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temata:17a-matematicka_analyza:derivace_jedne_promenne:main [2011/03/15 13:23] vagabund |
temata:17a-matematicka_analyza:derivace_jedne_promenne:main [2011/03/15 13:55] (aktuální) vagabund |
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|---|---|---|---|
| Řádek 239: | Řádek 239: | ||
| <m>f(x) = a_0 + a_1(x - x_0) + a_2(x - x_0)^2 + a_3(x - x_0)^3 + ...</m> | <m>f(x) = a_0 + a_1(x - x_0) + a_2(x - x_0)^2 + a_3(x - x_0)^3 + ...</m> | ||
| + | |||
| <m>f(x)\prime = 1.a_1 + 2.a_2(x - x_0) + 3.a_3(x - x_0)^2 + 4.a_4(x - x_0)^3 + ...</m> | <m>f(x)\prime = 1.a_1 + 2.a_2(x - x_0) + 3.a_3(x - x_0)^2 + 4.a_4(x - x_0)^3 + ...</m> | ||
| + | |||
| <m>f(x)\prime\prime = 2.1.a_2 + 3.2.a_3(x - x_0) + 4.3.a_4(x - x_0)^2 + 5.4.a_5(x - x_0)^3 + ...</m> | <m>f(x)\prime\prime = 2.1.a_2 + 3.2.a_3(x - x_0) + 4.3.a_4(x - x_0)^2 + 5.4.a_5(x - x_0)^3 + ...</m> | ||
| + | |||
| <m>f(x)\prime\prime\prime = 3.2.1.a_3(x - x_0) + 4.3.2.a_4(x - x_0) + 5.4.3.a_5(x - x_0)^2 + ...</m> | <m>f(x)\prime\prime\prime = 3.2.1.a_3(x - x_0) + 4.3.2.a_4(x - x_0) + 5.4.3.a_5(x - x_0)^2 + ...</m> | ||
| + | |||
| ... | ... | ||
| Řádek 247: | Řádek 251: | ||
| <m>f(x_0) = a_0</m> | <m>f(x_0) = a_0</m> | ||
| + | |||
| <m>f(x_0)\prime = 1.a_1</m> | <m>f(x_0)\prime = 1.a_1</m> | ||
| + | |||
| <m>f(x_0)\prime\prime = 2.1.a_2</m> | <m>f(x_0)\prime\prime = 2.1.a_2</m> | ||
| + | |||
| <m>f(x_0)\prime\prime\prime = 3.2.1.a_3</m> | <m>f(x_0)\prime\prime\prime = 3.2.1.a_3</m> | ||
| + | |||
| ... | ... | ||
| + | |||
| <m>f(x_0)^{(n)} = n!.a_3</m> | <m>f(x_0)^{(n)} = n!.a_3</m> | ||
| Řádek 259: | Řádek 268: | ||
| dosadíme do polynomu: | dosadíme do polynomu: | ||
| - | f(x) = f(x_0) + f(x_0)/{1!}(x - x_0) + f(x_0)/{2!}(x - x_0)^2 + f(x_0)/{3!}(x - x_0)^3 + ... = \sum{n = 0}{\infty}{{f^{(n)}}/{n!}(x - c)^n} | + | <m>f(x) = f(x_0) + f(x_0)/{1!}(x - x_0) + f(x_0)/{2!}(x - x_0)^2 + f(x_0)/{3!}(x - x_0)^3 + ... = \sum{n = 0}{\infty}{{f^{(n)}}/{n!}(x - x_0)^n}</m> |
| </box> | </box> | ||
| Řádek 267: | Řádek 276: | ||
| <m>f(x) = \sum{n = 0}{\infty}{{f^{(n)}}/{n!}(x - x_0)^n}</m> | <m>f(x) = \sum{n = 0}{\infty}{{f^{(n)}}/{n!}(x - x_0)^n}</m> | ||
| + | </box> | ||
| + | |||
| + | <box round 90% green|**Příklad**> | ||
| + | |||
| + | <m>f(x) = sin(x), x_0 = 0</m> | ||
| + | |||
| + | <m>f(x)\prime = cos(x)</m> | ||
| + | |||
| + | <m>f(x)\prime\prime = -sin(x)</m> | ||
| + | |||
| + | <m>f(x)\prime\prime\prime = -cos(x)</m> | ||
| + | |||
| + | <m>f(x)^{(4)} = sin(x)</m> | ||
| + | |||
| + | ... | ||
| + | |||
| + | <m>f(0) = 0</m> | ||
| + | |||
| + | <m>f(0)\prime = 1</m> | ||
| + | |||
| + | <m>f(0)\prime\prime = 0</m> | ||
| + | |||
| + | <m>f(0)\prime\prime\prime = -1</m> | ||
| + | |||
| + | <m>f(0)^{(4)} = 0</m> | ||
| + | |||
| + | ... | ||
| + | |||
| + | <m>sin(x) = x - {x^3}/{3!} + {x^5}/{5!} - {x^7}/{7!} + ... + (-1)^{m - 1} . {x^{2m - 1}}/{(2m - 1)!} + R_m{x}</m> | ||
| + | |||
| + | Dále: | ||
| + | |||
| + | <m>cos(x) = 1 - {x^2}/{2!} + {x^4}/{4!} - {x^6}/{6!} + ... + (-1)^m . {x^{2m}}/{(2m)!} + R_m{x}</m> | ||
| + | |||
| + | <m>e^x = 1 + x/{1!} + {x^2}/{2!} + {x^3}/{3!} + {x^4}/{4!} + ... + + {x^n}/{n!} + R_n{x}</m> | ||
| + | |||
| + | Perlička: | ||
| + | |||
| + | <m>e^{ix} = 1 + {ix}/{1!} + {{ix}^2}/{2!} + {{ix}^3}/{3!} + {{ix}^4}/{4!} + ... + + {{ix}^n}/{n!} + R_n{x}</m> | ||
| + | |||
| + | <m>e^{ix} = 1 + i{x}/{1!} - {{x}^2}/{2!} - i{{x}^3}/{3!} + {{x}^4}/{4!} + ... + + {{ix}^n}/{n!} + R_m{x}</m> | ||
| + | |||
| + | <m>e^{ix} = 1 - {{x}^2}/{2!} + {{x}^4}/{4!} + ... + i{x}/{1!} - i{{x}^3}/{3!} + ... + {{ix}^n}/{n!} + R_m{x}</m> | ||
| + | |||
| + | <m>e^{ix} = (1 - {{x}^2}/{2!} + {{x}^4}/{4!}) + ... + i({x}/{1!} - {{x}^3}/{3!}) + ... + {{ix}^n}/{n!} + R_m{x}</m> | ||
| + | |||
| + | <m>e^{ix} = cos(x) + isin(x)</m> | ||
| + | |||
| </box> | </box> | ||
| Řádek 323: | Řádek 380: | ||
| Analogicky pro <m>x \right -\infty^{+}}</m> | Analogicky pro <m>x \right -\infty^{+}}</m> | ||
| + | |||
| + | </box> | ||
| + | |||
| + | <box round 90% green|**Cvičení**> | ||
| + | |||
| + | http://mathonline.fme.vutbr.cz/download.aspx?id_file=870 | ||
| </box> | </box> | ||
| Řádek 328: | Řádek 391: | ||
| ===== Potvrzení ===== | ===== Potvrzení ===== | ||
| - | <doodle single login|XX> | + | <doodle single login|17a1> |
| ^ OK ^ !!! ^ | ^ OK ^ !!! ^ | ||
| </doodle> | </doodle> | ||
