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|
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\begin_layout Title
List s formulami za 2.
kolokvij Analize 1
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\begin_layout Author
\noun on
Anton Luka Šijanec
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\begin_inset ERT
status open
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\backslash
today
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euler{e}
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Izraz
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Odvod
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Odvod
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,
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\begin_inset Formula $\cos^{-2}x$
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</cell>
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\begin_inset Formula $\cot x$
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</cell>
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\begin_inset Formula $-sin^{-2}x$
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\begin_inset Formula $a^{x}$
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\end_inset
</cell>
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\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $a^{x}\text{\ensuremath{\ln a}}$
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\end_inset
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konveksna na
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konkavna na
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\[
ab>0\wedge a<b\Leftrightarrow a^{-1}>b^{-1},\quad ab<0\wedge a<b\Leftrightarrow a^{-1}<b^{-1}
\]
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\begin_inset Formula
\[
\lim_{x\to0}\frac{\sin x}{x}=1\quad\quad\tan\phi=\left|\frac{k_{1}-k_{2}}{1+k_{1}k_{2}}\right|
\]
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\begin_inset Formula
\[
\lim_{x\to0}x\ln x=0
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\begin_layout Standard
\begin_inset Formula
\[
f\text{ zv.+odv.@ }\left[a,b\right]\Rightarrow\exists\xi\in\left[a,b\right]\ni:f\left(b\right)-f\left(a\right)=f'\left(\xi\right)\left(b-a\right)
\]
\end_inset
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\begin_inset Formula
\[
T_{f,a,n}\left(x\right)=\sum_{k=0}^{n}\frac{f^{\left(k\right)}\left(a\right)}{k!}\left(x-a\right)^{k}
\]
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\end_inset
na odprtem
\begin_inset Formula $I\subset\mathbb{R}\Rightarrow\forall a,x\in I\exists c\in\left(\min\left\{ a,x\right\} ,\max\left\{ a,x\right\} \right)\ni:f\left(x\right)-T_{f,a,n}\left(x\right)=R_{f,a,n}\left(x\right)=\frac{f^{\left(n+1\right)}\left(c\right)}{\left(n+1\right)!}$
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\begin_inset Formula $\left(x-a\right)^{n+1}.\text{ Posledično velja tudi takale ocena:}$
\end_inset
\begin_inset Formula
\[
\exists M>0\forall x\in I:\left|f^{\left(n+1\right)}\right|\leq M\Rightarrow R_{f,a,n}\left(x\right)=\frac{M}{\left(n+1\right)!}\left|x-a\right|^{n+1}
\]
\end_inset
\end_layout
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\begin_inset Formula
\[
R=\lim_{n\to\infty}\left|\frac{c_{n}}{c_{n+1}}\right|,\quad R=\lim_{n\to\infty}\frac{1}{\sqrt[n]{\left|c_{n}\right|}}
\]
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status open
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end{multicols}
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|
