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 ( x 2 ) = 0 \cos{\left (x^{2} \right )} = 0 cos ( x 2 ) = 0 Решаем это уравнение Аналитическое решение x 1 = − 2 π 2 x_{1} = - \frac{\sqrt{2} \sqrt{\pi}}{2} x 1 = − 2 2 π x 2 = 2 π 2 x_{2} = \frac{\sqrt{2} \sqrt{\pi}}{2} x 2 = 2 2 π x 3 = − 6 π 2 x_{3} = - \frac{\sqrt{6} \sqrt{\pi}}{2} x 3 = − 2 6 π x 4 = 6 π 2 x_{4} = \frac{\sqrt{6} \sqrt{\pi}}{2} x 4 = 2 6 π Численное решение x 1 = 36.0423162721 x_{1} = 36.0423162721 x 1 = 36.0423162721 x 2 = 20.2479095537 x_{2} = 20.2479095537 x 2 = 20.2479095537 x 3 = − 74.2423350301 x_{3} = -74.2423350301 x 3 = − 74.2423350301 x 4 = 63.9435910678 x_{4} = 63.9435910678 x 4 = 63.9435910678 x 5 = 67.736962366 x_{5} = 67.736962366 x 5 = 67.736962366 x 6 = − 65.8556653469 x_{6} = -65.8556653469 x 6 = − 65.8556653469 x 7 = − 29.8962056112 x_{7} = -29.8962056112 x 7 = − 29.8962056112 x 8 = 70.1296178458 x_{8} = 70.1296178458 x 8 = 70.1296178458 x 9 = 23.5476347721 x_{9} = 23.5476347721 x 9 = 23.5476347721 x 10 = − 63.2520216621 x_{10} = -63.2520216621 x 10 = − 63.2520216621 x 11 = 41.9626343589 x_{11} = 41.9626343589 x 11 = 41.9626343589 x 12 = − 3.75994241195 x_{12} = -3.75994241195 x 12 = − 3.75994241195 x 13 = − 33.7930307842 x_{13} = -33.7930307842 x 13 = − 33.7930307842 x 14 = 58.1813207502 x_{14} = 58.1813207502 x 14 = 58.1813207502 x 15 = 80.1827248575 x_{15} = 80.1827248575 x 15 = 80.1827248575 x 16 = − 97.7504683155 x_{16} = -97.7504683155 x 16 = − 97.7504683155 x 17 = 114.58746221 x_{17} = 114.58746221 x 17 = 114.58746221 x 18 = − 92.1420324625 x_{18} = -92.1420324625 x 18 = − 92.1420324625 x 19 = 36.7757479512 x_{19} = 36.7757479512 x 19 = 36.7757479512 x 20 = − 67.8759576469 x_{20} = -67.8759576469 x 20 = − 67.8759576469 x 21 = 34.2088185043 x_{21} = 34.2088185043 x 21 = 34.2088185043 x 22 = − 68.2222069146 x_{22} = -68.2222069146 x 22 = − 68.2222069146 x 23 = − 37.7869535826 x_{23} = -37.7869535826 x 23 = − 37.7869535826 x 24 = 68.1530977897 x_{24} = 68.1530977897 x 24 = 68.1530977897 x 25 = − 81.7925519581 x_{25} = -81.7925519581 x 25 = − 81.7925519581 x 26 = − 71.6803654075 x_{26} = -71.6803654075 x 26 = − 71.6803654075 x 27 = − 93.7978156802 x_{27} = -93.7978156802 x 27 = − 93.7978156802 x 28 = 17.4115964538 x_{28} = 17.4115964538 x 28 = 17.4115964538 x 29 = 66.1412713071 x_{29} = 66.1412713071 x 29 = 66.1412713071 x 30 = 13.5566651591 x_{30} = 13.5566651591 x 30 = 13.5566651591 x 31 = 40.1647585281 x_{31} = 40.1647585281 x 31 = 40.1647585281 x 32 = 26.0797777886 x_{32} = 26.0797777886 x 32 = 26.0797777886 x 33 = − 47.8726627711 x_{33} = -47.8726627711 x 33 = − 47.8726627711 x 34 = 27.9969170994 x_{34} = 27.9969170994 x 34 = 27.9969170994 x 35 = 82.2330723423 x_{35} = 82.2330723423 x 35 = 82.2330723423 x 36 = 54.1251803643 x_{36} = 54.1251803643 x 36 = 54.1251803643 x 37 = 60.2503967723 x_{37} = 60.2503967723 x 37 = 60.2503967723 x 38 = 103.115082971 x_{38} = 103.115082971 x 38 = 103.115082971 x 39 = − 69.7703218907 x_{39} = -69.7703218907 x 39 = − 69.7703218907 x 40 = 31.3829459564 x_{40} = 31.3829459564 x 40 = 31.3829459564 x 41 = − 76.1639733395 x_{41} = -76.1639733395 x 41 = − 76.1639733395 x 42 = − 75.895389388 x_{42} = -75.895389388 x 42 = − 75.895389388 x 43 = − 79.6323098451 x_{43} = -79.6323098451 x 43 = − 79.6323098451 x 44 = 56.1199308198 x_{44} = 56.1199308198 x 44 = 56.1199308198 x 45 = − 46.338737345 x_{45} = -46.338737345 x 45 = − 46.338737345 x 46 = − 50.6779241318 x_{46} = -50.6779241318 x 46 = − 50.6779241318 x 47 = − 57.7477324385 x_{47} = -57.7477324385 x 47 = − 57.7477324385 x 48 = − 58.8524077722 x_{48} = -58.8524077722 x 48 = − 58.8524077722 x 49 = 23.3466555506 x_{49} = 23.3466555506 x 49 = 23.3466555506 x 50 = − 21.7441875996 x_{50} = -21.7441875996 x 50 = − 21.7441875996 x 51 = 32.1249620498 x_{51} = 32.1249620498 x 51 = 32.1249620498 x 52 = 92.0738169933 x_{52} = 92.0738169933 x 52 = 92.0738169933 x 53 = 84.0842093791 x_{53} = 84.0842093791 x 53 = 84.0842093791 x 54 = 22.3146238058 x_{54} = 22.3146238058 x 54 = 22.3146238058 x 55 = 46.0326432529 x_{55} = 46.0326432529 x 55 = 46.0326432529 x 56 = 16.0012437413 x_{56} = 16.0012437413 x 56 = 16.0012437413 x 57 = − 3.31595752198 x_{57} = -3.31595752198 x 57 = − 3.31595752198 x 58 = − 22.2441192889 x_{58} = -22.2441192889 x 58 = − 22.2441192889 x 59 = − 7.8269442789 x_{59} = -7.8269442789 x 59 = − 7.8269442789 x 60 = − 2.17080376367 x_{60} = -2.17080376367 x 60 = − 2.17080376367 x 61 = 96.1791015164 x_{61} = 96.1791015164 x 61 = 96.1791015164 x 62 = 62.2255007658 x_{62} = 62.2255007658 x 62 = 62.2255007658 x 63 = − 32.0270198644 x_{63} = -32.0270198644 x 63 = − 32.0270198644 x 64 = − 89.4079555021 x_{64} = -89.4079555021 x 64 = − 89.4079555021 x 65 = 2.17080376367 x_{65} = 2.17080376367 x 65 = 2.17080376367 x 66 = − 16.0012437413 x_{66} = -16.0012437413 x 66 = − 16.0012437413 x 67 = 6.26657068658 x_{67} = 6.26657068658 x 67 = 6.26657068658 x 68 = 4.15677273792 x_{68} = 4.15677273792 x 68 = 4.15677273792 x 69 = − 43.9375543884 x_{69} = -43.9375543884 x 69 = − 43.9375543884 x 70 = − 53.6295447898 x_{70} = -53.6295447898 x 70 = − 53.6295447898 x 71 = − 5.16754657023 x_{71} = -5.16754657023 x 71 = − 5.16754657023 x 72 = 89.4957567169 x_{72} = 89.4957567169 x 72 = 89.4957567169 x 73 = 18.1189522959 x_{73} = 18.1189522959 x 73 = 18.1189522959 x 74 = − 85.7673550856 x_{74} = -85.7673550856 x 74 = − 85.7673550856 x 75 = 10.2588183479 x_{75} = 10.2588183479 x 75 = 10.2588183479 x 76 = − 17.8569777493 x_{76} = -17.8569777493 x 76 = − 17.8569777493 x 77 = 57.475078732 x_{77} = 57.475078732 x 77 = 57.475078732 x 78 = − 60.5883736019 x_{78} = -60.5883736019 x 78 = − 60.5883736019 x 79 = − 13.7864555105 x_{79} = -13.7864555105 x 79 = − 13.7864555105 x 80 = − 8.77319896121 x_{80} = -8.77319896121 x 80 = − 8.77319896121 x 81 = − 42.0000508927 x_{81} = -42.0000508927 x 81 = − 42.0000508927
Точки пересечения с осью координат Y
График пересекает ось Y, когда x равняется 0:cos ( 0 2 ) \cos{\left (0^{2} \right )} cos ( 0 2 ) 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 − 2 x sin ( x 2 ) = 0 - 2 x \sin{\left (x^{2} \right )} = 0 − 2 x sin ( x 2 ) = 0 Решаем это уравнение x 1 = 0 x_{1} = 0 x 1 = 0 x 2 = − π x_{2} = - \sqrt{\pi} x 2 = − π x 3 = π x_{3} = \sqrt{\pi} x 3 = π (0, 1) ____
(-\/ pi, -1) ____
(\/ pi, -1) Интервалы возрастания и убывания функции: x 3 = − π x_{3} = - \sqrt{\pi} x 3 = − π x 3 = π x_{3} = \sqrt{\pi} x 3 = π [sqrt(pi), oo) (-oo, -sqrt(pi)]
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 ( 2 x 2 cos ( x 2 ) + sin ( x 2 ) ) = 0 - 2 \left(2 x^{2} \cos{\left (x^{2} \right )} + \sin{\left (x^{2} \right )}\right) = 0 − 2 ( 2 x 2 cos ( x 2 ) + sin ( x 2 ) ) = 0 Решаем это уравнение x 1 = − 53.6002486537 x_{1} = -53.6002486537 x 1 = − 53.6002486537 x 2 = − 38.3235554978 x_{2} = -38.3235554978 x 2 = − 38.3235554978 x 3 = − 91.8517671604 x_{3} = -91.8517671604 x 3 = − 91.8517671604 x 4 = 10.2590498848 x_{4} = 10.2590498848 x 4 = 10.2590498848 x 5 = − 83.5970868479 x_{5} = -83.5970868479 x 5 = − 83.5970868479 x 6 = 22.3146463051 x_{6} = 22.3146463051 x 6 = 22.3146463051 x 7 = 20.2479396697 x_{7} = 20.2479396697 x 7 = 20.2479396697 x 8 = 36.0423216116 x_{8} = 36.0423216116 x 8 = 36.0423216116 x 9 = 8.40790743486 x_{9} = 8.40790743486 x 9 = 8.40790743486 x 10 = − 70.1072165206 x_{10} = -70.1072165206 x 10 = − 70.1072165206 x 11 = − 8.95080183389 x_{11} = -8.95080183389 x 11 = − 8.95080183389 x 12 = 18.7997496854 x_{12} = 18.7997496854 x 12 = 18.7997496854 x 13 = 54.125181941 x_{13} = 54.125181941 x 13 = 54.125181941 x 14 = − 13.3230177429 x_{14} = -13.3230177429 x 14 = − 13.3230177429 x 15 = − 2.19450274956 x_{15} = -2.19450274956 x 15 = − 2.19450274956 x 16 = 96.0483569951 x_{16} = 96.0483569951 x 16 = 96.0483569951 x 17 = − 16.0013047615 x_{17} = -16.0013047615 x 17 = − 16.0013047615 x 18 = 91.8517671604 x_{18} = 91.8517671604 x 18 = 91.8517671604 x 19 = 46.0326458158 x_{19} = 46.0326458158 x 19 = 46.0326458158 x 20 = − 65.8556662222 x_{20} = -65.8556662222 x 20 = − 65.8556662222 x 21 = − 17.8570216542 x_{21} = -17.8570216542 x 21 = − 17.8570216542 x 22 = − 33.7930372624 x_{22} = -33.7930372624 x 22 = − 33.7930372624 x 23 = − 7.82746557123 x_{23} = -7.82746557123 x 23 = − 7.82746557123 x 24 = 6.26758611849 x_{24} = 6.26758611849 x 24 = 6.26758611849 x 25 = 27.9969284916 x_{25} = 27.9969284916 x 25 = 27.9969284916 x 26 = 35.4269200396 x_{26} = 35.4269200396 x 26 = 35.4269200396 x 27 = 1.35521112863 x_{27} = 1.35521112863 x 27 = 1.35521112863 x 28 = 70.1520134668 x_{28} = 70.1520134668 x 28 = 70.1520134668 x 29 = − 21.7442119165 x_{29} = -21.7442119165 x 29 = − 21.7442119165 x 30 = − 1.35521112863 x_{30} = -1.35521112863 x 30 = − 1.35521112863 x 31 = − 90.0556577728 x_{31} = -90.0556577728 x 31 = − 90.0556577728 x 32 = 37.0735370545 x_{32} = 37.0735370545 x 32 = 37.0735370545 x 33 = 94.215551259 x_{33} = 94.215551259 x 33 = 94.215551259 x 34 = 26.438704218 x_{34} = 26.438704218 x 34 = 26.438704218 x 35 = − 23.4138597867 x_{35} = -23.4138597867 x 35 = − 23.4138597867 x 36 = − 85.7673554819 x_{36} = -85.7673554819 x 36 = − 85.7673554819 x 37 = − 42.3724023394 x_{37} = -42.3724023394 x 37 = − 42.3724023394 x 38 = 58.2083140493 x_{38} = 58.2083140493 x 38 = 58.2083140493 x 39 = 6.01183407098 x_{39} = 6.01183407098 x 39 = 6.01183407098 x 40 = 90.2299142369 x_{40} = 90.2299142369 x 40 = 90.2299142369 x 41 = 5.16935647583 x_{41} = 5.16935647583 x 41 = 5.16935647583 x 42 = 2.19450274956 x_{42} = 2.19450274956 x 42 = 2.19450274956 x 43 = 12.7198707532 x_{43} = 12.7198707532 x 43 = 12.7198707532 x 44 = 33.0409428607 x_{44} = 33.0409428607 x 44 = 33.0409428607 x 45 = − 47.8726650497 x_{45} = -47.8726650497 x 45 = − 47.8726650497 x 46 = − 81.0594911844 x_{46} = -81.0594911844 x 46 = − 81.0594911844 x 47 = − 14.3449206559 x_{47} = -14.3449206559 x 47 = − 14.3449206559 x 48 = − 52.1143649403 x_{48} = -52.1143649403 x 48 = − 52.1143649403 x 49 = 40.1647623864 x_{49} = 40.1647623864 x 49 = 40.1647623864 x 50 = 60.2503979154 x_{50} = 60.2503979154 x 50 = 60.2503979154 x 51 = − 6.97889329813 x_{51} = -6.97889329813 x 51 = − 6.97889329813 x 52 = − 35.6038344867 x_{52} = -35.6038344867 x 52 = − 35.6038344867 x 53 = − 3.76462907533 x_{53} = -3.76462907533 x 53 = − 3.76462907533 x 54 = 18.2915584905 x_{54} = 18.2915584905 x 54 = 18.2915584905 x 55 = − 75.8953899599 x_{55} = -75.8953899599 x 55 = − 75.8953899599 x 56 = 18.1189943238 x_{56} = 18.1189943238 x 56 = 18.1189943238 x 57 = − 93.7978159832 x_{57} = -93.7978159832 x 57 = − 93.7978159832 x 58 = − 4.16024524967 x_{58} = -4.16024524967 x 58 = − 4.16024524967 x 59 = 58.5580782404 x_{59} = 58.5580782404 x 59 = 58.5580782404 x 60 = − 43.3979845305 x_{60} = -43.3979845305 x 60 = − 43.3979845305 x 61 = − 44.0803279658 x_{61} = -44.0803279658 x 61 = − 44.0803279658 x 62 = − 9.78896285609 x_{62} = -9.78896285609 x 62 = − 9.78896285609 x 63 = 41.2071732071 x_{63} = 41.2071732071 x 63 = 41.2071732071 x 64 = 34.2088247492 x_{64} = 34.2088247492 x 64 = 34.2088247492 x 65 = − 97.7504685831 x_{65} = -97.7504685831 x 65 = − 97.7504685831 x 66 = 84.2708150183 x_{66} = 84.2708150183 x 66 = 84.2708150183 x 67 = − 69.7703226268 x_{67} = -69.7703226268 x 67 = − 69.7703226268 x 68 = 80.1827253424 x_{68} = 80.1827253424 x 68 = 80.1827253424 x 69 = 77.3509100937 x_{69} = 77.3509100937 x 69 = 77.3509100937 x 70 = − 18.0320929835 x_{70} = -18.0320929835 x 70 = − 18.0320929835 x 71 = 26.6163455262 x_{71} = 26.6163455262 x 71 = 26.6163455262 x 72 = − 29.8962149672 x_{72} = -29.8962149672 x 72 = − 29.8962149672 x 73 = 14.1242217429 x_{73} = 14.1242217429 x 73 = 14.1242217429 x 74 = 56.1199322343 x_{74} = 56.1199322343 x 74 = 56.1199322343 x 75 = − 11.1398805605 x_{75} = -11.1398805605 x 75 = − 11.1398805605 x 76 = − 5.74472561217 x_{76} = -5.74472561217 x 76 = − 5.74472561217 x 77 = 62.2255018034 x_{77} = 62.2255018034 x 77 = 62.2255018034 x 78 = 32.1249695906 x_{78} = 32.1249695906 x 78 = 32.1249695906 x 79 = 0 x_{79} = 0 x 79 = 0 x 80 = 55.698500039 x_{80} = 55.698500039 x 80 = 55.698500039 x 81 = 4.16024524967 x_{81} = 4.16024524967 x 81 = 4.16024524967 x 82 = − 68.7040251219 x_{82} = -68.7040251219 x 82 = − 68.7040251219 x 83 = − 42.0000542671 x_{83} = -42.0000542671 x 83 = − 42.0000542671 x 84 = − 26.7928090701 x_{84} = -26.7928090701 x 84 = − 26.7928090701 x 85 = 82.5952088233 x_{85} = 82.5952088233 x 85 = 82.5952088233 Интервалы выпуклости и вогнутости: [96.0483569951, oo) (-oo, -97.7504685831]
Горизонтальные асимптоты
Горизонтальные асимптоты найдём с помощью пределов данной функции при x->+oo и x->-oolim x → − ∞ cos ( x 2 ) = ⟨ − 1 , 1 ⟩ \lim_{x \to -\infty} \cos{\left (x^{2} \right )} = \langle -1, 1\rangle x → − ∞ lim cos ( x 2 ) = ⟨ − 1 , 1 ⟩ Возьмём предел y = ⟨ − 1 , 1 ⟩ y = \langle -1, 1\rangle y = ⟨ − 1 , 1 ⟩ lim x → ∞ cos ( x 2 ) = ⟨ − 1 , 1 ⟩ \lim_{x \to \infty} \cos{\left (x^{2} \right )} = \langle -1, 1\rangle x → ∞ lim cos ( x 2 ) = ⟨ − 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(x^2), делённой на x при x->+oo и x ->-oolim x → − ∞ ( 1 x cos ( x 2 ) ) = 0 \lim_{x \to -\infty}\left(\frac{1}{x} \cos{\left (x^{2} \right )}\right) = 0 x → − ∞ lim ( x 1 cos ( x 2 ) ) = 0 Возьмём предел lim x → ∞ ( 1 x cos ( x 2 ) ) = 0 \lim_{x \to \infty}\left(\frac{1}{x} \cos{\left (x^{2} \right )}\right) = 0 x → ∞ lim ( x 1 cos ( x 2 ) ) = 0 Возьмём предел
Чётность и нечётность функции
Проверим функци чётна или нечётна с помощью соотношений f = f(-x) и f = -f(-x).cos ( x 2 ) = cos ( x 2 ) \cos{\left (x^{2} \right )} = \cos{\left (x^{2} \right )} cos ( x 2 ) = cos ( x 2 ) cos ( x 2 ) = − cos ( x 2 ) \cos{\left (x^{2} \right )} = - \cos{\left (x^{2} \right )} cos ( x 2 ) = − cos ( x 2 ) 