Funkcja ![{\displaystyle y=x^{3}-x^{2}-x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/be2b6271f8686bd16b0f43e89f3eeb5c94c349e4)
Dziedzina: R (Bo to wielomian)
Granice: w
w ![{\displaystyle -\infty ,-\infty }](https://wikimedia.org/api/rest_v1/media/math/render/svg/883239b5b3d4acdf6e4630d7979f5b4138c2b55c)
Parzystość i nieparzystość:
Funkcja nie jest parzysta ani nieparzysta
Monotoniczność i ekstrema:
Funkcja ![{\displaystyle y=x^{3}-x^{2}-x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/be2b6271f8686bd16b0f43e89f3eeb5c94c349e4)
Pochodna ![{\displaystyle y'=3x^{2}-2x-1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e3aae08799ce10539b703bd56a212fb27376e312)
![{\displaystyle y'=0\Leftrightarrow 0=3x^{2}-2x-1\Leftrightarrow x={\frac {-1}{3}}\lor x=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1b29ef719cd3840c6ca9762153108357022639a8)
![{\displaystyle y'>0\Leftrightarrow 0>3x^{2}-2x-1\Leftrightarrow x<{\frac {-1}{3}}\lor x>1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a1afb22aca7775ebbc3defc1a9414cfc985781b)
![{\displaystyle y'<0\Leftrightarrow 0<3x^{2}-2x-1\Leftrightarrow x>{\frac {-1}{3}}\land x<1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a8a5773e998eda2427692353fcbf3afad0a2ecb0)
czyli w punktach
i
Mamy ekstrema
Wypukłość i punkty przegięcia:
Druga pochodna
czyli w punkcie
mamy punkt przegięcia do tego punktu funkcja jest wypukła w górę a za nim w dół.
Miejsca zerowe
![{\displaystyle 0=x^{3}-x^{2}-x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/568439bf4e52c93ca92bd9e16626606153016052)
![{\displaystyle 0=x(x^{2}-x-1)\Leftrightarrow x={\frac {1-{\sqrt {5}}}{2}},x=0,x={\frac {1+{\sqrt {5}}}{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7733ad567b8377fe49ee4bc06fa360633103d3aa)
Przecięcie z osią OY
![{\displaystyle y=0^{3}-0^{2}-0\Leftrightarrow y=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/55e2ee9766ac328cf3196903b2358b5347992ec9)
Koniec teraz wszystko wkładamy w tabele
. | ![{\displaystyle -\infty }](https://wikimedia.org/api/rest_v1/media/math/render/svg/ca2608c4b5fd3bffc73585f8c67e379b4e99b6f1) | ![{\displaystyle (-\infty ,{\frac {1-{\sqrt {5}}}{2}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/695fbc24c43dbcd0569c879067da074cdb17d116) | ![{\displaystyle {\frac {1-{\sqrt {5}}}{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/44dca93af8441b7e02c212f81854cb8a5861e8dd) | ![{\displaystyle ({\frac {1-{\sqrt {5}}}{2}},{\frac {-1}{3}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aea33024a674ca352953bb5e4c1f5221b3859bf2) | ![{\displaystyle {\frac {-1}{3}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0623d28d48b620d582a9e905a0a7df843937b93a) | ![{\displaystyle ({\frac {-1}{3}},0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0af48325d31787abe07d2616c53f416cec6d6228) | ![{\displaystyle 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950) | ![{\displaystyle (0,1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c79c6838e423c1ed3c7ea532a56dc9f9dae8290b) | ![{\displaystyle 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf) | ![{\displaystyle (1,{\frac {1+{\sqrt {5}}}{2}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9f7936b71214bb487abdfc83583d141cfd351b67) | ![{\displaystyle {\frac {1+{\sqrt {5}}}{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2102ba6ed802cb9a98dc1a0fc1ac99b1a03b4047) | ![{\displaystyle ({\frac {1+{\sqrt {5}}}{2}},\infty }](https://wikimedia.org/api/rest_v1/media/math/render/svg/96c108e1b34bb88b985296c237942af2a9a9c988) | ![{\displaystyle \infty }](https://wikimedia.org/api/rest_v1/media/math/render/svg/c26c105004f30c27aa7c2a9c601550a4183b1f21) |
y' | + | + | + | + | 0 | - | - | - | 0 | + | + | + | + |
y | - | - | - | - | 0 | + | + | + | + | + | + | + | + |
y | ![{\displaystyle \infty }](https://wikimedia.org/api/rest_v1/media/math/render/svg/c26c105004f30c27aa7c2a9c601550a4183b1f21) | ![{\displaystyle \nearrow }](https://wikimedia.org/api/rest_v1/media/math/render/svg/13726ca48b64be8035bbf69dedc5de51b6c59b62) | 0 | ![{\displaystyle \nearrow }](https://wikimedia.org/api/rest_v1/media/math/render/svg/13726ca48b64be8035bbf69dedc5de51b6c59b62) | ![{\displaystyle {\tfrac {-10}{27}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fae765e7f6f501b0a911d389324716285552cb6e) | ![{\displaystyle \searrow }](https://wikimedia.org/api/rest_v1/media/math/render/svg/c086f3b8ec7a49977877c105da5f386531d5775a) | 0 | ![{\displaystyle \searrow }](https://wikimedia.org/api/rest_v1/media/math/render/svg/c086f3b8ec7a49977877c105da5f386531d5775a) | -1 | ![{\displaystyle \nearrow }](https://wikimedia.org/api/rest_v1/media/math/render/svg/13726ca48b64be8035bbf69dedc5de51b6c59b62) | 0 | ![{\displaystyle \nearrow }](https://wikimedia.org/api/rest_v1/media/math/render/svg/13726ca48b64be8035bbf69dedc5de51b6c59b62) | ![{\displaystyle \infty }](https://wikimedia.org/api/rest_v1/media/math/render/svg/c26c105004f30c27aa7c2a9c601550a4183b1f21) |
Funkcja:![{\displaystyle {\sqrt {x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d62b24be305beff66cba9bfbcc01a362ba390f44)
Dziedzina: ![{\displaystyle <0,\infty )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c2ce8046ad01ec7344fd7ef78c02bebae37c7cad)
Granice
![{\displaystyle \lim _{x\rightarrow 0}{\sqrt {x}}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/71014a03a2aaf3feaca51a4635ed80bfab187ce2)
![{\displaystyle \lim _{x\rightarrow \infty }{\sqrt {x}}=\infty }](https://wikimedia.org/api/rest_v1/media/math/render/svg/8849308f9181936e62a5bbe2259487cd7740e1d8)
Parzystość i nieparzystość Funkcja ma nie symetryczną dziedzinę więc nie może być ani parzysta ani nieparzysta
Monotoniczność i ekstrema
![{\displaystyle y'={\frac {1}{2{\sqrt {x}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4f91722beb770e9f275dcb0ee983b6d244cb61d6)
Funkcja stale rosnąca
wypukłość i pinkty przegięcia
![{\displaystyle y''=-{\frac {1}{4{\sqrt[{2}]{3}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d34d487870e562272a8928328b3b99e752d5ab09)
![{\displaystyle y''=-{\frac {1}{4{\sqrt[{2}]{x^{3}}}}}=0,\forall x=-{\frac {1}{4{\sqrt[{2}]{x^{3}}}}}<0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/224d08d84a8c6109dcc3d19f24709f27c68114eb)
Funkcja stale wypukła w górę
Miejsca zerowe
![{\displaystyle 0={\sqrt {x}}\Leftrightarrow x=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f1635008a5d6e8e135f0911cfad860365d6da249)
Miejsce przecięcia z osią OY
![{\displaystyle y={\sqrt {0}}\Leftrightarrow y=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d7e37ceda5d34a548a4bd3e69030b9861fd1c1bd)
I tabelka
. | 0 | ![{\displaystyle (0,\infty )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/da17102e4ed0886686094ee531df040d2e86352a) | ![{\displaystyle \infty }](https://wikimedia.org/api/rest_v1/media/math/render/svg/c26c105004f30c27aa7c2a9c601550a4183b1f21) |
y' | + | + | + |
y | - | - | - |
y | 0 | ![{\displaystyle \nearrow }](https://wikimedia.org/api/rest_v1/media/math/render/svg/13726ca48b64be8035bbf69dedc5de51b6c59b62) | ![{\displaystyle \infty }](https://wikimedia.org/api/rest_v1/media/math/render/svg/c26c105004f30c27aa7c2a9c601550a4183b1f21) |
![{\displaystyle {\frac {sinx}{x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/96a868c18c165e18db8702e9ece611d21bd53b82)
Dziedzina: R\{0} (mianownik musi być różny od zera)
Granice:
![{\displaystyle \lim _{x\rightarrow 0^{-}}{\frac {sinx}{x}}=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7af2624a3ab6e6901ffb67194145fc347b3b6d2b)
![{\displaystyle \lim _{x\rightarrow 0^{+}}{\frac {sinx}{x}}=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6077c0925e0b5cdff6920cb31fbf49b68c725fea)
![{\displaystyle \lim _{x\rightarrow -\infty }{\frac {sinx}{x}}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/758f1aee451399727118a75e845d374b10098498)
![{\displaystyle \lim _{x\rightarrow \infty }{\frac {sinx}{x}}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/29be9aaf2685b76b86ad7561d79e635b268f5801)
Parzystość i nieparzystość
![{\displaystyle f(-x)={\frac {-sinx}{-x}}={\frac {sinx}{x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b875416960d6863b53c010d675ddbf3c26f521ac)
Funkcja jest parzysta Nieparzystości sprawdzać nietrzeba
Ekstrema i monotoniczność
![{\displaystyle y'=cosx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/64f2a838fe538f3eea280a39b7f20ed71e09345b)
Ekstrema ![{\displaystyle y=k\pi +{\frac {\pi }{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/02f78832be18e2a0605679a44447afc0d373d8f4)
Funkcja rośnie ![{\displaystyle y\subset (2k\pi -{\frac {\pi }{2}},2k\pi +{\frac {\pi }{2}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e9e778db659a0d4c9b10ec128c5a97adf4c959cd)
Funkcja maleje ![{\displaystyle y\subset (2k\pi +{\frac {\pi }{2}},2k\pi -{\frac {\pi }{2}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/96fb4be5ae2c4552141d59b813b1fdde7c6e6486)
Punkty przegięcia
![{\displaystyle y''=-sinx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/09db147b9d418dd7550b4e8650f74e76957d710f)
Punkty przegięcia ![{\displaystyle k\pi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf859397db5c3d7bddebe20b20a69d8191f2448f)
Funkcja wygięta w górę ![{\displaystyle (2k\pi ,2x\pi +\pi )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/69b79b7ea4c79a58661f5136114e6963088823c7)
Funkcja wygięta w dół![{\displaystyle (2k\pi +\pi ,2x\pi )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7e83126263ca7f7860a5bb6fdb0020e81d04d35f)
Miejsca zerowe
![{\displaystyle y=k\pi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/bb717fc5605a39d0cb2bcc5a53b43b844033b28c)
Przecięcie z osią OY
Poza dziedziną
![{\displaystyle y=e^{\frac {1}{1-x^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ddcae060d2c577c89104a941146e66b52fc2d56b)
Dziedzina: R\{-1,1}
Granice:
![{\displaystyle \lim _{x\rightarrow \infty }e^{\frac {1}{1-x^{2}}}=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6cd1876f5a6ff5dd3aa472ed219c00e9c22db0e2)
![{\displaystyle \lim _{x\rightarrow -\infty }e^{\frac {1}{1-x^{2}}}=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/755ebf6071d4782f70a67c95748ca8e192581839)
![{\displaystyle \lim _{x\rightarrow -1^{-}}e^{\frac {1}{1-x^{2}}}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8a2a29e758aa78f88ecd69b4abec30a7b005daec)