Make coin par coeur fit into two pages
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2 changed files with 57 additions and 45 deletions
100
analyse/main.tex
100
analyse/main.tex
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@ -40,7 +40,7 @@
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\paragraph{Dérivées et Primitives}
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Usuelles~:
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\begin{multicols}{2}
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\begin{tabularx}{\linewidth}{YY}
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\toprule
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@ -82,48 +82,46 @@
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\bottomrule
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\end{tabularx}
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Composées~:
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\columnbreak
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\begin{multicols}{2}
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\begin{tabularx}{\linewidth}{lY}
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\toprule
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\multirow{2}{*}{Linéarité} & $(u + v)' = u' + v'$ \\
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& $(au)' = au'$ \\
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\midrule
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Produit & $(uv)' = u'v + uv'$ \\
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\midrule
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Inverse & $\left(\frac{1}{v}\right)' = - \frac{v'}{v^2}$ \\
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\midrule
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Quotient & $\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}$ \\
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\midrule
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Composée & $(f(u))' = u'f'(u)$ \\
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\bottomrule
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\end{tabularx}
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\begin{tabularx}{\linewidth}{lY}
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\toprule
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\multirow{2}{*}{Linéarité} & $(u + v)' = u' + v'$ \\
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& $(au)' = au'$ \\
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\midrule
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Produit & $(uv)' = u'v + uv'$ \\
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\midrule
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Inverse & $\left(\frac{1}{v}\right)' = - \frac{v'}{v^2}$ \\
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\midrule
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Quotient & $\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}$ \\
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\midrule
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Composée & $(f(u))' = u'f'(u)$ \\
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\bottomrule
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\end{tabularx}
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\begin{tabularx}{\linewidth}{YY}
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\toprule
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Fonction & Primitive \\
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\toprule
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$u'u^n$ & $\frac{u^{n+1}}{n+1}$ \\
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\midrule
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$\frac{u'}{u^2}$ & $-\frac{1}{u}$ \\
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\midrule
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$\frac{u'}{\sqrt{u}}$ & $2\sqrt{u}$ \\
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\midrule
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$u'\cos{u}$ & $\sin{u}$ \\
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\midrule
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$u'\sin{u}$ & $-\cos{u}$ \\
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\midrule
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$\frac{u'}{u}$ & $\ln{|u|}$ \\
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\midrule
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$u'e^u$ & $e^u$ \\
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\midrule
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$\frac{u'}{1 + u^2}$ & $\arctan{u}$ \\
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\bottomrule
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\end{tabularx}
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\begin{tabularx}{\linewidth}{YY}
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\toprule
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Fonction & Primitive \\
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\toprule
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$u'u^n$ & $\frac{u^{n+1}}{n+1}$ \\
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\midrule
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$\frac{u'}{u^2}$ & $-\frac{1}{u}$ \\
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\midrule
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$\frac{u'}{\sqrt{u}}$ & $2\sqrt{u}$ \\
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\midrule
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$u'\cos{u}$ & $\sin{u}$ \\
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\midrule
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$u'\sin{u}$ & $-\cos{u}$ \\
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\midrule
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$\frac{u'}{u}$ & $\ln{|u|}$ \\
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\midrule
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$u'e^u$ & $e^u$ \\
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\midrule
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$\frac{u'}{1 + u^2}$ & $\arctan{u}$ \\
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\bottomrule
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\end{tabularx}
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\end{multicols}
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\end{multicols}
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\paragraph{Intégrales}
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$\int_a^b f(x)\dif x = [F(x)]_a^b = F(b) - F(a)$
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@ -136,7 +134,6 @@
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\bottomrule
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\end{tabularx}
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\paragraph{Équations différentielles}
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\begin{tabularx}{\linewidth}{lllc}
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@ -147,11 +144,26 @@
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\midrule
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\multicolumn{2}{l}{$ay' + by = f(x)$} & $y_0 + \lambda e^{rx} \quad \text{ avec } r = \frac{-b}{a}$ & \makecell{$y_0$ solution particulière de \\ $ay' + by = f(x)$ \\ $f$ une fonction et $a, b, \lambda\in\mathbb{R}$} \\
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\midrule
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\multirow{3}{*}{$ay'' + by' + cy = 0$} & $\Delta > 0$ & $\lambda e^{r_1 x} + \mu e^{r_2 x} + y_1$ & \multirowcell{3}[0pt][c]{$\lambda, \mu \in \mathbb{R}$, $y_1$ solution particulière \\ $\alpha = \frac{-b}{2a} \quad \beta = \frac{\sqrt{|\Delta|}}{2a}$} \\
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\multirow{3}{*}{$ay'' + by' + cy = 0$} & $\Delta > 0$ & $\lambda e^{r_1 x} + \mu e^{r_2 x}$ & \multirowcell{3}[0pt][c]{$\lambda, \mu \in \mathbb{R}$ \\ $\alpha = \frac{-b}{2a} \quad \beta = \frac{\sqrt{|\Delta|}}{2a}$} \\
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\cline{2-3}
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& $\Delta = 0$ & $(\lambda x + \mu) e^{r_0 x} + y_1$ & \\
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& $\Delta = 0$ & $(\lambda x + \mu) e^{r_0 x}$ & \\
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\cline{2-3}
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& $\Delta < 0$ & $e^{\alpha x}(\lambda\cos{(\beta x)} + \mu\sin{(\beta x)}) + y_1$ & \\
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& $\Delta < 0$ & $e^{\alpha x}(\lambda\cos{(\beta x)} + \mu\sin{(\beta x)})$ & \\
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\bottomrule
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\end{tabularx}
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\paragraph{Solutions particulières des équations différentielles de 2\up{nd} ordre}
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\begin{tabularx}{\linewidth}{XX}
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\toprule
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\multicolumn{2}{l}{Second membre du type $e^{\alpha x}P(x)$} \\
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$\alpha$ non racine & $y_1 = e^{\alpha x} Q(x)$ \\
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$\alpha$ racine simple & $y_1 = x e^{\alpha x} Q(x)$ \\
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$\alpha$ racine double & $y_1 = x^2 e^{\alpha x} Q(x)$ \\
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\midrule
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\multicolumn{2}{l}{Second membre du type $e^{\alpha x}(P_1(x)\cos(\beta x) + P_2(x)\sin(\beta x))$} \\
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$\alpha + i\beta$ non racine & $y_1 = e^{\alpha x}(Q_1(x)\cos(\beta x) + Q_2(x)\sin(\beta x))$ \\
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$\alpha + i\beta$ racine & $y_1 = x e^{\alpha x}(Q_1(x)\cos(\beta x) + Q_2(x)\sin(\beta x))$ \\
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\bottomrule
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\end{tabularx}
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@ -2,7 +2,7 @@
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\usepackage[
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%showframe,
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a4paper,includeheadfoot,margin=2cm
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a4paper,includeheadfoot,margin=2cm,top=1cm,bottom=1cm
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]{geometry}
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\setcounter{tocdepth}{2}
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