gistfile1.tex

\documentclass[8pt, a4paper]{extarticle}

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\usepackage[fleqn]{amsmath}
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\newcommand{\arctg}{\textup{arctg}}
\newcommand{\arccotg}{\textup{arccotg}}
\newcommand{\diff}{\textup{d}}
\newcommand{\reals}{\mathbb{R}}


\begin{document}
\begin{multicols}{3}
\textbf{Goniometrické identity}
\begin{align*}
\sin x = -\sin(-x) \\
\cos x = \cos(-x) \\
\sin(x+\pi)=-\sin(x) \\
\cos(x+\pi)=-\cos(x) \\
\sin\left(x+\frac{\pi}{2}\right)=\cos x \\
\cos\left(x+\frac{\pi}{2}\right)=-\sin x \\
\sin^2x+\cos^2x=1 \\
\sin(x+y)=\sin x\cos y + \cos x\sin y \\
\cos(x+y)=\cos x\cos y - \sin x\sin y \\
\sin x + \sin y = 2\sin\left(\frac{x+y}{2}\right)\cos\left(\frac{x-y}{2}\right) \\
\sin x - \sin y = 2\cos\left(\frac{x+y}{2}\right)\sin\left(\frac{x-y}{2}\right) \\
\cos x + \cos y = 2\cos\left(\frac{x+y}{2}\right)\cos\left(\frac{x-y}{2}\right) \\
\cos x - \cos y = -2\sin\left(\frac{x+y}{2}\right)\sin\left(\frac{x-y}{2}\right) \\
\sin^2x=\frac{1}{2}(1-\cos(2x)) \\
\cos^2x=\frac{1}{2}(1+\cos(2x)) \\
\arccotg x = \arctg\left(\frac{1}{x}\right)
\end{align*}

\textbf{Exponenciela a logaritmus}
\begin{align*}
x^y=e^{y\log x} \\
\log_yx=\frac{\log x}{\log y} \\
\end{align*}

\textbf{Taylorova řada}
Pro $f(x)$ v $a$:\\ $\sum\limits_{n=0}^\infty \frac{f^{(n)}(a)}{n!}(x-a)^n$\\

\textbf{Limity}
\begin{align*}
\lim\limits_{x\rightarrow\infty} \sqrt[x]{x} = 1 \\
\lim\limits_{x\rightarrow 0} \frac{\sin x}{x} = 1 \\
\lim\limits_{x\rightarrow 0} \frac{1 - \cos x}{x^2} = 1 \\
\lim\limits_{x\rightarrow 0} \frac{e^x}{x} = 1 \\
\lim\limits_{x\rightarrow 0} \frac{\log(x+1)}{x} = 1 \\
\lim\limits_{x\rightarrow 1} \frac{\log(x)}{x-1} = 1 \\
e^x=\lim_{x\to\infty}(1+\frac{x}{n})^n \\
a>0, b\in \mathbb{R}: \lim\limits_{x\rightarrow \infty} x^a\log^bx = \infty \\
a<0, b\in \mathbb{R}: \lim\limits_{x\rightarrow \infty} x^a\log^bx = 0 \\
a>0, b\in \mathbb{R}: \lim\limits_{x\rightarrow 0} x^a\log^bx = 0 \\
a<0, b\in \mathbb{R}: \lim\limits_{x\rightarrow 0} x^a\log^bx = \infty \\
a>0, b\in \mathbb{R}: \lim\limits_{x\rightarrow \infty} \frac{e^{ax}}{x^b} = \infty \\
a<0, b\in \mathbb{R}: \lim\limits_{x\rightarrow \infty} \frac{e^{ax}}{x^b} =  \\
\end{align*}

\textbf{Derivace}
\begin{align*}
(f+g)'(a)=f'(a)+g'(a)\\
(f\cdot g)'(a)=f'(a)g(a)+f(a)g'(a) \\
\left(\frac{f}{g}\right)'(a)=\frac{f'(a)g(a)-f(a)g'(a)}{g^2(a)} \\
(f^{-1})'(f(a))=\frac{1}{f'(a)} \\
(f\cdot g)^{(n)}(a)=\sum\limits_{k=0}^n \binom{n}{k}f^{(k)}(x)g^{(n-k)}(a) \\
f'(a):=\lim\limits_{h\to 0}\frac{f(a+h)-f(a)}{h} \\
\end{align*}

\textbf{Tabulkové derivace}
\begin{align*}
(x^k)'=kx^{k-1}\\
(e^x)'=e^x\\
(\log|x|)'=\frac{1}{x}\\
(a^x)'=a^x \log a\\
(\log_a x)'=\frac{1}{x\log a}\\
(\sin x)'= \cos x\\
(\cos x)'= -\sin x\\
(\text{tg } x)'= \frac{1}{\cos^2x}\\
(\text{cotg }x)'=\frac{1}{\sin^2x}\\
(\arcsin x)'=\frac{1}{\sqrt{1-x^2}}\\
(\arccos x)'=\frac{-1}{\sqrt{1-x^2}}\\
(\text{arctg } x)'=\frac{1}{1+x^2}\\
(\text{arccotg }x)'=\frac{-1}{1+x^2} 
\end{align*}

\textbf{Limity posloupnosti}
\begin{align*}
\frac{a_{n+1}}{a_n}<1 \Rightarrow K
\end{align*}

\textbf{Průběh funkce}
\begin{enumerate}
\item $D(f), H(f)$
\item Limity v krajních bodech
\item Derivace
\item Monotonie
\item Extrémy: $f'(x)=0$
\item Druhá derivace
\item Konvexita $f''(x)\ge0$, konkávnost $f''(x)\le0$
\item Inflexní body
\item Asymptoty $y=kx+q \Leftrightarrow 
\lim\limits_{x \rightarrow\infty}\frac{f(x)}{x}=k,
\lim\limits_{x \rightarrow\infty}(f(x)-kx)=q$
\item Graf
\end{enumerate}


\textbf{Řady}

Nutná podmínka: $\sum a_n < \infty \Rightarrow a_n\rightarrow0$ \\
D'Alambert: $\forall a_n>0: \frac{a_{n+1}}{a_n}<1 \Rightarrow \\ \sum a_n < \infty$ \\
Limitně: $\forall a_n \ge 0, b_n >0: c:=\lim\limits_{x \rightarrow\infty}\frac{a_n}{b_n}$
\begin{enumerate}
\item $0<c<\infty: \sum a_n < \infty \Leftrightarrow \sum b_n < \infty$
\item $c=0: \sum b_n < \infty \Rightarrow \sum a_n < \infty$
\item $c=\infty: \sum b_n = \infty \Rightarrow \sum a_n = \infty$
Cauchy: $\lim\limits_{n \rightarrow\infty}\sqrt[n]{a_n}\le 1 \Rightarrow \sum a_n < \infty$
$\lim\limits_{n \rightarrow\infty}\sqrt[n]{a_n}\ge 1 \Rightarrow \sum a_n = \infty$\\
Leibnitz (neabsolutní): $a_n\rightarrow 0 \Rightarrow \sum (-1)^{n+1}a_n\le \infty$
\end{enumerate}


\textbf{Integrálky}\\
Ultimátní goniometrická substituce:
\begin{itemize}
\item $ y=\tan\left(x/2\right) $
\item $ dx=\frac{2dy}{1+y^2} $
\item $ \cos x = \frac{1-y^2}{1+y^2} $
\item $ \sin x = \frac{2y}{1+y^2} $
\end{itemize}
\begin{align*}
\int\limits_{0}^{\pi}\sin^2x=\frac{\pi}{2}\\
\int u\diff v = uv-\int v \diff u
\end{align*}
Aplikace
\begin{itemize}
\item $l(\phi)=\int\limits_{a}^{b}\sqrt{x'(t)+y'(t)+...}\diff x$
\item $S=\int\limits_{a}^{b}(g-f)\diff x$
\item $V= \pi \int\limits_{a}^{b}f^2(x)\diff x$
\end{itemize}
Konvergence
\begin{align*}
\int\limits_{a}^{b}\frac{1}{(b-x)^p} K \Leftrightarrow p < 1 \\
\int\limits_{a}^{b}\frac{1}{(x-a)^p} K \Leftrightarrow p < 1
\end{align*}


\textbf{Více proměnných}\\
Parciální derivace jsou zaměnitelné pouze na spojitém okolí\\
Diferenciál v $a$: $L(h_1,...,h_n)=\frac{\partial f}{\partial x_1}(a)h_1+...+\frac{\partial f}{\partial x_n}(a)h_n$\\
$\lim\limits_{h\to 0}\frac{f(a+h)-f(a)-L(h)}{||h||}=0$\\
Tečná nadrovina v $a$: $x_{n+1}=f(a)+\frac{\partial f}{\partial x_1}(a)(x_1-a_1)+...+\frac{\partial f}{\partial x_n}(a)(x_n-a_n)$\\
Směrová derivace: $D_vf(a)=\lim\limits_{t\to 0}\frac{f(a+tv)-f(a)}{t}$\\
$\nabla f(a)=\left(\frac{\partial f}{\partial x_1}(a), ..., \frac{\partial f}{\partial x_n}(a)\right)$ \\
$f: \reals^n\to\reals^m, g: \reals^m\to\reals^s: (g\circ f)'(a)=g'(f(a))\circ f'(a)$\\
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\end{multicols}
\end{document}