 x_{85} = 61.8503014279 x 85 = 61.8503014279 x 86 = 48.2707120485 x_{86} = 48.2707120485 x 86 = 48.2707120485 x 87 = − 75.9405298154 x_{87} = -75.9405298154 x 87 = − 75.9405298154 x 88 = − 93.750606485 x_{88} = -93.750606485 x 88 = − 93.750606485 x 89 = 51.9991459765 x_{89} = 51.9991459765 x 89 = 51.9991459765 x 90 = − 29.7477536262 x_{90} = -29.7477536262 x 90 = − 29.7477536262 x 91 = − 12.1100433533 x_{91} = -12.1100433533 x 91 = − 12.1100433533 x 92 = − 82.2747356821 x_{92} = -82.2747356821 x 92 = − 82.2747356821 x 93 = 54.2464029356 x_{93} = 54.2464029356 x 93 = 54.2464029356 x 94 = 29.9582245865 x_{94} = 29.9582245865 x 94 = 29.9582245865 x 95 = − 33.7549687282 x_{95} = -33.7549687282 x 95 = − 33.7549687282 x 96 = 64.339871441 x_{96} = 64.339871441 x 96 = 64.339871441 x 97 = 26.2707065293 x_{97} = 26.2707065293 x 97 = 26.2707065293 x 98 = 18.2210885593 x_{98} = 18.2210885593 x 98 = 18.2210885593 x 99 = − 20.0280612244 x_{99} = -20.0280612244 x 99 = − 20.0280612244 x 100 = − 8.0603720628 x_{100} = -8.0603720628 x 100 = − 8.0603720628 x 101 = 1.42068899323 x_{101} = 1.42068899323 x 101 = 1.42068899323 Интервалы выпуклости и вогнутости: [100.307134565, oo) [-1.42068899323, 1.42068899323]
Горизонтальные асимптоты
Горизонтальные асимптоты найдём с помощью пределов данной функции при x->+oo и x->-oolim x → − ∞ tan ( x 2 + 1 ) = ⟨ − ∞ , ∞ ⟩ \lim_{x \to -\infty} \tan{\left (x^{2} + 1 \right )} = \langle -\infty, \infty\rangle x → − ∞ lim tan ( x 2 + 1 ) = ⟨ − ∞ , ∞ ⟩ Возьмём предел y = ⟨ − ∞ , ∞ ⟩ y = \langle -\infty, \infty\rangle y = ⟨ − ∞ , ∞ ⟩ lim x → ∞ tan ( x 2 + 1 ) = ⟨ − ∞ , ∞ ⟩ \lim_{x \to \infty} \tan{\left (x^{2} + 1 \right )} = \langle -\infty, \infty\rangle x → ∞ lim tan ( x 2 + 1 ) = ⟨ − ∞ , ∞ ⟩ Возьмём предел y = ⟨ − ∞ , ∞ ⟩ y = \langle -\infty, \infty\rangle y = ⟨ − ∞ , ∞ ⟩
Наклонные асимптоты
Наклонную асимптоту можно найти, подсчитав предел функции tan(x^2 + 1), делённой на x при x->+oo и x ->-ooTrue Возьмём предел y = x lim x → − ∞ ( 1 x tan ( x 2 + 1 ) ) y = x \lim_{x \to -\infty}\left(\frac{1}{x} \tan{\left (x^{2} + 1 \right )}\right) y = x x → − ∞ lim ( x 1 tan ( x 2 + 1 ) ) True Возьмём предел y = x lim x → ∞ ( 1 x tan ( x 2 + 1 ) ) y = x \lim_{x \to \infty}\left(\frac{1}{x} \tan{\left (x^{2} + 1 \right )}\right) y = x x → ∞ lim ( x 1 tan ( x 2 + 1 ) )
Чётность и нечётность функции
Проверим функци чётна или нечётна с помощью соотношений f = f(-x) и f = -f(-x).tan ( x 2 + 1 ) = tan ( x 2 + 1 ) \tan{\left (x^{2} + 1 \right )} = \tan{\left (x^{2} + 1 \right )} tan ( x 2 + 1 ) = tan ( x 2 + 1 ) tan ( x 2 + 1 ) = − tan ( x 2 + 1 ) \tan{\left (x^{2} + 1 \right )} = - \tan{\left (x^{2} + 1 \right )} tan ( x 2 + 1 ) = − tan ( x 2 + 1 ) 