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D\[EAcute]signons-les par ", StyleBox["a", FontSlant->"Italic"], ", ", StyleBox["b", FontSlant->"Italic"], ", ", StyleBox["c", FontSlant->"Italic"], " et notons \[CapitalDelta]", StyleBox["a", FontSlant->"Italic"], ", \[CapitalDelta]", StyleBox["b", FontSlant->"Italic"], ", \[CapitalDelta]", StyleBox["c", FontSlant->"Italic"], " les incertitudes affectant ces mesures. Pour des incertitudes petites \ compar\[EAcute]es aux valeurs mesur\[EAcute]es, nous ne commettons qu'une tr\ \[EGrave]s petite erreur si nous rempla\[CCedilla]ons \ l\[CloseCurlyQuote]accroissement total de la fonction par sa \ diff\[EAcute]rentielle (voir document \[LeftGuillemet] Calcul d'erreur \ \[RightGuillemet]). Dor\[EAcute]navant, nous utiliserons donc, pour calculer \ l'incertitude qui affecte un r\[EAcute]sultat, la diff\[EAcute]rentielle de \ la fonction qui lie ce r\[EAcute]sultat aux mesures. Pour les mesures que \ vous avez effectu\[EAcute]es, ces fonctions sont les suivantes :\na) la \ longueur ", StyleBox["l", FontSlant->"Italic"], " des ar\[EHat]tes du parall\[EAcute]lipip\[EGrave]de est donn\[EAcute]e \ par ", StyleBox["l", FontSlant->"Italic"], " = 4", StyleBox["a", FontSlant->"Italic"], " + 4", StyleBox["b", FontSlant->"Italic"], " + 4", StyleBox["c\n", FontSlant->"Italic"], StyleBox["b) la surface ", FontVariations->{"CompatibilityType"->0}], StyleBox["S", FontSlant->"Italic", FontVariations->{"CompatibilityType"->0}], StyleBox[" ", FontVariations->{"CompatibilityType"->0}], "du parall\[EAcute]lipip\[EGrave]de est donn\[EAcute]e par ", StyleBox["S", FontSlant->"Italic"], " = 2", StyleBox["ab", FontSlant->"Italic"], " + 2", StyleBox["ac", FontSlant->"Italic"], " + 2", StyleBox["bc\n", FontSlant->"Italic"], StyleBox["c) le volume ", FontVariations->{"CompatibilityType"->0}], StyleBox["V", FontSlant->"Italic", FontVariations->{"CompatibilityType"->0}], StyleBox[" ", FontVariations->{"CompatibilityType"->0}], "du parall\[EAcute]lipip\[EGrave]de est donn\[EAcute] par ", StyleBox["V", FontSlant->"Italic"], " = ", StyleBox["abc", FontSlant->"Italic"], ".\n\n", StyleBox["Utilisation du logiciel ", FontWeight->"Bold"], StyleBox["Mathematica ", FontWeight->"Bold", FontSlant->"Italic"], "(ATTENTION, c'est \[AGrave] vous d'exprimer correctement les \ r\[EAcute]sultats !)\nD\[EAcute]finissons chacune de ces fonctions et \ utilisons sa diff\[EAcute]rentielle totale pour exprimer l'incertitude sur \ les r\[EAcute]sultats :" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(l[a_, b_, c_] := 4 \((a + b + c)\)\), "\[IndentingNewLine]", \(\[CapitalDelta]l = Abs[D[l[a, b, c], a]] \[CapitalDelta]a + Abs[D[l[a, b, c], b]] \[CapitalDelta]b + Abs[D[l[a, b, c], c]] \[CapitalDelta]c\)}], "Input"], Cell[BoxData[ \(TraditionalForm\`4\ \[CapitalDelta]a + 4\ \[CapitalDelta]b + 4\ \[CapitalDelta]c\)], "Output"] }, Open ]], Cell[TextData[{ "Nous constatons que la diff\[EAcute]rentielle totale fournit une \ expression identique \[AGrave] celle donn\[EAcute]e par la r\[EGrave]gle pour \ l'addition (voir document \[LeftGuillemet] Erreur et incertitude \ \[RightGuillemet]). Calculons la longueur ", StyleBox["l", FontSlant->"Italic"], " et l' incertitude \[CapitalDelta]", StyleBox["l ", FontSlant->"Italic"], "pour des valeurs ", StyleBox["a", FontSlant->"Italic"], ", ", StyleBox["b", FontSlant->"Italic"], ", ", StyleBox["c", FontSlant->"Italic"], ", \[CapitalDelta]", StyleBox["a", FontSlant->"Italic"], ", \[CapitalDelta]", StyleBox["b", FontSlant->"Italic"], ", \[CapitalDelta]", StyleBox["c", FontSlant->"Italic"], " :" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(l[a, b, c] /. {a \[Rule] 10.2, \[CapitalDelta]a \[Rule] 0.1, b \[Rule] 7.7, \[CapitalDelta]b \[Rule] 0.08, c \[Rule] 3.17, \[CapitalDelta]c \[Rule] 0.04}\), "\[IndentingNewLine]", \(\[CapitalDelta]l /. {a \[Rule] 10.2, \[CapitalDelta]a \[Rule] 0.1, b \[Rule] 7.7, \[CapitalDelta]b \[Rule] 0.08, c \[Rule] 3.17, \[CapitalDelta]c \[Rule] 0.04}\)}], "Input"], Cell[BoxData[ \(84.28`\)], "Output"], Cell[BoxData[ \(0.88`\)], "Output"] }, Open ]], Cell["Proc\[EAcute]dons de m\[EHat]me pour la surface :", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(s[a_, b_, c_] := 2 \((a*b + a*c + b*c)\)\), "\[IndentingNewLine]", \(\[CapitalDelta]s = Abs[D[s[a, b, c], a]] \[CapitalDelta]a + Abs[D[s[a, b, c], b]] \[CapitalDelta]b + Abs[D[s[a, b, c], c]] \[CapitalDelta]c\), "\[IndentingNewLine]", \(s[a, b, c] /. {a \[Rule] 10.2, \[CapitalDelta]a \[Rule] 0.1, b \[Rule] 7.7, \[CapitalDelta]b \[Rule] 0.08, c \[Rule] 3.17, \[CapitalDelta]c \[Rule] 0.04}\), "\[IndentingNewLine]", \(\[CapitalDelta]s /. {a \[Rule] 10.2, \[CapitalDelta]a \[Rule] 0.1, b \[Rule] 7.7, \[CapitalDelta]b \[Rule] 0.08, c \[Rule] 3.17, \[CapitalDelta]c \[Rule] 0.04}\)}], "Input"], Cell[BoxData[ \(TraditionalForm\`2\ \[CapitalDelta]c\ \[LeftBracketingBar]a + b\[RightBracketingBar] + 2\ \[CapitalDelta]b\ \[LeftBracketingBar]a + c\[RightBracketingBar] + 2\ \[CapitalDelta]a\ \[LeftBracketingBar]b + c\[RightBracketingBar]\)], "Output"], Cell[BoxData[ \(270.566`\)], "Output"], Cell[BoxData[ \(5.7452000000000005`\)], "Output"] }, Open ]], Cell["Et pour le volume :", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(v[a_, b_, c_] := a*b*c\), "\[IndentingNewLine]", \(\[CapitalDelta]v = Abs[D[v[a, b, c], a]] \[CapitalDelta]a + Abs[D[v[a, b, c], b]] \[CapitalDelta]b + Abs[D[v[a, b, c], c]] \[CapitalDelta]c\), "\[IndentingNewLine]", \(v[a, b, c] /. {a \[Rule] 10.2, \[CapitalDelta]a \[Rule] 0.1, b \[Rule] 7.7, \[CapitalDelta]b \[Rule] 0.08, c \[Rule] 3.17, \[CapitalDelta]c \[Rule] 0.04}\), "\[IndentingNewLine]", \(\[CapitalDelta]v /. {a \[Rule] 10.2, \[CapitalDelta]a \[Rule] 0.1, b \[Rule] 7.7, \[CapitalDelta]b \[Rule] 0.08, c \[Rule] 3.17, \[CapitalDelta]c \[Rule] 0.04}\)}], "Input"], Cell[BoxData[ \(TraditionalForm\`\[CapitalDelta]c\ \[LeftBracketingBar]a\ b\ \[RightBracketingBar] + \[CapitalDelta]b\ \[LeftBracketingBar]a\ c\ \[RightBracketingBar] + \[CapitalDelta]a\ \[LeftBracketingBar]b\ c\ \[RightBracketingBar]\)], "Output"], Cell[BoxData[ \(248.97179999999997`\)], "Output"], Cell[BoxData[ \(8.16922`\)], "Output"] }, Open ]], Cell[TextData[{ StyleBox["Exercices\n", FontSize->14, FontWeight->"Bold"], StyleBox["Exercice 1", FontWeight->"Bold"], "\nAttention, la diff\[EAcute]rence des diam\[EGrave]tres donne 2 fois l'\ \[EAcute]paisseur : 26.7 - 19.5 = 7.2 mm = 2", StyleBox["e", FontSlant->"Italic"], "\nEn appliquant la r\[EGrave]gle pour la soustraction, vous obtenez donc, \ pour l'\[EAcute]paisseur ", StyleBox["e", FontSlant->"Italic"], " = 3.6 \[PlusMinus] 0.1 mm et pour la pr\[EAcute]cision ", Cell[BoxData[ FormBox[ StyleBox[\(\[CapitalDelta]e\/\(\(e\)\(\ \)\)\), FontSize->12], TraditionalForm]], FontSize->14], " \[TildeTilde] 3%" }], "Text"], Cell[TextData[{ StyleBox["Exercice 2", FontWeight->"Bold"], "\nL'aire du cercle est donn\[EAcute]e par ", StyleBox["S", FontSlant->"Italic"], " = \[Pi]", Cell[BoxData[ \(TraditionalForm\`R\^2\)]], ". En appliquant la r\[EGrave]gle pour la multiplication, vous obtenez ", StyleBox["S", FontSlant->"Italic"], " = 85.3 \[PlusMinus] 3.3 c", Cell[BoxData[ \(TraditionalForm\`m\^2\)]], ". La pr\[EAcute]cision est fournie par l'incertitude relative ", Cell[BoxData[ FormBox[ StyleBox[\(\[CapitalDelta]S\/S\), FontSize->12], TraditionalForm]]], "\[TildeTilde] 3.9 %" }], "Text"], Cell[TextData[{ StyleBox["Exercice 3", FontWeight->"Bold"], "\nD\[EAcute]finissons les fonctions donnant le p\[EAcute]rim\[EGrave]tre \ ", StyleBox["p", FontSlant->"Italic"], ", la surface ", StyleBox["S", FontSlant->"Italic"], " du sol et le volume ", StyleBox["V", FontSlant->"Italic"], " de la salle et utilisons les diff\[EAcute]rentielles totales pour \ calculer l'incertitude sur les r\[EAcute]sultats (ATTENTION, \[AGrave] vous \ de les exprimer correctement !)" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(p[a_, b_] := 2 \((a + b)\)\), "\[IndentingNewLine]", \(\[CapitalDelta]p = Abs[D[p[a, b], a]] \[CapitalDelta]a + Abs[D[p[a, b], b]] \[CapitalDelta]b\), "\[IndentingNewLine]", \(p[a, b] /. {a \[Rule] 10.2, \[CapitalDelta]a \[Rule] 0.1, b \[Rule] 7.7, \[CapitalDelta]b \[Rule] 0.08}\), "\[IndentingNewLine]", \(\[CapitalDelta]p /. {a \[Rule] 10.2, \[CapitalDelta]a \[Rule] 0.1, b \[Rule] 7.7, \[CapitalDelta]b \[Rule] 0.08}\)}], "Input"], Cell[BoxData[ \(TraditionalForm\`2\ \[CapitalDelta]a + 2\ \[CapitalDelta]b\)], "Output"], Cell[BoxData[ \(35.8`\)], "Output"], Cell[BoxData[ \(0.36`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \(s[a_, b_] := a*b\), "\[IndentingNewLine]", \(\[CapitalDelta]s = Abs[D[s[a, b], a]] \[CapitalDelta]a + Abs[D[s[a, b], b]] \[CapitalDelta]b\), "\[IndentingNewLine]", \(s[a, b] /. {a \[Rule] 10.2, \[CapitalDelta]a \[Rule] 0.1, b \[Rule] 7.7, \[CapitalDelta]b \[Rule] 0.08}\), "\[IndentingNewLine]", \(\[CapitalDelta]s /. {a \[Rule] 10.2, \[CapitalDelta]a \[Rule] 0.1, b \[Rule] 7.7, \[CapitalDelta]b \[Rule] 0.08}\)}], "Input"], Cell[BoxData[ \(TraditionalForm\`\[CapitalDelta]b\ \[LeftBracketingBar]a\ \[RightBracketingBar] + \[CapitalDelta]a\ \[LeftBracketingBar]b\ \[RightBracketingBar]\)], "Output"], Cell[BoxData[ \(78.53999999999999`\)], "Output"], Cell[BoxData[ \(1.5859999999999999`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \(v[a_, b_, c_] := a*b*c\), "\n", \(\[CapitalDelta]v = Abs[D[v[a, b, c], a]] \[CapitalDelta]a + Abs[D[v[a, b, c], b]] \[CapitalDelta]b + Abs[D[v[a, b, c], c]] \[CapitalDelta]c\), "\n", \(v[a, b, c] /. {a \[Rule] 10.2, \[CapitalDelta]a \[Rule] 0.1, b \[Rule] 7.7, \[CapitalDelta]b \[Rule] 0.08, c \[Rule] 3.17, \[CapitalDelta]c \[Rule] 0.04}\), "\n", \(\[CapitalDelta]v /. {a \[Rule] 10.2, \[CapitalDelta]a \[Rule] 0.1, b \[Rule] 7.7, \[CapitalDelta]b \[Rule] 0.08, c \[Rule] 3.17, \[CapitalDelta]c \[Rule] 0.04}\)}], "Input"], Cell[BoxData[ \(TraditionalForm\`\[CapitalDelta]c\ \[LeftBracketingBar]a\ b\ \[RightBracketingBar] + \[CapitalDelta]b\ \[LeftBracketingBar]a\ c\ \[RightBracketingBar] + \[CapitalDelta]a\ \[LeftBracketingBar]b\ c\ \[RightBracketingBar]\)], "Output"], Cell[BoxData[ \(248.97179999999997`\)], "Output"], Cell[BoxData[ \(8.16922`\)], "Output"] }, Open ]], Cell[TextData[{ StyleBox["Exercice 4", FontWeight->"Bold"], "\nLa masse volumique \[Rho] de l'objet est donn\[EAcute]e par le quotient \ de sa masse par son volume. En appliquant la r\[EGrave]gle pour la division, \ vous obtenez \[Rho] = 1.91 \[PlusMinus] 0.09 g/", Cell[BoxData[ \(TraditionalForm\`cm\^3\)]] }], "Text"], Cell[TextData[{ StyleBox["Exercice 5", FontWeight->"Bold"], "\nLe volume du cylindre est donn\[EAcute] par ", StyleBox["V", FontSlant->"Italic"], " = \[Pi]", Cell[BoxData[ \(TraditionalForm\`R\^2\)]], StyleBox["h", FontSlant->"Italic"], ". En appliquant les r\[EGrave]gles, vous obtenez ", StyleBox["V", FontSlant->"Italic"], " = 50.27 \[PlusMinus] 0.19 ", Cell[BoxData[ \(TraditionalForm\`cm\^3\)]], " pour le volume et \[Rho] = 7.80 \[PlusMinus] 0.03 g/", Cell[BoxData[ \(TraditionalForm\`cm\^3\)]], " pour la masse volumique" }], "Text"], Cell[TextData[{ StyleBox["Exercice 6", FontWeight->"Bold"], "\nD\[EAcute]finissons la fonction reliant ", StyleBox["g", FontSlant->"Italic"], " aux grandeurs mesur\[EAcute]es, puis calculons ", StyleBox["g", FontSlant->"Italic"], " et \[CapitalDelta]", StyleBox["g", FontSlant->"Italic"], " :" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(f[d_, T_] := \ 4 Pi^2\ d/T^2\), "\[IndentingNewLine]", \(\[CapitalDelta]g = Abs[D[f[d, T], d]] \[CapitalDelta]d + Abs[D[f[d, T], T]] \[CapitalDelta]T\), "\[IndentingNewLine]", \(N[f[d, T]] /. {d -> 1, \[CapitalDelta]d -> 0.005, T -> 2, \[CapitalDelta]T -> 0.01}\), "\[IndentingNewLine]", \(\[CapitalDelta]g /. {d -> 1, \[CapitalDelta]d -> 0.005, T -> 2, \[CapitalDelta]T -> 0.01}\)}], "Input"], Cell[BoxData[ \(TraditionalForm\`\(4\ \[Pi]\^2\ \ \[CapitalDelta]d\)\/\[LeftBracketingBar]T\[RightBracketingBar]\^2 + 8\ \[Pi]\^2\ \[CapitalDelta]T\ \[LeftBracketingBar]d\/T\^3\ \[RightBracketingBar]\)], "Output"], Cell[BoxData[ \(9.869604401089358`\)], "Output"], Cell[BoxData[ \(0.14804406601634038`\)], "Output"] }, Open ]], Cell[TextData[{ "IMPORTANT. Le r\[EAcute]sultat se donne de la mani\[EGrave]re suivante : \ ", StyleBox["g", FontSlant->"Italic"], " = 9.87 \[PlusMinus] 0.15 m/", Cell[BoxData[ FormBox[ SuperscriptBox[ StyleBox["s", FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], "2"], TraditionalForm]]], ". L'erreur relative vaut environ 1.5%" }], "Text"] }, Open ]] }, FrontEndVersion->"5.2 for Macintosh", ScreenRectangle->{{0, 994}, {0, 746}}, ScreenStyleEnvironment->"Printout", WindowToolbars->{"RulerBar", "EditBar"}, CellGrouping->Automatic, WindowSize->{756, 641}, WindowMargins->{{62, Automatic}, {-98, Automatic}}, PrintingCopies->1, PrintingPageRange->{2, 2}, PrintingOptions->{"PrintingMargins"->{{85, 28.25}, {36, 42.5}}, "PrintCellBrackets"->False, "PrintRegistrationMarks"->False, "PrintMultipleHorizontalPages"->False}, PrivateNotebookOptions->{"ColorPalette"->{RGBColor, 128}}, ShowCellLabel->False, ShowCellTags->False, RenderingOptions->{"ObjectDithering"->True, "RasterDithering"->False}, CharacterEncoding->"MacintoshAutomaticEncoding", Magnification->1.5 ] (******************************************************************* Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. 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