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+\pagestyle{empty} + +\begin{document} + +\maketitle + +\setcounter{section}{3} +\section{Approximation de l'aire sous la courbe} + +Dans la pratique, calculer l'aire sous la courbe d'une fonction est une tache difficile. Quand les fonctions sont trop compliquées, on n'a d'autres choix que d'approximer l'aire sous la courbe. Ici on trouve un encadrement de cette quantité. + +\bigskip +\begin{tabular}{ccccc} + \begin{tikzpicture}[xscale=0.5, yscale=0.5] + \tkzInit[xmin=6,xmax=13, + ymin=0,ymax=250,ystep=50] + \tkzGrid + \tkzDrawX[label={\textit{Heure}},below= -18pt] + \tkzLabelX + \tkzDrawY[label={\textit{P}}] + \tkzLabelY + \tkzClip + \tkzFct[thick,color=red,domain=0:24]{1000*exp(-((\x-13)/2)**2/2)/(2*sqrt(2*pi))} + \draw[thick, fill=green, opacity=0.3] (1,0) -- (1,0.04) -- (2,0.04) -- (2,0) --cycle; + \draw[thick, fill=green, opacity=0.3] (2,0) -- (2,0.2) -- (3,0.2) -- (3,0) --cycle; + \draw[thick, fill=green, opacity=0.3] (3,0) -- (3,0.5) -- (4,0.5) -- (4,0) --cycle; + \draw[thick, fill=green, opacity=0.3] (4,0) -- (4,1.3) -- (5,1.3) -- (5,0) --cycle; + \draw[thick, fill=green, opacity=0.3] (5,0) -- (5,2.4) -- (6,2.4) -- (6,0) --cycle; + \draw[thick, fill=green, opacity=0.3] (6,0) -- (6,3.5) -- (7,3.5) -- (7,0) --cycle; + \end{tikzpicture} + && + \begin{tikzpicture}[xscale=0.5, yscale=0.5] + \tkzInit[xmin=6,xmax=13, + ymin=0,ymax=250,ystep=50] + \tkzGrid + \tkzDrawX[label={\textit{Heure}},below= -18pt] + \tkzLabelX + \tkzDrawY[label={\textit{P}}] + \tkzLabelY + \tkzClip + \tkzFct[thick,color=red,domain=0:24]{1000*exp(-((\x-13)/2)**2/2)/(2*sqrt(2*pi))} + \tkzDrawArea[pattern=north west lines,domain =6:13] + \end{tikzpicture} + && + \begin{tikzpicture}[xscale=0.5, yscale=0.5] + \tkzInit[xmin=6,xmax=13, + ymin=0,ymax=250,ystep=50] + \tkzGrid + \tkzDrawX[label={\textit{Heure}},below= -18pt] + \tkzLabelX + \tkzDrawY[label={\textit{P}}] + \tkzLabelY + \tkzClip + \tkzFct[thick,color=red,domain=0:24]{1000*exp(-((\x-13)/2)**2/2)/(2*sqrt(2*pi))} + \draw[thick, fill=green, opacity=0.3] (0,0) -- (0,0.04) -- (1,0.04) -- (1,0) --cycle; + \draw[thick, fill=green, opacity=0.3] (1,0) -- (1,0.2) -- (2,0.2) -- (2,0) --cycle; + \draw[thick, fill=green, opacity=0.3] (2,0) -- (2,0.5) -- (3,0.5) -- (3,0) --cycle; + \draw[thick, fill=green, opacity=0.3] (3,0) -- (3,1.3) -- (4,1.3) -- (4,0) --cycle; + \draw[thick, fill=green, opacity=0.3] (4,0) -- (4,2.4) -- (5,2.4) -- (5,0) --cycle; + \draw[thick, fill=green, opacity=0.3] (5,0) -- (5,3.5) -- (6,3.5) -- (6,0) --cycle; + \draw[thick, fill=green, opacity=0.3] (6,0) -- (6,4) -- (7,4) -- (7,0) --cycle; + \end{tikzpicture} + \\ + $400$ & < & $\displaystyle \int_6^{13} f(x)dx$ & < & $600$ +\end{tabular} + +\bigskip +Les valeurs obtenues ont été trouvée en sommant les aires des rectangles. +\bigskip + +Pour obtenir, une approximation plus précise, on peut diminuer la largeur des rectangles. C'est de cette façon, que l'on définir proprement l'intégrale que l'on nomme \textbf{intégrale de Riemman} (ce terme n'est pas au programme). + + + +\end{document} diff --git a/TST_sti2d/01_Aire_sous_la_courbe/3E_approximation.pdf b/TST_sti2d/01_Aire_sous_la_courbe/3E_approximation.pdf new file mode 100644 index 0000000000000000000000000000000000000000..fb17e7e375b4f902805aac0aec8aa817a9e32baf GIT binary patch literal 44343 zcmce-b97`+_x~H)wr$(CJ+W;k6Wg|JXJSumO>Apo>zDc9-tWb__xI29^jh8f)UJKb zslBSY`qcZ?MXDeoM$1UY3Pn1zFt`lGLcl;^XJ`qK(8ThXlY{P3`MW( zV(9#rN6gOA28v!t$j+VM8^c#TBg0>B9Vq%gHGiGO2t_Yz;OuDe6~Xuw@m2rl^D9Eh z#L3RZ(a6M!fc49x1Vt}nVr%AXPQb`Y@Xzb(Di+qxCXNL3V%7%ECL$(AcE%=9e0)$& z&WB> zc1{LbG13y?yr$j~KqQ~z?XB(Tg&PkOGnt_-ax5i0xb?(ielLi>#w*K98^?zyY{0Gu zdA^6-qF@HxphfspM zQ816Ey^aCj&G$n)yQFTf&*$hnH#_{zZU?u^Eu7?3x0^}d_v_CY`fryq1jBOD7pHn3 zCkL^|ExxzImmfE{iy7zdtRqMkwV?>RzCADk1^mrW2Ha56@Fz4 zHQikf58!vdepqe%c*yY)i++6XU?1DuC|#3df1}%af`7Do+eoFiDz%(j 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