From 53b31c6ea98363871c7c6bdb8e662ad825e3f47a Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Anton=20Luka=20=C5=A0ijanec?= Date: Fri, 2 Feb 2024 12:49:38 +0100 Subject: poprbot --- "\305\241ola/ana1/kolokvij2.lyx" | 658 +++++++++++++++++++++++++++++++++++++++ 1 file changed, 658 insertions(+) (limited to 'šola/ana1/kolokvij2.lyx') diff --git "a/\305\241ola/ana1/kolokvij2.lyx" "b/\305\241ola/ana1/kolokvij2.lyx" index 486a401..a057288 100644 --- "a/\305\241ola/ana1/kolokvij2.lyx" +++ "b/\305\241ola/ana1/kolokvij2.lyx" @@ -158,6 +158,580 @@ begin{multicols}{2} \end_inset +\begin_inset Formula $\log_{a}1=0$ +\end_inset + +, +\begin_inset Formula $\log_{a}a=1$ +\end_inset + +, +\begin_inset Formula $\log_{a}a^{x}=x$ +\end_inset + +, +\begin_inset Formula $a^{\log_{a}x}=x$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $\log_{a}x^{n}=n\log_{a}x$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $D=b^{2}-4ac$ +\end_inset + +, +\begin_inset Formula $x_{1,2}=\frac{-b\pm\sqrt{D}}{2a}$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right)$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $zw=\left(ac-bd\right)+\left(ad+bc\right)i$ +\end_inset + +, +\begin_inset Formula $\vert zw\vert=\vert z\vert\vert w\vert$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $\arg\left(zw\right)=\arg z+\arg w$ +\end_inset + + (kot) +\end_layout + +\begin_layout Standard +\begin_inset Formula $z\overline{z}=a^{2}-\left(bi\right)^{2}=a^{2}+b^{2}$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $\left(\cos\phi+i\sin\phi\right)$ +\end_inset + + +\begin_inset Formula $\left(\cos\psi+i\sin\psi\right)=\cos\left(\phi+\psi\right)+i\sin\left(\phi+\psi\right)$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $z^{2}=a^{2}+2abi-b^{2}$ +\end_inset + +, +\begin_inset Formula $z^{3}=a^{3}-3ab^{2}+\left(3a^{2}b-b^{3}\right)i$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $(a+b)^{n}=\sum_{k=0}^{n}{n \choose k}ab^{n-k}$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $z^{n}=r^{3}\left(\cos\left(3\phi\right)+i\sin\left(3\phi\right)\right)$ +\end_inset + +, +\begin_inset Formula $\phi=\arctan\frac{\Im z}{\Re z}$ +\end_inset + + +\end_layout + +\begin_layout Standard +Odprta množica ne vsebuje robnih točk. + Zaprta vsebuje vse. +\end_layout + +\begin_layout Standard +\begin_inset Formula $\sin\left(x\pm y\right)=\sin x\cdot\cos y\pm\sin y\cdot\cos x$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $\cos\left(x\pm y\right)=\cos x\cdot\cos y\mp\sin y\cdot\sin x$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $\tan\left(x\pm y\right)=\frac{\tan x\pm\tan y}{1\text{\ensuremath{\mp\tan}x\ensuremath{\cdot\tan y}}}$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $a_{n}$ +\end_inset + +je konv. + +\begin_inset Formula $\Longleftrightarrow$ +\end_inset + + +\begin_inset Formula $\forall\varepsilon>0:\exists n_{0}\ni:\forall n,m:n_{0}n_{0}$ +\end_inset + + +\begin_inset Formula $\wedge$ +\end_inset + + vrsti sta navzdol omejeni +\begin_inset Formula $\Longrightarrow$ +\end_inset + + +\begin_inset Formula $\sum_{1}^{\infty}b_{k}$ +\end_inset + + konv. + +\begin_inset Formula $\sum_{1}^{\infty}a_{k}$ +\end_inset + + rečemo +\shape italic +majoranta +\shape default +. +\end_layout + +\begin_layout Standard + +\series bold +Kvocientni +\series default +: +\begin_inset Formula $a_{k}>0$ +\end_inset + +, +\begin_inset Formula $D_{n}\coloneqq\frac{a_{n}+1}{a_{n}}$ +\end_inset + + +\end_layout + +\begin_layout Itemize +\begin_inset Formula $\forall n1\Longrightarrow div.$ +\end_inset + + +\end_layout + +\begin_layout Standard + +\series bold +Korenski +\series default +: Kot Kvocientni, le da +\begin_inset Formula $D_{n}\coloneqq\sqrt[n]{a_{n}}$ +\end_inset + +. +\end_layout + +\begin_layout Standard + +\series bold +Leibnizov +\series default +: +\begin_inset Formula $a_{n}\to0\Longrightarrow\sum_{1}^{\infty}\left(\left(-1\right)^{k}a_{k}\right)<\infty$ +\end_inset + + +\end_layout + +\begin_layout Standard +Absolutna konvergenca +\begin_inset Formula $\left(\sum_{1}^{\infty}\vert a_{n}\vert<\infty\right)$ +\end_inset + + +\begin_inset Formula $\Longrightarrow$ +\end_inset + + konvergenca +\end_layout + +\begin_layout Standard +Pri konv. + po točkah je +\begin_inset Formula $n_{0}$ +\end_inset + + odvisen od +\begin_inset Formula $x$ +\end_inset + +, pri enakomerni ni. +\end_layout + +\begin_layout Standard +Potenčna vrsta: +\begin_inset Formula $\sum_{j=1}^{\infty}b_{j}x^{j}$ +\end_inset + +. + +\begin_inset Formula $R^{-1}=\limsup_{k\to\infty}\sqrt[k]{\vert b_{k}\vert}$ +\end_inset + +. + +\begin_inset Formula $\vert x\vertR\Longrightarrow$ +\end_inset + +divergira +\end_layout + +\begin_layout Standard +\begin_inset Formula $\lim_{x\to a}\left(\alpha f\left(x\right)\right)=\alpha\lim_{x\to a}f\left(x\right)$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Tabular + + + + + + + + +\begin_inset Text + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $\sin$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $\cos$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $\tan$ +\end_inset + + +\end_layout + +\end_inset + + + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $30^{\circ}=\frac{\pi}{6}$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $\frac{1}{2}$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $\frac{\sqrt{3}}{2}$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $\frac{\sqrt{3}}{3}$ +\end_inset + + +\end_layout + +\end_inset + + + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $45^{\circ}=\frac{\pi}{4}$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $\frac{\sqrt{2}}{2}$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $\frac{\sqrt{2}}{2}$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +1 +\end_layout + +\end_inset + + + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $60^{\circ}=\frac{\pi}{3}$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $\frac{\sqrt{3}}{2}$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $\frac{1}{2}$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $\sqrt{3}$ +\end_inset + + +\end_layout + +\end_inset + + + + +\end_inset + + +\end_layout + +\begin_layout Standard +Krožnica: +\begin_inset Formula $\left(x-p\right)^{2}+\left(y-q\right)^{2}=r^{2}$ +\end_inset + + +\end_layout + +\begin_layout Standard +Elipsa: +\begin_inset Formula $\frac{\left(x-p\right)^{2}}{a^{2}}+\frac{\left(y-q\right)^{2}}{b^{2}}=1$ +\end_inset + + \end_layout \begin_layout Standard @@ -667,6 +1241,90 @@ R=\lim_{n\to\infty}\left|\frac{c_{n}}{c_{n+1}}\right|,\quad R=\lim_{n\to\infty}\ \end_inset +\end_layout + +\begin_layout Standard +Zvezna +\begin_inset Formula $\text{f}$ +\end_inset + + na zaprtem intervalu +\begin_inset Formula $\left[a,b\right]$ +\end_inset + + doseže +\begin_inset Formula $\inf$ +\end_inset + + in +\begin_inset Formula $\sup$ +\end_inset + +, je omejena in doseže vse funkcijske vrednosti na +\begin_inset Formula $\left[f\left(a\right),f\left(b\right)\right]$ +\end_inset + +. +\end_layout + +\begin_layout Standard +\begin_inset Formula $f:I\to\mathbb{R}$ +\end_inset + + je enakomerno zvezna na +\begin_inset Formula $I$ +\end_inset + +, če +\begin_inset Formula $\forall\varepsilon>0\exists\delta_{\left(\varepsilon\right)}>0\ni:\forall x,y\in I:\left|x-y\right|<\delta\Rightarrow\left|f\left(x\right)-f\left(y\right)\right|<\varepsilon$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula $f:I\to\mathbb{R}$ +\end_inset + + je zvezna na +\begin_inset Formula $I$ +\end_inset + +, če +\begin_inset Formula $\forall\varepsilon>0\forall x\in I\exists\delta_{\left(x,\varepsilon\right)}>0\ni:\forall x,y\in I:\left|x-y\right|<\delta\Rightarrow\left|f\left(x\right)-f\left(y\right)\right|<\varepsilon$ +\end_inset + + +\end_layout + +\begin_layout Standard +Zvezna +\begin_inset Formula $f$ +\end_inset + + na kompaktni množici je enakomerno zvezna. +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +f'\left(x\right)=\lim_{x\to0}\frac{f\left(x+h\right)-f\left(x\right)}{h} +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +\sinh x=\frac{e^{x}-e^{-x}}{2},\quad\cosh x=\frac{e^{x}+e^{-x}}{2} +\] + +\end_inset + + \end_layout \begin_layout Standard -- cgit v1.2.3