From a23a0d3bb008814a67709ceeb8dd120d892407c7 Mon Sep 17 00:00:00 2001 From: zacharias Date: Wed, 12 Nov 2025 15:01:58 +0100 Subject: [PATCH] vault backup: 2025-11-12 15:01:58 --- .obsidian/workspace.json | 30 +++++++++++++++++++++++------- Gräsvärde (1).md | 21 +++++++++++++++++++++ Komplexa tal.md | 10 ++++++++-- gv1.png | Bin 0 -> 31098 bytes 4 files changed, 52 insertions(+), 9 deletions(-) create mode 100644 Gräsvärde (1).md create mode 100644 gv1.png diff --git a/.obsidian/workspace.json b/.obsidian/workspace.json index 26ee7d9..fa8f765 100644 --- a/.obsidian/workspace.json +++ b/.obsidian/workspace.json @@ -76,9 +76,23 @@ "icon": "lucide-file", "title": "Komplexa tal" } + }, + { + "id": "76c8d943958d45bf", + "type": "leaf", + "state": { + "type": "markdown", + "state": { + "file": "Gräsvärde (1).md", + "mode": "source", + "source": false + }, + "icon": "lucide-file", + "title": "Gräsvärde (1)" + } } ], - "currentTab": 4 + "currentTab": 5 } ], "direction": "vertical" @@ -238,15 +252,17 @@ "obsidian-git:Open Git source control": false } }, - "active": "be47d5ede3a9176b", + "active": "76c8d943958d45bf", "lastOpenFiles": [ + "gv1.png", + "Funktioner.md", + "Funktioner Forts.md", + "Gräsvärde (1).md", + "Komplexa tal.md", + "Grafer.md", + "Trigonometri.md", "k2.png", "k1.png", - "Trigonometri.md", - "Komplexa tal.md", - "Funktioner Forts.md", - "Grafer.md", - "Funktioner.md", "f_inverse.png", "g2.png", "g1.png", diff --git a/Gräsvärde (1).md b/Gräsvärde (1).md new file mode 100644 index 0000000..fbe3ef1 --- /dev/null +++ b/Gräsvärde (1).md @@ -0,0 +1,21 @@ +- Gränsvärden + - **Def**: *Om för varje $\epsilon>0$ existerar $\delta>0$ så att $$\mid{x-a}\mid<\delta\Rightarrow\mid{f(x)-L}\mid<\epsilon$$är talet $L$ gransvärde till $f(x)$ då $x$ får mot $a$. Betekning: $f(x)\longrightarrow{L}$ då $x\longrightarrow{a}$, eller $$\lim_{x\to{a}} f(x)=L$$* + - **Def**: *Om för varje $\epsilon>0$ existerar $M>0$ så att$$x>M\;\Rightarrow\;\mid{f(x)-L}\mid<\epsilon$$är talet $L$ gränsvärde till $f(x) då $x$ går mot oändlighit. Beteckning: $f(x)\longrightarrow{L}$ då $x\longrightarrow\infty$, eller $$\lim_{x\to\infty}f(x)=L$$* +- Remarks + - *Om det inte fins sådant $L$ värde, saknar funktionen gränsvärde på punkten $a$,* + - **Ex**: $$\begin{align}f(x)=\sin x\\\lim_{x\to\infty}\sin x\\\text{Existerar inte}\end{align}$$ + - **Ex**: $$\begin{align}f(x)=\sin\frac1x\\\lim_{x\to0}\sin\frac1x\\\text{Existerar inte}\end{align}$$ + - *Punkten $a$ behöver inte vara i $D_f$.* + - *Beteende av funktionen kring "problempunkter" är intressant.* + - *Långsiktig beteende hos funktioner: $$\lim_{x\longrightarrow\infty}f(x)$$* + - *Derivator, integraler, asymptot etc definieras med hjälp av gränsvärde.* + - *Om $a$ int är "problempunkt" stoppar vi in $x=a$ i $f(x)$* + - **Def**: *"Problempunkt" t.ex $\lim_{x\to 0}\frac1x$ går inte att direkt lösa på grund av division med $0$* +- One sided limits + - ![[gv1.png]] +- Problem fall + - $\left[\frac00\right]$ form: **Ex**: $$\lim_{x\to1}\frac{x^2-3x+2}{x^2-1},\;\lim_{x\to0}\frac{e^x-1}x,\;\lim_{x\to\infty}\frac{\tan{x}}x$$ + - $\left[\frac\infty\infty\right]$ form: **Ex**: $$\lim_{x\to\infty}\frac{x^2-3x+2}{x^2-1},\;\lim_{x\to\infty}\frac{x^3}{2^x}$$ + - $\left[0\times\infty\right]$ form: **Ex**: $$\lim_{x\to\infty}x^2\ln\mid{x}\mid$$ + - $\left[0^0\right]$ form: **Ex**: $$\lim_{x\to0+}x^x$$ + - $\left[\infty^0\right]$ form **Ex**: $$\lim_{x\to\infty}$$ \ No newline at end of file diff --git a/Komplexa tal.md b/Komplexa tal.md index 54f8602..6d67d59 100644 --- a/Komplexa tal.md +++ b/Komplexa tal.md @@ -20,5 +20,11 @@ - **Eulers formel**: $e^{i\theta}=\cos\theta+i\sin\theta$ - Varje komplex tal $z=x+yi$ kan skrivas på pol'r form som $$z=re^{i\theta}$$ där $$r=\sqrt{x^2+y^2}$$ och $arg(z)=\theta$ är så att $$\cos\theta=\frac{x}{\sqrt{x^2+y^2}}\text{ och }\sin\theta=\frac{y}{x^2+y^2}$$ - **de Moivre**: $z=re^{i\theta}\Rightarrow z^n=r^ne^{in\theta}=r^n(\cos(n\theta)+i\sin(n\theta))$ - - **Ex**: Lös $z^3=1+i\sqrt3$ $$\begin{align}r=\sqrt{1^2+\sqrt3^2}\\=\sqrt{1+3}\\=\sqrt{4}\\=2\\\\\end{align}$$ - - **Ex 2**: $$\begin{align}z=-\frac{\sqrt3}{2}+\frac{1}{2}i\\z=ne^{i\theta}\\n=\sqrt{\left(-\frac{\sqrt{3}}{2}\right)^2+\left(\frac{1}{2}\right)^2}=1\\\theta\text{ är så att }\cos\theta=\frac{\sqrt{3}}{2},\sin\theta=\frac{1}{2}\\\text{En lösning}:\end{align}$$ \ No newline at end of file + - **Ex**: Lös $z^3=1+i\sqrt3$ $$\begin{align*}1+i\sqrt3=n_\circ e^{i\theta}, \theta\in\left[0,2\pi\right)\\n_\circ=\sqrt{1^2+\left(\sqrt3\right)^2}=2\\\theta\in\left[0,2\pi\right)\text{ uppfyller}\\\cos\theta=\frac12,\sin\theta=\frac{\sqrt3}2\\\Rightarrow\theta=\frac\pi3\\z^3=1+i\sqrt3=2e^{i\frac\pi3}\\\text{Låt }z=n_1e^{i\phi}\\\text{Då är }z^3=n_1^3e^{i3\phi}\\\Leftrightarrow\left\{\begin{aligned}n_1^3=2,n\in\mathbb{R}\\e\phi=\frac\pi3+2\pi{k},k\in\mathbb{Z},\phi\in\left[0,2\pi\right)\end{aligned}\right.\\\Leftrightarrow\left\{\begin{aligned}n_1=\sqrt[3]{2}\\\phi=\frac\pi9+\frac{2\pi k}{3},k=0,1,2\end{aligned}\right.\\k=0:\;\phi=\frac\pi9+\frac{2\pi}3\times0=\frac\pi9\\k=1:\;\phi=\frac\pi9+\frac{2\pi}3=\frac{7\pi}9\\k=2:\;\phi=\frac\pi9+\frac{2\pi}4\times2=\frac{13\pi}9\end{align*}$$ + - **Ex 2**: $$\begin{align}z=-\frac{\sqrt3}{2}+\frac{1}{2}i\\z=ne^{i\theta}\\n=\sqrt{\left(-\frac{\sqrt{3}}{2}\right)^2+\left(\frac{1}{2}\right)^2}=1\\\theta\text{ är så att }\cos\theta=\frac{\sqrt{3}}{2},\sin\theta=\frac{1}{2}\\\text{En lösning}:\theta=\pi-\frac{\pi}{6}=\frac{5\pi}{6}\\\text{Alla lösningar}:\theta=\frac{5\pi}{6}+2\pi{n},n\in\mathbb{Z}\\z=e^{i\left(\frac{5\pi}{6}+2\pi{n}\right)},n\in\mathbb{Z}\\\text{Svar: }z=e^{i\frac{5\pi}{6}}\end{align}$$ +- Polynom + - **Theorem**: *Algebrans huvudsats: Polynomet$$p(z)=c_nz^n+c_{n-1}z^{n-1}\dots+c_0,\;c_k\in\mathbb{C}$$har en rot i $\mathbb{C}$. D.v.s det finns en $z_1\in\mathbb{C}$ så att $p(z_1)=0$.* + - **Faktorsats**: $p(z)=(z-z_1)q(z)$ + - **Theorem**: *Polynomet ovan kan skrivas som $p(z)=c_n(z-z_1)\dots(z-z_n)$. Alla polynom har $n$ komplexa rötter (och faktorer).* + - **Theorem**: *Polynom med reella koefficienter:$p(x)=a_nz^n+a_{n-1}z^{n-1}\dots+a_0,\;a_k\in\mathbb{R}$. Om $z_0$ är en rot så är $\overline{z_0}$* + - **Ex**: $$\begin{align}p(z)=3z^3-7z^2+17z-5\\p(1+2i)=0\\\text{Polynomet har reella koefficienten. även konjugatet 1-2i är en rot.}\\\text{Enlight faktorsatsen}\\p(z)=(z-1-2i)(z-1+2i)q(z)\\\text{för något polynom }q(z)\\p(z)=\left(\left(z-1\right)^2-\left(2i\right)^2\right)q(z)\\=\left(z^2-2z+1+4\right)q(z)\\=\left(z^2-2z+5\right)q(z)\\\text{Polynomdivision: }\\\frac{3z-1}{z^2-2z+5}\\p(z)=\left(z-1-2i\right)\left(z-1+2i\right)\left(3z-1\right)\\\text{Rötter: }1+2i,1-2i,\frac13\end{align}$$ \ No newline at end of file diff --git a/gv1.png b/gv1.png new file mode 100644 index 0000000000000000000000000000000000000000..0e2aaab9af2335824baedc4b96552eea539e6703 GIT binary patch literal 31098 zcmeAS@N?(olHy`uVBq!ia0y~yVEoI#z}UpW#=yYvYKPPu1_lPs0*}aI1_r((Aj~*b zn@^g7fq|#QHKHUZKRq)!F(-n-$il!s@_ZkfcqD_dg+;9bi#1HVIJqb_HIE_HB|z6u z)H`Ys1A_vCr;B4qMckXauZyL>C+huUkGS(boJoQ6o@>_>kD1$^xNS>0{zT~tcqoBO|Cg@lGWy12Nw&)fHtPf1D1YL>(= 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