107 lines
4.0 KiB
TeX
107 lines
4.0 KiB
TeX
\documentclass[a4paper,10pt]{article}
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%\documentclass[a4paper,10pt]{scrartcl}
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\input{../Common.tex}
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\begin{document}
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\begin{tabularx}{\textwidth}{ |X|X| }
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\hline
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\textbf{Polynômes de Taylor} \newline
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$\begin{aligned}
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e^x &= \sum\limits_{k=0}^\infty \frac{x^k}{k!}, &x \in \mathbb{R} \\
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\sinh(x) &= \sum\limits_{k=0}^\infty \frac{x^{2k+1}}{(2k+1)!}, &x \in \mathbb{R} \\
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\cosh(x) &= \sum\limits_{k=0}^\infty \frac{x^{2k}}{(2k)!}, &x \in \mathbb{R} \\
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\sin(x) &= \sum\limits_{k=0}^\infty (-1)^k \frac{x^{2k+1}}{(2k+1)!}, &x \in \mathbb{R} \\
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\cos(x) &= \sum\limits_{k=0}^\infty (-1)^k \frac{x^{2k}}{(2k)!}, &x \in \mathbb{R} \\
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\ln(1+x) &= \sum\limits_{k=0}^\infty (-1)^{k+1} \frac{x^k}{k}, &x \in {]-1,1[} \\
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\frac{1}{1+x} &= \sum\limits_{k=0}^\infty (-1)^{k} x^k, &x \in {]-1,1[} \\
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\arctan(x) &= \sum\limits_{k=0}^\infty (-1)^k \frac{x^{2k+1}}{2k+1}, &x \in {]-1,1[} \\
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\end{aligned}$ \newline
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&
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\textbf{Intégrales} \newline
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$\begin{aligned}
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&\int \frac{f'(x)}{f(x)}\mathrm{d}x &&= \ln \left|f(x)\right| + C \\
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&\int \frac{f'(x)}{1+f^2(x)}\mathrm{d}x &&= \arctan \left[f(x)\right] + C \\
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&\int \left[f(x)\right]^\alpha f'(x) \mathrm{d}x &&= \frac{\left[f(x)\right]^{\alpha+1}}{\alpha + 1} + C, &\forall \alpha \neq -1 \\
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&\int e^{f(x)} f'(x) \mathrm{d}x &&= e^{f(x)} + C \\
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&\int \frac{f'(x)}{\sqrt{1-f^2(x)}}\mathrm{d}x &&= \arcsin \left[f(x)\right] + C \\
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\end{aligned}$ \newline
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\\
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\end{tabularx}
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\offinterlineskip
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\begin{tabularx}{\textwidth}{ |X|X|X| }
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\hline
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\textbf{Racine carrée complexe} \newline
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$\begin{aligned}
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w = u + vi, z = a + bi, z^2 = w \\
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\begin{cases}
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a^2 - b^2 &= u \\
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2ab &= v \\
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a^2 + b^2 &= \sqrt{u^2 + v^2} \\
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\end{cases}
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\end{aligned}$
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&
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\textbf{Somme géométrique} \newline
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$\begin{aligned}
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\sum\limits_{k=0}^n q^k &= \frac{1-q^{n+1}}{1-q} \\
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\sum\limits_{k=0}^\infty q^k &= \frac{1}{1-q} \\
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\end{aligned}$ \newline
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&
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\\ \hline
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\textbf{Exponentielle} \newline
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$\begin{aligned}
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\cos(\theta) &= \frac{e^{i\theta} + e^{-i\theta}}{2} \\
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\sin(\theta) &= \frac{e^{i\theta} - e^{-i\theta}}{2i} \\
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\cosh(\theta) &= \frac{e^{\theta} + e^{-\theta}}{2} \\
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\sinh(\theta) &= \frac{e^{\theta} - e^{-\theta}}{2} \\
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\end{aligned}$ \newline
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&
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\textbf{Exponentielle} \newline
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$\begin{aligned}
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\lim_{n \to \infty} \left(1 + \frac{L}{n}\right)^n &= e^L \\
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\text{De manière\ générale :} \\
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\lim_{x \to \infty} f(x) &= +\infty \\
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\lim_{x \to \infty} f(x)h(x) &= L \\
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\lim_{x \to \infty} \left[1 + h(x)\right]^{f(x)} &= e^L \\
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\end{aligned}$ \newline
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&
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\textbf{Trigonométrie} \newline
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$\begin{aligned}
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\cosh^2(x) - \sinh^2(x) = 1 \\
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\cos^2(x) + \sin^2(x) = 1 \\
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\sin(x+y) = \sin x \cos y + \cos x \sin y \\
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\cos(x+y) = \cos x \cos y + \sin x \sin y \\
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\sin x + \sin y = 2 \sin(\frac{x+y}{2})\cos(\frac{x-y}{2}) \\
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\sin x - \sin y = 2 \sin(\frac{x-y}{2})\cos(\frac{x+y}{2}) \\
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\cos x + \cos y = 2 \cos(\frac{x+y}{2})\cos(\frac{x-y}{2}) \\
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\cos x - \cos y = -2 \sin(\frac{x+y}{2})\sin(\frac{x-y}{2}) \\
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\end{aligned}$ \newline
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\\ \hline
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\multicolumn{2}{|X|}{
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\textbf{Angles particuliers} \newline
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$\begin{aligned}
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\cos(0) = 1 \quad &\cos(\frac{\pi}{6}) = \frac{1}{2} \quad &\cos(\frac{\pi}{4}) = \frac{1}{2}\sqrt{2} \quad &\cos(\frac{\pi}{3}) = \frac{1}{2}\sqrt{3} \quad &\cos(\frac{\pi}{2}) = 0 \\
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\sin(0) = 0 \quad &\sin(\frac{\pi}{6}) = \frac{1}{2}\sqrt{3} \quad &\sin(\frac{\pi}{4}) = \frac{1}{2}\sqrt{2} \quad &\sin(\frac{\pi}{3}) = \frac{1}{2} \quad &\sin(\frac{\pi}{2}) = 1 \\
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\end{aligned}$ \newline
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}
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&
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\\ \hline
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\multicolumn{2}{|X|}{
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\textbf{Convergence} \newline
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$\begin{aligned}
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&\int_M^\infty x^a e^{-bx} \mathrm{d}x &\text{ converge pour tout } a \in \mathbb{R} \text{ et tout } b > 0 \\
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&\int_a^\infty \frac{1}{x^p} \mathrm{d}x &\text{ converge si et seulement si } p > 1 \quad (a > 0) \\
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&\int_0^b \frac{1}{x^p} \mathrm{d}x &\text{ converge si et seulement si } p < 1 \\
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\end{aligned}$ \newline
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}
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&
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\\ \hline
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\end{tabularx}
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\end{document} |