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Indiquer les algorithmes qui permettent de déterminer par dichotomie un encadrement d'amplitude inférieure à 10-1 de la solution de l'équation f(x)=k sur un intervalle sur lequel f est une fonction continue strictement monotone    ressource 4325

Soit f une fonction continue et strictement croissante sur l'intervalle [ 2 ; 3 ] prenant ses valeurs dans l'intervalle [ - 2 ; 1 ] .

Indiquez, pour chacun des trois algorithmes suivants, s'il permet de déterminer, par dichotomie, un encadrement d'amplitude inférieure ou égale à 10 - 1 de la solution de l'équation f ( x ) = 0 sur l'intervalle [ 2 ; 3 ] :

Algorithme 1

            a 2
            b 3
 
            Tant que b - a < 0,1 TemplateBox[List[RowBox[List["b", "-", "a"]], "\"\[ThinSpace]\"", "<", "\"\[ThinSpace]\"", TagBox[InterpretationBox["\"0,1\"", 0.100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001`100., Rule[AutoDelete, True]], Function[NumberForm[Slot[1], List[1, 1], Rule[ExponentFunction, Function[Null]], Rule[NumberPoint, ","], Rule[DigitBlock, 3], Rule[NumberSeparator, " "]]]]], "Row", Rule[DisplayFunction, Function[RowBox[List[Slot[1], "\[InvisibleSpace]", Slot[2], "\[InvisibleSpace]", Slot[3], "\[InvisibleSpace]", Slot[4], "\[InvisibleSpace]", Slot[5]]]]], Rule[InterpretationFunction, Function[RowBox[List["Row", "[", RowBox[List["{", RowBox[List[Slot[1], ",", Slot[2], ",", Slot[3], ",", Slot[4], ",", Slot[5]]], "}"]], "]"]]]]]
                  m a + b 2
                     Si f ( a )  ×  f ( m ) > 0 TemplateBox[List[RowBox[List["f", "(", "a", ")"]], "\"\[ThinSpace]\[Times]\[ThinSpace]\"", RowBox[List["f", "(", "m", ")"]], "\"\[ThinSpace]\"", ">", "\"\[ThinSpace]\"", "0"], "Row", Rule[DisplayFunction, Function[RowBox[List[Slot[1], "\[InvisibleSpace]", Slot[2], "\[InvisibleSpace]", Slot[3], "\[InvisibleSpace]", Slot[4], "\[InvisibleSpace]", Slot[5], "\[InvisibleSpace]", Slot[6], "\[InvisibleSpace]", Slot[7]]]]], Rule[InterpretationFunction, Function[RowBox[List["Row", "[", RowBox[List["{", RowBox[List[Slot[1], ",", Slot[2], ",", Slot[3], ",", Slot[4], ",", Slot[5], ",", Slot[6], ",", Slot[7]]], "}"]], "]"]]]]]
                            Alors b m
                            Sinon a m
                     Fin de Si
            Fin de Tant que
Algorithme 2

            a 2
            b 3
 
            Tant que b - a < 0,1 TemplateBox[List[RowBox[List["b", "-", "a"]], "\"\[ThinSpace]\"", "<", "\"\[ThinSpace]\"", TagBox[InterpretationBox["\"0,1\"", 0.100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001`100., Rule[AutoDelete, True]], Function[NumberForm[Slot[1], List[1, 1], Rule[ExponentFunction, Function[Null]], Rule[NumberPoint, ","], Rule[DigitBlock, 3], Rule[NumberSeparator, " "]]]]], "Row", Rule[DisplayFunction, Function[RowBox[List[Slot[1], "\[InvisibleSpace]", Slot[2], "\[InvisibleSpace]", Slot[3], "\[InvisibleSpace]", Slot[4], "\[InvisibleSpace]", Slot[5]]]]], Rule[InterpretationFunction, Function[RowBox[List["Row", "[", RowBox[List["{", RowBox[List[Slot[1], ",", Slot[2], ",", Slot[3], ",", Slot[4], ",", Slot[5]]], "}"]], "]"]]]]]
                  m a + b 2
                     Si f ( m ) < 0 TemplateBox[List[RowBox[List["f", "(", "m", ")"]], "<", "0"], "Row", Rule[DisplayFunction, Function[RowBox[List[Slot[1], "\[InvisibleSpace]", Slot[2], "\[InvisibleSpace]", Slot[3]]]]], Rule[InterpretationFunction, Function[RowBox[List["Row", "[", RowBox[List["{", RowBox[List[Slot[1], ",", Slot[2], ",", Slot[3]]], "}"]], "]"]]]]]
                            Alors b m
                            Sinon a m
                     Fin de Si
            Fin de Tant que
Algorithme 3

            a 2
            b 3
 
            Tant que b - a > 0,1 TemplateBox[List[RowBox[List["b", "-", "a"]], "\"\[ThinSpace]\"", ">", "\"\[ThinSpace]\"", TagBox[InterpretationBox["\"0,1\"", 0.100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001`100., Rule[AutoDelete, True]], Function[NumberForm[Slot[1], List[1, 1], Rule[ExponentFunction, Function[Null]], Rule[NumberPoint, ","], Rule[DigitBlock, 3], Rule[NumberSeparator, " "]]]]], "Row", Rule[DisplayFunction, Function[RowBox[List[Slot[1], "\[InvisibleSpace]", Slot[2], "\[InvisibleSpace]", Slot[3], "\[InvisibleSpace]", Slot[4], "\[InvisibleSpace]", Slot[5]]]]], Rule[InterpretationFunction, Function[RowBox[List["Row", "[", RowBox[List["{", RowBox[List[Slot[1], ",", Slot[2], ",", Slot[3], ",", Slot[4], ",", Slot[5]]], "}"]], "]"]]]]]
                  m a + b 2
                     Si f ( a )  ×  f ( m ) < 0 TemplateBox[List[RowBox[List["f", "(", "a", ")"]], "\"\[ThinSpace]\[Times]\[ThinSpace]\"", RowBox[List["f", "(", "m", ")"]], "\"\[ThinSpace]\"", "<", "\"\[ThinSpace]\"", "0"], "Row", Rule[DisplayFunction, Function[RowBox[List[Slot[1], "\[InvisibleSpace]", Slot[2], "\[InvisibleSpace]", Slot[3], "\[InvisibleSpace]", Slot[4], "\[InvisibleSpace]", Slot[5], "\[InvisibleSpace]", Slot[6], "\[InvisibleSpace]", Slot[7]]]]], Rule[InterpretationFunction, Function[RowBox[List["Row", "[", RowBox[List["{", RowBox[List[Slot[1], ",", Slot[2], ",", Slot[3], ",", Slot[4], ",", Slot[5], ",", Slot[6], ",", Slot[7]]], "}"]], "]"]]]]]
                            Alors a m
                            Sinon b m
                     Fin de Si
            Fin de Tant que