Область определения функции
Точки, в которых функция точно неопределена:x 1 = 2 x_{1} = 2 x 1 = 2
Точки пересечения с осью координат X
График функции пересекает ось X при f = 0 значит надо решить уравнение:x 3 x − 2 = 0 \sqrt{\frac{x^{3}}{x - 2}} = 0 x − 2 x 3 = 0 Решаем это уравнение Точки пересечения с осью X:Численное решение x 1 = − 1.64293746144 ⋅ 1 0 − 10 x_{1} = -1.64293746144 \cdot 10^{-10} x 1 = − 1.64293746144 ⋅ 1 0 − 10 x 2 = − 1.65448774399 ⋅ 1 0 − 10 x_{2} = -1.65448774399 \cdot 10^{-10} x 2 = − 1.65448774399 ⋅ 1 0 − 10 x 3 = − 1.65286421441 ⋅ 1 0 − 10 x_{3} = -1.65286421441 \cdot 10^{-10} x 3 = − 1.65286421441 ⋅ 1 0 − 10 x 4 = − 3.18626324537 ⋅ 1 0 − 10 x_{4} = -3.18626324537 \cdot 10^{-10} x 4 = − 3.18626324537 ⋅ 1 0 − 10 x 5 = − 1.65601555413 ⋅ 1 0 − 10 x_{5} = -1.65601555413 \cdot 10^{-10} x 5 = − 1.65601555413 ⋅ 1 0 − 10 x 6 = − 1.64732002372 ⋅ 1 0 − 10 x_{6} = -1.64732002372 \cdot 10^{-10} x 6 = − 1.64732002372 ⋅ 1 0 − 10 x 7 = − 3.17930012362 ⋅ 1 0 − 10 x_{7} = -3.17930012362 \cdot 10^{-10} x 7 = − 3.17930012362 ⋅ 1 0 − 10 x 8 = − 1.51507950317 ⋅ 1 0 − 10 x_{8} = -1.51507950317 \cdot 10^{-10} x 8 = − 1.51507950317 ⋅ 1 0 − 10 x 9 = − 1.66461951834 ⋅ 1 0 − 10 x_{9} = -1.66461951834 \cdot 10^{-10} x 9 = − 1.66461951834 ⋅ 1 0 − 10 x 10 = − 3.19452933474 ⋅ 1 0 − 10 x_{10} = -3.19452933474 \cdot 10^{-10} x 10 = − 3.19452933474 ⋅ 1 0 − 10 x 11 = − 1.59197223212 ⋅ 1 0 − 10 x_{11} = -1.59197223212 \cdot 10^{-10} x 11 = − 1.59197223212 ⋅ 1 0 − 10 x 12 = − 1.67280755415 ⋅ 1 0 − 10 x_{12} = -1.67280755415 \cdot 10^{-10} x 12 = − 1.67280755415 ⋅ 1 0 − 10 x 13 = − 3.53645655733 ⋅ 1 0 − 10 x_{13} = -3.53645655733 \cdot 10^{-10} x 13 = − 3.53645655733 ⋅ 1 0 − 10 x 14 = − 3.70967259465 ⋅ 1 0 − 10 x_{14} = -3.70967259465 \cdot 10^{-10} x 14 = − 3.70967259465 ⋅ 1 0 − 10 x 15 = − 3.23222979925 ⋅ 1 0 − 10 x_{15} = -3.23222979925 \cdot 10^{-10} x 15 = − 3.23222979925 ⋅ 1 0 − 10 x 16 = − 3.27945684549 ⋅ 1 0 − 10 x_{16} = -3.27945684549 \cdot 10^{-10} x 16 = − 3.27945684549 ⋅ 1 0 − 10 x 17 = − 3.16985080357 ⋅ 1 0 − 10 x_{17} = -3.16985080357 \cdot 10^{-10} x 17 = − 3.16985080357 ⋅ 1 0 − 10 x 18 = − 3.18885683415 ⋅ 1 0 − 10 x_{18} = -3.18885683415 \cdot 10^{-10} x 18 = − 3.18885683415 ⋅ 1 0 − 10 x 19 = − 1.49093965058 ⋅ 1 0 − 10 x_{19} = -1.49093965058 \cdot 10^{-10} x 19 = − 1.49093965058 ⋅ 1 0 − 10 x 20 = − 3.16105381224 ⋅ 1 0 − 10 x_{20} = -3.16105381224 \cdot 10^{-10} x 20 = − 3.16105381224 ⋅ 1 0 − 10 x 21 = − 3.16236745644 ⋅ 1 0 − 10 x_{21} = -3.16236745644 \cdot 10^{-10} x 21 = − 3.16236745644 ⋅ 1 0 − 10 x 22 = − 1.574275406 ⋅ 1 0 − 10 x_{22} = -1.574275406 \cdot 10^{-10} x 22 = − 1.574275406 ⋅ 1 0 − 10 x 23 = − 1.53435318361 ⋅ 1 0 − 10 x_{23} = -1.53435318361 \cdot 10^{-10} x 23 = − 1.53435318361 ⋅ 1 0 − 10 x 24 = − 1.35965289929 ⋅ 1 0 − 10 x_{24} = -1.35965289929 \cdot 10^{-10} x 24 = − 1.35965289929 ⋅ 1 0 − 10 x 25 = − 1.6699511404 ⋅ 1 0 − 10 x_{25} = -1.6699511404 \cdot 10^{-10} x 25 = − 1.6699511404 ⋅ 1 0 − 10 x 26 = − 4.59937486039 ⋅ 1 0 − 10 x_{26} = -4.59937486039 \cdot 10^{-10} x 26 = − 4.59937486039 ⋅ 1 0 − 10 x 27 = − 1.64520721954 ⋅ 1 0 − 10 x_{27} = -1.64520721954 \cdot 10^{-10} x 27 = − 1.64520721954 ⋅ 1 0 − 10 x 28 = − 1.6349896351 ⋅ 1 0 − 10 x_{28} = -1.6349896351 \cdot 10^{-10} x 28 = − 1.6349896351 ⋅ 1 0 − 10 x 29 = − 1.63187805919 ⋅ 1 0 − 10 x_{29} = -1.63187805919 \cdot 10^{-10} x 29 = − 1.63187805919 ⋅ 1 0 − 10 x 30 = − 3.21677008472 ⋅ 1 0 − 10 x_{30} = -3.21677008472 \cdot 10^{-10} x 30 = − 3.21677008472 ⋅ 1 0 − 10 x 31 = − 1.64929161005 ⋅ 1 0 − 10 x_{31} = -1.64929161005 \cdot 10^{-10} x 31 = − 1.64929161005 ⋅ 1 0 − 10 x 32 = − 3.20829984166 ⋅ 1 0 − 10 x_{32} = -3.20829984166 \cdot 10^{-10} x 32 = − 3.20829984166 ⋅ 1 0 − 10 x 33 = − 1.5500974298 ⋅ 1 0 − 10 x_{33} = -1.5500974298 \cdot 10^{-10} x 33 = − 1.5500974298 ⋅ 1 0 − 10 x 34 = − 3.1651658503 ⋅ 1 0 − 10 x_{34} = -3.1651658503 \cdot 10^{-10} x 34 = − 3.1651658503 ⋅ 1 0 − 10 x 35 = − 1.27123370524 ⋅ 1 0 − 10 x_{35} = -1.27123370524 \cdot 10^{-10} x 35 = − 1.27123370524 ⋅ 1 0 − 10 x 36 = − 3.16665819379 ⋅ 1 0 − 10 x_{36} = -3.16665819379 \cdot 10^{-10} x 36 = − 3.16665819379 ⋅ 1 0 − 10 x 37 = − 1.62066912211 ⋅ 1 0 − 10 x_{37} = -1.62066912211 \cdot 10^{-10} x 37 = − 1.62066912211 ⋅ 1 0 − 10 x 38 = − 3.439688331 ⋅ 1 0 − 10 x_{38} = -3.439688331 \cdot 10^{-10} x 38 = − 3.439688331 ⋅ 1 0 − 10 x 39 = − 3.18149592022 ⋅ 1 0 − 10 x_{39} = -3.18149592022 \cdot 10^{-10} x 39 = − 3.18149592022 ⋅ 1 0 − 10 x 40 = − 1.63785151116 ⋅ 1 0 − 10 x_{40} = -1.63785151116 \cdot 10^{-10} x 40 = − 1.63785151116 ⋅ 1 0 − 10 x 41 = − 3.35469766785 ⋅ 1 0 − 10 x_{41} = -3.35469766785 \cdot 10^{-10} x 41 = − 3.35469766785 ⋅ 1 0 − 10 x 42 = − 3.2009565521 ⋅ 1 0 − 10 x_{42} = -3.2009565521 \cdot 10^{-10} x 42 = − 3.2009565521 ⋅ 1 0 − 10 x 43 = − 1.66247703213 ⋅ 1 0 − 10 x_{43} = -1.66247703213 \cdot 10^{-10} x 43 = − 1.66247703213 ⋅ 1 0 − 10 x 44 = 0 x_{44} = 0 x 44 = 0 x 45 = − 1.65881599136 ⋅ 1 0 − 10 x_{45} = -1.65881599136 \cdot 10^{-10} x 45 = − 1.65881599136 ⋅ 1 0 − 10 x 46 = − 8.17365865285 ⋅ 1 0 − 11 x_{46} = -8.17365865285 \cdot 10^{-11} x 46 = − 8.17365865285 ⋅ 1 0 − 11 x 47 = − 4.10911035276 ⋅ 1 0 − 10 x_{47} = -4.10911035276 \cdot 10^{-10} x 47 = − 4.10911035276 ⋅ 1 0 − 10 x 48 = − 3.26940698592 ⋅ 1 0 − 10 x_{48} = -3.26940698592 \cdot 10^{-10} x 48 = − 3.26940698592 ⋅ 1 0 − 10 x 49 = − 1.67143773343 ⋅ 1 0 − 10 x_{49} = -1.67143773343 \cdot 10^{-10} x 49 = − 1.67143773343 ⋅ 1 0 − 10 x 50 = − 1.65745586755 ⋅ 1 0 − 10 x_{50} = -1.65745586755 \cdot 10^{-10} x 50 = − 1.65745586755 ⋅ 1 0 − 10 x 51 = − 1.66561410307 ⋅ 1 0 − 10 x_{51} = -1.66561410307 \cdot 10^{-10} x 51 = − 1.66561410307 ⋅ 1 0 − 10 x 52 = − 1.65113567902 ⋅ 1 0 − 10 x_{52} = -1.65113567902 \cdot 10^{-10} x 52 = − 1.65113567902 ⋅ 1 0 − 10 x 53 = − 3.25231260369 ⋅ 1 0 − 10 x_{53} = -3.25231260369 \cdot 10^{-10} x 53 = − 3.25231260369 ⋅ 1 0 − 10 x 54 = − 3.17721674659 ⋅ 1 0 − 10 x_{54} = -3.17721674659 \cdot 10^{-10} x 54 = − 3.17721674659 ⋅ 1 0 − 10 x 55 = − 3.31818141528 ⋅ 1 0 − 10 x_{55} = -3.31818141528 \cdot 10^{-10} x 55 = − 3.31818141528 ⋅ 1 0 − 10 x 56 = − 3.20450186823 ⋅ 1 0 − 10 x_{56} = -3.20450186823 \cdot 10^{-10} x 56 = − 3.20450186823 ⋅ 1 0 − 10 x 57 = − 3.17156095581 ⋅ 1 0 − 10 x_{57} = -3.17156095581 \cdot 10^{-10} x 57 = − 3.17156095581 ⋅ 1 0 − 10 x 58 = − 1.62848261283 ⋅ 1 0 − 10 x_{58} = -1.62848261283 \cdot 10^{-10} x 58 = − 1.62848261283 ⋅ 1 0 − 10 x 59 = − 1.62476259249 ⋅ 1 0 − 10 x_{59} = -1.62476259249 \cdot 10^{-10} x 59 = − 1.62476259249 ⋅ 1 0 − 10 x 60 = − 1.66010244105 ⋅ 1 0 − 10 x_{60} = -1.66010244105 \cdot 10^{-10} x 60 = − 1.66010244105 ⋅ 1 0 − 10 x 61 = − 3.33504193124 ⋅ 1 0 − 10 x_{61} = -3.33504193124 \cdot 10^{-10} x 61 = − 3.33504193124 ⋅ 1 0 − 10 x 62 = − 6.0135346622 ⋅ 1 0 − 10 x_{62} = -6.0135346622 \cdot 10^{-10} x 62 = − 6.0135346622 ⋅ 1 0 − 10 x 63 = − 1.61614299351 ⋅ 1 0 − 10 x_{63} = -1.61614299351 \cdot 10^{-10} x 63 = − 1.61614299351 ⋅ 1 0 − 10 x 64 = − 1.60548595104 ⋅ 1 0 − 10 x_{64} = -1.60548595104 \cdot 10^{-10} x 64 = − 1.60548595104 ⋅ 1 0 − 10 x 65 = − 3.1733543994 ⋅ 1 0 − 10 x_{65} = -3.1733543994 \cdot 10^{-10} x 65 = − 3.1733543994 ⋅ 1 0 − 10 x 66 = − 1.66746740883 ⋅ 1 0 − 10 x_{66} = -1.66746740883 \cdot 10^{-10} x 66 = − 1.66746740883 ⋅ 1 0 − 10 x 67 = − 1.67070994721 ⋅ 1 0 − 10 x_{67} = -1.67070994721 \cdot 10^{-10} x 67 = − 1.67070994721 ⋅ 1 0 − 10 x 68 = − 1.58375938397 ⋅ 1 0 − 10 x_{68} = -1.58375938397 \cdot 10^{-10} x 68 = − 1.58375938397 ⋅ 1 0 − 10 x 69 = − 3.17523737117 ⋅ 1 0 − 10 x_{69} = -3.17523737117 \cdot 10^{-10} x 69 = − 3.17523737117 ⋅ 1 0 − 10 x 70 = − 1.41819841996 ⋅ 1 0 − 10 x_{70} = -1.41819841996 \cdot 10^{-10} x 70 = − 1.41819841996 ⋅ 1 0 − 10 x 71 = − 1.61111178387 ⋅ 1 0 − 10 x_{71} = -1.61111178387 \cdot 10^{-10} x 71 = − 1.61111178387 ⋅ 1 0 − 10 x 72 = − 1.4598242194 ⋅ 1 0 − 10 x_{72} = -1.4598242194 \cdot 10^{-10} x 72 = − 1.4598242194 ⋅ 1 0 − 10 x 73 = − 1.64049258518 ⋅ 1 0 − 10 x_{73} = -1.64049258518 \cdot 10^{-10} x 73 = − 1.64049258518 ⋅ 1 0 − 10 x 74 = − 3.19160730453 ⋅ 1 0 − 10 x_{74} = -3.19160730453 \cdot 10^{-10} x 74 = − 3.19160730453 ⋅ 1 0 − 10 x 75 = − 1.56320042919 ⋅ 1 0 − 10 x_{75} = -1.56320042919 \cdot 10^{-10} x 75 = − 1.56320042919 ⋅ 1 0 − 10 x 76 = − 1.66833217863 ⋅ 1 0 − 10 x_{76} = -1.66833217863 \cdot 10^{-10} x 76 = − 1.66833217863 ⋅ 1 0 − 10 x 77 = − 3.24498272637 ⋅ 1 0 − 10 x_{77} = -3.24498272637 \cdot 10^{-10} x 77 = − 3.24498272637 ⋅ 1 0 − 10 x 78 = − 1.67213636318 ⋅ 1 0 − 10 x_{78} = -1.67213636318 \cdot 10^{-10} x 78 = − 1.67213636318 ⋅ 1 0 − 10 x 79 = − 1.66915928652 ⋅ 1 0 − 10 x_{79} = -1.66915928652 \cdot 10^{-10} x 79 = − 1.66915928652 ⋅ 1 0 − 10 x 80 = − 1.66132104488 ⋅ 1 0 − 10 x_{80} = -1.66132104488 \cdot 10^{-10} x 80 = − 1.66132104488 ⋅ 1 0 − 10 x 81 = − 1.66357510814 ⋅ 1 0 − 10 x_{81} = -1.66357510814 \cdot 10^{-10} x 81 = − 1.66357510814 ⋅ 1 0 − 10 x 82 = − 3.29075793368 ⋅ 1 0 − 10 x_{82} = -3.29075793368 \cdot 10^{-10} x 82 = − 3.29075793368 ⋅ 1 0 − 10 x 83 = − 3.86023063749 ⋅ 1 0 − 10 x_{83} = -3.86023063749 \cdot 10^{-10} x 83 = − 3.86023063749 ⋅ 1 0 − 10 x 84 = − 3.26041145393 ⋅ 1 0 − 10 x_{84} = -3.26041145393 \cdot 10^{-10} x 84 = − 3.26041145393 ⋅ 1 0 − 10 x 85 = − 3.18381348799 ⋅ 1 0 − 10 x_{85} = -3.18381348799 \cdot 10^{-10} x 85 = − 3.18381348799 ⋅ 1 0 − 10 x 86 = − 3.23831726581 ⋅ 1 0 − 10 x_{86} = -3.23831726581 \cdot 10^{-10} x 86 = − 3.23831726581 ⋅ 1 0 − 10 x 87 = − 3.6087801314 ⋅ 1 0 − 10 x_{87} = -3.6087801314 \cdot 10^{-10} x 87 = − 3.6087801314 ⋅ 1 0 − 10 x 88 = − 3.16821827201 ⋅ 1 0 − 10 x_{88} = -3.16821827201 \cdot 10^{-10} x 88 = − 3.16821827201 ⋅ 1 0 − 10 x 89 = − 3.22151211925 ⋅ 1 0 − 10 x_{89} = -3.22151211925 \cdot 10^{-10} x 89 = − 3.22151211925 ⋅ 1 0 − 10 x 90 = − 1.6665623448 ⋅ 1 0 − 10 x_{90} = -1.6665623448 \cdot 10^{-10} x 90 = − 1.6665623448 ⋅ 1 0 − 10 x 91 = − 1.59915345744 ⋅ 1 0 − 10 x_{91} = -1.59915345744 \cdot 10^{-10} x 91 = − 1.59915345744 ⋅ 1 0 − 10 x 92 = − 3.40572789477 ⋅ 1 0 − 10 x_{92} = -3.40572789477 \cdot 10^{-10} x 92 = − 3.40572789477 ⋅ 1 0 − 10 x 93 = − 3.21237847893 ⋅ 1 0 − 10 x_{93} = -3.21237847893 \cdot 10^{-10} x 93 = − 3.21237847893 ⋅ 1 0 − 10 x 94 = − 3.22664826952 ⋅ 1 0 − 10 x_{94} = -3.22664826952 \cdot 10^{-10} x 94 = − 3.22664826952 ⋅ 1 0 − 10 x 95 = − 3.37790657922 ⋅ 1 0 − 10 x_{95} = -3.37790657922 \cdot 10^{-10} x 95 = − 3.37790657922 ⋅ 1 0 − 10 x 96 = − 3.19763949304 ⋅ 1 0 − 10 x_{96} = -3.19763949304 \cdot 10^{-10} x 96 = − 3.19763949304 ⋅ 1 0 − 10 x 97 = − 3.48207199237 ⋅ 1 0 − 10 x_{97} = -3.48207199237 \cdot 10^{-10} x 97 = − 3.48207199237 ⋅ 1 0 − 10 x 98 = − 3.30355943675 ⋅ 1 0 − 10 x_{98} = -3.30355943675 \cdot 10^{-10} x 98 = − 3.30355943675 ⋅ 1 0 − 10 x 99 = − 1.12220893855 ⋅ 1 0 − 10 x_{99} = -1.12220893855 \cdot 10^{-10} x 99 = − 1.12220893855 ⋅ 1 0 − 10 x 100 = − 3.16373692394 ⋅ 1 0 − 10 x_{100} = -3.16373692394 \cdot 10^{-10} x 100 = − 3.16373692394 ⋅ 1 0 − 10
Точки пересечения с осью координат Y
График пересекает ось Y, когда x равняется 0: подставляем x = 0 в sqrt(x^3/(x - 2)).0 3 − 2 \sqrt{\frac{0^{3}}{-2}} − 2 0 3 Результат:f ( 0 ) = 0 f{\left (0 \right )} = 0 f ( 0 ) = 0 Точка:(0, 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 (вторая производная равняется нулю), корни полученного уравнения будут точками перегибов для указанного графика функции: 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 1 x x 3 x − 2 ( − x x − 2 − 3 2 x − 4 + 1 4 x ( x x − 2 − 3 ) 2 + 1 2 x ( 3 x x − 2 − 9 ) + 1 x ( x 2 ( x − 2 ) 2 − 3 x x − 2 + 3 ) ) = 0 \frac{1}{x} \sqrt{\frac{x^{3}}{x - 2}} \left(- \frac{\frac{x}{x - 2} - 3}{2 x - 4} + \frac{1}{4 x} \left(\frac{x}{x - 2} - 3\right)^{2} + \frac{1}{2 x} \left(\frac{3 x}{x - 2} - 9\right) + \frac{1}{x} \left(\frac{x^{2}}{\left(x - 2\right)^{2}} - \frac{3 x}{x - 2} + 3\right)\right) = 0 x 1 x − 2 x 3 ( − 2 x − 4 x − 2 x − 3 + 4 x 1 ( x − 2 x − 3 ) 2 + 2 x 1 ( x − 2 3 x − 9 ) + x 1 ( ( x − 2 ) 2 x 2 − x − 2 3 x + 3 ) ) = 0 Решаем это уравнение Решения не найдены, возможно перегибов у функции нет
Горизонтальные асимптоты
Горизонтальные асимптоты найдём с помощью пределов данной функции при x->+oo и x->-oolim x → − ∞ x 3 x − 2 = ∞ \lim_{x \to -\infty} \sqrt{\frac{x^{3}}{x - 2}} = \infty x → − ∞ lim x − 2 x 3 = ∞ Возьмём предел значит, горизонтальной асимптоты слева не существуетlim x → ∞ x 3 x − 2 = ∞ \lim_{x \to \infty} \sqrt{\frac{x^{3}}{x - 2}} = \infty x → ∞ lim x − 2 x 3 = ∞ Возьмём предел значит, горизонтальной асимптоты справа не существует
Наклонные асимптоты
Наклонную асимптоту можно найти, подсчитав предел функции sqrt(x^3/(x - 2)), делённой на x при x->+oo и x ->-oolim x → − ∞ ( 1 x x 3 x − 2 ) = − 1 \lim_{x \to -\infty}\left(\frac{1}{x} \sqrt{\frac{x^{3}}{x - 2}}\right) = -1 x → − ∞ lim ( x 1 x − 2 x 3 ) = − 1 Возьмём предел значит, уравнение наклонной асимптоты слева:y = − x y = - x y = − x lim x → ∞ ( 1 x x 3 x − 2 ) = 1 \lim_{x \to \infty}\left(\frac{1}{x} \sqrt{\frac{x^{3}}{x - 2}}\right) = 1 x → ∞ lim ( x 1 x − 2 x 3 ) = 1 Возьмём предел значит, уравнение наклонной асимптоты справа:y = x y = x y = x
Чётность и нечётность функции
Проверим функци чётна или нечётна с помощью соотношений f = f(-x) и f = -f(-x). Итак, проверяем:x 3 x − 2 = − x 3 − x − 2 \sqrt{\frac{x^{3}}{x - 2}} = \sqrt{- \frac{x^{3}}{- x - 2}} x − 2 x 3 = − − x − 2 x 3 - Нетx 3 x − 2 = − − x 3 − x − 2 \sqrt{\frac{x^{3}}{x - 2}} = - \sqrt{- \frac{x^{3}}{- x - 2}} x − 2 x 3 = − − − x − 2 x 3 - Нет значит, функция не является ни чётной ни нечётной