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diff --git a/šola/ana1/teor3.lyx b/šola/ana1/teor3.lyx index 107763e..97befd1 100644 --- a/šola/ana1/teor3.lyx +++ b/šola/ana1/teor3.lyx @@ -1,4 +1,4 @@ -#LyX 2.4 created this file. For more info see https://www.lyx.org/ +#LyX 2.3 created this file. For more info see http://www.lyx.org/ \lyxformat 544 \begin_document \begin_header @@ -76,14 +76,22 @@ \begin_body \begin_layout Title -Rešen tretji izpit teorije Analize 1 — - IŠRM 2023/24 +Rešen tretji izpit teorije Analize 1 — IŠRM 2023/24 \end_layout \begin_layout Abstract -Izpit je potekal v petek, - 30. - avgusta 2024. +Izpit je potekal v petek, 30. + avgusta 2024 od desete +\begin_inset Foot +status open + +\begin_layout Plain Layout +Avtor tega besedila je na izpit zamudil poldrugo uro. +\end_layout + +\end_inset + + do dvanajste ure. Nosilec predmeta je \noun on Oliver Dragičević @@ -109,46 +117,429 @@ Podaj natančne definicije naslednjih pojmov: \begin_deeper \begin_layout Enumerate -limita zaporedja, - stekališče zaporedja +limita zaporedja, stekališče zaporedja +\end_layout + +\begin_deeper +\begin_layout Standard +Naj bo +\begin_inset Formula $\left(a_{n}\right)_{n\in\mathbb{N}}$ +\end_inset + + realno zaporedje in +\begin_inset Formula $L\in\mathbb{R}$ +\end_inset + +. +\end_layout + +\begin_layout Enumerate +\begin_inset Formula $L$ +\end_inset + + je limita +\begin_inset Formula $\left(a_{n}\right)_{n\in\mathbb{N}}\sim L=\lim_{n\to\infty}a_{n}\Leftrightarrow\forall\varepsilon>0\exists n_{0}\in\mathbb{N}\forall n>n_{0}:\left|a_{n}-L\right|<\varepsilon$ +\end_inset + +. +\end_layout + +\begin_layout Enumerate +\begin_inset Formula $L$ +\end_inset + + je stekališče +\begin_inset Formula $\left(a_{n}\right)_{n\in\mathbb{N}}\Leftrightarrow\forall\varepsilon>0\exists\mathcal{A}\subseteq\mathbb{N},\left|\mathcal{A}\right|=\left|\mathcal{\mathbb{N}}\right|\ni:\left\{ a_{n};n\in\mathcal{A}\right\} \subseteq\left(L-\varepsilon,L+\varepsilon\right)$ +\end_inset + + \end_layout +\end_deeper \begin_layout Enumerate vsota (neskončne) konvergentne vrste \end_layout +\begin_deeper +\begin_layout Standard +Naj bo +\begin_inset Formula $\left(a_{n}\right)_{n\in\mathbb{N}}$ +\end_inset + + poljubno zaporedje. + +\begin_inset Formula $\sum_{n=1}^{\infty}a_{n}\coloneqq\lim_{n\to\infty}\sum_{k=1}^{n}a_{n}$ +\end_inset + +. + Če limita obstaja, je vrsta +\begin_inset Formula $\sum_{n=1}^{\infty}a_{n}$ +\end_inset + + konvergentna in njena vsota je enaka tej limiti. +\end_layout + +\end_deeper \begin_layout Enumerate Cauchyjev pogoj za zaporedja \end_layout +\begin_deeper +\begin_layout Standard +Naj bo +\begin_inset Formula $\left(a_{n}\right)_{n\in\mathbb{N}}$ +\end_inset + + realno zaporedje. + Konvergentno je natanko tedaj, ko ustreza Cauchyjevemu pogoju: +\begin_inset Formula $\forall\varepsilon>0\exists n_{0}\in\mathbb{N}\forall m,n\geq n_{0}:\left|a_{n}-a_{m}\right|<\varepsilon$ +\end_inset + +. +\end_layout + +\end_deeper \begin_layout Enumerate -odprte, - zaprte, - omejene, - kompaktne množice v +odprte, zaprte, omejene, kompaktne množice v \begin_inset Formula $\mathbb{R}$ \end_inset \end_layout +\begin_deeper +\begin_layout Enumerate +Množica +\begin_inset Formula $\mathcal{A}$ +\end_inset + + je odprta, ko +\begin_inset Formula $\forall a\in\mathcal{A}\exists\varepsilon>0\ni:\left(a-\varepsilon,a+\varepsilon\right)\subseteq\mathcal{A}$ +\end_inset + +, ko za vsako točko množice obstaja neka njena okolica, ki je podmnožica + množice +\begin_inset Formula $\mathcal{A}$ +\end_inset + +. +\end_layout + +\begin_layout Enumerate +Množica +\begin_inset Formula $\mathcal{A}$ +\end_inset + + je zaprta, ko je +\begin_inset Formula $\mathcal{A}^{\mathcal{C}}\coloneqq\mathbb{R}\setminus\mathcal{A}$ +\end_inset + + odprta. +\end_layout + +\begin_layout Enumerate +Množica +\begin_inset Formula $\mathcal{A}$ +\end_inset + + je omejena, ko +\begin_inset Formula $\exists m,M\in\mathbb{R}\forall a\in\mathcal{A}:a\leq M\wedge a\geq m$ +\end_inset + +. +\end_layout + +\begin_layout Enumerate +Množica +\begin_inset Formula $\mathcal{A}$ +\end_inset + + je kompaktna +\begin_inset Formula $\Leftrightarrow\mathcal{A}$ +\end_inset + + zaprta +\begin_inset Formula $\wedge$ +\end_inset + + +\begin_inset Formula $\mathcal{A}$ +\end_inset + + omejena. +\end_layout + +\end_deeper \begin_layout Enumerate limita funkcije v dani točki \end_layout +\begin_deeper +\begin_layout Standard +Naj bodo +\begin_inset Formula $a\in\mathbb{R}$ +\end_inset + +, +\begin_inset Formula $\mathcal{D}$ +\end_inset + + okolica +\begin_inset Formula $a$ +\end_inset + + in +\begin_inset Formula $f:\mathcal{D}\setminus\left\{ a\right\} \to\mathbb{R}$ +\end_inset + + poljubne. + +\begin_inset Formula $L\in\mathbb{R}$ +\end_inset + + je limita +\begin_inset Formula $f$ +\end_inset + + v točki +\begin_inset Formula $a\sim L=\lim_{x\to a}f\left(x\right)\Leftrightarrow\forall\varepsilon>0\exists\delta>0\forall x\in\mathcal{D}\setminus\left\{ a\right\} :\left|x-a\right|<\delta\Rightarrow\left|f\left(x\right)-L\right|<\varepsilon$ +\end_inset + +. +\end_layout + +\end_deeper \begin_layout Enumerate zveznost funkcije \end_layout +\begin_deeper +\begin_layout Standard +Naj bodo +\begin_inset Formula $\mathcal{D}\subseteq\mathbb{R}$ +\end_inset + +, +\begin_inset Formula $a\in\mathcal{D}$ +\end_inset + + in +\begin_inset Formula $f:\mathcal{D}\to\mathbb{R}$ +\end_inset + + poljubne. + +\begin_inset Formula $f$ +\end_inset + + je zvezna v +\begin_inset Formula $a\Leftrightarrow\forall\varepsilon>0\exists\delta>0\forall x\in\mathcal{D}:\left|x-a\right|<\delta\Rightarrow\left|f\left(x\right)-f\left(a\right)\right|<\varepsilon$ +\end_inset + + . + +\begin_inset Formula $f$ +\end_inset + + je zvezna na množici +\begin_inset Formula $\mathcal{A}$ +\end_inset + +, če je zvezna na vsaki točki množice +\begin_inset Formula $\mathcal{A}$ +\end_inset + +. +\end_layout + +\end_deeper \begin_layout Enumerate odvedljivost funkcije \end_layout +\begin_deeper +\begin_layout Standard +Naj bodo +\begin_inset Formula $a\in\mathbb{R}$ +\end_inset + +, +\begin_inset Formula $\mathcal{D}\subseteq\mathbb{R}$ +\end_inset + +, +\begin_inset Formula $f:\mathcal{D}\to\mathbb{R}$ +\end_inset + + poljubne. + +\begin_inset Formula $f$ +\end_inset + + je odvedljiva v +\begin_inset Formula $a\text{\ensuremath{\Leftrightarrow\lim_{h\to0}\frac{f\left(a+h\right)-f\left(a\right)}{h}}}\in\mathbb{R}$ +\end_inset + +, ZDB ko obstaja slednja limita. + Tedaj definiramo +\begin_inset Quotes eld +\end_inset + +odvod funkcije +\begin_inset Formula $f$ +\end_inset + + v točki +\begin_inset Formula $a$ +\end_inset + + +\begin_inset Quotes erd +\end_inset + +: +\begin_inset Formula $f'\left(a\right)=\lim_{h\to0}\frac{f\left(a+h\right)-f\left(a\right)}{h}$ +\end_inset + +. + +\begin_inset Formula $f$ +\end_inset + + je odvedljiva na množici +\begin_inset Formula $\mathcal{A}$ +\end_inset + +, če je odvedljiva na vsaki točki množice +\begin_inset Formula $\mathcal{A}$ +\end_inset + +. +\end_layout + +\end_deeper +\begin_layout Enumerate +določen integral realne funkcije na zaprtem omejenem intervalu. +\end_layout + +\begin_deeper +\begin_layout Enumerate +Naj bodo +\begin_inset Formula $a,b\in\mathbb{R}$ +\end_inset + + in +\begin_inset Formula $f:\left[a,b\right]\to\mathbb{R}$ +\end_inset + + poljubne. +\end_layout + +\begin_layout Enumerate +Definirajmo pojem delitve +\begin_inset Formula $\left[a,b\right]$ +\end_inset + +. + Delitev so točke +\begin_inset Formula $t_{0},\dots,t_{n}$ +\end_inset + +, da velja +\begin_inset Formula $a=t_{0}<t_{1}<\cdots<t_{n}=b$ +\end_inset + + za nek +\begin_inset Formula $n\in\mathbb{N}$ +\end_inset + +. + Točke identificiramo z delilnimi intervali takole: +\begin_inset Formula $D_{n}=\left[t_{n-1},t_{n}\right]$ +\end_inset + +. + Delitev torej identificiramo z množico teh dedlilnih intervalov: +\begin_inset Formula $D=\left\{ D_{k};\forall k\in\left\{ 1..n\right\} \right\} $ +\end_inset + +. + Definiramo tudi velikost delitve: +\begin_inset Formula $\left|D_{\infty}\right|=\max_{k\in\left\{ 1..n\right\} }\left|D_{k}\right|$ +\end_inset + +. +\end_layout + +\begin_layout Enumerate +Definirajmo pojem izbire za dano delitev. + Naj bo +\begin_inset Formula $D$ +\end_inset + + delitev. + Pripadajoča izbira so take izbirne točke +\begin_inset Formula $\xi_{1},\dots,\xi_{n}$ +\end_inset + +, da velja +\begin_inset Formula $\forall k\in\left\{ 1..n\right\} :\xi_{k}\in D_{k}$ +\end_inset + +. + Množico teh izbirnih točk označimo z +\begin_inset Formula $\xi\coloneqq\left\{ \xi_{k};\forall k\in\left\{ 1..n\right\} \right\} $ +\end_inset + +. +\end_layout + \begin_layout Enumerate -določeni integral realne funkcije na zaprtem omejenem intervalu. +\begin_inset Formula $f$ +\end_inset + + je integrabilna na +\begin_inset Formula $\left[a,b\right]$ +\end_inset + +, če +\begin_inset Formula $\exists I\in\mathbb{R}\forall\varepsilon>0\exists\delta>0\forall$ +\end_inset + + delitev +\begin_inset Formula $D\forall$ +\end_inset + + izbiro +\begin_inset Formula $\xi$ +\end_inset + +, pripadajočo delitvi +\begin_inset Formula $D:\left|D_{\infty}\right|<\delta\Rightarrow\left|\sum_{k=1}^{n}\left|D_{k}\right|f\left(\xi\right)-I\right|<\varepsilon$ +\end_inset + +. + Tedaj pravimo, da je +\begin_inset Formula $I$ +\end_inset + + določen integral +\begin_inset Formula $f$ +\end_inset + + na +\begin_inset Formula $\left[a,b\right]$ +\end_inset + + in pišemo +\begin_inset Formula $I\eqqcolon\int_{a}^{b}f\left(x\right)dx$ +\end_inset + +. \end_layout \end_deeper +\end_deeper \begin_layout Enumerate \begin_inset Formula $\left[15\right]$ \end_inset @@ -161,11 +552,177 @@ določeni integral realne funkcije na zaprtem omejenem intervalu. Pojasni princip matematične indukcije. \end_layout +\begin_deeper +\begin_layout Standard +Naj bo +\begin_inset Formula $\left(P_{n}\right)_{n\in\mathbb{N}}$ +\end_inset + + zaporedje logičnih vrednosti/izjav/izrazov. + Če velja +\end_layout + +\begin_layout Enumerate +\begin_inset Formula $P_{1}$ +\end_inset + + drži in hkrati +\end_layout + +\begin_layout Enumerate +\begin_inset Formula $\forall n\in\mathbb{N}:P_{n}$ +\end_inset + + drži +\begin_inset Formula $\Rightarrow P_{n+1}$ +\end_inset + + drži, +\end_layout + +\begin_layout Standard +potem velja +\begin_inset Formula $\forall n\in\mathbb{N}:P_{n}$ +\end_inset + + drži. +\end_layout + +\end_deeper \begin_layout Enumerate Z matematično indukcijo dokaži \begin_inset Formula \[ -1+2+\cdots+n=\frac{n\left(n+1\right)}{2} +\forall n\in\mathbb{N}:1+2+\cdots+n=\frac{n\left(n+1\right)}{2} +\] + +\end_inset + + +\end_layout + +\begin_deeper +\begin_layout Enumerate +Baza +\begin_inset Formula $n=1$ +\end_inset + +: +\begin_inset Formula $1=\frac{1\left(1+1\right)}{2}$ +\end_inset + + Velja. +\end_layout + +\begin_layout Enumerate +Indukcijska predpostavka: +\begin_inset Formula $1+2+\cdots+n=\frac{n\left(n+1\right)}{2}$ +\end_inset + +. +\end_layout + +\begin_layout Enumerate +Korak +\begin_inset Formula $n\to n+1$ +\end_inset + +: +\begin_inset Formula +\[ +1+2+\cdots+n+\cancel{n+1}\overset{?}{=}\frac{\left(n+1\right)\left(n+1+1\right)}{2}=\frac{n^{2}+2n+n+2}{2}=\frac{n\left(n+1\right)}{2}+\cancel{n+1} +\] + +\end_inset + + +\begin_inset Formula +\[ +1+2+\cdots+n\overset{\text{I.P.}}{=}\frac{n\left(n+1\right)}{2} +\] + +\end_inset + + +\end_layout + +\begin_layout Enumerate +Sklep: +\begin_inset Formula $\forall n\in\mathbb{N}:1+2+\cdots+n=\frac{n\left(n+1\right)}{2}$ +\end_inset + +. +\end_layout + +\end_deeper +\end_deeper +\begin_layout Enumerate +\begin_inset Formula $\left[25\right]$ +\end_inset + + +\begin_inset Newline newline +\end_inset + +Naj bosta +\begin_inset Formula $\left(a_{n}\right)_{n\in\mathbb{N}}$ +\end_inset + + in +\begin_inset Formula $\left(b_{n}\right)_{n\in\mathbb{N}}$ +\end_inset + + realni konvergentni zaporedji. + Dokaži, da je +\begin_inset Formula $c_{n}\coloneqq a_{n}b_{n}$ +\end_inset + + prav tako konvergentno zaporedje. +\end_layout + +\begin_deeper +\begin_layout Itemize +Označimo +\begin_inset Formula $\lim_{n\to\infty}a_{n}\eqqcolon A$ +\end_inset + + in +\begin_inset Formula $\lim_{n\to\infty}b_{n}\eqqcolon B$ +\end_inset + +. +\end_layout + +\begin_layout Itemize +Uganemo, da je +\begin_inset Formula $\lim_{n\to\infty}a_{n}b_{n}=AB$ +\end_inset + +. + To moramo sedaj dokazati. +\end_layout + +\begin_layout Itemize +Dokazujemo, da +\begin_inset Formula $\forall\varepsilon>0\exists n_{0}\in\mathbb{N}\forall n\geq n_{0}:\left|a_{n}b_{n}-AB\right|<\varepsilon\sim\left|a_{n}b_{n}+a_{n}B-a_{n}B-AB\right|=\left|a_{n}\left(b_{n}-B\right)+B\left(a_{n}-A\right)\right|<\varepsilon$ +\end_inset + + +\end_layout + +\begin_layout Itemize +Ker po trikotniški neenakosti velja +\begin_inset Formula $\left|a_{n}\left(b_{n}-B\right)+B\left(a_{n}-A\right)\right|\leq\left|a_{n}\right|\left|b_{n}-B\right|+\left|B\right|\left|a_{n}-A\right|$ +\end_inset + +, je dovolj za poljuben +\begin_inset Formula $\varepsilon>0$ +\end_inset + + dokazati +\begin_inset Formula +\[ +\exists n_{0}\in\mathbb{N}\forall n\geq n_{0}:\left|a_{n}\right|\left|b_{n}-B\right|+\left|B\right|\left|a_{n}-A\right|<\varepsilon \] \end_inset @@ -173,10 +730,509 @@ Z matematično indukcijo dokaži \end_layout +\begin_layout Itemize +Ker je +\begin_inset Formula $\left(a_{n}\right)_{n\in\mathbb{N}}$ +\end_inset + + konvergentno, +\begin_inset Formula $\exists n_{1}\in\mathbb{N}\forall n\geq n_{1}:\left|a_{n}-A\right|<\frac{\varepsilon}{2\left|a\right|}$ +\end_inset + +, kjer je +\begin_inset Formula $a$ +\end_inset + + zgornja meja zaporedja +\begin_inset Formula $a_{n}$ +\end_inset + +. + Slednje je omejeno, ker je konvergentno. +\end_layout + +\begin_layout Itemize +Ker je +\begin_inset Formula $\left(b_{n}\right)_{n\in\mathbb{N}}$ +\end_inset + + konvergentno, +\begin_inset Formula $\exists n_{2}\in\mathbb{N}\forall n\geq n_{1}:\left|b_{n}-B\right|<\frac{\varepsilon}{2\left|B\right|}$ +\end_inset + +. +\end_layout + +\begin_layout Itemize +Tedaj za +\begin_inset Formula $n_{0}\coloneqq\max\left\{ n_{1},n_{2}\right\} $ +\end_inset + + velja +\begin_inset Formula +\[ +\left|a_{n}\right|\left|b_{n}-B\right|+\left|B\right|\left|a_{n}-A\right|<\frac{\varepsilon\left|a\right|}{2\left|a_{n}\right|}+\frac{\varepsilon}{2}\leq\frac{\varepsilon}{2}+\frac{\varepsilon}{2}=\varepsilon +\] + +\end_inset + +in izrek je dokazan. +\end_layout + \end_deeper \begin_layout Enumerate -Naj bo +\begin_inset Formula $\left[?\right]$ +\end_inset + + +\begin_inset Newline newline +\end_inset + +Dokaži, da je zvezna realna funkcija na zaprtem intervalu omejena. + Natančno navedi vse izreke, ki jih pri tem dokazu uporabiš. +\end_layout + +\begin_deeper +\begin_layout Standard +Naj bodo +\begin_inset Formula $a,b\in\mathbb{R}$ +\end_inset + + in zvezna +\begin_inset Formula $f:\left[a,b\right]\to\mathbb{R}$ +\end_inset + + poljubne. +\end_layout + +\begin_layout Itemize +Dokaz, da je +\begin_inset Formula $f$ +\end_inset + + omejena navzgor. +\end_layout + +\begin_deeper +\begin_layout Itemize +PDDRAA +\begin_inset Formula $f$ +\end_inset + + ni navzgor omejena. + Tedaj +\begin_inset Formula $\forall n\in\mathbb{N}\exists x_{n}\in\left[a,b\right]\ni:f\left(x_{n}\right)\geq n$ +\end_inset + +. \end_layout +\begin_layout Itemize +Ker je +\begin_inset Formula $\left(a_{n}\right)_{n\in\mathbb{N}}$ +\end_inset + + na zaprti množici, je omejeno zaporedje, torej ima stekališče. + Recimo mu +\begin_inset Formula $s\in\mathbb{R}$ +\end_inset + +. +\end_layout + +\begin_layout Itemize +Ker je +\begin_inset Formula $\left[a,b\right]$ +\end_inset + + zaprta, je +\begin_inset Formula $s\in\left[a,b\right]$ +\end_inset + +. +\end_layout + +\begin_layout Itemize +Ker je +\begin_inset Formula $f$ +\end_inset + + zvezna na +\begin_inset Formula $\left[a,b\right]$ +\end_inset + + in s tem v +\begin_inset Formula $s$ +\end_inset + +, velja +\begin_inset Formula $\lim_{n\to\infty}f\left(x_{n}\right)=f\left(s\right)$ +\end_inset + +. +\end_layout + +\begin_layout Itemize +Po konstrukciji +\begin_inset Formula $\left(x_{n}\right)_{n\in\mathbb{N}}$ +\end_inset + + velja +\begin_inset Formula $\lim_{n\to\infty}f\left(x_{n}\right)=\infty$ +\end_inset + +, torej +\begin_inset Formula $f\left(s\right)=\infty$ +\end_inset + +, kar ni mogoče, saj +\begin_inset Formula $f\left(s\right)\in\mathbb{R}$ +\end_inset + + po predpostavki. + +\begin_inset Formula $\rightarrow\!\leftarrow$ +\end_inset + +. +\end_layout + +\begin_layout Standard +Predpostavka +\begin_inset Quotes eld +\end_inset + + +\begin_inset Formula $f$ +\end_inset + + ni navzgor omejena +\begin_inset Quotes erd +\end_inset + + ne velja, torej smo dokazali, da je +\begin_inset Formula $f$ +\end_inset + + navzgor omejena. +\end_layout + +\end_deeper +\begin_layout Itemize +Dokaz, da je +\begin_inset Formula $f$ +\end_inset + + omejena navzdol. +\end_layout + +\begin_deeper +\begin_layout Itemize +PDDRAA +\begin_inset Formula $f$ +\end_inset + + ni navzdol omejena. + Tedaj +\begin_inset Formula $\forall n\in\mathbb{N}\exists x_{n}\in\left[a,b\right]\ni:f\left(x_{n}\right)\leq-n$ +\end_inset + +. +\end_layout + +\begin_layout Itemize +Ker je +\begin_inset Formula $\left(a_{n}\right)_{n\in\mathbb{\mathbb{N}}}$ +\end_inset + + na zaprti množici, je omejeno zaporedje, torej ima stekališče. + Recimo mu +\begin_inset Formula $s\in\mathbb{R}$ +\end_inset + +. +\end_layout + +\begin_layout Itemize +Ker je +\begin_inset Formula $\left[a,b\right]$ +\end_inset + + zaprta, je +\begin_inset Formula $s\in\left[a,b\right]$ +\end_inset + +. +\end_layout + +\begin_layout Itemize +Ker je +\begin_inset Formula $f$ +\end_inset + + zvezna na +\begin_inset Formula $\left[a,b\right]$ +\end_inset + + in s tem v +\begin_inset Formula $s$ +\end_inset + +, velja +\begin_inset Formula $\lim_{n\to\infty}f\left(x_{n}\right)=f\left(s\right)$ +\end_inset + +. +\end_layout + +\begin_layout Itemize +Po konstrukciji +\begin_inset Formula $\left(x_{n}\right)_{n\in\mathbb{N}}$ +\end_inset + + velja +\begin_inset Formula $\lim_{n\to\infty}f\left(x_{n}\right)=-\infty$ +\end_inset + +, torej +\begin_inset Formula $f\left(s\right)=-\infty$ +\end_inset + +, kar ni mogoče, saj +\begin_inset Formula $f\left(s\right)\in\mathbb{R}$ +\end_inset + + po predpostavki. + +\begin_inset Formula $\rightarrow\!\leftarrow$ +\end_inset + +. +\end_layout + +\begin_layout Standard +Predpostavka +\begin_inset Quotes eld +\end_inset + + +\begin_inset Formula $f$ +\end_inset + + ni navzdol omejena +\begin_inset Quotes erd +\end_inset + + ne velja, torej smo dokazali, da je +\begin_inset Formula $f$ +\end_inset + + navzdol omejena. +\end_layout + +\end_deeper +\begin_layout Itemize +Ker je +\begin_inset Formula $f$ +\end_inset + + omejena navzgor in navzdol, je omejena. +\end_layout + +\begin_layout Itemize +Uporabljeni izreki. +\end_layout + +\begin_deeper +\begin_layout Itemize +Zaporedje +\begin_inset Formula $\left(a_{n}\right)_{n\in\mathbb{N}}$ +\end_inset + + s členi na kompaktni množici je omejeno. +\end_layout + +\begin_layout Itemize +Omejeno zaporedje ima stekališče. +\end_layout + +\begin_layout Itemize +Če je +\begin_inset Formula $s\in\mathbb{R}$ +\end_inset + + stekališče zaporedja +\begin_inset Formula $\left(a_{n}\right)_{n\in\mathbb{N}}$ +\end_inset + +, obstaja konvergentno podzaporedje +\begin_inset Formula $\left(a_{n_{k}}\right)_{k\in\mathbb{N}}$ +\end_inset + +, da je +\begin_inset Formula $s$ +\end_inset + + njegova limita. +\end_layout + +\begin_layout Itemize +Množica je kompaktna natanko tedaj, ko vsebuje limite vseh konvergentnih + zaporedij s členi v njej. +\end_layout + +\begin_layout Itemize +Funkcija +\begin_inset Formula $f$ +\end_inset + + je zvezna v +\begin_inset Formula $s$ +\end_inset + +, če za vsako k +\begin_inset Formula $s$ +\end_inset + + konvergentno zaporedje velja, da njegovi s +\begin_inset Formula $f$ +\end_inset + + preslikani členi konvergirajo v +\begin_inset Formula $f\left(s\right)$ +\end_inset + +. +\end_layout + +\end_deeper +\end_deeper +\begin_layout Enumerate +\begin_inset Formula $\left[?\right]$ +\end_inset + + +\begin_inset Newline newline +\end_inset + +Za realno funkcijo ene spremenljivke dokaži verižno pravilo. +\end_layout + +\begin_deeper +\begin_layout Itemize +Naj bodo +\begin_inset Formula $\mathcal{D},\mathcal{E},\mathcal{F}\subseteq\mathbb{R}$ +\end_inset + +, +\begin_inset Formula $x\in\mathcal{D}$ +\end_inset + + in +\begin_inset Formula $f:\mathcal{D}\to\mathcal{E}$ +\end_inset + +, +\begin_inset Formula $g:\mathcal{E}\to\mathcal{F}$ +\end_inset + + poljubne. + Naj bo +\begin_inset Formula $f$ +\end_inset + + odvedljiva v +\begin_inset Formula $x$ +\end_inset + + in +\begin_inset Formula $g$ +\end_inset + + odvedljiva v +\begin_inset Formula $f\left(x\right)$ +\end_inset + +. +\end_layout + +\begin_layout Itemize +Dokažimo, da je +\begin_inset Formula $g\circ f$ +\end_inset + + odvedljiva v +\begin_inset Formula $x$ +\end_inset + + in da velja +\begin_inset Formula +\[ +\left(g\circ f\right)'\left(x\right)=g'\left(f\left(x\right)\right)f'\left(x\right). +\] + +\end_inset + + +\end_layout + +\begin_layout Itemize +Označimo +\begin_inset Formula $a\coloneqq f\left(x\right)$ +\end_inset + + in +\begin_inset Formula $\delta_{h}\coloneqq f\left(x+h\right)-f\left(x\right)$ +\end_inset + +. + Potemtakem +\begin_inset Formula $f\left(x+h\right)=\delta_{h}+a$ +\end_inset + +. +\begin_inset Formula +\[ +\left(g\circ f\right)'\left(x\right)=\lim_{h\to0}\frac{g\left(f\left(x+h\right)\right)-g\left(f\left(x\right)\right)=g\left(\delta_{h}+a\right)-g\left(a\right)}{h}= +\] + +\end_inset + + +\begin_inset Formula +\[ +=\lim_{h\to0}\frac{g\left(\delta_{h}+a\right)-g\left(a\right)}{\delta_{h}}\cdot\frac{\delta_{h}}{h}=\lim_{h\to0}\frac{g\left(\delta_{h}+a\right)-g\left(a\right)}{\delta_{h}}\cdot\frac{f\left(x+h\right)-f\left(x\right)}{h}=\cdots +\] + +\end_inset + +Ker je +\begin_inset Formula $f$ +\end_inset + + v +\begin_inset Formula $x$ +\end_inset + + odvedljiva, je v +\begin_inset Formula $x$ +\end_inset + + zvezna, zato sledi +\begin_inset Formula $h\to0\Rightarrow\delta_{h}\to0$ +\end_inset + +. +\begin_inset Formula +\[ +\cdots=g'\left(a\right)\cdot f'\left(x\right)=g'\left(f\left(x\right)\right)\cdot f'\left(x\right) +\] + +\end_inset + + +\end_layout + +\end_deeper \end_body \end_document |