(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 7.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 28404, 770] NotebookOptionsPosition[ 25425, 664] NotebookOutlinePosition[ 26199, 692] CellTagsIndexPosition[ 26073, 686] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["\<\ Calcul exact et r\[EAcute]solution d' \[EAcute]quations\ \>", "Subtitle", CellChangeTimes->{{3.56748915634375*^9, 3.567489196046875*^9}}], Cell["\<\ Mathematica peut donnner des r\[EAcute]sultats exacts. Pour obtenir un tel r\ \[EAcute]sultat, aucune des donn\[EAcute]es num\[EAcute]riques qui figurent \ dans le calcul ne doit comporter de s\[EAcute]parateur d\[EAcute]cimal \ (point). Exemple :\ \>", "Text", CellChangeTimes->{{3.56748921225*^9, 3.567489272703125*^9}, {3.567489503*^9, 3.567489573703125*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"1", "/", "3"}], "+", RowBox[{"1", "/", "7"}]}]], "Input", CellChangeTimes->{{3.567489576828125*^9, 3.56748958915625*^9}}], Cell[BoxData[ FractionBox["10", "21"]], "Output", CellChangeTimes->{3.567489590640625*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"1.", "/", "3"}], "+", RowBox[{"1", "/", "7"}]}]], "Input", CellChangeTimes->{{3.5674895964375*^9, 3.567489603828125*^9}}], Cell[BoxData["0.47619047619047616`"], "Output", CellChangeTimes->{3.567489604921875*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"3", "^", "2.5"}]], "Input", CellChangeTimes->{{3.567489700015625*^9, 3.567489703765625*^9}}], Cell[BoxData["15.588457268119896`"], "Output", CellChangeTimes->{3.567489706671875*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"3", "^", RowBox[{"(", RowBox[{"5", "/", "2"}], ")"}]}]], "Input", CellChangeTimes->{{3.567489729265625*^9, 3.5674897371875*^9}}], Cell[BoxData[ RowBox[{"9", " ", SqrtBox["3"]}]], "Output", CellChangeTimes->{3.567489737671875*^9}] }, Open ]], Cell[CellGroupData[{ Cell["R\[EAcute]solution d' \[EAcute]quation", "Subsubtitle", CellChangeTimes->{{3.567489761578125*^9, 3.567489772421875*^9}}], Cell["\<\ Pour r\[EAcute]soudre une \[EAcute]quation polynomiale, on utilise la \ commande Solve\ \>", "Text", CellChangeTimes->{{3.5674897951875*^9, 3.567489835296875*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"?", "Solve"}]], "Input", CellChangeTimes->{{3.567489856171875*^9, 3.56748985815625*^9}}], Cell[BoxData[ RowBox[{ StyleBox["\<\"\\!\\(\\*RowBox[{\\\"Solve\\\", \\\"[\\\", \ RowBox[{StyleBox[\\\"eqns\\\", \\\"TI\\\"], \\\",\\\", StyleBox[\\\"vars\\\", \ \\\"TI\\\"]}], \\\"]\\\"}]\\) attempts to solve an equation or set of \ equations for the variables \\!\\(\\*StyleBox[\\\"vars\\\", \\\"TI\\\"]\\). \ \\n\\!\\(\\*RowBox[{\\\"Solve\\\", \\\"[\\\", RowBox[{StyleBox[\\\"eqns\\\", \ \\\"TI\\\"], \\\",\\\", StyleBox[\\\"vars\\\", \\\"TI\\\"], \\\",\\\", \ StyleBox[\\\"elims\\\", \\\"TI\\\"]}], \\\"]\\\"}]\\) attempts to solve the \ equations for \\!\\(\\*StyleBox[\\\"vars\\\", \\\"TI\\\"]\\), eliminating the \ variables \\!\\(\\*StyleBox[\\\"elims\\\", \\\"TI\\\"]\\). \"\>", "MSG"], "\[NonBreakingSpace]", ButtonBox[ StyleBox["\[RightSkeleton]", "SR"], Active->True, BaseStyle->"Link", ButtonData->"paclet:ref/Solve"]}]], "Print", "PrintUsage", CellChangeTimes->{3.56748985940625*^9}, CellTags->"Info3567493459-4940378"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Solve", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{"x", "^", "2"}], "+", RowBox[{"a", "*", "x"}], "+", "1"}], "==", "0"}], ",", "x"}], "]"}]], "Input", CellChangeTimes->{{3.56748991709375*^9, 3.567489926859375*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"x", "\[Rule]", RowBox[{ FractionBox["1", "2"], " ", RowBox[{"(", RowBox[{ RowBox[{"-", "a"}], "-", SqrtBox[ RowBox[{ RowBox[{"-", "4"}], "+", SuperscriptBox["a", "2"]}]]}], ")"}]}]}], "}"}], ",", RowBox[{"{", RowBox[{"x", "\[Rule]", RowBox[{ FractionBox["1", "2"], " ", RowBox[{"(", RowBox[{ RowBox[{"-", "a"}], "+", SqrtBox[ RowBox[{ RowBox[{"-", "4"}], "+", SuperscriptBox["a", "2"]}]]}], ")"}]}]}], "}"}]}], "}"}]], "Output", CellChangeTimes->{3.567489935265625*^9}] }, Open ]], Cell["\<\ La syntaxe de la commande peut s'obtenir en faisant pr\[EAcute]c\[EAcute]der \ son nom d'un point d'interrogation. A partir de l'information obtenue, vous \ pouvez acc\[EAcute]der \[AGrave] une page d'exemples ex\[EAcute]cutables. La ou les solutions sont obtenues sont la forme de r\[EGrave]gles de \ substitution x->sol. S'il y a plusieurs solutions, elles sont donn\[EAcute]es \ dans une liste.\ \>", "Text", CellChangeTimes->{{3.567489953265625*^9, 3.567490138171875*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Solve", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{"a", "*", " ", RowBox[{"x", "^", "2"}]}], "+", RowBox[{"b", "*", "x"}], "+", "c"}], "\[Equal]", "0"}], ",", "x"}], "]"}]], "Input", CellChangeTimes->{{3.567490147125*^9, 3.567490179390625*^9}, { 3.567490415328125*^9, 3.567490437796875*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"x", "\[Rule]", FractionBox[ RowBox[{ RowBox[{"-", "b"}], "-", SqrtBox[ RowBox[{ SuperscriptBox["b", "2"], "-", RowBox[{"4", " ", "a", " ", "c"}]}]]}], RowBox[{"2", " ", "a"}]]}], "}"}], ",", RowBox[{"{", RowBox[{"x", "\[Rule]", FractionBox[ RowBox[{ RowBox[{"-", "b"}], "+", SqrtBox[ RowBox[{ SuperscriptBox["b", "2"], "-", RowBox[{"4", " ", "a", " ", "c"}]}]]}], RowBox[{"2", " ", "a"}]]}], "}"}]}], "}"}]], "Output", CellChangeTimes->{ 3.567490185765625*^9, {3.56749041896875*^9, 3.56749043878125*^9}}] }, Open ]], Cell[TextData[{ "Attention, l'\[EAcute]galit\[EAcute] dans ", StyleBox["Mathematica", FontSlant->"Italic"], " se donne \[AGrave] l'aide de 2 signes \[EAcute]gal. Une \[EAcute]quation \ donn\[EAcute]e dans Solve comportera donc toujours == (2 signes \[EAcute]gal)" }], "Text", CellChangeTimes->{{3.567490202390625*^9, 3.56749025028125*^9}, { 3.567490285859375*^9, 3.567490343125*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"2", "\[Equal]", "2"}]], "Input", CellChangeTimes->{{3.567490258703125*^9, 3.567490262734375*^9}}], Cell[BoxData["True"], "Output", CellChangeTimes->{3.5674902635625*^9}] }, Open ]], Cell["\<\ Dans l' exemple ci - dessus, la solution consiste en une liste qui comporte \ deux sous-listes. On peut extraire la solution qui nous int\[EAcute]resse de \ la mani\[EGrave]re suivante 1\[Degree] on affecte la liste des solutions \[AGrave] une variable\ \>", "Text", CellChangeTimes->{{3.56749036359375*^9, 3.56749039265625*^9}, { 3.567490447046875*^9, 3.567490526578125*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"sol", "=", RowBox[{"Solve", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{"a", "*", " ", RowBox[{"x", "^", "2"}]}], "+", RowBox[{"b", "*", "x"}], "+", "c"}], "\[Equal]", "0"}], ",", "x"}], "]"}]}], ";"}]], "Input", CellChangeTimes->{{3.5674905484375*^9, 3.567490569484375*^9}}], Cell[CellGroupData[{ Cell[BoxData["sol"], "Input", CellChangeTimes->{{3.567490575953125*^9, 3.567490576359375*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"x", "\[Rule]", FractionBox[ RowBox[{ RowBox[{"-", "b"}], "-", SqrtBox[ RowBox[{ SuperscriptBox["b", "2"], "-", RowBox[{"4", " ", "a", " ", "c"}]}]]}], RowBox[{"2", " ", "a"}]]}], "}"}], ",", RowBox[{"{", RowBox[{"x", "\[Rule]", FractionBox[ RowBox[{ RowBox[{"-", "b"}], "+", SqrtBox[ RowBox[{ SuperscriptBox["b", "2"], "-", RowBox[{"4", " ", "a", " ", "c"}]}]]}], RowBox[{"2", " ", "a"}]]}], "}"}]}], "}"}]], "Output", CellChangeTimes->{3.5674905770625*^9, 3.56749096934375*^9}] }, Open ]], Cell["\<\ 2 \[Degree] on extrait de liste obtenue la partie qui nous int\[EAcute]resse \ en utilisant la syntaxe suivante\ \>", "Text", CellChangeTimes->{{3.5674905881875*^9, 3.5674906421875*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"sol", "[", RowBox[{"[", "1", "]"}], "]"}]], "Input", CellChangeTimes->{{3.567490645171875*^9, 3.567490660546875*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{"x", "\[Rule]", FractionBox[ RowBox[{ RowBox[{"-", "b"}], "-", SqrtBox[ RowBox[{ SuperscriptBox["b", "2"], "-", RowBox[{"4", " ", "a", " ", "c"}]}]]}], RowBox[{"2", " ", "a"}]]}], "}"}]], "Output", CellChangeTimes->{3.567490662328125*^9, 3.567490969390625*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"sol", "[", RowBox[{"[", "2", "]"}], "]"}]], "Input", CellChangeTimes->{3.567490673078125*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{"x", "\[Rule]", FractionBox[ RowBox[{ RowBox[{"-", "b"}], "+", SqrtBox[ RowBox[{ SuperscriptBox["b", "2"], "-", RowBox[{"4", " ", "a", " ", "c"}]}]]}], RowBox[{"2", " ", "a"}]]}], "}"}]], "Output", CellChangeTimes->{3.56749067378125*^9, 3.567490969453125*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"sol", "[", RowBox[{"[", RowBox[{"1", ",", "1"}], "]"}], "]"}]], "Input", CellChangeTimes->{{3.567490702328125*^9, 3.567490702765625*^9}, { 3.567490858984375*^9, 3.56749086340625*^9}}], Cell[BoxData[ RowBox[{"x", "\[Rule]", FractionBox[ RowBox[{ RowBox[{"-", "b"}], "-", SqrtBox[ RowBox[{ SuperscriptBox["b", "2"], "-", RowBox[{"4", " ", "a", " ", "c"}]}]]}], RowBox[{"2", " ", "a"}]]}]], "Output", CellChangeTimes->{3.567490864*^9, 3.567490969515625*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"sol", "[", RowBox[{"[", RowBox[{"1", ",", "1", ",", "2"}], "]"}], "]"}]], "Input", CellChangeTimes->{{3.567490891890625*^9, 3.5674908924375*^9}}], Cell[BoxData[ FractionBox[ RowBox[{ RowBox[{"-", "b"}], "-", SqrtBox[ RowBox[{ SuperscriptBox["b", "2"], "-", RowBox[{"4", " ", "a", " ", "c"}]}]]}], RowBox[{"2", " ", "a"}]]], "Output", CellChangeTimes->{3.56749089309375*^9, 3.567490969640625*^9}] }, Open ]], Cell[TextData[{ "Si on souhaite r\[EAcute]soudre l'\[EAcute]quation ", StyleBox["sin", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ") = ", StyleBox["x", FontSlant->"Italic"], " qui n'est pas une \[EAcute]quation polynomiale, il faut utiliser une autre \ commande qui est FindRoot" }], "Text", CellChangeTimes->{{3.56749100853125*^9, 3.5674911135*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"?", "FindRoot"}]], "Input", CellChangeTimes->{{3.567491120140625*^9, 3.567491121046875*^9}}], Cell[BoxData[ RowBox[{ StyleBox["\<\"\\!\\(\\*RowBox[{\\\"FindRoot\\\", \\\"[\\\", \ RowBox[{StyleBox[\\\"f\\\", \\\"TI\\\"], \\\",\\\", RowBox[{\\\"{\\\", \ RowBox[{StyleBox[\\\"x\\\", \\\"TI\\\"], \\\",\\\", \ SubscriptBox[StyleBox[\\\"x\\\", \\\"TI\\\"], StyleBox[\\\"0\\\", \ \\\"TR\\\"]]}], \\\"}\\\"}]}], \\\"]\\\"}]\\) searches for a numerical root \ of \\!\\(\\*StyleBox[\\\"f\\\", \\\"TI\\\"]\\), starting from the point \ \\!\\(\\*RowBox[{StyleBox[\\\"x\\\", \\\"TI\\\"], \\\"=\\\", \ SubscriptBox[StyleBox[\\\"x\\\", \\\"TI\\\"], StyleBox[\\\"0\\\", \ \\\"TR\\\"]]}]\\).\\n\\!\\(\\*RowBox[{\\\"FindRoot\\\", \\\"[\\\", \ RowBox[{RowBox[{StyleBox[\\\"lhs\\\", \\\"TI\\\"], \\\"==\\\", \ StyleBox[\\\"rhs\\\", \\\"TI\\\"]}], \\\",\\\", RowBox[{\\\"{\\\", \ RowBox[{StyleBox[\\\"x\\\", \\\"TI\\\"], \\\",\\\", \ SubscriptBox[StyleBox[\\\"x\\\", \\\"TI\\\"], StyleBox[\\\"0\\\", \ \\\"TR\\\"]]}], \\\"}\\\"}]}], \\\"]\\\"}]\\) searches for a numerical \ solution to the equation \\!\\(\\*RowBox[{StyleBox[\\\"lhs\\\", \\\"TI\\\"], \ \\\"==\\\", StyleBox[\\\"rhs\\\", \\\"TI\\\"]}]\\). \ \\n\\!\\(\\*RowBox[{\\\"FindRoot\\\", \\\"[\\\", RowBox[{RowBox[{\\\"{\\\", \ RowBox[{SubscriptBox[StyleBox[\\\"f\\\", \\\"TI\\\"], StyleBox[\\\"1\\\", \ \\\"TR\\\"]], \\\",\\\", SubscriptBox[StyleBox[\\\"f\\\", \\\"TI\\\"], \ StyleBox[\\\"2\\\", \\\"TR\\\"]], \\\",\\\", StyleBox[\\\"\[Ellipsis]\\\", \\\ \"TR\\\"]}], \\\"}\\\"}], \\\",\\\", RowBox[{\\\"{\\\", RowBox[{RowBox[{\\\"{\ \\\", RowBox[{StyleBox[\\\"x\\\", \\\"TI\\\"], \\\",\\\", \ SubscriptBox[StyleBox[\\\"x\\\", \\\"TI\\\"], StyleBox[\\\"0\\\", \ \\\"TR\\\"]]}], \\\"}\\\"}], \\\",\\\", RowBox[{\\\"{\\\", \ RowBox[{StyleBox[\\\"y\\\", \\\"TI\\\"], \\\",\\\", \ SubscriptBox[StyleBox[\\\"y\\\", \\\"TI\\\"], StyleBox[\\\"0\\\", \ \\\"TR\\\"]]}], \\\"}\\\"}], \\\",\\\", StyleBox[\\\"\[Ellipsis]\\\", \ \\\"TR\\\"]}], \\\"}\\\"}]}], \\\"]\\\"}]\\) searches for a simultaneous \ numerical root of all the \\!\\(\\*SubscriptBox[StyleBox[\\\"f\\\", \ \\\"TI\\\"], StyleBox[\\\"i\\\", \ \\\"TI\\\"]]\\).\\n\\!\\(\\*RowBox[{\\\"FindRoot\\\", \\\"[\\\", \ RowBox[{RowBox[{\\\"{\\\", RowBox[{SubscriptBox[StyleBox[\\\"eqn\\\", \ \\\"TI\\\"], StyleBox[\\\"1\\\", \\\"TR\\\"]], \\\",\\\", \ SubscriptBox[StyleBox[\\\"eqn\\\", \\\"TI\\\"], StyleBox[\\\"2\\\", \ \\\"TR\\\"]], \\\",\\\", StyleBox[\\\"\[Ellipsis]\\\", \\\"TR\\\"]}], \\\"}\\\ \"}], \\\",\\\", RowBox[{\\\"{\\\", RowBox[{RowBox[{\\\"{\\\", \ RowBox[{StyleBox[\\\"x\\\", \\\"TI\\\"], \\\",\\\", \ SubscriptBox[StyleBox[\\\"x\\\", \\\"TI\\\"], StyleBox[\\\"0\\\", \ \\\"TR\\\"]]}], \\\"}\\\"}], \\\",\\\", RowBox[{\\\"{\\\", \ RowBox[{StyleBox[\\\"y\\\", \\\"TI\\\"], \\\",\\\", \ SubscriptBox[StyleBox[\\\"y\\\", \\\"TI\\\"], StyleBox[\\\"0\\\", \ \\\"TR\\\"]]}], \\\"}\\\"}], \\\",\\\", StyleBox[\\\"\[Ellipsis]\\\", \ \\\"TR\\\"]}], \\\"}\\\"}]}], \\\"]\\\"}]\\) searches for a numerical \ solution to the simultaneous equations \\!\\(\\*SubscriptBox[StyleBox[\\\"eqn\ \\\", \\\"TI\\\"], StyleBox[\\\"i\\\", \\\"TI\\\"]]\\). \"\>", "MSG"], "\[NonBreakingSpace]", ButtonBox[ StyleBox["\[RightSkeleton]", "SR"], Active->True, BaseStyle->"Link", ButtonData->"paclet:ref/FindRoot"]}]], "Print", "PrintUsage", CellChangeTimes->{3.567491122109375*^9}, CellTags->"Info3567494721-7847478"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"FindRoot", "[", RowBox[{ RowBox[{ RowBox[{"Cos", "[", "x", "]"}], "\[Equal]", "x"}], ",", RowBox[{"{", RowBox[{"x", ",", "1"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.567491139734375*^9, 3.567491192828125*^9}, { 3.567491278828125*^9, 3.567491279875*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{"x", "\[Rule]", "0.7390851332151607`"}], "}"}]], "Output", CellChangeTimes->{{3.56749118434375*^9, 3.567491193265625*^9}, 3.567491280890625*^9}] }, Open ]], Cell["\<\ Cette commande ne fournit pas une valeur exacte. Comme il peut y avoir \ plusieurs solutions, il faut indiquer une valeur proche de celle qu'on \ souhaite obtnenir. Comment obtenir cette valeur proche ? \ \>", "Text", CellChangeTimes->{{3.56749156275*^9, 3.56749166996875*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Plot", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"Cos", "[", "x", "]"}], ",", "x"}], "}"}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "Pi"}], ",", "Pi"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.56749169540625*^9, 3.567491756125*^9}}], Cell[BoxData[ GraphicsBox[{{}, {}, {Hue[0.67, 0.6, 0.6], LineBox[CompressedData[" 1:eJw12Xk0Vd/bAHDi3ntuSpJSCg2akJTSQPZJI6WkQshQGVJpQIOhKCQkMiRl ioSIylSG5xCRyDyGDJnvvdtcpvzOd633Pf/c9Vln3bP3fp5n73P22qvOXdUx n8PHx/eQn4/vv19NDeF9U6vYVPrsfxcGlTSnQUKKTQkqz10+TVt+1UDYEgk2 lZFx5fIgbcnHZ47ILGFTYUabRptpC08U/d0iyqYy27U+5NGevaAci4TZlFJf bWw47cGKmJNac9mU9WRZpS3tNtVF/IZMNtXmSirtpV0Z5/rOag6b2lu56zuT dr7YoOHNfwRlIl38vOAfhg8uxmy3SYKS85wKd6AdzSlN9x8nqLR53U0baAfq q1yIGCaodQEvtMpnMLgXxC9M4hHU5+4tf67Qvqm4FD73ExRkUPWCtPWJsWV1 HQRlM09Rbfk0Bg2780WdrQRVcFGjIHQKw+62SruhJoKq/VPqKkp7Rea78nnV 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