From 1bd452ec8ffe5b607d09236369a690f168e59444 Mon Sep 17 00:00:00 2001 From: zacharias Date: Tue, 25 Nov 2025 16:05:22 +0100 Subject: [PATCH] vault backup: 2025-11-25 16:05:22 --- .obsidian/workspace.json | 12 ++++++------ Derivata.md | 17 +++++++++++++---- MVT.png | Bin 0 -> 31024 bytes 3 files changed, 19 insertions(+), 10 deletions(-) create mode 100644 MVT.png diff --git a/.obsidian/workspace.json b/.obsidian/workspace.json index 97347b4..7727610 100644 --- a/.obsidian/workspace.json +++ b/.obsidian/workspace.json @@ -159,8 +159,7 @@ "title": "Bookmarks" } } - ], - "currentTab": 1 + ] } ], "direction": "horizontal", @@ -267,17 +266,18 @@ "obsidian-git:Open Git source control": false } }, - "active": "e616c86f78b96cf1", + "active": "ad6eb280b4b8718c", "lastOpenFiles": [ + "MVT.png", + "Gräsvärde (1).md", + "Trigonometri.md", + "Derivata.md", "Pasted image 20251119134315.png", "d_ex_1.png", "d1.png", - "Gräsvärde (1).md", - "Derivata.md", "Funktioner.md", "Funktioner Forts.md", "Komplexa tal.md", - "Trigonometri.md", "Grafer.md", "conflict-files-obsidian-git.md", "gv1.png", diff --git a/Derivata.md b/Derivata.md index 23e8c4a..4fa081d 100644 --- a/Derivata.md +++ b/Derivata.md @@ -3,18 +3,18 @@ - **Defs**: - $Df$: *Oendlig liten ändrig i $f$* - $Dx$: *Oendlig liten ändrig i $x$* - - $f[\bullet]=f'$ - - + - $\overset{\bullet}f=f'$ ![[d1.png]] - Egenskaper och regler - $f$ deriverbar $\Rightarrow$ $f$ kontinuerlig. **Obs!** Inte alla kontinuerliga funktioner är deriverbara - Derivering är linjär avbildning: $\left(\alpha f+\beta g\right)'=\alpha f'+\beta g'$ - **Produkt regel** (*Leibniz*): $\left(f\left(x\right)g\left(x\right)\right)'=f'\left(x\right)g\left(x\right)+f\left(x\right)g'\left(x\right)$ - **Sammansatt funktion**: $\left(f\circ g\right)'\left(x\right)=f'\circ g\left(x\right)g'\left(x\right)$ + - **Kjedje regel**: $(f(g(x)))'=f'(f(x))g'(x)$ - **Division**: $\left(\frac{f\left(x\right)}{g\left(x\right)}\right)'=\frac{f'\left(x\right)g\left(x\right)-f\left(x\right)g'\left(x\right)}{g\left(x\right)^2}$ - **Ex**: ![[d_ex_1.png]]$$\begin{align}f(x)=\mid x\mid\\f\text{ är kontinuerlig på }\mathbb{R}.\\f\text{ är inte deriverbar i }0.\\\lim_{x\to0+}\frac{f\left(x\right)-f\left(0\right)}{x-0}=\lim_{x\to0+}\frac{\mid x\mid-0}x=\lim_{x\to0+}\frac xx=1\\\lim_{x\to0-}\frac{f\left(x\right)-f\left(0\right)}{x-0}=\lim_{x\to0-}\frac{\mid x\mid-0}x=\lim_{x\to0-}\frac{-x}x=-1\\\lim_{x\to0}\frac{f\left(x\right)-f\left(0\right)}{x-0}=f'(0)\text{ existerar inte-}\end{align}$$ - **Ex**: $$\begin{align}\text{Leibniz regel}\\\left(f\left(x\right)g\left(x\right)\right)'=\lim_{h\to0}\frac{f\left(x+h\right)-f\left(x\right)g\left(x\right)}h\\=\lim_{h\to0}\frac{f(x+h)g(x+h)-f(x)g(x+h)+f(x)g(x+h)-f(x)g(x)}{h}\\=\lim_{h\to0}\left(g(x+h)\frac{f(x+h)-f(x)}{h}+f(x)\frac{g(x+h)-g(x)}{h}\right)\\=g(x)f'(x)+f(x)g'(x)\end{align}$$ - - **Ex**: $$\begin{align}h(x)=\frac1x\\h'(x)=-\frac1{x^2}\\h\circ g(x)=h(g(x))=\frac1{g(x)}\\(g\circ g)'(x)=\left(\frac1{g(x)}\right)^2\\=h'\circ g(x)g'(x)=h'(g(x))g'(x)\frac{-1}{(g(x))^2}g'(x)\end{align}$$ + - **Ex**: $$\begin{align}h(x)=\frac1x\\h'(x)=-\frac1{x^2}\\h\circ g(x)=h(g(x))=\frac1{g(x)}\\(g\circ g)'(x)=\left(\frac1{g(x)}\right)^2\\=h'\circ g(x)g'(x)=h'(g(x))g'(x)=\frac{-1}{(g(x))^2}g'(x)\end{align}$$ - Standerd derivarives 1. $f(x)=c\;\;\Rightarrow\;\;f'(x)=0$ 2. $f(x)=n^n\;\;\Rightarrow\;\;f'(x)=nx^{n-1},\;n\in\mathbb{Z}$ @@ -26,4 +26,13 @@ 8. $f(x)=\tan x\;\;\Rightarrow\;\;f'(x)=\sec^2x=1+\tan^2x$ 9. $f(x)=a^x\;\;\Rightarrow\;\;f'(x)=a^x\ln a,\;a>0$ 10. $f(x)=\log_a\mid x\mid\;\;\Rightarrow\;\;f'(x)=(x\ln a)^-1,\;a>0,\;x\neq0$ - 11. \ No newline at end of file +- Implicit derviering + - **Ex**: Bestäm tangent & normal ekvation till kurvan $x^3y^2-x^2y^3=12$ i punkten $(2,-1)$ $$\begin{align}\text{Antag att }y=f(x)\text{ för någon funktion }f\text{ nära punkten }(2,-1)\\x^3(f(x))^2-x^2(f(x))^3=12\\\text{Derivera m.a.p. }x\\(x^3(f(x))^2)'-(x^2(f(x))^3)'=(12)'\\\Leftrightarrow(x^3)'(f(x))^2+x^3((f(x))^2)'\\-(x^2)'(f(x))^3-x^2((f(x))^3)'=0\\\text{(produkt regeln)}\\\Rightarrow3x^2(f(x))^2+x^3\times2f(x)f'(x)\\-2x(f(x))^3-x^2\times3(f(x))^2\times f'(x)=0\\\Leftrightarrow(2x^3f(x)-3x^2(f(x))^2)f'(x)\\=2x(f(x))^3-3x^2(f(x))^2\\\text{På punkten }(2,-1)\text{ har vi}\\y=f(2)=-1\\\text{sätt in }x=2\\\left(2\times2^3f(2)-3\times2^2\times(f(2))^2)f'(x)\right)=2\times2\times(f(2))^3-2\times2^2(f(2))^2\\\Leftrightarrow(-16-12)f'(2)=-4-12\Leftrightarrow f'(2)=\frac{-16}{-28}=\frac47\\\text{Tangent ekv: }y=f'(a)(x-a)+f(a)\\\Leftrightarrow y=\frac47(x-2)-1\Leftrightarrow4x-7y=15\\\text{Normal ekv: }y=-\frac1{f'(a)}(x-a)+f(a)\\\Leftrightarrow y=-\frac74(x-2)-1\Leftrightarrow7x+4y=10\end{align}$$ + - **Kedje regeln**: $$\begin{align}\frac{df(y(x))}{dx}=\frac{df(y(x))}{dy}\times\frac{dy(x)}{dx}\\(f(y(x)))'=f'(y(x))y'(x)\end{align}$$ +- Invers + - **Theorem**: *Om $f$ är inverterbar och deriverbar i punkten $a$ så att $f'(a)\neq0$ då är inversen $f^{-1}$ deriverbar i punkten $b=f(a)$ med derivatan* $$\left(f^{-1}\right)'\left(b\right)=\frac1{f'(a)}$$ + - Följdsats: + - **Theorem**: $$\begin{gather}\text{För }-1\;f(x)=\arcsin x\;\;\;\;\;\Rightarrow f'(x)=\frac1{\sqrt{1-x^2}}\\>\;f(x)=\arccos x\;\;\Rightarrow f'(x)=-\frac1{\sqrt{1-x^2}}\\>\;f(x)=\arctan x\;\;\;\;\;\;\;\;\Rightarrow f'(x)=\frac1{1+x^2}\\>\;f(x)=\arccot x\;\;\;\;\Rightarrow f'(x)=-\frac1{1+x^2}\end{gather}$$ +- Medelvärdessats + - **Theorem** *Om $f$ är kontinuerlig på slutet intervall $\left[a,b\right]$ och deriverbar på öppet intervall $\left(a,b\right)$, dår fins det minst en punkt $\xi\in\left(a,b\right)$ så att* $$f'(\xi)=\frac{f(b)-f(a)}{b-a}$$![[MVT.png]] + - \ No newline at end of file diff --git a/MVT.png b/MVT.png new file mode 100644 index 0000000000000000000000000000000000000000..793460ca3d1ab50a7b97650258532e0fdb83b105 GIT binary patch literal 31024 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