(* 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[ 3070299, 52873] NotebookOptionsPosition[ 3034378, 51953] NotebookOutlinePosition[ 3053412, 52316] CellTagsIndexPosition[ 3052693, 52297] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["\<\ TEMA 2 Definici\[OAcute]n y representaci\[OAcute]n de funciones\ \>", "Title", CellChangeTimes->{ 3.427365121439425*^9, 3.428127464015625*^9, {3.4589714283125*^9, 3.458971434609375*^9}, {3.47167104190625*^9, 3.471671050375*^9}}, TextAlignment->Center, TextJustification->0], Cell[CellGroupData[{ Cell["Introducci\[OAcute]n", "Section 1"], Cell[TextData[{ "Dos de las capacidades m\[AAcute]s importantes de ", StyleBox["Mathematica ", FontSlant->"Italic"], "tanto en la ense\[NTilde]anza de las Matem\[AAcute]ticas como en sus \ aplicaciones cientifico-t\[EAcute]cnicas son la realizaci\[OAcute]n de c\ \[AAcute]lculos simb\[OAcute]licos y num\[EAcute]ricos, y la realizaci\ \[OAcute]n de representaciones gr\[AAcute]ficas.\nEn este tema vamos a \ realizar c\[AAcute]lculos simb\[OAcute]licos con expresiones \ algebr\[AAcute]icas y a hacer las primeras definiciones y representaciones gr\ \[AAcute]ficas de funciones." }], "Text", CellChangeTimes->{ 3.425710648609375*^9, {3.456646338359375*^9, 3.456646481515625*^9}, { 3.45664651303125*^9, 3.456646574234375*^9}}, TextAlignment->Left, TextJustification->1] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Operaciones algebr\[AAcute]icas elementales\ \>", "Section 1"], Cell[TextData[{ "Las expresiones algebr\[AAcute]icas se introducen igual que los \ n\[UAcute]meros. ", StyleBox["Mathematica", FontSlant->"Italic"], " permite manejar expresiones de todo tipo, desde polinomios con muchos t\ \[EAcute]rminos a complejas combinaciones de funciones matem\[AAcute]ticas y \ proporciona comandos que nos permiten la transformaci\[OAcute]n y \ simplificaci\[OAcute]n de estas expresiones. Entre los comandos que realizan \ estas operaciones est\[AAcute]n: Expand, Factor, Together, Apart, Cancel y \ Simplify. Se puede obtener informaci\[OAcute]n sobre cada una de ellos con \ el s\[IAcute]mbolo ?.\n" }], "Text", CellChangeTimes->{{3.425710804546875*^9, 3.425710982578125*^9}, { 3.4257110509375*^9, 3.4257110558125*^9}, {3.4273652006277742`*^9, 3.4273652051184263`*^9}, {3.427367351820075*^9, 3.427367387864448*^9}, { 3.45664660075*^9, 3.456646638671875*^9}}, TextAlignment->Left, TextJustification->1], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"?", "Expand"}]], "Input"], Cell[BoxData[ RowBox[{ StyleBox["\<\"\!\(\*RowBox[{\\\"Expand\\\", \\\"[\\\", \ StyleBox[\\\"expr\\\", \\\"TI\\\"], \\\"]\\\"}]\) expands out products and \ positive integer powers in \!\(\*StyleBox[\\\"expr\\\", \\\"TI\\\"]\). \ \\n\!\(\*RowBox[{\\\"Expand\\\", \\\"[\\\", RowBox[{StyleBox[\\\"expr\\\", \\\ \"TI\\\"], \\\",\\\", StyleBox[\\\"patt\\\", \\\"TI\\\"]}], \\\"]\\\"}]\) \ leaves unexpanded any parts of \!\(\*StyleBox[\\\"expr\\\", \\\"TI\\\"]\) \ that are free of the pattern \!\(\*StyleBox[\\\"patt\\\", \\\"TI\\\"]\). \ \"\>", "MSG"], "\[NonBreakingSpace]", ButtonBox[ StyleBox["\[RightSkeleton]", "SR"], Active->True, BaseStyle->"Link", ButtonData->"paclet:ref/Expand"]}]], "Print", "PrintUsage", CellChangeTimes->{3.469794901953125*^9}, CellTags->"Info3469798501-8602478"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"?", "Factor"}]], "Input"], Cell[BoxData[ RowBox[{ StyleBox["\<\"\!\(\*RowBox[{\\\"Factor\\\", \\\"[\\\", \ StyleBox[\\\"poly\\\", \\\"TI\\\"], \\\"]\\\"}]\) factors a polynomial over \ the integers. \\n\!\(\*RowBox[{\\\"Factor\\\", \\\"[\\\", RowBox[{StyleBox[\\\ \"poly\\\", \\\"TI\\\"], \\\",\\\", RowBox[{\\\"Modulus\\\", \\\"->\\\", \ StyleBox[\\\"p\\\", \\\"TI\\\"]}]}], \\\"]\\\"}]\) factors a polynomial \ modulo a prime \!\(\*StyleBox[\\\"p\\\", \\\"TI\\\"]\). \ \\n\!\(\*RowBox[{\\\"Factor\\\", \\\"[\\\", RowBox[{StyleBox[\\\"poly\\\", \\\ \"TI\\\"], \\\",\\\", RowBox[{\\\"Extension\\\", \\\"->\\\", \ RowBox[{\\\"{\\\", RowBox[{SubscriptBox[StyleBox[\\\"a\\\", \\\"TI\\\"], \ StyleBox[\\\"1\\\", \\\"TR\\\"]], \\\",\\\", SubscriptBox[StyleBox[\\\"a\\\", \ \\\"TI\\\"], StyleBox[\\\"2\\\", \\\"TR\\\"]], \\\",\\\", StyleBox[\\\"\ \[Ellipsis]\\\", \\\"TR\\\"]}], \\\"}\\\"}]}]}], \\\"]\\\"}]\) factors a \ polynomial allowing coefficients that are rational combinations of the \ algebraic numbers \!\(\*SubscriptBox[StyleBox[\\\"a\\\", \\\"TI\\\"], \ StyleBox[\\\"i\\\", \\\"TI\\\"]]\). \"\>", "MSG"], "\[NonBreakingSpace]", ButtonBox[ StyleBox["\[RightSkeleton]", "SR"], Active->True, BaseStyle->"Link", ButtonData->"paclet:ref/Factor"]}]], "Print", "PrintUsage", CellChangeTimes->{3.469794902203125*^9}, CellTags->"Info3469798502-6579079"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"?", "Together"}]], "Input"], Cell[BoxData[ RowBox[{ StyleBox["\<\"\!\(\*RowBox[{\\\"Together\\\", \\\"[\\\", \ StyleBox[\\\"expr\\\", \\\"TI\\\"], \\\"]\\\"}]\) puts terms in a sum over a \ common denominator, and cancels factors in the result. \"\>", "MSG"], "\[NonBreakingSpace]", ButtonBox[ StyleBox["\[RightSkeleton]", "SR"], Active->True, BaseStyle->"Link", ButtonData->"paclet:ref/Together"]}]], "Print", "PrintUsage", CellChangeTimes->{3.469794902390625*^9}, CellTags->"Info3469798502-2224153"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"?", "Apart"}]], "Input"], Cell[BoxData[ RowBox[{ StyleBox["\<\"\!\(\*RowBox[{\\\"Apart\\\", \\\"[\\\", \ StyleBox[\\\"expr\\\", \\\"TI\\\"], \\\"]\\\"}]\) rewrites a rational \ expression as a sum of terms with minimal denominators. \ \\n\!\(\*RowBox[{\\\"Apart\\\", \\\"[\\\", RowBox[{StyleBox[\\\"expr\\\", \ \\\"TI\\\"], \\\",\\\", StyleBox[\\\"var\\\", \\\"TI\\\"]}], \\\"]\\\"}]\) \ treats all variables other than \!\(\*StyleBox[\\\"var\\\", \\\"TI\\\"]\) as \ constants. \"\>", "MSG"], "\[NonBreakingSpace]", ButtonBox[ StyleBox["\[RightSkeleton]", "SR"], Active->True, BaseStyle->"Link", ButtonData->"paclet:ref/Apart"]}]], "Print", "PrintUsage", CellChangeTimes->{3.469794902578125*^9}, CellTags->"Info3469798502-3914342"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"?", "Cancel"}]], "Input"], Cell[BoxData[ RowBox[{ StyleBox["\<\"\!\(\*RowBox[{\\\"Cancel\\\", \\\"[\\\", \ StyleBox[\\\"expr\\\", \\\"TI\\\"], \\\"]\\\"}]\) cancels out common factors \ in the numerator and denominator of \!\(\*StyleBox[\\\"expr\\\", \ \\\"TI\\\"]\). \"\>", "MSG"], "\[NonBreakingSpace]", ButtonBox[ StyleBox["\[RightSkeleton]", "SR"], Active->True, BaseStyle->"Link", ButtonData->"paclet:ref/Cancel"]}]], "Print", "PrintUsage", CellChangeTimes->{3.46979490278125*^9}, CellTags->"Info3469798502-8867515"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"?", "Simplify"}]], "Input"], Cell[BoxData[ RowBox[{ StyleBox["\<\"\!\(\*RowBox[{\\\"Simplify\\\", \\\"[\\\", \ StyleBox[\\\"expr\\\", \\\"TI\\\"], \\\"]\\\"}]\) performs a sequence of \ algebraic and other transformations on \!\(\*StyleBox[\\\"expr\\\", \ \\\"TI\\\"]\), and returns the simplest form it finds. \ \\n\!\(\*RowBox[{\\\"Simplify\\\", \\\"[\\\", RowBox[{StyleBox[\\\"expr\\\", \ \\\"TI\\\"], \\\",\\\", StyleBox[\\\"assum\\\", \\\"TI\\\"]}], \\\"]\\\"}]\) \ does simplification using assumptions. \"\>", "MSG"], "\[NonBreakingSpace]", ButtonBox[ StyleBox["\[RightSkeleton]", "SR"], Active->True, BaseStyle->"Link", ButtonData->"paclet:ref/Simplify"]}]], "Print", "PrintUsage", CellChangeTimes->{3.469794902984375*^9}, CellTags->"Info3469798502-2677681"] }, Open ]], Cell["\<\ Estos comandos se pueden introducir de dos formas, tecle\[AAcute]ndolos \ directamente o utilizando la paleta ALGEBRAIC MAIPULATIONS. Los siguientes ejemplos muestran la utilizaci\[OAcute]n de estos comandos.\ \>", "Text", CellChangeTimes->{{3.456646652*^9, 3.456646658953125*^9}}], Cell[CellGroupData[{ Cell[TextData[{ "Ejemplo 1\n\ta) Descomponer en factores el polinomio ", Cell[BoxData[ RowBox[{"\t\t", RowBox[{ SuperscriptBox["x", "5"], "-", RowBox[{"5", " ", SuperscriptBox["x", "4"], " ", "y"}], "-", RowBox[{"5", " ", SuperscriptBox["x", "3"], " ", SuperscriptBox["y", "2"]}], "+", RowBox[{"45", " ", SuperscriptBox["x", "2"], " ", SuperscriptBox["y", "3"]}], "-", RowBox[{"108", " ", SuperscriptBox["y", "5"]}]}]}]]], "\n\tb) Desarrollar el producto ", Cell[BoxData[ RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"x", "-", RowBox[{"3", " ", "y"}]}], ")"}], "3"], " ", SuperscriptBox[ RowBox[{"(", RowBox[{"x", "+", RowBox[{"2", " ", "y"}]}], ")"}], "2"]}]]], "\n\tc) Sumar, poniendo denominador comun las fracciones: \t\t", Cell[BoxData[ RowBox[{ FractionBox[ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "2"}], "+", "x"}], ")"}], " ", RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", "x"}], ")"}]}], RowBox[{ RowBox[{"-", "4"}], "+", SuperscriptBox["x", "2"]}]], "+", FractionBox[ SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", "x"}], ")"}], "2"], RowBox[{ RowBox[{"-", "1"}], "+", SuperscriptBox["x", "2"]}]]}]]] }], "Subsection", CellChangeTimes->{{3.427367973704398*^9, 3.4273679743779907`*^9}, { 3.456646745234375*^9, 3.456646767390625*^9}, {3.45664685428125*^9, 3.456646889265625*^9}}, TextAlignment->Left, TextJustification->0.], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Factor", "[", RowBox[{ SuperscriptBox["x", "5"], "-", RowBox[{"5", " ", SuperscriptBox["x", "4"], " ", "y"}], "-", RowBox[{"5", " ", SuperscriptBox["x", "3"], " ", SuperscriptBox["y", "2"]}], "+", RowBox[{"45", " ", SuperscriptBox["x", "2"], " ", SuperscriptBox["y", "3"]}], "-", RowBox[{"108", " ", SuperscriptBox["y", "5"]}]}], "]"}]], "Input", CellChangeTimes->{3.456646900328125*^9}], Cell[BoxData[ RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"x", "-", RowBox[{"3", " ", "y"}]}], ")"}], "3"], " ", SuperscriptBox[ RowBox[{"(", RowBox[{"x", "+", RowBox[{"2", " ", "y"}]}], ")"}], "2"]}]], "Output", CellChangeTimes->{ 3.461397624093303*^9, {3.4697948845625*^9, 3.4697949030625*^9}}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Expand", "[", RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"x", "-", RowBox[{"3", " ", "y"}]}], ")"}], "3"], " ", SuperscriptBox[ RowBox[{"(", RowBox[{"x", "+", RowBox[{"2", " ", "y"}]}], ")"}], "2"]}], "]"}]], "Input", CellChangeTimes->{3.456646952609375*^9}], Cell[BoxData[ RowBox[{ SuperscriptBox["x", "5"], "-", RowBox[{"5", " ", SuperscriptBox["x", "4"], " ", "y"}], "-", RowBox[{"5", " ", SuperscriptBox["x", "3"], " ", SuperscriptBox["y", "2"]}], "+", RowBox[{"45", " ", SuperscriptBox["x", "2"], " ", SuperscriptBox["y", "3"]}], "-", RowBox[{"108", " ", SuperscriptBox["y", "5"]}]}]], "Output", CellChangeTimes->{ 3.461397624108928*^9, {3.46979488459375*^9, 3.46979490309375*^9}}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Together", "[", RowBox[{ FractionBox[ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "2"}], "+", "x"}], ")"}], " ", RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", "x"}], ")"}]}], RowBox[{ RowBox[{"-", "4"}], "+", SuperscriptBox["x", "2"]}]], "+", FractionBox[ SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", "x"}], ")"}], "2"], RowBox[{ RowBox[{"-", "1"}], "+", SuperscriptBox["x", "2"]}]]}], "]"}]], "Input", CellChangeTimes->{3.45664696315625*^9}], Cell[BoxData[ FractionBox[ RowBox[{ RowBox[{"-", "3"}], "+", "x", "+", RowBox[{"2", " ", SuperscriptBox["x", "2"]}]}], RowBox[{ RowBox[{"(", RowBox[{"1", "+", "x"}], ")"}], " ", RowBox[{"(", RowBox[{"2", "+", "x"}], ")"}]}]]], "Output", CellChangeTimes->{ 3.461397624140178*^9, {3.469794884625*^9, 3.469794903125*^9}}] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "Ejemplo 2\n\ta) Descomponer en fracciones parciales : ", Cell[BoxData[ FractionBox[ RowBox[{ RowBox[{"2", " ", SuperscriptBox["x", "2"]}], "+", "x", "-", "3"}], RowBox[{ SuperscriptBox["x", "2"], "+", RowBox[{"3", "x"}], "+", "2"}]]]], "\n\tb) Simplificar : ", Cell[BoxData[ FractionBox[ RowBox[{ SuperscriptBox["x", "2"], "-", RowBox[{"3", "x"}], "+", "2"}], RowBox[{ SuperscriptBox["x", "2"], "-", RowBox[{"2", "x"}], "+", "1"}]]]] }], "Subsection", CellChangeTimes->{{3.456646991359375*^9, 3.4566470475*^9}, { 3.45664707959375*^9, 3.456647108921875*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Apart", "[", FractionBox[ RowBox[{ RowBox[{"2", " ", SuperscriptBox["x", "2"]}], "+", "x", "-", "3"}], RowBox[{ SuperscriptBox["x", "2"], "+", RowBox[{"3", "x"}], "+", "2"}]], "]"}]], "Input", CellChangeTimes->{3.45664709221875*^9}], Cell[BoxData[ RowBox[{"2", "-", FractionBox["2", RowBox[{"1", "+", "x"}]], "-", FractionBox["3", RowBox[{"2", "+", "x"}]]}]], "Output", CellChangeTimes->{ 3.461397624171428*^9, {3.469794884671875*^9, 3.46979490315625*^9}}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Cancel", "[", FractionBox[ RowBox[{ SuperscriptBox["x", "2"], "-", RowBox[{"3", "x"}], "+", "2"}], RowBox[{ SuperscriptBox["x", "2"], "-", RowBox[{"2", "x"}], "+", "1"}]], "]"}]], "Input", CellChangeTimes->{{3.456647101609375*^9, 3.456647112859375*^9}}], Cell[BoxData[ FractionBox[ RowBox[{ RowBox[{"-", "2"}], "+", "x"}], RowBox[{ RowBox[{"-", "1"}], "+", "x"}]]], "Output", CellChangeTimes->{ 3.461397624218303*^9, {3.46979488471875*^9, 3.469794903203125*^9}}] }, Open ]], Cell["\<\ Adem\[AAcute]s, Mathematica nos permite manipular fracciones mediante los \ comandos: Numerator, ExpandNumerator, Denominator y ExpandAll. Las siguientes \ entradas nos permiten obtener informaci\[OAcute]n sobre estos comandos.\ \>", "Text", CellChangeTimes->{ 3.425711099421875*^9, {3.45664713228125*^9, 3.456647144515625*^9}}, TextAlignment->Left, TextJustification->1], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"?", "Numerator"}]], "Input"], Cell[BoxData[ RowBox[{ StyleBox["\<\"\!\(\*RowBox[{\\\"Numerator\\\", \\\"[\\\", StyleBox[\\\"expr\ \\\", \\\"TI\\\"], \\\"]\\\"}]\) gives the numerator of \ \!\(\*StyleBox[\\\"expr\\\", \\\"TI\\\"]\). \"\>", "MSG"], "\[NonBreakingSpace]", ButtonBox[ StyleBox["\[RightSkeleton]", "SR"], Active->True, BaseStyle->"Link", ButtonData->"paclet:ref/Numerator"]}]], "Print", "PrintUsage", CellChangeTimes->{3.469794903359375*^9}, CellTags->"Info3469798503-9656795"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"?", "ExpandNumerator"}]], "Input"], Cell[BoxData[ RowBox[{ StyleBox["\<\"\!\(\*RowBox[{\\\"ExpandNumerator\\\", \\\"[\\\", StyleBox[\\\ \"expr\\\", \\\"TI\\\"], \\\"]\\\"}]\) expands out products and powers that \ appear in the numerator of \!\(\*StyleBox[\\\"expr\\\", \\\"TI\\\"]\). \"\>", "MSG"], "\[NonBreakingSpace]", ButtonBox[ StyleBox["\[RightSkeleton]", "SR"], Active->True, BaseStyle->"Link", ButtonData->"paclet:ref/ExpandNumerator"]}]], "Print", "PrintUsage", CellChangeTimes->{3.46979490359375*^9}, CellTags->"Info3469798503-7412798"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"?", "Denominator"}]], "Input"], Cell[BoxData[ RowBox[{ StyleBox["\<\"\!\(\*RowBox[{\\\"Denominator\\\", \\\"[\\\", \ StyleBox[\\\"expr\\\", \\\"TI\\\"], \\\"]\\\"}]\) gives the denominator of \!\ \(\*StyleBox[\\\"expr\\\", \\\"TI\\\"]\). \"\>", "MSG"], "\[NonBreakingSpace]", ButtonBox[ StyleBox["\[RightSkeleton]", "SR"], Active->True, BaseStyle->"Link", ButtonData->"paclet:ref/Denominator"]}]], "Print", "PrintUsage", CellChangeTimes->{3.4697949038125*^9}, CellTags->"Info3469798503-2382094"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"?", "ExpandDenominator"}]], "Input"], Cell[BoxData[ RowBox[{ StyleBox["\<\"\!\(\*RowBox[{\\\"ExpandDenominator\\\", \\\"[\\\", StyleBox[\ \\\"expr\\\", \\\"TI\\\"], \\\"]\\\"}]\) expands out products and powers that \ appear as denominators in \!\(\*StyleBox[\\\"expr\\\", \\\"TI\\\"]\). \"\>", "MSG"], "\[NonBreakingSpace]", ButtonBox[ StyleBox["\[RightSkeleton]", "SR"], Active->True, BaseStyle->"Link", ButtonData->"paclet:ref/ExpandDenominator"]}]], "Print", "PrintUsage", CellChangeTimes->{3.46979490403125*^9}, CellTags->"Info3469798503-1662603"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"?", "ExpandAll"}]], "Input"], Cell[BoxData[ RowBox[{ StyleBox["\<\"\!\(\*RowBox[{\\\"ExpandAll\\\", \\\"[\\\", StyleBox[\\\"expr\ \\\", \\\"TI\\\"], \\\"]\\\"}]\) expands out all products and integer powers \ in any part of \!\(\*StyleBox[\\\"expr\\\", \\\"TI\\\"]\). \ \\n\!\(\*RowBox[{\\\"ExpandAll\\\", \\\"[\\\", RowBox[{StyleBox[\\\"expr\\\", \ \\\"TI\\\"], \\\",\\\", StyleBox[\\\"patt\\\", \\\"TI\\\"]}], \\\"]\\\"}]\) \ avoids expanding parts of \!\(\*StyleBox[\\\"expr\\\", \\\"TI\\\"]\) that do \ not contain terms matching the pattern \!\(\*StyleBox[\\\"patt\\\", \ \\\"TI\\\"]\). \"\>", "MSG"], "\[NonBreakingSpace]", ButtonBox[ StyleBox["\[RightSkeleton]", "SR"], Active->True, BaseStyle->"Link", ButtonData->"paclet:ref/ExpandAll"]}]], "Print", "PrintUsage", CellChangeTimes->{3.46979490425*^9}, CellTags->"Info3469798504-2503816"] }, Open ]], Cell[TextData[{ "En Mathematica podemos guardar en la memoria expresiones, gr\[AAcute]ficos, \ etc. como distintos tipos de objetos. Es como ponerles un nombre para poder \ utilizarlos cuando sea necesario. Esto se hace simplemente utilizando el \ signo = que asigna la expresi\[OAcute]n, gr\[AAcute]fico, etc. a la variable \ utilizada como nombre. Como las funciones y comandos disponibles en el Kernel \ comienzan siempre con may\[UAcute]sculas es una buena pr\[AAcute]ctica que \ los nombres de los objetos comiencen con min\[UAcute]sculas para evitar \ errores.\nPara evaluar una expresi\[OAcute]n algebr\[AAcute]ica para valores \ concretos de las variables se puede asignar el valor a la variable o utilizar \ el operador ", StyleBox["reemplazamiento. ", FontWeight->"Bold"], "Por ejemplo, en la siguiente entrada se asigna a la variable ", StyleBox["a", FontWeight->"Bold"], " el desarrollo de ", Cell[BoxData[ FormBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"x", "+", "y"}], ")"}], "3"], TraditionalForm]]], " y a continuaci\[OAcute]n se hace ", StyleBox["x", FontWeight->"Bold"], " igual a 3 con lo que ", StyleBox["a", FontWeight->"Bold"], " resulta igual a ", Cell[BoxData[ RowBox[{"27", "+", RowBox[{"27", " ", "y"}], "+", RowBox[{"9", " ", SuperscriptBox["y", "2"]}], "+", SuperscriptBox["y", "3"]}]], CellChangeTimes->{3.4589728744375*^9}], ".\n" }], "Text", CellChangeTimes->{{3.42571117628125*^9, 3.425711234546875*^9}, { 3.425711293328125*^9, 3.425711361140625*^9}, {3.425711398859375*^9, 3.42571145034375*^9}, {3.428127543828125*^9, 3.428127544109375*^9}, { 3.456647171671875*^9, 3.456647319390625*^9}, {3.458972373296875*^9, 3.45897239303125*^9}, {3.458972774625*^9, 3.45897283475*^9}, { 3.458972877875*^9, 3.45897296471875*^9}}, TextJustification->1], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"a", "=", RowBox[{"Expand", "[", RowBox[{ RowBox[{"(", RowBox[{"x", "+", "y"}], ")"}], "^", "3"}], "]"}]}], ";", RowBox[{"x", "=", "3"}], ";", "a"}]], "Input", CellChangeTimes->{{3.4589728395625*^9, 3.458972871828125*^9}}], Cell[BoxData[ RowBox[{"27", "+", RowBox[{"27", " ", "y"}], "+", RowBox[{"9", " ", SuperscriptBox["y", "2"]}], "+", SuperscriptBox["y", "3"]}]], "Output", CellChangeTimes->{ 3.461397625374553*^9, {3.469794886046875*^9, 3.46979490434375*^9}}] }, Open ]], Cell[TextData[{ "El problema es que en la variable x queda almacenado el valor 3. Se puede \ obtener el mismo resultado sin que en la variable x quede almacenado el 3 \ utilizando el operador ", StyleBox["reemplazamiento.", FontWeight->"Bold"] }], "Text", CellChangeTimes->{{3.4589729684375*^9, 3.458973056984375*^9}}] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Operador Reemplazamiento", "Section", CellChangeTimes->{{3.45897308896875*^9, 3.4589730945*^9}}], Cell[TextData[{ "El operador reemplazamiento , ", StyleBox["/.", FontWeight->"Bold"], ", tiene la siguiente sintaxis:\n", StyleBox["Expr /. reglas ", FontWeight->"Bold"], "\nAplicando las ", StyleBox["reglas", FontWeight->"Bold"], " en la expresi\[OAcute]n o expresiones, ", StyleBox["Expr.", FontWeight->"Bold"], "\nEste operador aplica una regla o serie de reglas intentando transformar \ cada subparte de ", StyleBox["Expr", FontWeight->"Bold"], ". Por ejemplo, supongamos que se quiere evaluar la expresi\[OAcute]n:\n\t", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["x", "2"], "+", "x", "+", "1"}], TraditionalForm]]], " para x = 1.\nPodemos hacerlo de dos formas. En primer lugar, asignando a x \ el valor 1:" }], "Text", CellChangeTimes->{{3.458973114328125*^9, 3.4589731235*^9}, { 3.45897318415625*^9, 3.458973292484375*^9}, {3.45897334134375*^9, 3.45897336409375*^9}, {3.458973403953125*^9, 3.458973430421875*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"Clear", "[", "a", "]"}], ";", RowBox[{"a", "=", RowBox[{ RowBox[{"x", "^", "2"}], "+", "x", "+", "1"}]}], ";", RowBox[{"x", "=", "1"}], ";", "a"}]], "Input", CellChangeTimes->{{3.427693181420295*^9, 3.427693191766214*^9}, { 3.427693287188525*^9, 3.427693301571433*^9}, {3.45897343646875*^9, 3.45897343975*^9}, {3.4589734921875*^9, 3.45897349534375*^9}}], Cell[BoxData["13"], "Output", CellChangeTimes->{ 3.461397625405803*^9, {3.469794886078125*^9, 3.469794904375*^9}}] }, Open ]], Cell["\<\ De esta forma en x ha quedado almacenado un valor igual a 1. Utilizando el \ operador reemplazamiento podemos evitar esto haciendo,\ \>", "Text", CellChangeTimes->{{3.42769333528968*^9, 3.427693403106722*^9}, 3.45897350453125*^9}], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"Clear", "[", "x", "]"}], ";", RowBox[{"a", "/.", RowBox[{"x", "\[Rule]", "1"}]}]}], "\[IndentingNewLine]", "x"}], "Input", CellChangeTimes->{{3.427693408330215*^9, 3.4276934163262444`*^9}, { 3.45897351784375*^9, 3.458973537703125*^9}}], Cell[BoxData["13"], "Output", CellChangeTimes->{ 3.461397625421428*^9, {3.469794886109375*^9, 3.46979490440625*^9}}], Cell[BoxData["x"], "Output", CellChangeTimes->{ 3.461397625421428*^9, {3.469794886109375*^9, 3.469794904421875*^9}}] }, Open ]], Cell[TextData[{ "La regla aplicada en este caso es \[UAcute]nica y muy sencilla. Se pueden \ aplicar varias reglas y m\[AAcute]s complejas. Sea, por ejemplo,\n\ta=", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["x", "3"], "+", RowBox[{ SuperscriptBox["x", "2"], "y"}], "+", SuperscriptBox["xy", "2"], "+", SuperscriptBox["y", "3"]}], TraditionalForm]]], "\nVamos a sustituir x por a+b, e y por a-b" }], "Text", CellChangeTimes->{{3.4276934373070993`*^9, 3.427693584798161*^9}}], Cell[BoxData[ RowBox[{"Clear", "[", "a", "]"}]], "Input", CellChangeTimes->{{3.4276936518898478`*^9, 3.427693656237906*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"expre", "=", RowBox[{ RowBox[{"x", "^", "3"}], "+", RowBox[{ RowBox[{"x", "^", "2"}], "*", "y"}], "+", RowBox[{"x", "*", RowBox[{"y", "^", "2"}]}], "+", RowBox[{"y", "^", "3"}]}]}]], "Input", CellChangeTimes->{{3.4276935910372868`*^9, 3.427693606924473*^9}, { 3.427693637939601*^9, 3.4276936416953278`*^9}}], Cell[BoxData[ RowBox[{ SuperscriptBox["x", "3"], "+", RowBox[{ SuperscriptBox["x", "2"], " ", "y"}], "+", RowBox[{"x", " ", SuperscriptBox["y", "2"]}], "+", SuperscriptBox["y", "3"]}]], "Output", CellChangeTimes->{ 3.461397625452678*^9, {3.46979488615625*^9, 3.46979490446875*^9}}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"expre", "/.", RowBox[{"{", RowBox[{ RowBox[{"x", "\[Rule]", RowBox[{"a", "+", "b"}]}], ",", RowBox[{"y", "\[Rule]", RowBox[{"a", "-", "b"}]}]}], "}"}]}]], "Input", CellChangeTimes->{{3.427693613677162*^9, 3.4276936308374033`*^9}, { 3.427693663463416*^9, 3.427693665917017*^9}}], Cell[BoxData[ RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"a", "-", "b"}], ")"}], "3"], "+", RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"a", "-", "b"}], ")"}], "2"], " ", RowBox[{"(", RowBox[{"a", "+", "b"}], ")"}]}], "+", RowBox[{ RowBox[{"(", RowBox[{"a", "-", "b"}], ")"}], " ", SuperscriptBox[ RowBox[{"(", RowBox[{"a", "+", "b"}], ")"}], "2"]}], "+", SuperscriptBox[ RowBox[{"(", RowBox[{"a", "+", "b"}], ")"}], "3"]}]], "Output", CellChangeTimes->{ 3.461397625483928*^9, {3.4697948861875*^9, 3.4697949045*^9}}] }, Open ]], Cell[TextData[{ "Con el comando ", StyleBox["Expand", FontWeight->"Bold"], " podemos desarrollar esta expresi\[OAcute]n" }], "Text", CellChangeTimes->{{3.427693674921989*^9, 3.4276936892887077`*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"expre", "/.", RowBox[{"{", RowBox[{ RowBox[{"x", "\[Rule]", RowBox[{"a", "+", "b"}]}], ",", RowBox[{"y", "\[Rule]", RowBox[{"a", "-", "b"}]}]}], "}"}]}], "//", "Expand"}]], "Input", CellChangeTimes->{{3.427693613677162*^9, 3.4276936308374033`*^9}, { 3.427693663463416*^9, 3.427693665917017*^9}, {3.427693714209968*^9, 3.4276937168999653`*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"4", " ", SuperscriptBox["a", "3"]}], "+", RowBox[{"4", " ", "a", " ", SuperscriptBox["b", "2"]}]}]], "Output", CellChangeTimes->{ 3.461397625499553*^9, {3.469794886234375*^9, 3.46979490453125*^9}}] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "Ejemplo 3\n Evaluar la fracci\[OAcute]n ", Cell[BoxData[ FractionBox[ RowBox[{ SuperscriptBox["x", "2"], "-", RowBox[{"3", "x"}], "+", "2"}], RowBox[{ SuperscriptBox["x", "2"], "-", RowBox[{"2", "x"}], "+", "1"}]]]], " para x=2 y para x=-4\n Evaluar el polinomio ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SuperscriptBox["x", "2"], SuperscriptBox["y", "2"]}], " ", "-", " ", SuperscriptBox["xy", "3"], "+", RowBox[{ SuperscriptBox["x", "3"], "y"}]}], TraditionalForm]]], " para x=1 e y=2." }], "Subsection", CellChangeTimes->{{3.425711457734375*^9, 3.425711523734375*^9}, { 3.45897361703125*^9, 3.458973651828125*^9}, {3.45897369996875*^9, 3.458973730140625*^9}, {3.45897377328125*^9, 3.458973775296875*^9}}, TextAlignment->Left, TextJustification->1], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"fraccion", "=", FractionBox[ RowBox[{ SuperscriptBox["x", "2"], "-", RowBox[{"3", "x"}], "+", "2"}], RowBox[{ SuperscriptBox["x", "2"], "-", RowBox[{"2", "x"}], "+", "1"}]]}]], "Input", CellChangeTimes->{{3.427365330067586*^9, 3.427365332030587*^9}, 3.458973661203125*^9}], Cell[BoxData[ FractionBox[ RowBox[{"2", "-", RowBox[{"3", " ", "x"}], "+", SuperscriptBox["x", "2"]}], RowBox[{"1", "-", RowBox[{"2", " ", "x"}], "+", SuperscriptBox["x", "2"]}]]], "Output", CellChangeTimes->{ 3.461397625530803*^9, {3.469794886265625*^9, 3.4697949045625*^9}}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"fraccion", " ", "/.", " ", RowBox[{"x", "\[Rule]", "2"}]}]], "Input", CellChangeTimes->{{3.427365335816494*^9, 3.427365336751343*^9}, { 3.4589736775*^9, 3.45897367803125*^9}}], Cell[BoxData["0"], "Output", CellChangeTimes->{ 3.461397625546428*^9, {3.469794886296875*^9, 3.46979490459375*^9}}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"fraccion", " ", "/.", " ", RowBox[{"x", "->", RowBox[{"-", "4"}]}]}]], "Input", CellChangeTimes->{{3.427365340398691*^9, 3.42736534106573*^9}, { 3.458973684859375*^9, 3.458973685359375*^9}}], Cell[BoxData[ FractionBox["6", "5"]], "Output", CellChangeTimes->{ 3.461397625577678*^9, {3.46979488634375*^9, 3.469794904640625*^9}}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{ RowBox[{"x", "^", "2"}], "*", RowBox[{"y", "^", "2"}]}], "-", RowBox[{"x", "*", RowBox[{"y", "^", "3"}]}], "+", RowBox[{ RowBox[{"x", "^", "3"}], "*", "y"}]}], "/.", RowBox[{"{", RowBox[{ RowBox[{"x", "\[Rule]", "1"}], ",", RowBox[{"y", "\[Rule]", "2"}]}], "}"}]}]], "Input", CellChangeTimes->{{3.425711527828125*^9, 3.425711550203125*^9}, { 3.4589737510625*^9, 3.458973780171875*^9}}], Cell[BoxData[ RowBox[{"-", "2"}]], "Output", CellChangeTimes->{ 3.461397625593303*^9, {3.469794886375*^9, 3.469794904671875*^9}}] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Definiendo y evaluando funciones", "Section 1"], Cell["\<\ La definici\[OAcute]n de funciones consiste en asociar a un s\[IAcute]mbolo \ que representa a la funci\[OAcute]n una regla o conjunto de reglas que se \ aplicar\[AAcute]n autom\[AAcute]ticamente cuando se llame a la \ funci\[OAcute]n. Por ejemplo para definir una funci\[OAcute]n sencilla como \ f(x)=sinx+cosx, haremos:\ \>", "Text", CellChangeTimes->{{3.4257115635*^9, 3.425711615859375*^9}, 3.42736742495022*^9, 3.427367455350143*^9, {3.45897379784375*^9, 3.458973886765625*^9}}, TextAlignment->Left, TextJustification->1], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"f", "[", "x_", "]"}], "=", RowBox[{ RowBox[{"Sin", "[", "x", "]"}], "+", RowBox[{"Cos", "[", "x", "]"}]}]}]], "Input", CellChangeTimes->{{3.45897388959375*^9, 3.458973896953125*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"Cos", "[", "x", "]"}], "+", RowBox[{"Sin", "[", "x", "]"}]}]], "Output", CellChangeTimes->{ 3.461397625624553*^9, {3.469794886421875*^9, 3.469794904703125*^9}}] }, Open ]], Cell["\<\ El argumento debe ir entre corchetes y seguido del gui\[OAcute]n bajo. Este \ gui\[OAcute]n bajo garantiza que la variable o argumento podr\[AAcute] ser \ sustitu\[IAcute]do por cualquier valor o expresi\[OAcute]n y solo se coloca \ en la definici\[OAcute]n y en el t\[EAcute]rmino de la izquierda.\ \>", "Text", CellChangeTimes->{{3.458973904171875*^9, 3.458973992546875*^9}}], Cell[CellGroupData[{ Cell[TextData[{ "Ejemplo 4: \nDefinir las funciones f(x)=", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["x", "3"], "+", "2"}], TraditionalForm]]], " , g(x)=", Cell[BoxData[ FormBox[ RadicalBox[ SuperscriptBox["x", "2"], "3"], TraditionalForm]]], " y h(x)=", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["e", "x"], "cosx"}], TraditionalForm]]], ". Calcular f(-1), g(1) y h(\[Pi]/4). En primer lugar es conveniente \ utilizar el comando Clear para borrar anteriores deficiones si es que las \ hubiera. Se puede preguntar a ", StyleBox["Mathematica", FontSlant->"Italic"], " sobre una determinada funci\[OAcute]n con el ?." }], "Subsection", CellChangeTimes->{{3.42571163265625*^9, 3.4257116408125*^9}, { 3.458974008046875*^9, 3.4589740571875*^9}}, TextAlignment->Left, TextJustification->1], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{"Clear", "[", RowBox[{"f", ",", "g", ",", "h"}], "]"}], "\n", RowBox[{ RowBox[{"f", "[", "x_", "]"}], "=", RowBox[{ SuperscriptBox["x", "3"], "+", "2"}]}]}], "Input", CellChangeTimes->{{3.458974069640625*^9, 3.45897407303125*^9}}], Cell[BoxData[ RowBox[{"2", "+", SuperscriptBox["x", "3"]}]], "Output", CellChangeTimes->{ 3.461397625640178*^9, {3.469794886453125*^9, 3.469794904734375*^9}}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"f", "[", RowBox[{"-", "1"}], "]"}]], "Input", CellChangeTimes->{{3.458974080515625*^9, 3.458974082328125*^9}}], Cell[BoxData["1"], "Output", CellChangeTimes->{ 3.461397625671428*^9, {3.4697948865*^9, 3.46979490478125*^9}}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"f", "[", "x", "]"}], " ", "/.", " ", RowBox[{"x", "\[Rule]", RowBox[{"-", "1"}]}]}]], "Input", CellChangeTimes->{{3.45897408746875*^9, 3.45897408825*^9}}], Cell[BoxData["1"], "Output", CellChangeTimes->{ 3.461397625687053*^9, {3.46979488653125*^9, 3.4697949048125*^9}}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"g", "[", "x_", "]"}], "=", RadicalBox[ RowBox[{"x", "^", "2"}], "3"]}]], "Input", CellChangeTimes->{{3.45897409409375*^9, 3.458974105625*^9}}], Cell[BoxData[ SuperscriptBox[ RowBox[{"(", SuperscriptBox["x", "2"], ")"}], RowBox[{"1", "/", "3"}]]], "Output", CellChangeTimes->{ 3.461397625718303*^9, {3.4697948865625*^9, 3.46979490484375*^9}}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"g", "[", "1", "]"}]], "Input", CellChangeTimes->{3.458974114671875*^9}], Cell[BoxData["1"], "Output", CellChangeTimes->{ 3.461397625749553*^9, {3.46979488659375*^9, 3.469794904875*^9}}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"g", "[", "x", "]"}], " ", "/.", " ", RowBox[{"x", "\[Rule]", "1"}]}]], "Input", CellChangeTimes->{3.45897412434375*^9}], Cell[BoxData["1"], "Output", CellChangeTimes->{ 3.461397625765178*^9, {3.469794886671875*^9, 3.46979490496875*^9}}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"h", "[", "x_", "]"}], "=", RowBox[{ RowBox[{"Exp", "[", "x", "]"}], "*", RowBox[{"Cos", "[", "x", "]"}]}]}]], "Input", CellChangeTimes->{3.425711668828125*^9, {3.4589741295*^9, 3.458974133875*^9}} ], Cell[BoxData[ RowBox[{ SuperscriptBox["\[ExponentialE]", "x"], " ", RowBox[{"Cos", "[", "x", "]"}]}]], "Output", CellChangeTimes->{ 3.461397625796428*^9, {3.469794886703125*^9, 3.469794905*^9}}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{"?", "h"}], "\n", RowBox[{"h", "[", "x", "]"}]}], "Input"], Cell["Global`h", "Print", "PrintUsage", CellChangeTimes->{3.4697949051875*^9}, CellTags->"Info3469798505-6777737"], Cell[BoxData[ InterpretationBox[GridBox[{ {GridBox[{ { RowBox[{ RowBox[{"h", "[", "x_", "]"}], "=", RowBox[{ SuperscriptBox["\[ExponentialE]", "x"], " ", RowBox[{"Cos", "[", "x", "]"}]}]}]} }, BaselinePosition->{Baseline, {1, 1}}, GridBoxAlignment->{ "Columns" -> {{Left}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxItemSize->{"Columns" -> {{ Scaled[0.999]}}, "ColumnsIndexed" -> {}, "Rows" -> {{1.}}, "RowsIndexed" -> {}}]} }, BaselinePosition->{Baseline, {1, 1}}, GridBoxAlignment->{ "Columns" -> {{Left}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}], Definition[$CellContext`h], Editable->False]], "Print", CellChangeTimes->{3.469794905203125*^9}, CellTags->"Info3469798505-6777737"], Cell[BoxData[ RowBox[{ SuperscriptBox["\[ExponentialE]", "x"], " ", RowBox[{"Cos", "[", "x", "]"}]}]], "Output", CellChangeTimes->{ 3.461397625968303*^9, {3.469794886984375*^9, 3.46979490528125*^9}}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"h", "[", FractionBox["\[Pi]", "4"], "]"}]], "Input", CellChangeTimes->{3.458974143890625*^9}], Cell[BoxData[ FractionBox[ SuperscriptBox["\[ExponentialE]", RowBox[{"\[Pi]", "/", "4"}]], SqrtBox["2"]]], "Output", CellChangeTimes->{ 3.461397625999553*^9, {3.46979488703125*^9, 3.469794905390625*^9}}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"h", "[", "x", "]"}], "/.", RowBox[{"x", "\[Rule]", FractionBox["\[Pi]", "4"]}]}]], "Input", CellChangeTimes->{3.45897414915625*^9}], Cell[BoxData[ FractionBox[ SuperscriptBox["\[ExponentialE]", RowBox[{"\[Pi]", "/", "4"}]], SqrtBox["2"]]], "Output", CellChangeTimes->{ 3.461397626015178*^9, {3.4697948870625*^9, 3.469794905421875*^9}}] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Definici\[OAcute]n inmediata y diferida\ \>", "Section", CellChangeTimes->{{3.458974823875*^9, 3.45897483209375*^9}}], Cell[TextData[{ "Una funci\[OAcute]n se puede definir con = o bien :=. En el primer caso se \ trata de la ", StyleBox["definici\[OAcute]n inmediata", FontSlant->"Italic"], " y se pide a Mathematica que la defina y seguidamente haga las operaciones \ que se indican. En el segundo caso se trata de la ", StyleBox["definici\[OAcute]n diferida", FontSlant->"Italic"], " y, en ese momento solo se define la funci\[OAcute]n , las operaciones se \ har\[AAcute]n despu\[EAcute]s cuando se vaya a utilizar la funci\[OAcute]n. \ En este caso ", StyleBox["Mathematica", FontSlant->"Italic"], " no devuelve ninguna salida. \nSe pueden definir funciones de una o de \ varias variables. En este caso cada variable deber\[AAcute] ir seguida del \ gui\[OAcute]n bajo pero \[UAcute]nicamente en la definici\[OAcute]n de la \ funci\[OAcute]n y a la izquierda del signo igual. \nLas siguientes entradas \ muestran definiciones de funciones de varias variables." }], "Text", CellChangeTimes->{{3.42571168921875*^9, 3.425711826671875*^9}, { 3.4257118704375*^9, 3.425711961578125*^9}, {3.427365434324916*^9, 3.427365440941324*^9}, {3.45897418909375*^9, 3.4589742539375*^9}}, TextAlignment->Left, TextJustification->1], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"f1", "[", RowBox[{"x_", ",", "y_"}], "]"}], "=", RowBox[{ RowBox[{"Cos", "[", "x", "]"}], "+", RowBox[{"Sin", "[", "y", " ", "]"}]}]}]], "Input", CellChangeTimes->{{3.458974256765625*^9, 3.45897427059375*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"Cos", "[", "x", "]"}], "+", RowBox[{"Sin", "[", "y", "]"}]}]], "Output", CellChangeTimes->{ 3.461397626046428*^9, {3.46979488709375*^9, 3.469794905453125*^9}}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"f2", "[", RowBox[{"x_", ",", "y_", ",", "z_"}], "]"}], "=", RowBox[{ RowBox[{"x", "^", "2"}], "+", RowBox[{"y", "^", "2"}], "+", RowBox[{"z", "^", "2"}]}]}]], "Input", CellChangeTimes->{{3.4589742738125*^9, 3.458974291578125*^9}}], Cell[BoxData[ RowBox[{ SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"], "+", SuperscriptBox["z", "2"]}]], "Output", CellChangeTimes->{ 3.461397626077678*^9, {3.469794887125*^9, 3.469794905484375*^9}}] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "Ejemplo 5\nUtilizar las dos formas de definici\[OAcute]n de funciones, = y \ := , para definir una funci\[OAcute]n igual al desarrollo de ", Cell[BoxData[ FormBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"t", "+", "1"}], ")"}], "2"], TraditionalForm]]], ", f1 para la definici\[OAcute]n inmediata y f2 para la diferida, y obtener \ el valor de f(a+b). \nSea f(t)=", Cell[BoxData[ RowBox[{"4", "-", SuperscriptBox["t", "2"]}]]], ", se pide:\n\ta) Calcular y desarrollar ", Cell[BoxData[ RowBox[{"f", RowBox[{"(", RowBox[{"a", "-", SuperscriptBox["b", "2"]}], ")"}]}]]], " .\n\tb) Calcular y simplificar (f(x+h)-f(x))/h" }], "Subsection", CellChangeTimes->{{3.427365451377993*^9, 3.427365458746715*^9}, { 3.45897430553125*^9, 3.458974434515625*^9}, {3.45897451503125*^9, 3.4589745279375*^9}}], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{"Clear", "[", "f", "]"}], "\n", RowBox[{ RowBox[{"f1", "[", "t_", "]"}], "=", RowBox[{"Expand", "[", SuperscriptBox[ RowBox[{"(", RowBox[{"t", "+", "1"}], ")"}], "2"], "]"}]}]}], "Input", CellChangeTimes->{{3.458974444828125*^9, 3.45897446828125*^9}, { 3.458974532484375*^9, 3.458974538421875*^9}}], Cell[BoxData[ RowBox[{"1", "+", RowBox[{"2", " ", "t"}], "+", SuperscriptBox["t", "2"]}]], "Output", CellChangeTimes->{ 3.461397626093303*^9, {3.46979488715625*^9, 3.46979490553125*^9}}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"f1", "[", RowBox[{"a", "+", "b"}], "]"}]], "Input", 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3.461397626155803*^9, {3.469794887234375*^9, 3.46979490559375*^9}}] }, Open ]], Cell[TextData[{ "En la definici\[OAcute]n inmediata se asigna a f1(t) la expresi\[OAcute]n ", Cell[BoxData[ RowBox[{"1", "+", RowBox[{"2", " ", "t"}], "+", SuperscriptBox["t", "2"]}]], CellChangeTimes->{3.45897446990625*^9, 3.45897453953125*^9}], ", mientras que en la diferida se asigna a f2(t) el desarrollo de ", Cell[BoxData[ FormBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"t", "+", "1"}], ")"}], "2"], TraditionalForm]]], " pero no en el momento de la definici\[OAcute]n sino cuando se llame a la \ funci\[OAcute]n, por lo que al pedir f2(a+b) se sustituye t por a+b y a \ continuaci\[OAcute]n se hace el desarrollo." }], "Text", CellChangeTimes->{{3.425711993078125*^9, 3.42571200153125*^9}, { 3.45897460365625*^9, 3.45897474159375*^9}}, TextAlignment->Left, TextJustification->1], Cell["\<\ Tambi\[EAcute]n se pueden definir funciones vectoriales\ \>", "Text", CellChangeTimes->{{3.458974850875*^9, 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Una funci\ \[OAcute]n sencilla de este tipo es la siguiente:\n\t\tf(t) = ", Cell[BoxData[ FormBox[ TagBox[ StyleBox[ RowBox[{"{", StyleBox[GridBox[{ { RowBox[{ RowBox[{"1", "+", RowBox[{"cost", " ", "t"}]}], " ", "<", " ", "0"}]}, { RowBox[{ RowBox[{"2", "+", RowBox[{"sint", " ", "t"}]}], "\[GreaterEqual]", " ", "0"}]} }], ShowAutoStyles->True]}], ShowAutoStyles->False], #& ], TraditionalForm]]], ".\nLas funciones continuas a trozos se pueden definir de varias formas. El \ comando Piecewise nos proporciona una forma sencilla para la \ definici\[OAcute]n de este tipo de funciones. La sintaxis del comando es:\n \t\ \tPiecewise[{{val1,cond1},{val2,cond2},...}].\nPara definir la \ funci\[OAcute]n anterior," }], "Text", CellChangeTimes->{{3.425713289015625*^9, 3.42571332425*^9}, { 3.425713377375*^9, 3.425713423640625*^9}, {3.4589749378125*^9, 3.458975201546875*^9}, {3.45897525159375*^9, 3.458975254984375*^9}}, TextAlignment->Left, TextJustification->1], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"f", "[", "t_", "]"}], "=", RowBox[{"Piecewise", "[", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"1", "+", RowBox[{"Cos", "[", "t", "]"}]}], ",", RowBox[{"t", "<", "0"}]}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"2", "+", RowBox[{"Sin", "[", "t", "]"}]}], ",", RowBox[{"0", "\[LessEqual]", "t"}]}], "}"}]}], "}"}], "]"}]}]], "Input", CellChangeTimes->{{3.4589752038125*^9, 3.458975311234375*^9}}], Cell[BoxData[ RowBox[{"\[Piecewise]", GridBox[{ { RowBox[{"1", "+", RowBox[{"Cos", "[", "t", "]"}]}], RowBox[{"t", "<", "0"}]}, { RowBox[{"2", "+", RowBox[{"Sin", "[", "t", "]"}]}], RowBox[{"0", "\[LessEqual]", "t"}]}, {"0", TagBox["True", "PiecewiseDefault", AutoDelete->False, DeletionWarning->True]} }, GridBoxAlignment->{ "Columns" -> {{Left}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxItemSize->{ "Columns" -> {{Automatic}}, "ColumnsIndexed" -> {}, "Rows" -> {{1.}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.84]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}]}]], "Output", CellChangeTimes->{ 3.461397626265178*^9, {3.469794887375*^9, 3.46979490571875*^9}}] }, Open ]], Cell[TextData[{ "Se pueden definir funciones \[UAcute]nicamente en un intervalo finito, por \ ejemplo, el [-10, 10)\n\t\tf(t) = ", Cell[BoxData[ FormBox[ TagBox[ StyleBox[ RowBox[{"{", StyleBox[GridBox[{ { RowBox[{ RowBox[{"1", "+", "cost", " ", "-", "10"}], "\[LessEqual]", " ", "t", " ", "<", " ", "0"}]}, { RowBox[{ RowBox[{"2", "+", RowBox[{"sint", " ", "0"}]}], "\[LessEqual]", " ", "t", "<", "10"}]} }], ShowAutoStyles->True]}], ShowAutoStyles->False], #& ], TraditionalForm]]] }], "Text", CellChangeTimes->{{3.4589753361875*^9, 3.458975444609375*^9}, { 3.4589755070625*^9, 3.4589755144375*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"Clear", "[", "f", "]"}], ";", RowBox[{ RowBox[{"f", "[", "t_", "]"}], "=", RowBox[{"Piecewise", "[", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"1", "+", RowBox[{"Cos", "[", "t", "]"}]}], ",", RowBox[{ RowBox[{"-", "10"}], "<=", "t", "<", "0"}]}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"2", "+", RowBox[{"Sin", "[", "t", "]"}]}], ",", RowBox[{"0", "\[LessEqual]", "t", "<", "10"}]}], "}"}]}], "}"}], "]"}]}]}]], "Input", CellChangeTimes->{{3.458975449765625*^9, 3.45897547375*^9}}], Cell[BoxData[ RowBox[{"\[Piecewise]", GridBox[{ { RowBox[{"1", "+", RowBox[{"Cos", "[", "t", "]"}]}], RowBox[{ RowBox[{"-", "10"}], "\[LessEqual]", "t", "<", "0"}]}, { RowBox[{"2", "+", RowBox[{"Sin", "[", "t", "]"}]}], RowBox[{"0", "\[LessEqual]", "t", "<", "10"}]}, {"0", TagBox["True", "PiecewiseDefault", AutoDelete->False, DeletionWarning->True]} }, GridBoxAlignment->{ "Columns" -> {{Left}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxItemSize->{ "Columns" -> {{Automatic}}, "ColumnsIndexed" -> {}, "Rows" -> {{1.}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.84]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}]}]], "Output", CellChangeTimes->{ 3.461397626296428*^9, {3.46979488746875*^9, 3.46979490575*^9}}] }, Open ]], Cell["\<\ En este caso si se pide el valor de la funci\[OAcute]n para un valor de t que \ no pertenezca al intervalo de definici\[OAcute]n se obtiene 0 como resultado.\ \>", "Text", CellChangeTimes->{{3.45897547921875*^9, 3.45897555415625*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"f", "[", "20", "]"}]], "Input", CellChangeTimes->{{3.458975556328125*^9, 3.458975558171875*^9}}], Cell[BoxData["0"], "Output", CellChangeTimes->{ 3.461397626327678*^9, {3.4697948875*^9, 3.469794905796875*^9}}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"f", "[", RowBox[{"-", "15"}], "]"}]], "Input", CellChangeTimes->{{3.458975560171875*^9, 3.458975563171875*^9}}], Cell[BoxData["0"], "Output", CellChangeTimes->{ 3.461397626343303*^9, {3.46979488753125*^9, 3.469794905828125*^9}}] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Composici\[OAcute]n de funciones", "Section", CellChangeTimes->{{3.458976621265625*^9, 3.45897662821875*^9}}], Cell[TextData[{ "La composici\[OAcute]n de funciones es una operaci\[OAcute]n que puede \ resultar complicada en ocasiones. Mathematica proporciona un comando para \ efectuar la composici\[OAcute]n de funciones, el comando ", StyleBox["Composition", FontWeight->"Bold"], ". Por ejemplo si f(x) = ", Cell[BoxData[ FormBox[ SuperscriptBox["x", "2"], TraditionalForm]]], " y g(x) = sinx , f\[SmallCircle]g(x) = f[g(x)], se obtendr\[AAcute]," }], "Text", CellChangeTimes->{{3.458976638703125*^9, 3.458976842890625*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"f", "[", "x_", "]"}], "=", RowBox[{"x", "^", "2"}]}], ";", RowBox[{ RowBox[{"g", "[", "x_", "]"}], "=", RowBox[{"Sin", "[", "x", "]"}]}], ";", RowBox[{ RowBox[{"Composition", "[", RowBox[{"f", ",", "g"}], "]"}], "[", "x", "]"}]}]], "Input", CellChangeTimes->{{3.458976846421875*^9, 3.458976880328125*^9}}], Cell[BoxData[ SuperscriptBox[ RowBox[{"Sin", "[", "x", "]"}], "2"]], "Output", CellChangeTimes->{ 3.461397626374553*^9, 3.461397761921428*^9, {3.46979488759375*^9, 3.469794905859375*^9}}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"g", "[", RowBox[{"f", "[", "x", "]"}], "]"}]], "Input", CellChangeTimes->{{3.461397750765178*^9, 3.461397753640178*^9}}], Cell[BoxData[ RowBox[{"Sin", "[", SuperscriptBox["x", "2"], "]"}]], "Output", CellChangeTimes->{{3.461397754218303*^9, 3.461397764046428*^9}, { 3.469794887625*^9, 3.469794905890625*^9}}] }, Open ]], Cell["\<\ Del mismo modo, g\[SmallCircle]f(x) = g[f(x)],\ \>", "Text", CellChangeTimes->{{3.458976893984375*^9, 3.45897692634375*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"Composition", "[", RowBox[{"g", ",", "f"}], "]"}], "[", "x", "]"}]], "Input", CellChangeTimes->{{3.458976932890625*^9, 3.45897693628125*^9}}], Cell[BoxData[ RowBox[{"Sin", "[", SuperscriptBox["x", "2"], "]"}]], "Output", CellChangeTimes->{ 3.461397626390178*^9, {3.4697948876875*^9, 3.469794905921875*^9}}] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[" Representaci\[OAcute]n de Funciones ", "Section 1", CellChangeTimes->{{3.458975682390625*^9, 3.45897568753125*^9}, { 3.458976616734375*^9, 3.458976617453125*^9}}], Cell[TextData[{ "Antes de comenzar con la representaci\[OAcute]n gr\[AAcute]fica de \ funciones con ", StyleBox["Mathematica", FontSlant->"Italic"], " repasaremos algunas cuestiones b\[AAcute]sicas " }], "Text", CellChangeTimes->{{3.427366557088963*^9, 3.4273666274606953`*^9}, 3.428127607984375*^9}] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Representaci\[OAcute]n de curvas en el Plano. Repaso de conceptos previos.\ \>", "Section", CellChangeTimes->{{3.427369999570315*^9, 3.427370020787835*^9}, { 3.428128119078125*^9, 3.4281281294375*^9}}], Cell[TextData[{ "Una curva en el plano puede venir dada de distintas formas, como por \ ejemplo, la forma expl\[IAcute]cita, la \[IAcute]mplicita o la \ param\[EAcute]trica. A continuaci\[OAcute]n se indican los comandos a \ utilizar seg\[UAcute]n venga dada la curva. En su forma m\[AAcute]s sencilla \ hay que darle a ", StyleBox["Mathematica", FontSlant->"Italic"], " la funci\[OAcute]n y el rango de variaci\[OAcute]n de la variable o \ variables.\n" }], "Text", CellChangeTimes->{{3.427366941320663*^9, 3.4273669546102552`*^9}, { 3.45897575128125*^9, 3.4589757578125*^9}, {3.4589758033125*^9, 3.45897599434375*^9}}, TextJustification->1.], Cell[CellGroupData[{ Cell[TextData[{ "1) ", StyleBox["FORMA EXPLICITA : y=f(x), ", FontColor->RGBColor[1, 0, 0]], "Ejemplo: y = ", Cell[BoxData[ FormBox[ SuperscriptBox["x", "2"], TraditionalForm]]], StyleBox[" ", FontColor->RGBColor[1, 0, 0]], "\t\t\n\tComando: ", StyleBox["Plot\t\t", FontColor->RGBColor[0, 0, 1]], "\n\t\t" }], "Subsection", CellChangeTimes->{{3.427366983656033*^9, 3.4273670119017687`*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Plot", "[", RowBox[{ RowBox[{"x", "^", "2"}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "2"}], ",", "2"}], "}"}]}], "]"}]], "Input", 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Los distintos \ estilos se obtienen con la opci\[OAcute]n PlotStyle a la que se le asignan \ tonos de gris, color o tipos de trazo. Una gr\[AAcute]fica se puede representar en varios tonos de gris \ (GrayLevel[w], con w entre 0 y 1), varios colores (Red, Green, Blue, \ RGBColor[r,g,b], con r,g,b comprendidos entre 0 y 1) o distintos tipos de \ trazo (Dashing). Obtener informaci\[OAcute]n las opciones GrayLevel , \ RGBColor y Dashing correspondientes a PlotStyle y que ser\[AAcute]n de una \ gran utilidad.\ \>", "Text", CellChangeTimes->{{3.458976229484375*^9, 3.458976413125*^9}}, TextAlignment->Left, TextJustification->1], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"?", "Dashing"}]], "Input"], Cell[BoxData[ RowBox[{ StyleBox["\<\"\!\(\*RowBox[{\\\"Dashing\\\", \\\"[\\\", RowBox[{\\\"{\\\", \ RowBox[{SubscriptBox[StyleBox[\\\"r\\\", \\\"TI\\\"], StyleBox[\\\"1\\\", \ \\\"TR\\\"]], \\\",\\\", SubscriptBox[StyleBox[\\\"r\\\", \\\"TI\\\"], \ StyleBox[\\\"2\\\", \\\"TR\\\"]], \\\",\\\", StyleBox[\\\"\[Ellipsis]\\\", \\\ \"TR\\\"]}], \\\"}\\\"}], \\\"]\\\"}]\) is a two-dimensional graphics \ directive which specifies that lines which follow are to be drawn dashed, \ with successive segments of lengths \!\(\*SubscriptBox[StyleBox[\\\"r\\\", \\\ \"TI\\\"], StyleBox[\\\"1\\\", \\\"TR\\\"]]\), \!\(\*SubscriptBox[StyleBox[\\\ \"r\\\", \\\"TI\\\"], StyleBox[\\\"2\\\", \\\"TR\\\"]]\), \[Ellipsis] \ (repeated cyclically). 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A continuaci\[OAcute]n, el \ comando Show permite mostrar conjuntamente mostrar varios gr\[AAcute]ficos .\ \>", "Text", CellChangeTimes->{{3.45897642728125*^9, 3.458976531390625*^9}}, TextAlignment->Left, TextJustification->1], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"g1", "=", RowBox[{"Plot", "[", RowBox[{ RowBox[{"Sin", "[", "x", "]"}], ",", RowBox[{"{", RowBox[{"x", ",", "0", ",", RowBox[{"2", "Pi"}]}], "}"}], ",", RowBox[{"PlotStyle", "\[Rule]", "Red"}]}], "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"g2", "=", RowBox[{"Plot", "[", RowBox[{ RowBox[{"Sin", "[", RowBox[{"2", "x"}], "]"}], ",", RowBox[{"{", RowBox[{"x", ",", "0", ",", RowBox[{"2", "Pi"}]}], "}"}], ",", RowBox[{"PlotStyle", "\[Rule]", "Blue"}]}], "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{"Show", "[", RowBox[{"g1", ",", "g2"}], "]"}]}], "Input", CellChangeTimes->{{3.45897653525*^9, 3.45897659028125*^9}}], Cell[BoxData[ GraphicsBox[{{{}, {}, {RGBColor[1, 0, 0], LineBox[CompressedData[" 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Definir f(x) utilizando el comando Piecewise y representarla en intervalo \ [-2,6] comprobando que se le ha asignado el valor cero en los intervalos en \ que no se ha definido. 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Tiene las mismas opciones que el comando Plot. \ Vamos a utilizar la ayuda para completar la informaci\[OAcute]n referente a \ este comando.\ \>", "Text", CellChangeTimes->{{3.42571401053125*^9, 3.425714028125*^9}, { 3.458977779890625*^9, 3.45897778353125*^9}}, TextAlignment->Left, TextJustification->1.], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"?", "ParametricPlot"}]], "Input", CellChangeTimes->{{3.45897779446875*^9, 3.4589777994375*^9}}], Cell[BoxData[ RowBox[{ StyleBox["\<\"\!\(\*RowBox[{\\\"ParametricPlot\\\", \\\"[\\\", \ RowBox[{RowBox[{\\\"{\\\", RowBox[{SubscriptBox[StyleBox[\\\"f\\\", \ \\\"TI\\\"], StyleBox[\\\"x\\\", \\\"TI\\\"]], \\\",\\\", \ SubscriptBox[StyleBox[\\\"f\\\", \\\"TI\\\"], StyleBox[\\\"y\\\", \ \\\"TI\\\"]]}], \\\"}\\\"}], \\\",\\\", RowBox[{\\\"{\\\", \ RowBox[{StyleBox[\\\"u\\\", \\\"TI\\\"], \\\",\\\", \ SubscriptBox[StyleBox[\\\"u\\\", \\\"TI\\\"], StyleBox[\\\"min\\\", \ \\\"TI\\\"]], \\\",\\\", SubscriptBox[StyleBox[\\\"u\\\", \\\"TI\\\"], \ 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"Section", CellChangeTimes->{{3.459059124203125*^9, 3.459059133*^9}}], Cell[TextData[{ "El comando ContourPlot representa b\[AAcute]sicamente curvas de nivel de \ una funci\[OAcute]n f (x, y). Este comando tambi\[EAcute]n se puede utilizar \ para representar una curva plana dada en forma impl\[IAcute]cita. \nLa \ sintaxis es :\n", StyleBox["\tContourPlot[f[x,y],{x,xmin,xmax},{y,ymin,ymax}]\n\t\ ContourPlot[f[x,y]==g[x,y],{x,xmin,xmax},{y,ymin,ymax}]\n", FontWeight->"Bold"], "El comando ContourPlot tiene muchas opciones como : AspectRatio, \ ColorFunction, Contours, Frame, Mesh, PlotRange...\nPor ejemplo, la siguiente \ entrada representa las curvas de nivel de la superficie, z=cos(x)+cos(y)" }], "Text", CellChangeTimes->{{3.4278165826533337`*^9, 3.427816992347825*^9}, { 3.4278170288756857`*^9, 3.427817174659053*^9}, {3.4278172465111923`*^9, 3.427817267018812*^9}, {3.427817300554113*^9, 3.427817334886786*^9}, 3.427817375155424*^9, {3.45905923984375*^9, 3.4590592690625*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"ContourPlot", "[", RowBox[{ RowBox[{ RowBox[{"Cos", "[", "x", "]"}], "+", RowBox[{"Cos", "[", "y", "]"}]}], ",", RowBox[{"{", RowBox[{"x", ",", "0", ",", 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Repaso de \ conceptos previos.\ \>", "Section", CellChangeTimes->{{3.427369999570315*^9, 3.427370020787835*^9}}], Cell[TextData[StyleBox["Una curva en el espacio puede venir dada en forma \ param\[EAcute]trica o como intersecci\[OAcute]n de dos superficies.", FontFamily->"Times New Roman"]], "Text", CellChangeTimes->{{3.4589780771875*^9, 3.45897810240625*^9}}], Cell[CellGroupData[{ Cell["A) CURVAS\t\t", "Subsection", CellChangeTimes->{ 3.427370037345087*^9, {3.427370192659731*^9, 3.4273703334554653`*^9}, { 3.427370399611333*^9, 3.427370404898705*^9}, {3.428128313125*^9, 3.428128323140625*^9}, {3.45897814821875*^9, 3.458978226953125*^9}}], Cell[BoxData[{ RowBox[{ RowBox[{"1", ")"}], StyleBox["FORMA", FontFamily->"Times New Roman", FontColor->RGBColor[1, 0, 0]], StyleBox[" ", FontFamily->"Times New Roman"], StyleBox[ RowBox[{"PARAM\[CapitalEAcute]TRICA", " ", ":"}], FontFamily->"Times New Roman", FontColor->RGBColor[1, 0, 0]]}], "\[IndentingNewLine]", RowBox[{ StyleBox[" ", FontColor->RGBColor[1, 0, 0]], StyleBox[ RowBox[{ FormBox[ TagBox[ StyleBox[ RowBox[{"{", GridBox[{ { RowBox[{"x", "=", RowBox[{"x", "(", "t", ")"}]}]}, { RowBox[{"y", "=", RowBox[{"y", "(", "t", ")"}]}]}, { RowBox[{"z", "=", RowBox[{"z", "(", "t", ")"}]}]} }]}], ShowAutoStyles->False], #& ], TraditionalForm], StyleBox["\t", FontColor->RGBColor[1, 0, 0]], RowBox[{"Ejemplo", ":", " ", RowBox[{ RowBox[{"(", RowBox[{"La", " ", "h\[EAcute]lice"}], ")"}], " ", FormBox[ TagBox[ StyleBox[ RowBox[{"{", GridBox[{ { RowBox[{"x", "=", "cost"}]}, { RowBox[{"y", "=", "sent"}]}, { RowBox[{"z", "=", "t"}]} }]}], ShowAutoStyles->False], #& ], TraditionalForm]}]}]}], FontFamily->"Times New Roman"]}], "\n", StyleBox[ RowBox[{"Comando", ":", " ", StyleBox["ParametricPlot3D", FontColor->RGBColor[0, 0, 1]], StyleBox[" ", FontColor->RGBColor[0, 0, 1]]}], FontFamily->"Times New Roman"]}], "Subsubsection", CellChangeTimes->{{3.45897826621875*^9, 3.458978295203125*^9}, 3.458978424734375*^9}], Cell[TextData[{ "2) ", StyleBox["INTERSECCI\[CapitalOAcute]N DE DOS SUPERFICIES", FontFamily->"Times New Roman", FontColor->RGBColor[1, 0, 0]], "\nUna recta, por ejemplo, puede venir dada como intersecci\[OAcute]n de dos \ planos:\n\t", Cell[BoxData[ RowBox[{ FormBox[ TagBox[ StyleBox[ RowBox[{"{", StyleBox[GridBox[{ { RowBox[{ RowBox[{"x", "+", "y", "+", "z"}], "=", "4"}]}, { RowBox[{ RowBox[{ RowBox[{"3", "x"}], "+", RowBox[{"2", "y"}], "-", "z"}], "=", "2"}]} }], ShowAutoStyles->True]}], ShowAutoStyles->False], #& ], TraditionalForm], ".", " "}]], CellChangeTimes->{{3.458978325234375*^9, 3.45897834915625*^9}, { 3.45897844321875*^9, 3.45897850325*^9}}, TextJustification->1., FontFamily->"Courier New"], "\nPara representarla, se crean dos gr\[AAcute]ficos, uno por cada plano, \ que despu\[EAcute]s se muestran conjuntamente con el comando Show. El ejemplo \ correspondiente se ver\[AAcute] despu\[EAcute]s de la representaci\[OAcute]n \ de superficies." }], "Subsubsection", CellChangeTimes->{{3.4589785154375*^9, 3.45897854878125*^9}, { 3.4589786096875*^9, 3.4589787008125*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["B) SUPERFICIES\t\t\t", "Subsection", CellChangeTimes->{ 3.427370037345087*^9, {3.427370192659731*^9, 3.4273703334554653`*^9}, { 3.427370399611333*^9, 3.427370404898705*^9}, {3.428128313125*^9, 3.428128323140625*^9}, {3.45897814821875*^9, 3.458978226953125*^9}, { 3.458978748171875*^9, 3.458978757203125*^9}}], Cell["\<\ Una superficie en el espacio puede venir dada en forma expl\[IAcute]cita, \ param\[EAcute]trica, dependinte de dos par\[AAcute]metros o en forma impl\ \[IAcute]cita.\ \>", "Text", CellChangeTimes->{{3.45897878890625*^9, 3.458978831953125*^9}}], Cell[TextData[{ "1) ", StyleBox["FORMA EXPL\[CapitalIAcute]CITA : z=z(x,y)", FontColor->RGBColor[1, 0, 0]], "\t\nComando : ", StyleBox["Plot3D", FontColor->RGBColor[0, 0, 1]], "\nEjemplo: (paraboloide) z=", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"]}], TraditionalForm]]] }], "Subsubsection", CellChangeTimes->{{3.427370415118232*^9, 3.427370453087318*^9}, { 3.427370495667507*^9, 3.4273705158279047`*^9}, {3.45897872753125*^9, 3.458978729671875*^9}, {3.45897884903125*^9, 3.458978859296875*^9}}, FontFamily->"Times New Roman"], Cell[TextData[{ "2) ", StyleBox["FORMA PARAMETRICA : ", FontColor->RGBColor[1, 0, 0]], Cell[BoxData[ FormBox[ TagBox[ StyleBox[ RowBox[{"{", GridBox[{ { RowBox[{"x", "=", RowBox[{"x", "(", RowBox[{"u", ",", "v"}], ")"}]}]}, { RowBox[{"y", "=", RowBox[{"y", "(", RowBox[{"u", ",", "v"}], ")"}]}]}, { RowBox[{"z", "=", RowBox[{"z", "(", RowBox[{"u", ",", "v"}], ")"}]}]} }]}], ShowAutoStyles->False], #& ], TraditionalForm]], FontColor->RGBColor[1, 0, 0]], "\nComando : ", StyleBox["ParametricPlot3D ", FontColor->RGBColor[0, 0, 1]], "\n\t\nEjemplo: ", Cell[BoxData[ FormBox[ TagBox[ StyleBox[ RowBox[{"{", GridBox[{ { RowBox[{"x", "=", RowBox[{"3", RowBox[{"sent", ".", "cosu"}]}]}]}, { RowBox[{"y", "=", RowBox[{"3", RowBox[{"sent", ".", "senu"}]}]}]}, { RowBox[{"z", "=", RowBox[{"3", "cost"}]}]} }]}], ShowAutoStyles->False], #& ], TraditionalForm]]], " , t\[Element][0,\[Pi]] , u\[Element][0,2\[Pi]] (esfera)" }], "Subsubsection", CellChangeTimes->{{3.427370415118232*^9, 3.427370453087318*^9}, { 3.427370495667507*^9, 3.4273705158279047`*^9}, {3.45897872753125*^9, 3.458978729671875*^9}, {3.458978873703125*^9, 3.458978884375*^9}}, FontFamily->"Times New Roman"], Cell[TextData[{ "3) ", StyleBox["FORMA IMPLICITA ; F(x,y,z)=C", FontColor->RGBColor[1, 0, 0]], "\nComando : ", StyleBox["ContourPlot3D ", FontColor->RGBColor[0, 0, 1]], "\nEjemplo : ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"], "+", SuperscriptBox["z", "2"]}], "=", " ", "1"}], TraditionalForm]]], "\n" }], "Subsubsection", CellChangeTimes->{{3.427370415118232*^9, 3.427370453087318*^9}, { 3.427370495667507*^9, 3.4273705158279047`*^9}, {3.45897872753125*^9, 3.458978729671875*^9}, {3.4589788960625*^9, 3.45897890440625*^9}}, FontFamily->"Times New Roman"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["El comando Plot3D", "Section", CellChangeTimes->{{3.458978935171875*^9, 3.458978939453125*^9}}], Cell[TextData[{ "El comando Plot3D genera un gr\[AAcute]fico tridimensional de una funci\ \[OAcute]n de dos variables, f (x, y), es decir, representa la superficie de \ ecuaci\[OAcute]n z = f (x, y). Su sintaxis es la siguiente:\n\t\t\ Plot3D[f[x,y], ", Cell[BoxData[ FormBox[ RowBox[{"{", RowBox[{"x", ",", " ", SubscriptBox["x", "min"], ",", SubscriptBox["x", "max"]}]}], TraditionalForm]]], "}]\n" }], "Text", CellChangeTimes->{{3.458978996625*^9, 3.458979045265625*^9}, { 3.458979087734375*^9, 3.458979181671875*^9}}], Cell["\<\ Podemos consultar sobre las distintas opciones de este comando en la \ siguiente entrada :\ \>", "Text", CellChangeTimes->{{3.458979203859375*^9, 3.45897923240625*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Options", "[", "Plot3D", "]"}]], "Input", CellChangeTimes->{{3.4589791876875*^9, 3.458979193859375*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"AlignmentPoint", "\[Rule]", "Center"}], ",", RowBox[{"AspectRatio", "\[Rule]", "Automatic"}], ",", RowBox[{"Axes", "\[Rule]", "True"}], ",", RowBox[{"AxesEdge", "\[Rule]", "Automatic"}], ",", RowBox[{"AxesLabel", "\[Rule]", "None"}], ",", RowBox[{"AxesOrigin", "\[Rule]", "Automatic"}], ",", 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RowBox[{"True", "&"}], ")"}]}], ",", RowBox[{"RotationAction", "\[Rule]", "\<\"Fit\"\>"}], ",", RowBox[{"SphericalRegion", "\[Rule]", "False"}], ",", RowBox[{"Ticks", "\[Rule]", "Automatic"}], ",", RowBox[{"TicksStyle", "\[Rule]", RowBox[{"{", "}"}]}], ",", RowBox[{"ViewAngle", "\[Rule]", "Automatic"}], ",", RowBox[{"ViewCenter", "\[Rule]", "Automatic"}], ",", RowBox[{"ViewMatrix", "\[Rule]", "Automatic"}], ",", RowBox[{"ViewPoint", "\[Rule]", RowBox[{"{", RowBox[{"1.3`", ",", RowBox[{"-", "2.4`"}], ",", "2.`"}], "}"}]}], ",", RowBox[{"ViewRange", "\[Rule]", "All"}], ",", RowBox[{"ViewVector", "\[Rule]", "Automatic"}], ",", RowBox[{"ViewVertical", "\[Rule]", RowBox[{"{", RowBox[{"0", ",", "0", ",", "1"}], "}"}]}], ",", RowBox[{"WorkingPrecision", "\[Rule]", "MachinePrecision"}]}], "}"}]], "Output", CellChangeTimes->{ 3.461397630280803*^9, {3.4697948916875*^9, 3.46979490925*^9}}] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "Ejemplo 14\nRepresentar el paraboloide de ecuaci\[OAcute]n 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Otras opciones que se pueden \ utilizar son la BoxRatios, para establecer la relaci\[OAcute]n entre las \ longitudes de la caja, Boxed\[Rule]False, para que no aparezca la caja y Axes", "\[Rule]", "Automatic" }], "Text", CellChangeTimes->{{3.42737061593717*^9, 3.427370624323668*^9}, { 3.458979364515625*^9, 3.458979391546875*^9}}, TextAlignment->Left, TextJustification->1] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["El comando ContourPlot3D", "Section", CellChangeTimes->{{3.458979416921875*^9, 3.4589794238125*^9}}], Cell[TextData[{ "El comando ContourPlot3D se puede utilizar de varias formas. \n\t\ ContourPlot3D[f(x,y,z) , ", Cell[BoxData[ FormBox[ RowBox[{"{", RowBox[{"x", ",", " ", SubscriptBox["x", "min"], ",", SubscriptBox["x", "max"]}]}], TraditionalForm]]], "} , ", Cell[BoxData[ FormBox[ RowBox[{"{", RowBox[{"y", ",", " ", SubscriptBox["y", "min"], ",", SubscriptBox["y", "max"]}]}], TraditionalForm]]], "},", Cell[BoxData[ FormBox[ RowBox[{"{", RowBox[{"z", ",", " ", 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En el caso de una \ curva las tres funciones correpondientes a la x, y y z de los puntos de la \ curva vendr\[AAcute]n dadas en funci\[OAcute]n de un \[UAcute]nico par\ \[AAcute]metro. En el caso de una superfice en funci\[OAcute]n de dos. 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3.469794910203125*^9}}] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["El comando MANIPULATE", "Section", CellChangeTimes->{{3.458971471453125*^9, 3.458971476390625*^9}}], Cell[TextData[{ "Este comando genera varias versiones de una expresi\[OAcute]n o un gr\ \[AAcute]fico a\[NTilde]adiendo controles que permiten la \ manipulaci\[OAcute]n interactiva de la expresi\[OAcute]n o gr\[AAcute]fico. \ Su sintaxis es la siguiente:\n\tManipulate[expr, {u, ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SubscriptBox["u", "min"], ",", " ", SubscriptBox["u", "max"]}], "}"}], TraditionalForm]]], "].\nEn el siguiente ejemplo se representa gr\[AAcute]ficamente la funci\ \[OAcute]n Sin[a*x] en el intervalo [0,2\[Pi]], variando entre 1 y 3 de forma \ continua los valores del par\[AAcute]metro a. El valor del par\[AAcute]metro \ se muestra sobre la barra deslizadora y tambi\[EAcute]n en la ventana de la \ derecha.Pinchando en el + al final de la barra deslizadora aparecencen los \ controles que permiten desplegar la animaci\[OAcute]n: Play y Pause. Tambi\ \[EAcute]n aparecen otros dos botones, Faster y Slower para que la animaci\ \[OAcute]n sea m\[AAcute]s r\[AAcute]pida o lenta.\nSe trata de un comando \ muy interesante para la visualizaci\[OAcute]n de gr\[AAcute]ficos." }], "Text", CellChangeTimes->{{3.45897160315625*^9, 3.4589717001875*^9}, { 3.458971792578125*^9, 3.45897204803125*^9}, 3.459059404765625*^9, { 3.45905955825*^9, 3.459059625359375*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Manipulate", "[", RowBox[{ RowBox[{"Plot", "[", RowBox[{ RowBox[{"Sin", "[", RowBox[{"a", "*", "x"}], "]"}], ",", RowBox[{"{", RowBox[{"x", ",", "0", ",", RowBox[{"2", "Pi"}]}], "}"}]}], "]"}], ",", RowBox[{"{", RowBox[{"a", ",", "1", ",", "3"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.45897172915625*^9, 3.458971763484375*^9}}], Cell[BoxData[ TagBox[ StyleBox[ DynamicModuleBox[{$CellContext`a$$ = 1, Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{ Hold[$CellContext`a$$], 1, 3}}, Typeset`size$$ = {540., {154., 163.}}, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = True, $CellContext`a$1071$$ = 0}, DynamicBox[Manipulate`ManipulateBoxes[ 1, StandardForm, "Variables" :> {$CellContext`a$$ = 1}, "ControllerVariables" :> { Hold[$CellContext`a$$, $CellContext`a$1071$$, 0]}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, Typeset`skipInitDone$$}, "Body" :> Plot[ Sin[$CellContext`a$$ $CellContext`x], {$CellContext`x, 0, 2 Pi}], "Specifications" :> {{$CellContext`a$$, 1, 3}}, "Options" :> {}, "DefaultOptions" :> {}], ImageSizeCache->{605., {219., 226.}}, SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, SynchronousInitialization->True, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], Manipulate`InterpretManipulate[1]]], "Output", CellChangeTimes->{ 3.45905938271875*^9, 3.459059537328125*^9, 3.461397631202678*^9, { 3.469794893125*^9, 3.46979491025*^9}}] }, Open ]], Cell[TextData[{ "Este comando tambi\[EAcute]n se puede utilizar con expresiones, por ejemplo \ algebr\[AAcute]icas. En el siguiente ejemplo se desarrolla el binomio ", Cell[BoxData[ FormBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"1", "+", "x"}], ")"}], "n"], TraditionalForm]]], " para valores de n de 1 a 10." }], "Text", CellChangeTimes->{{3.458972051765625*^9, 3.458972072390625*^9}, { 3.458972130765625*^9, 3.458972165125*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Manipulate", "[", RowBox[{ RowBox[{"Expand", "[", SuperscriptBox[ RowBox[{"(", RowBox[{"1", "+", "x"}], ")"}], "n"], "]"}], ",", RowBox[{"{", RowBox[{"n", ",", "1", ",", "10", ",", "1"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.458972079546875*^9, 3.458972114578125*^9}, { 3.459059410421875*^9, 3.459059413109375*^9}}], Cell[BoxData[ TagBox[ StyleBox[ DynamicModuleBox[{$CellContext`n$$ = 1, Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{ Hold[$CellContext`n$$], 1, 10, 1}}, Typeset`size$$ = {41., {0., 11.}}, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = True, $CellContext`n$1113$$ = 0}, DynamicBox[Manipulate`ManipulateBoxes[ 1, StandardForm, "Variables" :> {$CellContext`n$$ = 1}, "ControllerVariables" :> { Hold[$CellContext`n$$, $CellContext`n$1113$$, 0]}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, Typeset`skipInitDone$$}, "Body" :> Expand[(1 + $CellContext`x)^$CellContext`n$$], "Specifications" :> {{$CellContext`n$$, 1, 10, 1}}, "Options" :> {}, "DefaultOptions" :> {}], ImageSizeCache->{378., {71., 78.}}, SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, SynchronousInitialization->True, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], Manipulate`InterpretManipulate[1]]], "Output", CellChangeTimes->{{3.459059392984375*^9, 3.45905941453125*^9}, 3.461397631358928*^9, {3.46979489346875*^9, 3.46979491034375*^9}}] }, Open ]], Cell["\<\ Los valores del par\[AAcute]metro pueden ir tambi\[EAcute]n de dos en dos o \ de tres en tres. Por ejemplo :\ \>", "Text", CellChangeTimes->{{3.458972170765625*^9, 3.458972218296875*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Manipulate", "[", RowBox[{ RowBox[{"Expand", "[", SuperscriptBox[ RowBox[{"(", RowBox[{"1", "+", "x"}], ")"}], "n"], "]"}], ",", RowBox[{"{", RowBox[{"n", ",", "1", ",", "11", ",", "2"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.458972079546875*^9, 3.458972114578125*^9}, { 3.45897222603125*^9, 3.45897225046875*^9}, {3.4590594228125*^9, 3.459059425578125*^9}}], Cell[BoxData[ TagBox[ StyleBox[ DynamicModuleBox[{$CellContext`n$$ = 1, Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{ Hold[$CellContext`n$$], 1, 11, 2}}, Typeset`size$$ = {41., {0., 11.}}, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = True, $CellContext`n$1135$$ = 0}, DynamicBox[Manipulate`ManipulateBoxes[ 1, StandardForm, "Variables" 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Manipulate permite realizar un amplio conjunto de \ operaciones y demostraciones con ", StyleBox["Mathematica,", FontSlant->"Italic"], " eeste curso dispone de un tema espec\[IAcute]fico de estudio de las \ distintas posibilidades que proporciona dicho comando.\nEl comando Animate es \ similar al comando Manipulate con la \[UAcute]nica diferencia de que la \ animaci\[OAcute]n se genera autom\[AAcute]ticamente." }], "Text", CellChangeTimes->{{3.45905949753125*^9, 3.459059531546875*^9}, 3.459059644421875*^9, {3.46979495503125*^9, 3.46979506421875*^9}}] }, Open ]] }, Open ]] }, WindowToolbars->"EditBar", WindowSize->{1672, 933}, WindowMargins->{{0, Automatic}, {Automatic, 0}}, DockedCells->FEPrivate`If[ FEPrivate`SameQ[FEPrivate`$ProductIDName, "MathematicaPlayer"], FEPrivate`Join[{ Cell[ BoxData[ GraphicsBox[ RasterBox[CompressedData[" 1:eJztXXdYFNfaz3O/f+79vudek3vTLKBIEaQpTRC7BgtqLCTWGxNjYsFookE0 NoxRY6oFuyIiWJCO9N4WWHrZpSy7VKkiBo2iy+5+Z/fsHg4zO7NLUZJ4fs/I 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