(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 6.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 20406, 620] NotebookOptionsPosition[ 17419, 540] NotebookOutlinePosition[ 19464, 590] CellTagsIndexPosition[ 19421, 587] WindowFrame->Normal ContainsDynamic->False*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["Notions de base sur les fonctions", "Subsection", CellChangeTimes->{{3.399724913115114*^9, 3.399724918007456*^9}}], Cell[CellGroupData[{ Cell["1. Domaine de d\[EAcute]finition", "Subsubsection", CellChangeTimes->{{3.3997249257775097`*^9, 3.399724932888383*^9}}], Cell[TextData[{ StyleBox["D\[EAcute]finition", FontVariations->{"Underline"->True}], " : soit f, une fonction r\[EAcute]elle\n\n", StyleBox["dom f = {x \[Element] \[DoubleStruckCapitalR] : f(x) \[Element] \ \[DoubleStruckCapitalR] }", FontColor->RGBColor[0, 0, 1]] }], "definition", CellChangeTimes->{{3.3997249401964607`*^9, 3.3997250285122423`*^9}}], Cell[TextData[{ "D\[EAcute]terminer le domaine de d\[EAcute]finition, c'est trouver les r\ \[EAcute]els x qui ont une image par f, c'est-\[AGrave]-dire pour lesquels on \ sait calculer f(x)\n(pour lesquels f(x) est un r\[EAcute]el).\n\nPour cela, \ il suffit de r\[EAcute]soudre les conditions d'existence.\nLes 2 types de \ conditions rencontr\[EAcute]es en ce d\[EAcute]but de cours correspondent \ \[AGrave] deux op\[EAcute]rations impossibles dans les r\[EAcute]els:\n\n", StyleBox["- La division par 0", FontWeight->"Bold"], "\nDans \[DoubleStruckCapitalR], il est impossible de diviser par 0. ", StyleBox["Peux-tu expliquer pourquoi?", FontColor->RGBColor[1, 0, 0]], "\nCalculer le quotient ", Cell[BoxData[ FormBox[ StyleBox[ FractionBox["a", "0"], FontSize->16], TraditionalForm]]], " est effectivement impossible dans l'ensemble des r\[EAcute]els.\nDonc, si \ une fonction comporte un d\[EAcute]nominateur, il faut que ce dernier soit \ diff\[EAcute]rent de 0." }], "Text", CellChangeTimes->{{3.399725060135434*^9, 3.399725199583449*^9}, { 3.399725260880971*^9, 3.3997252869915743`*^9}, {3.399725424121127*^9, 3.3997255499594727`*^9}, {3.399725586881123*^9, 3.3997256679848747`*^9}, { 3.399725716202712*^9, 3.39972571767274*^9}, {3.399725919114477*^9, 3.399725920227268*^9}}], Cell[TextData[{ "exemple:\tf(x) =", StyleBox[" ", FontSize->16], Cell[BoxData[ FormBox[ FractionBox[ RowBox[{ RowBox[{"3", "x"}], "-", "1"}], RowBox[{ RowBox[{"2", SuperscriptBox["x", "2"]}], "-", " ", RowBox[{"3", "x"}], " ", "+", " ", "1"}]], TraditionalForm]], FontSize->16], "\n\tCE: ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{"2", SuperscriptBox["x", "2"]}], "-", RowBox[{"3", "x"}], "+", "1"}], " ", "\[NotEqual]", " ", "0"}], TraditionalForm]]], " c'est-\[AGrave]-dire x \[NotEqual] 1 et x \[NotEqual] -", Cell[BoxData[ FormBox[ StyleBox[ FractionBox["1", "2"], FontSize->16], TraditionalForm]]], "\n\tle domaine est donc\n\tDom f = \[DoubleStruckCapitalR] \\ {-", Cell[BoxData[ FormBox[ StyleBox[ FractionBox["1", "2"], FontSize->16], TraditionalForm]]], " , 1}\n\t" }], "Text", CellChangeTimes->{{3.399725669456045*^9, 3.39972587598356*^9}, 3.399725946241452*^9}, FontSlant->"Italic"], Cell[TextData[{ StyleBox["- La racine carr\[EAcute]e d'un nombre n\[EAcute]gatif", FontWeight->"Bold"], "\nDans \[DoubleStruckCapitalR], il est impossible de calculer la racine \ d'un nombre n\[EAcute]gatif. ", StyleBox["Peux-tu expliquer pourquoi?", FontColor->RGBColor[1, 0, 0]], "\nCalculer par exemple ", Cell[BoxData[ FormBox[ SqrtBox[ RowBox[{"-", "2"}]], TraditionalForm]]], " est impossible dans l'ensemble des r\[EAcute]els.\nDonc, si une fonction \ comporte une racine carr\[EAcute]e, il faut que le radicant (l'expression \ sous la racine) soit positif." }], "Text", CellChangeTimes->{{3.399725940505103*^9, 3.399726026831636*^9}, { 3.399726112961916*^9, 3.399726144151661*^9}}], Cell[TextData[{ "exemple:\tf(x) =", StyleBox[" ", FontSize->16], Cell[BoxData[ FormBox[ SqrtBox[ RowBox[{"5", "-", RowBox[{"7", "x"}]}]], TraditionalForm]]], "\n\tCE: ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"5", "-", RowBox[{"7", "x"}]}], " ", "\[GreaterEqual]", " ", "0"}], TraditionalForm]]], " c'est-\[AGrave]-dire x \[LessEqual] ", Cell[BoxData[ FormBox[ StyleBox[ FractionBox["5", "7"], FontSize->16], TraditionalForm]]], "\n\tle domaine est donc\n\tDom f = \[LeftArrow] , ", Cell[BoxData[ FormBox[ StyleBox[ FractionBox["5", "7"], FontSize->16], TraditionalForm]]], "]\n\t" }], "exemple", CellChangeTimes->{{3.399725669456045*^9, 3.39972587598356*^9}, 3.399725946241452*^9, {3.3997261589849358`*^9, 3.399726290560903*^9}}, FontSlant->"Italic"], Cell["\<\ Si la racine carr\[EAcute]e se trouve \[EHat]tre \[EAcute]galement le d\ \[EAcute]nominateur, alors le radicant doit \[EHat]tre \[AGrave] la fois diff\ \[EAcute]rent de 0 et positif, donc strictement positif.\ \>", "Text", CellChangeTimes->{{3.3997264214549294`*^9, 3.399726499074342*^9}, 3.399726616299666*^9}], Cell[TextData[{ "exemple:\tf(x) =", StyleBox[" ", FontSize->16], Cell[BoxData[ FormBox[ StyleBox[ FractionBox[ RowBox[{ RowBox[{"3", "x"}], "-", "5"}], SqrtBox[ RowBox[{ RowBox[{"2", "x"}], "+", " ", "6"}]]], FontSize->16], TraditionalForm]]], "\n\tCE: ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{"2", "x"}], "+", " ", "6"}], " ", ">", " ", "0"}], TraditionalForm]]], " c'est-\[AGrave]-dire x > 3\[LineSeparator]\n\tle domaine est donc\n\t\ Dom f = ] 3,\[Rule]\n\t\t" }], "exemple", CellChangeTimes->{{3.399725669456045*^9, 3.39972587598356*^9}, 3.399725946241452*^9, {3.3997261589849358`*^9, 3.399726290560903*^9}, { 3.399726410012903*^9, 3.399726417608182*^9}, {3.3997265153424873`*^9, 3.399726603816761*^9}}, FontSlant->"Italic"], Cell[TextData[{ "D'une mani\[EGrave]re plus g\[EAcute]n\[EAcute]rale, si l'on \ consid\[EGrave]re une racine ni\[EGrave]me ", Cell[BoxData[ FormBox[ StyleBox[ RadicalBox["f", "n"], FontSize->16], TraditionalForm]]], " o\[UGrave] n est un naturel >1 et un r\[EAcute]el quelconque,\n\t- si n \ est pair , ", Cell[BoxData[ FormBox[ StyleBox[ RadicalBox["f", "n"], FontSize->16], TraditionalForm]]], " est un r\[EAcute]el si f \[GreaterEqual] 0\n\t- si n est impair , ", Cell[BoxData[ FormBox[ StyleBox[ RadicalBox["f", "n"], FontSize->16], TraditionalForm]]], " est un r\[EAcute]el quel que soit le r\[EAcute]el f" }], "Text", CellChangeTimes->{{3.3997266321981688`*^9, 3.399726809864415*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Exercices - 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