From 5c6c7ee11eb73ada9df8ded450bcf8e741f2955d Mon Sep 17 00:00:00 2001 From: zacharias Date: Wed, 26 Nov 2025 16:53:03 +0100 Subject: [PATCH] vault backup: 2025-11-26 16:53:02 --- .obsidian/workspace.json | 73 +++++++++++++++++++++++++++++++-------- Def_graf1.png | Bin 0 -> 5113 bytes Definitioner.md | 8 +++++ Derivata.md | 15 +++++++- Differential.md | 3 ++ TE1.png | Bin 0 -> 20960 bytes Tenta Example.md | 18 ++++++++++ 7 files changed, 102 insertions(+), 15 deletions(-) create mode 100644 Def_graf1.png create mode 100644 Definitioner.md create mode 100644 Differential.md create mode 100644 TE1.png create mode 100644 Tenta Example.md diff --git a/.obsidian/workspace.json b/.obsidian/workspace.json index 7727610..e396b46 100644 --- a/.obsidian/workspace.json +++ b/.obsidian/workspace.json @@ -69,12 +69,12 @@ "state": { "type": "markdown", "state": { - "file": "Trigonometri.md", + "file": "Gräsvärde (1).md", "mode": "source", "source": false }, "icon": "lucide-file", - "title": "Trigonometri" + "title": "Gräsvärde (1)" } }, { @@ -83,12 +83,54 @@ "state": { "type": "markdown", "state": { - "file": "Komplexa tal.md", + "file": "Derivata.md", "mode": "source", "source": false }, "icon": "lucide-file", - "title": "Komplexa tal" + "title": "Derivata" + } + }, + { + "id": "68837a4aa2729e39", + "type": "leaf", + "state": { + "type": "markdown", + "state": { + "file": "Grafer.md", + "mode": "source", + "source": false + }, + "icon": "lucide-file", + "title": "Grafer" + } + }, + { + "id": "6226fb790fca274b", + "type": "leaf", + "state": { + "type": "markdown", + "state": { + "file": "Tenta Example.md", + "mode": "source", + "source": false + }, + "icon": "lucide-file", + "title": "Tenta Example" + } + }, + { + "id": "46f7f5c1bc5d29b2", + "type": "leaf", + "state": { + "type": "markdown", + "state": { + "file": "Differential.md", + "mode": "source", + "source": false + }, + "icon": "lucide-file", + "title": "Differential" } }, { @@ -106,7 +148,7 @@ } } ], - "currentTab": 1 + "currentTab": 8 } ], "direction": "vertical" @@ -266,26 +308,29 @@ "obsidian-git:Open Git source control": false } }, - "active": "ad6eb280b4b8718c", + "active": "46f7f5c1bc5d29b2", "lastOpenFiles": [ - "MVT.png", + "Tenta Example.md", + "Differential.md", "Gräsvärde (1).md", - "Trigonometri.md", + "Komplexa tal.md", + "Grafer.md", + "Funktioner Forts.md", + "Funktioner.md", "Derivata.md", + "Definitioner.md", + "Def_graf1.png", + "TE1.png", + "Trigonometri.md", + "MVT.png", "Pasted image 20251119134315.png", "d_ex_1.png", "d1.png", - "Funktioner.md", - "Funktioner Forts.md", - "Komplexa tal.md", - "Grafer.md", "conflict-files-obsidian-git.md", "gv1.png", "k2.png", "k1.png", "f_inverse.png", - "g2.png", - "g1.png", "Untitled.canvas" ] } \ No newline at end of file diff --git a/Def_graf1.png b/Def_graf1.png new file mode 100644 index 0000000000000000000000000000000000000000..78576ade785340f33ed2815b9a27819804437aa6 GIT binary patch literal 5113 zcmeAS@N?(olHy`uVBq!ia0y~yV60_eU^vRb#=yX^W4h8`1_lPs0*}aI1_r((Aj~*b zn@^g7fq|#QHKHUZKRq)!F(-n-$il+0$$bx+cqD_Vp^?Wft$Q%>;^d;#)I5e%mjGQu zQSYcl3=G1_o-U3d6>)Fx=J$ktoqPP_`MHt_4Li?N99>#^z&;{SQ1-urWB9ZyQ%fgy z@~<#WS-K(9XUY!WYtsa7$KJl1lk@gS&aAhu#GJDhWlK9=4Dva1K}6>YZ|T;MlIg5d zA2skCIx|=D*&&THXY3lf%j@PGzEgef{k`AT&-cW4@>jk4IQwwnr*|iJE{^9F&%C_; z)>&uO*RM{^YG>+vEF=BByyC{WtvvjzTK=y?y4+P;+G_dxJ*A z=4Y9E|9sNU%F8peXDX1hd8s^U*6i6^>;7_kEtfviU&GrgyV$?)>&*5>7sX#4ORilz z>b3mxvqXQ{n|if>XR2SVJv=q)-mTlHsbSUq~KEQ{aDKbxKLy6?O>7w=S>XP=x@ z{pG{q`t_05&s~zv-;+6S*-Cx7Mwb=Q zcXnN{dG_+E@9fYwJL`7suAXwzWM;d!*l|mY68ft+&x%?)2q8yU%t+?xOn}5w`jdJ|DvMB35jywfuYf z?)O*Ry}qYvtsH(#^l*{i_(=RuXYz5+!-sdhdz9HC-|6;6FjLxW&x)U)^Cz2leK@~j z^;*?G9wv+WBQ(^0RJO=_x_uF>T*_kK|3P_WhnChQ#$De z7HoN;T$IBGQmNTtr?gb&;Cbag9!!h+PiUlr{9|*d-$MCU!M%g`a*j65 ze=h#7)Vh4ea;Aw+UmkAh%b6~heLv#jgXfj4|96_#9~ZV)TDxxd?&CJkR!=G3`{nD` z+3if7ujg#|9=@qu*X&u#+`T`Pk9}cb7rHV^Cok5uKNDh zUO)MNPcK?0FKxV};@n0<=b~)C3zD)o^=v=>%Dy$-R%vNaRN3hoZT}yR>(wV`UV7Y{ zY5VEY=T}zqF6;l>;4G}}$FZeDRsY4-88NexliQwLlaV%7DS!5%iv677{w=YWjBa~< zIDBINtDom&%RfKWQ|f(jSTpUz^m98y@6Xu2VAa9%t5>hRdPY-x{^zH958bB-)LIC~w>_3Ojv+0F!Uv!TQDSkXPdk=&= zd$Z04Q}s80r@o2xo?*GoJg#-8Sn!Q%nP*XxHZ*F3#k+eg&C)!+J@a_@-f;G28BiF7 z+_hYnyT|C}4B<34HR1QXF=lBVd1paEpU$_wJfZjJ(zV>u>thbH>4ivNY!^-Y5PPfQ zsgzm4!dV@5PM^8ii?n&qaQ*(vPX` zKLYh!$%0p*Qrrf!6AQdI0&D+29Fq6pkjg8!nf`3~+bJ=kRENbM9;b?lHJ|q4^ z<)MDQ_4?(U_UGDPsEZuz=RedRB=DC->BmRsAAvfH<{Pls@8huVa{MCP^5D4q!{Zqp zbxs0*o+$jWkY6;PfyKU(!~T!J-x6SOseumBa&>ZKlC~d5={n?EfilOy>YQp?emw4f;m&f+^1{Il7kn>|xcH@7{6l{p#9ve$Jut zA9ruc*%lm+Z?8MVU!@BR{h4NOjY>=YoV1A%ioPzi+;TY^`_|Ww`D{0Q)8y@4efdb? 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(Eller gränsvärde)* + 3. *Värde på punkter där derivata saknas(Kritiska punkter)* + 4. *Jämför 1,2,3.* + - **Ex**: $$\begin{align}f(x)=1-\mid{x}\mid\\f'(0)\text{ Existerar inte}\end{align}$$![[Def_graf1.png]] \ No newline at end of file diff --git a/Derivata.md b/Derivata.md index 4fa081d..7ced331 100644 --- a/Derivata.md +++ b/Derivata.md @@ -35,4 +35,17 @@ - **Theorem**: $$\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}$$![[MVT.png]] - - \ No newline at end of file +- Egenskaper + - *Låt $f$ vara deriverbar i intevallet $\left(a,b\right). följande gäller$* + 1. *$f'(c)=0$ för något $c\in\left(a,b\right)\;\Rightarrow\;f$ har lokal extremvärde eller sadelpunkt i punkten $x=c$.* + 2. *$f'(x)=0\;\forall{x}\in\left(a,b\right)\Rightarrow f(x)=C$, konstant funktion* + 3. *$f'(x)=g'(x)\;\forall{x}\in\left(a,b\right)\Rightarrow f(x)=g(x)+C$, Där $C$ är någon konstant.* + 4. *$f'(x)>0\;\forall{x}\in\left(a,b\right)\Rightarrow f(x)$ är strängt växande i $\left(a,b\right)$.* + 5. *$f'(x)<0\;\forall{x}\in\left(a,b\right)\Rightarrow f(x)$ är strängt avtagande i $\left(a,b\right)$.* + 6. *$f'(x)\geq0\;\forall{x}\in\left(a,b\right)$ med likhet i ändligt antal punkter $\Rightarrow f(x)$ är stängt växande i $\left(a,b\right)$.* + 7. *$f'(x)\leq0\;\forall{x}\in\left(a,b\right)$ med likhet i ändligt antal punkter $\Rightarrow f(x)$ är stängt avtagande i $\left(a,b\right)$.* +- Andra derivata + - **Betäkning**: $f''(x)$ + - **Definition**: 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0000000..a3a3380 --- /dev/null +++ b/Tenta Example.md @@ -0,0 +1,18 @@ +**Rita graf till** $f(x)=-\frac{\ln x}x$ +$$\begin{align}1.\;\;D_f=\left(0,\infty\right)\\2.\;\;\text{Lodrät asymptot }x=0\\\text{Vågrät asymtot }y=0\\3.\;\;\text{Stationära punkten:}\\f(x)=\frac{\ln x}x\\\text{derivera m.a.p. }x\\f'(x)=-\frac{x(\ln x)'-(\ln x)(x)'}{x^2}\\=-\frac{x\times\frac1x-(\ln x)(1)}{x^2}\\=\frac{\ln x-1}{x^2}\\\text{Stationär punkten uppfyller }f'(x)=0\\\Leftrightarrow\frac{\ln x-1}{x^2}=0\\\Leftrightarrow\ln x=1\\x=e\end{align}$$ +*Täkentabell* + +| | $e$ | +| --------- | ----------------------------- | +| $\ln x-1$ | $\;\;\;\;0\;\;+$ | +| $x^2$ | $+++$ | +| $f'(x)$ | $-\;0\;\;+$ | +| $f(x)$ | $\searrow\rightarrow\nearrow$ | +*Enlight tabellen har $f$ en lokal minimum punkt på $\left(e,-\frac1e\right)$ Punkten är också en global minimum* +*Graf*![[TE1.png]] +**Visa att** $x^{\frac1x}\leq e^{\frac1e}$ +*Lösning: Från ovan:* +$$\begin{align}-\frac{\ln x}x\geq-\frac1e\\\Leftrightarrow\frac{\ln x}x\leq\frac1e\Leftrightarrow\ln x^{\frac1x}\leq\frac1e\Leftrightarrow x^{\frac1x}\leq e^{\frac1e}\\\text{(ty ln är strängt vexande)}\end{align}$$ +**Koraste avtåndet av**: $\left(0,1\right)$ till kurvan $x^2-y^2=1$ +$$\begin{align}\text{Lösn: Avståndet av }\left(0,1\right)\text{ till en punkt }\left(x,y\right)\text{ ges av}\\d)\sqrt{\left(x-0\right)^2+\left(y-1\right)^2}=\sqrt{x^2+y^2-2y+1}\\\text{Punkten }\left(x,y\right)\text{ ligger på kurvan om }x^2-y^2=1\\\text{Avståndet av }\left(0,1\right)\text{ till }\left(x,y\right)\text{ på kurvan är }\\d=\sqrt{1+y^2+y^2-2y+1}=\sqrt{2y^2-2y+2}\\\Rightarrow d^2=2y^2-2y+2\end{align}$$ +*Notera att $d$ och $d^2$ har minimum värde på samma punkt. Definiera* $$\begin{align}f(y)=d^2=2y^2-2y+2\\\text{Derivera m.a.p. }y\\f''(y)=4>0\\\text{Stationär punkt:}\\f'(x)=0\Leftrightarrow4y-2=0\Leftrightarrow y=\frac12\\f''(\frac12)=4>0\\\text{sum: }y=\frac12\text{ ger minimum värde för }f\\\text{sum: avståndet är minst då }y=\frac12\text{ Mista avståndet är}\\d_{min}=\sqrt{s\times\left(\frac12\right)^2-\cancel{2\times\frac12}+2}=\sqrt\frac32\\\text{Närmaste punkten}\\x-\left(\frac12\right)^2=1\Leftrightarrow x^2=\frac54\Leftrightarrow x=\pm\frac{\sqrt5}2\\\text{sum: }\left(-\frac{\sqrt5}2,\frac12\right)\&\left(\frac{\sqrt5}2,\frac12\right)\\\text{Kontroll: }\sqrt{\frac52+\left(\frac12-1\right)^2}=\sqrt{\frac54+\frac14}=+\sqrt\frac32\end{align}$$ \ No newline at end of file