diff --git a/.obsidian/workspace.json b/.obsidian/workspace.json index 55eab12..f9318fb 100644 --- a/.obsidian/workspace.json +++ b/.obsidian/workspace.json @@ -34,9 +34,37 @@ "icon": "lucide-file", "title": "Funktioner Forts" } + }, + { + "id": "54baa3edd65a7c5d", + "type": "leaf", + "state": { + "type": "markdown", + "state": { + "file": "Grafer.md", + "mode": "source", + "source": false + }, + "icon": "lucide-file", + "title": "Grafer" + } + }, + { + "id": "4eef5f8feb086f9e", + "type": "leaf", + "state": { + "type": "markdown", + "state": { + "file": "Trigonometri.md", + "mode": "source", + "source": false + }, + "icon": "lucide-file", + "title": "Trigonometri" + } } ], - "currentTab": 1 + "currentTab": 3 } ], "direction": "vertical" @@ -93,7 +121,8 @@ } ], "direction": "horizontal", - "width": 300 + "width": 300, + "collapsed": true }, "right": { "id": "b700e0cd0f882a5c", @@ -195,14 +224,15 @@ "obsidian-git:Open Git source control": false } }, - "active": "66704e0159322e3f", + "active": "4eef5f8feb086f9e", "lastOpenFiles": [ - "g2.png", - "Grafer.md", - "g1.png", "Funktioner Forts.md", - "f_inverse.png", + "Trigonometri.md", + "Grafer.md", "Funktioner.md", + "f_inverse.png", + "g2.png", + "g1.png", "Untitled.canvas" ] } \ No newline at end of file diff --git a/Funktioner Forts.md b/Funktioner Forts.md index 3c8cc16..cea38c3 100644 --- a/Funktioner Forts.md +++ b/Funktioner Forts.md @@ -36,3 +36,19 @@ - $f(x)=x^2,\;x\in[0,1]$ $D_f=[0,1]$ - ![[g2.png]] - $$\begin{align}f(x)=3x+5\\g(x)=\frac{x-5}{3}\end{align}$$ +- Exponential och logarithm + - Exponential: $f(x)=a^x$ för något $a>0$. + - Logaritm: $g(x)=\log_a(x)$ för något $a>0$ + - $f$ och $g$ inverse till varandra: $y=a^x\Leftrightarrow\log_a(y)=x$. + - $D_f=\mathbb{R}=V_g,\;\;V_f=(0,\infty)=D_g$. + - Om $a>1,\;f,\;g$ är strängt växande. + - $\log_a{(xy)}=\log_a(x)+\log_a(y),\;\log_a(x/y)=\log_a(x)-\log_a(y)$ + - $\log_a(x^b)=b\log_a(x)$ + - Basbyte: $\log_a(x)=\frac{\log_b(x)}{\log_b(a)}\Leftrightarrow\log_b(x)=\log_b(a)\log_a(x)$. $a^x=b^{x\log_b(a)}$ + - Ex: $$\begin{align}\text{Räkna }D_f\text{ för }f(x)=\log_{10}(x^2+2x-3)\\f\text{ är definierad för }x^2+2x-3>0\\\Leftrightarrow(x+3)(x-1)>0\\\Leftrightarrow x\in(-\infty,-3)\cup(1,\infty)\\D_f=(-\infty,-3)\cup(1,\infty)\\\\2^{x+3}>4\\\Leftrightarrow\log_2(2^{x+3})>\log_24\\\Leftrightarrow x+3>2\\\Leftrightarrow x>-1\\\\\log_{10}36\\=\log_{10}(2^2\times3^2)\\=\log_{10}(2^2)+\log_{10}(3^2)\\=2\log_{10}2+2\log_{10}3\\\\2^x=e^{x\log_e2}=e^{x\ln2}\\\log_2x=(\log_2e)\ln{x}\\=\frac{\ln x}{\ln 2}\end{align}$$ + - **Def**: $\log{x}=\log_{10}x$ + - **Def**: $\ln{x}=\log_ex$ + - **Def**: $a^x=e^{x\log_ea}=e^{x\ln a},\;a\in(0,\infty)$ + - **Def**: $\log_a1=0$ + - **Def**: $\log_aa=1$ + - **Def**: $\log_ab=\frac{1}{\log_ba}$ \ No newline at end of file diff --git a/Trigonometri.md b/Trigonometri.md new file mode 100644 index 0000000..67d393a --- /dev/null +++ b/Trigonometri.md @@ -0,0 +1,25 @@ +- Radian: + - **Def**: *It is the SI unit for measuring angles (in the plane).* + - **Def**: *$1$ radian is defined as the angle subtended at the center by a circular arc of length equal to the radius* + - **Def**: *A general angle is measured in radians as the ration of the length an associated circular arc and the corresponding radius. That is $\theta=\frac{s}{r}\text{rad}$* + - **Def**: *Usually "$rad$" is omitted.* + - Ex: $$\begin{align}180^\circ=\pi\text{ rad}\\\frac{\pi}{3}\text{ rad}=30^\circ\\\frac{\pi}{4}\text{ rad}=45^\circ\\\frac{\pi}{3}\text{ rad}=60^\circ\\\frac{\pi}{2}\text{ rad}=90^\circ\\2\pi\text{ rad}=360^\circ\end{align}$$ +- The right angled triangle + - **Def**: *The trigonometric functions: *$$\begin{align}\sin\theta=\frac{\text{perpendicular}}{\text{hypotenuse}}\\\cos\theta=\frac{\text{base}}{\text{hypotenuse}}\\\tan\theta=\frac{\text{perpendicular}}{\text{base}}\end{align}$$ + - Dominains and ranges: + - $D_{\sin}=\mathbb{R}\;\;R_{\sin}=[-1,1]$ + - $D_{\cos}=\mathbb{R}\;\;R_{\cos}=[-1,1]$ + - $D_{\tan}=\mathbb{R}\setminus\{n\pi+\frac{\pi}{2}:n\in\mathbb{Z}\}\;\;R_{\tan}=(-\infty,\infty)$ +- Useful relations + - $\sin(-\theta)=-\sin(\text{odd}),\cos(-\theta)=\cos\theta(\text{even})$ + - Periodicity: $\sin(\theta+2n\pi)=\sin\theta,\cos(\theta+2n\pi)=\cos\theta,\tan(\theta+n\pi)=\tan\theta$ + - Complementary angles: $\sin(\frac{\pi}{2}-\theta)=\cos\theta,\cos(\frac{\pi}{2}-\theta)=\sin\theta$ + - Sift by $\pi$: $\sin(\theta\pm\pi)=-\sin\theta,\cos(\theta\pm\pi=-\cos\theta$ + - Sum of angles: $\sin(\theta+\phi)=\sin\theta\times\cos\phi+\cos\theta\times\sin\phi,\cos(\theta+\phi)=\cos\theta\times\cos\phi-\sin\theta\times\sin\phi,\tan(\theta+\phi)=\frac{\tan\theta+\tan\phi}{1-\tan\theta\times\tan\phi}$ + - Double angle: $\sin(2\theta)=2\sin\theta\cos\theta,\tan2\theta=\frac{2\tan\theta}{1-\tan^2\theta},\cos(2\theta)=\cos^2-\sin^2\theta=2\cos^2\theta-1=1-2\sin^2\theta$ + - Half angle: $2\sin^2\frac{\theta}{2}=1-\cos\theta,2\cos^2\frac{\theta}{2}=1+\cos\theta$ +- Solving trigonometric equations + - $\sin\theta=\sin{a}\Leftrightarrow\theta=\left\{\begin{align}a+2n\pi,n\in\mathbb{Z}\\\pi-a+2n\pi,n\in\mathbb{Z}\end{align}\right.$ + - $\cos\theta=\cos{a}\Leftrightarrow\theta=\left\{\begin{align}a+2n\pi,n\in\mathbb{Z}\\-a+2n\pi,n\in{Z}\end{align}\right.$ + - $\tan\theta=\tan{a}\Leftrightarrow\theta=a+n\pi,n\in\mathbb{Z}$ + - Ex: Solve $\sin(x+\frac{\pi}{6})=\frac{\sqrt{3}}{2}$ \ No newline at end of file