(* 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[ 913559, 17302] NotebookOptionsPosition[ 888118, 16448] NotebookOutlinePosition[ 889561, 16496] CellTagsIndexPosition[ 889402, 16489] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell[TextData[{ "Introducci\[OAcute]n a ", StyleBox["Mathematica", FontSlant->"Italic"], " " }], "Title", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, TextAlignment->Center, TextJustification->0, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " es un paquete de software matem\[AAcute]tico, de uso f\[AAcute]cil y gran \ capacidad, que nos permitir\[AAcute] realizar las operaciones necesarias para \ resolver problemas que ya se abordaron en su momento en las diferentes \ asignaturas de matem\[AAcute]ticas en cuatrimestres anteriores. En este \ primer apartado nos iniciaremos en el conocimiento b\[AAcute]sico de las \ funciones m\[AAcute]s importantes de Mathematica que nos ser\[AAcute]n \ imprescindibles para el posterior desarrollo de las pr\[AAcute]cticas de las \ diversas asignaturas. En cada pr\[AAcute]ctica se har\[AAcute] un estudio de \ los comandos espec\[IAcute]ficos que son necesarios para el desarrollo de las \ mismas\n" }], "Text", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, CellChangeTimes->{3.5057120104950604`*^9}, TextJustification->1, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}] }, Open ]], Cell[CellGroupData[{ Cell["1.- Introducci\[OAcute]n ", "Subtitle", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, CellChangeTimes->{{3.505712046198185*^9, 3.505712061354435*^9}}, TextJustification->1, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell["\<\ Cuando se comienza a utilizar el programa se activa el Front End con lo que \ se puede comenzar a introducir datos y expresiones que posteriormente ser\ \[AAcute]n evaluadas. En el momento en que se desea realizar la primera \ operaci\[OAcute]n se activa el Kernel que es el m\[OAcute]dulo que realiza el \ c\[AAcute]lculo. En el Kernel se encuentran introducidos los procedimientos \ de c\[AAcute]lculo relacionados con una gran cantidad de operaciones ( las m\ \[AAcute]s habituales ). Sin embargo, dada la potencia de c\[AAcute]lculo del \ paquete, operaciones y procedimientos m\[AAcute]s complejos se encuentran \ almacenados en diferentes Packages que deben ser activados antes de realizar \ los c\[AAcute]lculos con ese tipo de sentencias. Muchos de los comandos que ejecuta Mathematica han sido introducidos en \ ventanas gr\[AAcute]ficas llamadas Paletas que permiten una m\[AAcute]s r\ \[AAcute]pida implementaci\[OAcute]n de las sentencias y evitan memorizar \ parte de los comandos utilizados. Todas estas caracter\[IAcute]sticas permiten un manejo sencillo y \ r\[AAcute]pido de modo que, con una peque\[NTilde]a introducci\[OAcute]n a \ las funciones b\[AAcute]sicas, al modo de introducir datos y la utilizaci\ \[OAcute]n de la ayuda, se puede manejar con soltura en un corto espacio de \ tiempo.\ \>", "Text", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, CellChangeTimes->{{3.5057120705575604`*^9, 3.505712191416935*^9}}, TextJustification->1, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}] }, Open ]], Cell[CellGroupData[{ Cell["\<\ 2.- Introducci\[OAcute]n de datos y operaciones\ \>", "Subtitle", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, TextJustification->1, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell["\<\ Comenzar a trabajar con Mathematica es muy sencillo, basta con introducir la \ operaci\[OAcute]n que se desea realizar y pulsar las teclas Shift + Intro. As\ \[IAcute] Mathematica realiza la operaci\[OAcute]n. Tambi\[EAcute]n ejecuta \ las operaciones pulsando el boton de Intro del teclado num\[EAcute]rico.\ \>", "Text", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, CellChangeTimes->{{3.505713944541935*^9, 3.5057139671200604`*^9}}, TextJustification->1, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"2", "*", "3"}]], "Input", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, TextJustification->1, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[BoxData["6"], "Output", CellChangeTimes->{3.505715460448185*^9, 3.5057158706825604`*^9, 3.506340724078125*^9}] }, Open ]], Cell["\<\ De este modo las operaciones quedan numeradas en el orden en que se van \ realizando, con lo que se pueden ir utilizando los resultados previos con \ solo indicar en que momento se obtuvieron. El resultado de la \[UAcute]ltima \ operaci\[OAcute]n se puede recuperar utilizando el s\[IAcute]mbolo %, el pen\ \[UAcute]ltimo mediante %% y en general el resultado de la k-\[EAcute]sima \ operaci\[OAcute]n con el s\[IAcute]mbolo %k.Por tanto, la siguiente operaci\ \[OAcute]n hace referencia al resultado obtenido en la primera operaci\ \[OAcute]n\ \>", "Text", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, TextJustification->1, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"(", RowBox[{"3", "+", "9"}], ")"}], "/", "%1"}]], "Input", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, TextJustification->1, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[BoxData["2"], "Output", CellChangeTimes->{3.505715461916935*^9, 3.505715870729435*^9, 3.506340724109375*^9}] }, Open ]], Cell[TextData[{ "En ", StyleBox["Mathematica", FontSlant->"Italic"], " el producto de dos factores se puede representar tanto por el asterisco \ como por un espacio en blanco. Es importante recordar que cuando se opera con \ variables no es lo mismo escribir x y que xy ya que ", StyleBox["Mathematica", FontSlant->"Italic"], " en el primer caso entiende x*y mientras que en el segundo considera una \ nueva variable llamada (xy).\nLas operaciones algebraicas comunes y el orden \ de evaluaci\[OAcute]n de los diferentes factores sigue el mismo criterio que \ cualquier lenguaje de programaci\[OAcute]n. En cualquier caso siempre se \ puede controlar la operaci\[OAcute]n que se realiza colocando \ par\[EAcute]ntesis. " }], "Text", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, TextJustification->1, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}] }, Open ]], Cell[CellGroupData[{ Cell["3.- Precisi\[OAcute]n en el c\[AAcute]lculo", "Subtitle", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, TextJustification->1, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell["\<\ Mathem\[AAcute]tica intenta siempre llegar al resultado m\[AAcute]s \ aproximado en cada operacion que realiza y siempre que puede llega al \ resultado exacto. De este modo cuando se realiza el cociente de dos \ n\[UAcute]meros enteros matem\[AAcute]tica da como resultado dicho cociente \ puesto que si sacara un n\[UAcute]mero decimal perder\[IAcute]a precisi\ \[OAcute]n.\ \>", "Text", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, TextJustification->1, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"3", "/", "5"}]], "Input", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, TextJustification->1, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[BoxData[ FractionBox["3", "5"]], "Output", CellChangeTimes->{3.5057154640263104`*^9, 3.5057158707450604`*^9, 3.506340724140625*^9}] }, Open ]], Cell["\<\ Lo mismo sucede cuando se realizan operaciones con variables. Siempre que no \ se pueda simplificar Mathematica presenta el valor que se ha introducido.\ \>", "Text", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, TextAlignment->Left, TextJustification->1, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"x", "^", "3"}], "+", "5"}], ")"}], "/", RowBox[{"x", "^", "2"}]}]], "Input", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, TextJustification->1, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[BoxData[ FractionBox[ RowBox[{"5", "+", SuperscriptBox["x", "3"]}], SuperscriptBox["x", "2"]]], "Output", CellChangeTimes->{3.505715466010685*^9, 3.5057158707763104`*^9, 3.506340724171875*^9}] }, Open ]], Cell["\<\ En ocasiones puede interesar llegar a un resultado aproximado en vez de al \ resultado exacto, para hacerse una idea del orden de magnitud del resultado \ obtenido. Para ello se pueden aplicar tres m\[EAcute]todos\ \>", "Text", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, TextAlignment->Left, TextJustification->1, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell["\<\ a) Introducir alguno de los factores de la operaci\[OAcute]n que se desea \ realizar de forma aproximada. Para ello basta con poner en forma decimal alg\ \[UAcute]n factor. Ej: 10 = 10.0 ( con poner tan solo el punto decimal es \ suficiente)\ \>", "Text", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, TextJustification->1, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"3.", "/", "5"}]], "Input", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, TextJustification->1, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[BoxData["0.6000000000000001`"], "Output", CellChangeTimes->{3.505715467698185*^9, 3.505715870791935*^9, 3.506340724203125*^9}] }, Open ]], Cell["\<\ b) Utilizar el comando N[ ]. Con este comando se puede obtener el resultado \ de forma aproximada e incluso indicar el n\[UAcute]mero de decimales que se \ desea que aparezcan. Primero se introduce la operaci\[OAcute]n a realizar y \ separado con una coma el n\[UAcute]mero de d\[IAcute]gitos significativos.\ \>", "Text", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, CellChangeTimes->{{3.505713985541935*^9, 3.505713990260685*^9}}, TextAlignment->Left, TextJustification->1, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"N", "[", RowBox[{ RowBox[{"20", "/", "17"}], ",", "5"}], "]"}]], "Input", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, TextJustification->1, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[BoxData["1.1764705882352941176`5."], "Output", CellChangeTimes->{3.5057154695575604`*^9, 3.505715870823185*^9, 3.506340724234375*^9}] }, Open ]], Cell[TextData[{ "c) Utilizar el comando N pero al final de la expresi\[OAcute]n que se desea \ evaluar de la siguiente manera expr //N ( Por defecto ", StyleBox["Mathematica", FontSlant->"Italic"], " presenta 6 d\[IAcute]gitos significativos )." }], "Text", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"20", "/", "17"}], " ", "//", "N"}]], "Input", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[BoxData["1.1764705882352942`"], "Output", CellChangeTimes->{3.5057154731825604`*^9, 3.5057158708388104`*^9, 3.506340724265625*^9}] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["4.- Comandos y variables predefinidas", "Subtitle", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, TextJustification->1, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " incorpora tanto sus propios comandos como algunas variables de uso muy \ frecuente de modo que la primera letra es siempre may\[UAcute]scula. Para \ evitar confusiones o errores entre las variables definidas por el usuario y \ las ya definidas por el programa se recomienda definir variables propias que \ comiencen por letras min\[UAcute]sculas. Entre las variables definidas en el \ programa se encuentran Pi, E, I." }], "Text", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, TextAlignment->Left, TextJustification->1, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[CellGroupData[{ Cell[BoxData[ StyleBox[ RowBox[{"N", "[", "Pi", "]"}], FontFamily->"Courier", FontWeight->"Bold"]], "Input", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, TextJustification->1, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[BoxData["3.141592653589793`"], "Output", CellChangeTimes->{3.5057154753075604`*^9, 3.5057158708700604`*^9, 3.50634072434375*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ StyleBox[ RowBox[{"N", "[", "E", "]"}], FontFamily->"Courier", FontWeight->"Bold"]], "Input", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, TextJustification->1, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[BoxData["2.718281828459045`"], "Output", CellChangeTimes->{3.5057154767763104`*^9, 3.505715870885685*^9, 3.506340724359375*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ SqrtBox[ RowBox[{"(", RowBox[{"-", "1"}], ")"}]]], "Input", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, CellChangeTimes->{{3.505714034198185*^9, 3.5057140432138104`*^9}}, TextJustification->1, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[BoxData["\[ImaginaryI]"], "Output", CellChangeTimes->{3.505715477354435*^9, 3.5057158709013104`*^9, 3.50634072440625*^9}] }, Open ]], Cell["\<\ En cuanto a los comandos, todos ellos comienzan tambi\[EAcute]n con may\ \[UAcute]scula y adem\[AAcute]s el argumento que introducimos as\[IAcute] \ como el resto de datos para el control de la operaci\[OAcute]n van entre \ corchetes. Comando[arg , control_1, control_2, . . . ]\ \>", "Text", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, TextAlignment->Left, TextJustification->1, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[CellGroupData[{ Cell[BoxData[ StyleBox[ RowBox[{"Cos", "[", RowBox[{"Pi", "/", "3"}], "]"}], FontFamily->"Courier", FontWeight->"Bold"]], "Input", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, TextJustification->1, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[BoxData[ FractionBox["1", "2"]], "Output", CellChangeTimes->{3.5057154784013104`*^9, 3.5057158709325604`*^9, 3.5063407244375*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ StyleBox[ RowBox[{"Plot3D", "[", RowBox[{ RowBox[{"Sin", "[", RowBox[{"x", "*", "y"}], "]"}], ",", " ", RowBox[{"{", RowBox[{"x", ",", " ", RowBox[{"-", "2"}], ",", " ", "2"}], "}"}], ",", RowBox[{"{", " ", 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}, Open ]], Cell[CellGroupData[{ Cell["\<\ 4.1.- Funciones Matem\[AAcute]ticas Comunes\ \>", "Section", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[TextData[{ "Las funciones matem\[AAcute]ticas m\[AAcute]s comunmente utilizadas pueden \ introducirse directamente atendiendo al comando o tambi\[EAcute]n mediante \ las paletas que proporciona ", StyleBox["Mathematica", FontSlant->"Italic"], ".\nPara uitlizar estas Paletas basta con situarse en la barra superior en \ el comando File y en Palettes seleccionar la paleta Basic Calculations. En \ esta paleta aparecen tanto funciones como comandos muy utilizados.\nPor \ ejemplo, para indicar que se desea realizar una raiz cuadrada se puede \ utilizar el comando Sqrt[ ] o tambi\[EAcute]n se puede utilizar el icono \ correspondiente que aparece en la paleta ", Cell[BoxData[ FormBox[ SqrtBox["\[Placeholder]"], TraditionalForm]]], " en este caso basta con rellenar el recuadro con la expresi\[OAcute]n que \ se desea evaluar.\nEn esta paleta tambi\[EAcute]n aparecen funciones \ trigonometricas tanto directas como inversas Sin[ ] , ArcCos[ ] , y las \ correspondientes expresiones Hiperb\[OAcute]licas Cosh[ ], ArcTanh[ ].\n\ Repasando esta paleta se pueden encontrar funciones como el factorial !, \ el entero mas cercano a x Round[x], etc." }], "Text", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Ejercicios de las secci\[OAcute]nes 2,3,4", Background->None]], "Subtitle", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[TextData[{ "- Utilice ", StyleBox["N", FontWeight->"Bold"], " para calcular \[Pi] con 50 decimales.\n- Utilice N para ver a que entero \ se aproxima ", Cell[BoxData[ FormBox[ SuperscriptBox["E", RowBox[{" ", "\[Pi]"}]], TraditionalForm]]], ". A continuaci\[OAcute]n utilizar los comandos Floor[", Cell[BoxData[ FormBox[ SuperscriptBox["E", RowBox[{" ", "\[Pi]"}]], TraditionalForm]]], "] y Ceiling[", Cell[BoxData[ FormBox[ SuperscriptBox["E", RowBox[{" ", "\[Pi]"}]], TraditionalForm]]], "].\n- Calcule dos n\[UAcute]meros aleatorios con Random[ ] y a continuaci\ \[OAcute]n multipl\[IAcute]quelos" }], "Text", CellMargins->{{10.375, Inherited}, {Inherited, Inherited}}, PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, TextAlignment->Left, TextJustification->1, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}] }, Open ]], Cell[CellGroupData[{ Cell["5.- Listas", "Subtitle", PageBreakAbove->Automatic, PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, TextJustification->1, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell["\<\ En muchas ocasiones es necesario utilizar datos que se encuentran \ relacionados entre si y que pueden ser todos ellos el argumento de una funci\ \[OAcute]n. En esas ocasiones utilizamos listas para definir todos estos \ elementos. Una lista contiene elementos que se encuentran separados por comas de la \ siguiente manera:\ \>", "Text", PageBreakAbove->Automatic, PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, TextAlignment->Left, TextJustification->1, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[CellGroupData[{ Cell[BoxData[ StyleBox[ RowBox[{" ", RowBox[{"{", RowBox[{ "1", ",", " ", "3", ",", " ", "6", ",", " ", "9", ",", " ", "12"}], "}"}]}], FontFamily->"Courier", FontWeight->"Bold"]], "Input", PageBreakAbove->Automatic, PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, TextJustification->1, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[BoxData[ RowBox[{"{", RowBox[{"1", ",", "3", ",", "6", ",", "9", ",", "12"}], "}"}]], "Output", CellChangeTimes->{3.505715483104435*^9, 3.5057158710575604`*^9, 3.5063407249375*^9}] }, Open ]], Cell["\<\ De este modo todos estos elementos se pueden utilizar como una \[UAcute]nica \ variable.\ \>", "Text", PageBreakAbove->Automatic, PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, TextJustification->1, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[CellGroupData[{ Cell[BoxData[ StyleBox[ RowBox[{" ", RowBox[{"2", "^", "%"}]}], FontFamily->"Courier", FontWeight->"Bold"]], "Input", PageBreakAbove->Automatic, PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, TextJustification->1, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[BoxData[ RowBox[{"{", RowBox[{"2", ",", "8", ",", "64", ",", "512", ",", "4096"}], "}"}]], "Output", CellChangeTimes->{3.505715484198185*^9, 3.5057158710888104`*^9, 3.50634072496875*^9}] }, Open ]], Cell["\<\ En una lista se pueden introducir elementos que no sean del mismo tipo, como \ valores num\[EAcute]ricos, variables, texto, etc. En estos casos deberemos \ referirnos a uno espec\[IAcute]ficamente para poder realizar la operaci\ \[OAcute]n correspondiente. Para ello se considera la posici\[OAcute]n que \ ocupa dicho elemento en la lista y se hace referencia a \[EAcute]l de la \ siguiente manera:\ \>", "Text", PageBreakAbove->Automatic, PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, TextJustification->1, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[CellGroupData[{ Cell[BoxData[ StyleBox[ RowBox[{"a", " ", "=", RowBox[{"{", RowBox[{"2.4", ",", " ", "x", ",", RowBox[{"{", " ", RowBox[{"2", ",", " ", "6"}], "}"}], ",", " ", "Pi"}], "}"}]}], FontFamily->"Courier", FontWeight->"Bold"]], "Input", PageBreakAbove->Automatic, PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, TextJustification->1, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[BoxData[ RowBox[{"{", RowBox[{"2.4`", ",", "x", ",", RowBox[{"{", RowBox[{"2", ",", "6"}], "}"}], ",", "\[Pi]"}], "}"}]], "Output", CellChangeTimes->{3.5057154856200604`*^9, 3.505715871104435*^9, 3.506340725*^9}] }, Open ]], Cell["\<\ Una vez definida la lista pasamos a operar con los elementos de la misma. La \ expresi\[OAcute]n a[[i]] hace referencia al elemento i-esimo de la lista.\ \>", "Text", PageBreakAbove->Automatic, PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, TextJustification->1, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[CellGroupData[{ Cell[BoxData[ StyleBox[ RowBox[{" ", RowBox[{"3", "*", RowBox[{"a", "[", RowBox[{"[", "1", "]"}], "]"}]}]}], FontFamily->"Courier", FontWeight->"Bold"]], "Input", PageBreakAbove->Automatic, PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, TextJustification->1, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[BoxData["7.199999999999999`"], "Output", CellChangeTimes->{3.505715486854435*^9, 3.505715871135685*^9, 3.506340725078125*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ StyleBox[ RowBox[{ RowBox[{"a", "[", RowBox[{"[", "2", "]"}], "]"}], "^", "3"}], FontFamily->"Courier", FontWeight->"Bold"]], "Input", PageBreakAbove->Automatic, PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, TextJustification->1, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[BoxData[ SuperscriptBox["x", "3"]], "Output", CellChangeTimes->{3.5057154875888104`*^9, 3.5057158711513104`*^9, 3.50634072509375*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ StyleBox[ RowBox[{" ", RowBox[{"Cos", "[", RowBox[{"a", "[", RowBox[{"[", "4", "]"}], "]"}], "]"}]}], FontFamily->"Courier", FontWeight->"Bold"]], "Input", PageBreakAbove->Automatic, PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, TextJustification->1, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[BoxData[ RowBox[{"-", "1"}]], "Output", CellChangeTimes->{3.505715488260685*^9, 3.505715871166935*^9, 3.506340725125*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ StyleBox[ RowBox[{"2", "*", RowBox[{"a", "[", RowBox[{"[", "3", "]"}], "]"}]}], FontFamily->"Courier", FontWeight->"Bold"]], "Input", PageBreakAbove->Automatic, PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, TextJustification->1, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[BoxData[ RowBox[{"{", RowBox[{"4", ",", "12"}], "}"}]], "Output", CellChangeTimes->{3.5057154900263104`*^9, 3.5057158711825604`*^9, 3.50634072515625*^9}] }, Open ]], Cell["\<\ El tercer elemento de la lista es a su vez una lista por lo que se puede \ hacer referencia al elemento en si, a[[3]], o a cada uno de los elementos que \ lo componen a[[3,1]] y a[[3,2]] pudiendo operar con ellos por separado\ \>", "Text", PageBreakAbove->Automatic, PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, TextJustification->1, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[CellGroupData[{ Cell[BoxData[ StyleBox[ RowBox[{" ", RowBox[{ RowBox[{ RowBox[{"a", "[", RowBox[{"[", "2", "]"}], "]"}], "^", RowBox[{"a", "[", RowBox[{"[", RowBox[{"3", ",", " ", "1"}], "]"}], "]"}]}], "\n"}]}], FontFamily->"Courier", FontWeight->"Bold"]], "Input", PageBreakAbove->Automatic, PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, TextJustification->1, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[BoxData[ SuperscriptBox["x", "2"]], "Output", CellChangeTimes->{3.5057154911825604`*^9, 3.5057158712138104`*^9, 3.5063407251875*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ StyleBox[ RowBox[{ RowBox[{"a", "[", RowBox[{"[", "2", "]"}], "]"}], "^", RowBox[{"a", "[", RowBox[{"[", RowBox[{"3", ",", " ", "2"}], "]"}], "]"}]}], FontFamily->"Courier", FontWeight->"Bold"]], "Input", PageBreakAbove->Automatic, PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, TextJustification->1, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[BoxData[ SuperscriptBox["x", "6"]], "Output", CellChangeTimes->{3.5057154924638104`*^9, 3.505715871229435*^9, 3.50634072521875*^9}] }, Open ]], Cell[TextData[{ "Este tipo de listas tienen gran aplicaci\[OAcute]n, para dar \ l\[IAcute]mites de integraci\[OAcute]n, zonas donde se desea dibujar una \ funci\[OAcute]n y fundamentalmente para el trabajo con matrices. Toda matriz \ se expresa como una lista, de modo que se puede definir un vector como una \ lista unidimensional {1,2,3}mientras que una matriz es una lista \ bidimensional {{1,3,5},{2,4,6}} (matriz 2x3). ", StyleBox["Mathematica", FontSlant->"Italic"], " cuando devuelve resultados tambi\[EAcute]n los devuelve en forma de lista, \ por ejemplo cuando proporciona las raices de un polinomio.\nEjemplos:" }], "Text", PageBreakAbove->Automatic, PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, TextAlignment->Left, TextJustification->1, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[CellGroupData[{ Cell[BoxData[ StyleBox[ RowBox[{"Plot", "[", RowBox[{ RowBox[{"x", "^", "2"}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "3"}], ",", "3"}], "}"}]}], "]"}], FontFamily->"Courier", FontWeight->"Bold"]], "Input", PageBreakAbove->Automatic, PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, TextJustification->1, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[BoxData[ GraphicsBox[{{}, {}, {Hue[0.67, 0.6, 0.6], LineBox[CompressedData[" 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{"1", "3", "5"}, {"2", "4", "6"} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]], "Output", CellChangeTimes->{3.505715499479435*^9, 3.505715871416935*^9, 3.506340726203125*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Chop", "[", RowBox[{ RowBox[{"Solve", " ", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{"x", "^", "3"}], "+", RowBox[{"2", RowBox[{"x", "^", "2"}]}], "-", RowBox[{"5", "x"}], "-", "4"}], "==", "0"}], ",", "x"}], "]"}], "//", "N"}], "]"}]], "Input", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"x", "\[Rule]", 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La 24 \ define una matriz de dos filas y tres columnas mientras que MatrixForm[ ] \ representa la Lista anterior en forma matricial. La \[UAcute]ltima operaci\ \[OAcute]n calcula las raices de la ecuaci\[OAcute]n y devuelve una lista de \ tres listas cada una de ellas con un elemento ( una soluci\[OAcute]n de la \ ecuaci\[OAcute]n).\nComo veremos posteriormente, no ser\[AAcute] necesario \ memorizar todo este tipo de comandos puesto que una de las herramientas m\ \[AAcute]s \[UAcute]tiles de ", StyleBox["Mathematica", FontSlant->"Italic"], " 3.0 (las paletas de comandos) nos permitir\[AAcute] introducir estas \ sentencias de otra forma" }], "Text", PageBreakAbove->Automatic, PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, TextAlignment->Left, TextJustification->1, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell["\<\ Utilizando el comandoTable[ ] se pueden generar listas sin dar \ expl\[IAcute]citamente cada uno de los elementos que la componen. Basta con \ indicar la expresi\[OAcute]n a trav\[EAcute]s de la que se genera y el n\ \[UAcute]mero de elementos que la componen de la siguiente manera: Table[expresi\[OAcute]n,{i, min,max,paso}] donde la expresi\[OAcute]n es una \ expresi\[OAcute]n en la que i variar\[AAcute] entre los valores \ m\[IAcute]nimo y m\[AAcute]ximo especificados y considerando el paso indicado \ en la lista.\ \>", "Text", PageBreakAbove->Automatic, PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, CellChangeTimes->{{3.505715507916935*^9, 3.5057155105575604`*^9}}, TextJustification->1, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[CellGroupData[{ Cell[BoxData[ StyleBox[ RowBox[{" ", RowBox[{"Table", "[", RowBox[{ RowBox[{"x", "^", "i"}], ",", " ", RowBox[{"{", RowBox[{"i", ",", " ", "2", ",", " ", "9", ",", " ", "2"}], "}"}]}], "]"}]}], FontFamily->"Courier", FontWeight->"Bold"]], "Input", PageBreakAbove->Automatic, PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, TextJustification->1, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[BoxData[ RowBox[{"{", RowBox[{ SuperscriptBox["x", "2"], ",", SuperscriptBox["x", "4"], ",", SuperscriptBox["x", "6"], ",", SuperscriptBox["x", "8"]}], "}"}]], "Output", CellChangeTimes->{3.505715502698185*^9, 3.5057158714638104`*^9, 3.5063407263125*^9}] }, Open ]], Cell["\<\ Se pueden generar de esta manera listas de varias dimensiones, introduciendo \ en vez de una \[UAcute]nica expresi\[OAcute]n una lista de expresiones \ \>", "Text", PageBreakAbove->Automatic, PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, TextJustification->1, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[CellGroupData[{ Cell[BoxData[ StyleBox[ RowBox[{"Table", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"E", "^", RowBox[{"(", RowBox[{"j", " ", "-", " ", "2"}], ")"}]}], ",", " ", RowBox[{"j", "*", RowBox[{"2", "/", "3"}]}]}], "}"}], ",", RowBox[{"{", " ", RowBox[{"j", ",", " ", "3", ",", " ", "5"}], "}"}]}], "]"}], FontFamily->"Courier", FontWeight->"Bold"]], "Input", PageBreakAbove->Automatic, PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, TextJustification->1, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"\[ExponentialE]", ",", "2"}], "}"}], ",", RowBox[{"{", RowBox[{ SuperscriptBox["\[ExponentialE]", "2"], ",", FractionBox["8", "3"]}], "}"}], ",", RowBox[{"{", RowBox[{ SuperscriptBox["\[ExponentialE]", "3"], ",", FractionBox["10", "3"]}], "}"}]}], "}"}]], "Output", CellChangeTimes->{3.505715513604435*^9, 3.5057158714950604`*^9, 3.50634072634375*^9}] }, Open ]], Cell["\<\ Si no se indica el paso se considera que vale uno. De igual manera, si no se \ indica el valor m\[IAcute]nimo se comienza desde uno\ \>", "Text", PageBreakAbove->Automatic, PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, TextJustification->1, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[CellGroupData[{ Cell[BoxData[ StyleBox[ RowBox[{" ", RowBox[{"Table", "[", RowBox[{"i", ",", RowBox[{"{", " ", RowBox[{"i", ",", " ", "4"}], "}"}]}], "]"}]}], FontFamily->"Courier", FontWeight->"Bold"]], "Input", PageBreakAbove->Automatic, PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, TextJustification->1, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[BoxData[ RowBox[{"{", RowBox[{"1", ",", "2", ",", "3", ",", "4"}], "}"}]], "Output", CellChangeTimes->{3.505715515791935*^9, 3.505715871510685*^9, 3.506340726359375*^9}] }, Open ]], Cell["\<\ Asi como conseguimos listas dimensiones introduciendo varias \ expresiones,tambi\[EAcute]n se pueden conseguir listas de este tipo \ introduciendo varios \[IAcute]ndices.\ \>", "Text", PageBreakAbove->Automatic, PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, TextJustification->1, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[CellGroupData[{ Cell[BoxData[ StyleBox[ RowBox[{" ", RowBox[{"Table", "[", RowBox[{ RowBox[{"i", " ", "+", " ", "j"}], ",", RowBox[{"{", " ", RowBox[{"i", ",", " ", "1", ",", " ", "3"}], "}"}], ",", RowBox[{"{", " ", RowBox[{"j", ",", " ", "2", ",", " ", "5"}], "}"}]}], "]"}]}], FontFamily->"Courier", FontWeight->"Bold"]], "Input", PageBreakAbove->Automatic, PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, TextJustification->1, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"3", ",", "4", ",", "5", ",", "6"}], "}"}], ",", RowBox[{"{", RowBox[{"4", ",", "5", ",", "6", ",", "7"}], "}"}], ",", RowBox[{"{", RowBox[{"5", ",", "6", ",", "7", ",", "8"}], "}"}]}], "}"}]], "Output", CellChangeTimes->{3.5057155168075604`*^9, 3.505715871541935*^9, 3.506340726390625*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"MatrixForm", "[", "%", "]"}]], "Input", PageBreakAbove->Automatic, PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, TextJustification->1, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"3", "4", "5", "6"}, {"4", "5", "6", "7"}, {"5", "6", "7", "8"} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]], "Output", CellChangeTimes->{3.505715517291935*^9, 3.5057158715575604`*^9, 3.506340726421875*^9}] }, Open ]], Cell["\<\ Se ha obtenido una matriz bidimensional en las columnas hacen referencia a \ todos distintos valores de j (2,3,4,5) mientras que las filas hacen \ referencia a los valores de i (1,2,3) \ \>", "Text", PageBreakAbove->Automatic, PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, TextAlignment->Left, TextJustification->1, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}] }, Open ]], Cell[CellGroupData[{ Cell["Ejercicios de la secci\[OAcute]n 5", "Subtitle", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[TextData[{ "- Utilice Table para hacer una lista de cinco nueves\n- Utilice Table para \ hacer una lista conjunta de los cuadrados y los cubos de los numeros pares \ del uno al nueve\n- Utilice Table para crear una lista con las potencias de x \ de 2 a 9 con paso 3\n- Seleccione de la lista anterior ", Cell[BoxData[ FormBox[ SuperscriptBox["x", "5"], TraditionalForm]]], " y eval\[UAcute]elo para x = 2\n- Crear con el comando Table una matriz 3 x \ 2 " }], "Text", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, TextAlignment->Left, TextJustification->1, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}] }, Open ]], Cell[CellGroupData[{ Cell["6.- Funciones definidas por el usuario", "Subtitle", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[TextData[{ "En general los ejercicios que realizaremos con ", StyleBox["Mathematica", FontSlant->"Italic"], " no ser\[AAcute]n \[UAcute]nicamente operaciones aritm\[EAcute]ticas sino \ que se necesitar\[AAcute] utilizar funciones definidas por el usuario para \ realizar el an\[AAcute]lisis de las mismas asi como para operar con elllas. \ Por ello se debe aprender a definir funciones de una o varias variables, as\ \[IAcute] como evaluar las mismas para cualquier punto." }], "Text", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, TextAlignment->Left, TextJustification->1, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[CellGroupData[{ Cell["6.1.-Definici\[OAcute]n de funciones", "Section", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[TextData[{ "Para definir funciones, igual que cualquier otra variable definida por el \ usuario, se recomienda utilizar palabras que empiecen por min\[UAcute]scula o \ letras min\[UAcute]sculas, para evitar que las variables definidas por el \ usuario se confundan con variables o comandos propios de ", StyleBox["Mathematica", FontSlant->"Italic"], ".\nLa definici\[OAcute]n de funciones es muy sencilla basta con indicar la \ expresi\[OAcute]n correspondiente de la siguiente manera:" }], "Text", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, TextAlignment->Left, TextJustification->1, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[BoxData[ RowBox[{ RowBox[{"f", "[", "x_", "]"}], ":=", RowBox[{ RowBox[{"x", "^", "3"}], "+", RowBox[{"2", " ", RowBox[{"x", "^", "2"}]}], "+", RowBox[{"3", " ", "x"}], "+", "5"}]}]], "Input", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell["\<\ Esta sentencia no da ning\[UAcute]n resultado puesto que tan solo se trata de \ una definici\[OAcute]n. Para ver que efectivamente la expresi\[OAcute]n que \ se ha introducido se corresponde con la deseada basta con preguntar cual es \ la funci\[OAcute]n f que se ha definido, de la siguiente manera:\ \>", "Text", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, TextAlignment->Left, TextJustification->1, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"?", "f"}]], "Input", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell["Global`f", "Print", "PrintUsage", CellChangeTimes->{3.506340726609375*^9}, CellTags->"Info3506344326-2632942"], Cell[BoxData[ InterpretationBox[GridBox[{ {GridBox[{ { RowBox[{ RowBox[{"f", "[", "x_", "]"}], ":=", RowBox[{ SuperscriptBox["x", "3"], "+", RowBox[{"2", " ", SuperscriptBox["x", "2"]}], "+", RowBox[{"3", " ", "x"}], "+", "5"}]}]} }, 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`f], Editable->False]], "Print", CellChangeTimes->{3.506340726640625*^9}, CellTags->"Info3506344326-2632942"] }, Open ]], Cell["\<\ Es fundamental definir de esta manera las funciones, colocando como variable \ independiente la variable x_, este s\[IAcute]mbolo permite considerar la \ variable x como una variable global que puede ser tanto un valor \ num\[EAcute]rico como una expresi\[OAcute]n en la que haya definidas otras \ variables\ \>", "Text", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, TextAlignment->Left, TextJustification->1, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"f", "[", RowBox[{"y", "-", "z"}], "]"}]], "Input", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[BoxData[ RowBox[{"5", "+", RowBox[{"3", " ", RowBox[{"(", RowBox[{"y", "-", "z"}], ")"}]}], "+", RowBox[{"2", " ", SuperscriptBox[ RowBox[{"(", RowBox[{"y", "-", "z"}], ")"}], "2"]}], "+", SuperscriptBox[ RowBox[{"(", RowBox[{"y", "-", "z"}], ")"}], "3"]}]], "Output", CellChangeTimes->{3.5057155243700604`*^9, 3.5057158718388104`*^9, 3.506340726734375*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"f", "[", "3", "]"}]], "Input", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[BoxData["59"], "Output", CellChangeTimes->{3.5057155249638104`*^9, 3.505715871854435*^9, 3.50634072675*^9}] }, Open ]], Cell["\<\ Del mismo modo se pueden asignar valores a las variables de las funciones a \ trav\[EAcute]s de la expresi\[OAcute]n /. de la forma siguiente:\ \>", "Text", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"f", "[", "x", "]"}], " ", "/.", " ", RowBox[{"x", "->", "3"}]}]], "Input", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic], Cell[BoxData["59"], "Output", CellChangeTimes->{3.505715525948185*^9, 3.505715871885685*^9, 3.50634072678125*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"f", "[", "x", "]"}], " ", "/.", " ", RowBox[{"x", "->", RowBox[{"u", "+", "v"}]}]}]], "Input", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic], Cell[BoxData[ RowBox[{"5", "+", RowBox[{"3", " ", RowBox[{"(", RowBox[{"u", "+", "v"}], ")"}]}], "+", RowBox[{"2", " ", SuperscriptBox[ RowBox[{"(", RowBox[{"u", "+", "v"}], ")"}], "2"]}], "+", SuperscriptBox[ RowBox[{"(", RowBox[{"u", "+", "v"}], ")"}], "3"]}]], "Output", CellChangeTimes->{3.505715526510685*^9, 3.5057158719013104`*^9, 3.5063407268125*^9}] }, Open ]], Cell["\<\ El s\[IAcute]mbolo /. se puede considerar como \"Tal que\" de forma que la \ \[UAcute]ltima operaci\[OAcute]n se puede traducir como: calcular cuanto vale \ f[x] tal que x=a+b. Lo mismo que se ha realizado para una funci\[OAcute]n de \ una variable se podr\[IAcute]a rea\[NTilde]izar para funciones de varias \ variables.\ \>", "Text", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic], Cell[BoxData[ RowBox[{ RowBox[{"m", "[", RowBox[{"x_", ",", "y_"}], "]"}], ":=", RowBox[{ RowBox[{"x", "^", "2"}], "+", RowBox[{"2", "y", " ", "x"}], "+", RowBox[{"5", "x"}], "+", "2"}]}]], "Input", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"m", "[", RowBox[{"x", ",", "y"}], "]"}], " ", "/.", " ", RowBox[{"{", RowBox[{ RowBox[{"x", "->", "2"}], ",", RowBox[{"y", "->", "5"}]}], "}"}]}]], "Input", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic], Cell[BoxData["36"], "Output", CellChangeTimes->{3.5057155280888104`*^9, 3.5057158719325604`*^9, 3.50634072684375*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"m", "[", RowBox[{"x", ",", "y"}], "]"}], " ", "/.", " ", RowBox[{"{", RowBox[{ RowBox[{"x", "->", RowBox[{"r", "-", "s"}]}], ",", RowBox[{"y", "->", "t"}]}], "}"}]}]], "Input", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic], Cell[BoxData[ RowBox[{"2", "+", RowBox[{"5", " ", RowBox[{"(", RowBox[{"r", "-", "s"}], ")"}]}], "+", SuperscriptBox[ RowBox[{"(", RowBox[{"r", "-", "s"}], ")"}], "2"], "+", RowBox[{"2", " ", RowBox[{"(", RowBox[{"r", "-", "s"}], ")"}], " ", "t"}]}]], "Output", CellChangeTimes->{3.5057155284638104`*^9, 3.505715871948185*^9, 3.506340726875*^9}] }, Open ]], Cell["\<\ Las funciones que se han definido hasta el momento tienen un dominio que para \ las funciones polin\[OAcute]micas, por ejemplo, es todo el conjunto de los n\ \[UAcute]meros reales. Sin embargo se pueden definir tambi\[EAcute]n \ funciones a trozos, definiendo en cada caso en que dominio esta definida la \ funci\[OAcute]n.\ \>", "Text", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, CellChangeTimes->{{3.5057155605263104`*^9, 3.505715583010685*^9}}, TextAlignment->Left, TextJustification->1, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[TextData[{ "Realmente la f\[OAcute]rma mas sencilla de definir una funci\[OAcute]n a \ trozos es utilizando el comando Piecewise[{{", Cell[BoxData[ FormBox[ SubscriptBox["val", "1"], TraditionalForm]]], ",", Cell[BoxData[ FormBox[ SubscriptBox["cond", "1"], TraditionalForm]]], "},{", Cell[BoxData[ FormBox[ SubscriptBox["val", "2"], TraditionalForm]]], ",", Cell[BoxData[ FormBox[ SubscriptBox["cond", "2"], TraditionalForm]]], "},...,}]" }], "Text", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, 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Sin embargo las \ expresiones que se obtienen pueden parecer muy complejas puesto que en ese \ proceso el paquete no se preocupa de reducir al m\[AAcute]ximo la expresi\ \[OAcute]n. Por ello siempre que las expresiones resulten complejas es muy \ \[UGrave]til considerar el comando Simplify[ ] que simplifica al \ m\[AAcute]ximo la expresi\[OAcute]n con que se trabaja. Por ejemplo:" }], "Text", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, TextAlignment->Left, TextJustification->1, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"x", "^", "3"}], "+", RowBox[{"3", RowBox[{"x", "^", "2"}]}], "+", RowBox[{"3", "x"}], "+", "1"}]], "Input", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[BoxData[ RowBox[{"1", "+", RowBox[{"3", " ", "x"}], "+", RowBox[{"3", " ", SuperscriptBox["x", "2"]}], "+", SuperscriptBox["x", "3"]}]], "Output", CellChangeTimes->{3.43227248904029*^9, 3.505715595323185*^9, 3.505715872073185*^9, 3.506340726984375*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Simplify", "[", "%", "]"}]], "Input", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[BoxData[ SuperscriptBox[ RowBox[{"(", RowBox[{"1", "+", "x"}], ")"}], "3"]], "Output", CellChangeTimes->{3.4322724899153852`*^9, 3.505715595948185*^9, 3.5057158720888104`*^9, 3.506340727015625*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"\[Integral]", RowBox[{ FractionBox["1", RowBox[{ SuperscriptBox["x", "4"], "-", "1"}]], RowBox[{"\[DifferentialD]", "x"}]}]}]], "Input", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[BoxData[ RowBox[{ RowBox[{"-", FractionBox[ RowBox[{"ArcTan", "[", "x", "]"}], "2"]}], "+", RowBox[{ FractionBox["1", "4"], " ", RowBox[{"Log", "[", RowBox[{"1", "-", "x"}], "]"}]}], "-", RowBox[{ FractionBox["1", "4"], " ", RowBox[{"Log", "[", RowBox[{"1", "+", "x"}], "]"}]}]}]], "Output", CellChangeTimes->{3.432272490727974*^9, 3.5057155965575604`*^9, 3.505715872135685*^9, 3.506340727125*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"D", "[", RowBox[{"%", ",", "x"}], "]"}]], "Input", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[BoxData[ RowBox[{ RowBox[{"-", FractionBox["1", RowBox[{"4", " ", RowBox[{"(", RowBox[{"1", "-", "x"}], ")"}]}]]}], "-", FractionBox["1", RowBox[{"4", " ", RowBox[{"(", RowBox[{"1", "+", "x"}], ")"}]}]], "-", FractionBox["1", RowBox[{"2", " ", RowBox[{"(", RowBox[{"1", "+", SuperscriptBox["x", "2"]}], ")"}]}]]}]], "Output", CellChangeTimes->{3.432272491399922*^9, 3.5057155969013104`*^9, 3.5057158721513104`*^9, 3.5063407271875*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Simplify", "[", "%", "]"}]], "Input", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[BoxData[ FractionBox["1", RowBox[{ RowBox[{"-", "1"}], "+", SuperscriptBox["x", "4"]}]]], "Output", CellChangeTimes->{3.43227249211875*^9, 3.505715597510685*^9, 3.505715872166935*^9, 3.50634072721875*^9}] }, Open ]], Cell["\<\ En las tres \[UGrave]ltimas operaciones se ha realizado una integral \ indefinida, a continuaci\[OAcute]n se ha realizado la derivada del resultado \ y por \[UGrave]ltimo se ha simplificado la expresi\[OAcute]n En este \[UAcute]ltimo ejemplo vemos como obtenemos el resultado inicial al \ derivar el resultado obtenido de la integral anterior, aunque si no \ hubieramos utilizado el comando Simplify[ ] resultar\[IAcute]a \ dif\[IAcute]cil darse cuenta de ello.\ \>", "Text", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, TextAlignment->Left, TextJustification->1, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell["\<\ Otra operaci\[OAcute]n que se puede realizar es la de obtener los factores de \ una expresi\[OAcute]n algebraica, muy \[UGrave]til para obtener las raices \ de un polinomio, por ejemplo. 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est\[AAcute] simplificada. Para ello se \ utiliza le comando Expand[ ] como veremos en el siguiente ejemplo\ \>", "Text", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, TextAlignment->Left, TextJustification->1, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"x", "^", "2"}], "+", "1"}], ")"}], "^", "2"}], "*", RowBox[{ RowBox[{"(", RowBox[{"x", "+", "2"}], ")"}], "^", "3"}]}]], "Input", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[BoxData[ RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"2", "+", "x"}], ")"}], "3"], " ", SuperscriptBox[ RowBox[{"(", RowBox[{"1", "+", SuperscriptBox["x", "2"]}], ")"}], "2"]}]], "Output", CellChangeTimes->{3.505715601729435*^9, 3.5057158723388104`*^9, 3.506340727390625*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Expand", "[", "%", "]"}]], "Input", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[BoxData[ RowBox[{"8", "+", RowBox[{"12", " ", "x"}], "+", RowBox[{"22", " ", SuperscriptBox["x", "2"]}], "+", RowBox[{"25", " ", SuperscriptBox["x", "3"]}], "+", RowBox[{"20", " ", SuperscriptBox["x", "4"]}], "+", RowBox[{"14", " ", SuperscriptBox["x", "5"]}], "+", RowBox[{"6", " ", SuperscriptBox["x", "6"]}], "+", SuperscriptBox["x", "7"]}]], "Output", CellChangeTimes->{3.505715602979435*^9, 3.505715872354435*^9, 3.506340727421875*^9}] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Ejercicios de la secci\[OAcute]n 6", "Subtitle", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[TextData[{ "- Representar como f(x) la funci\[OAcute]n ( x+2 )", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"x", "-", "1"}], ")"}], "2"], SuperscriptBox[ RowBox[{"(", RowBox[{"x", "-", "2"}], ")"}], "3"]}], TraditionalForm]]], "\n- Dibujar dicha funci\[OAcute]n con el comando Plot entre x = -3 , x = 3\n\ - Expandir la expresi\[OAcute]n de f(x) hasta obtener un polinomio de grado 6 \ en x\n- Aplicar la expresi\[OAcute]n Factor [ ] para llegar a la expresi\ \[OAcute]n de partida." }], "Text", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, TextAlignment->Left, TextJustification->1, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}] }, Open ]], Cell[CellGroupData[{ Cell["7.- Paletas", "Subtitle", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[TextData[{ "Una de las mejoras sustanciales de Mathematica 3.0 en cuanto a la \ comunicaci\[OAcute]n con el usuario es la presentaci\[OAcute]n de paletas que \ permiten introducir sentencias y realizar operaciones de forma m\[AAcute]s \ sencilla. Para algunas sentencias de ", StyleBox["Mathematica", FontSlant->"Italic"], " como Sqrt[ ] ( Raiz Cuadrada ), Sum[ ] ( Sumatorio ), Product[ ] ( \ Producto ),etc. resulta complicado escribir en el notebook cual es la operaci\ \[OAcute]n que se desea realizar, mientras que con las paletas estas \ operaciones vienen representadas por sus s\[IAcute]mbolos matem\[AAcute]ticos \ por lo que basta con introducir los datos de la operaci\[OAcute]n para que \ \[EAcute]sta se pueda llevar a cabo. Los s\[IAcute]mbolos que utilizan estas \ paletas para realizar las operaciones anteriores son: ", Cell[BoxData[ FormBox[ SqrtBox["\[Placeholder]"], TraditionalForm]]], ",", Cell[BoxData[ FormBox[ RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"\[Placeholder]", "=", "\[Placeholder]"}], "\[Placeholder]"], "\[Placeholder]"}], TraditionalForm]]], ", ", Cell[BoxData[ FormBox[ RowBox[{ UnderoverscriptBox["\[Product]", RowBox[{"\[Placeholder]", "=", "\[Placeholder]"}], "\[Placeholder]"], "\[Placeholder]"}], TraditionalForm]]], " . Introduciendo en los recuadros los valores deseados se completa la \ sentencia con lo que se puede realizar la operaci\[OAcute]n. 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TextJustification->0.], Cell[TextData[{ "En este sentido ", StyleBox["Mathematica", FontSlant->"Italic"], " introduce ya como predefinidas una serie de paletas en las que se \ representan tanto comandos, como operaciones, caracteres, matrices,etc. \nUna \ de las aplicaciones mas comodas de las paletas es la definici\[OAcute]n de \ matrices puesto que partiendo de matrices 2 x 2 que aparecen en la paleta se \ puede llegar a conseguir matrices de cualquier dimensi\[OAcute]n. Se aumenta \ el n\[UAcute]mero de filas con el comando \" Control + Intro\" mientras que \ el n\[UAcute]mero de columnas aumenta al pulsar \"Control + ,\"" }], "Text", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, CellChangeTimes->{{3.505714258260685*^9, 3.505714274916935*^9}}, TextAlignment->Left, TextJustification->1, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "8.- Gr\[AAcute]ficos con ", StyleBox["Mathematica", FontSlant->"Italic"], " " }], "Subtitle", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[TextData[{ "Una herramienta de apoyo muy importante a la hora de resolver una gran \ cantidad de problemas matem\[AAcute]ticos ( sobre todo de una y dos variables \ ) es la representaci\[OAcute]n de funciones. De esta manera se pueden ver \ cuales son los extremos de una funci\[OAcute]n, sus as\[IAcute]ntotas, etc.\n", StyleBox["Mathematica", FontSlant->"Italic"], " dispone de una gran cantidad de comandos que permiten representar curvas y \ superficies. En este apartado se har\[AAcute] una exposici\[OAcute]n general \ de los comandos que se utilizar\[AAcute]n posteriormente en las \ pr\[AAcute]cticas de las diferentes asignaturas. Debido a que la nomenclatura \ es exactamente igual se introducir\[AAcute]n de forma conjunta las \ expresiones de dibujo en dos y tres variables." }], "Text", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[CellGroupData[{ Cell[TextData[StyleBox["8.1.- Plot, Plot3D", "Section"]], "Section", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell["\<\ Estos comandos generan la funci\[OAcute]n f en un recinto definido por el \ usuario. La forma de expresarlo es la siguiente: Plot[ f , {x,x_min,x_max}] Plot3D[f , {x,x_min,x_max},{y,y_min,y_max}] Ejemplos: Definimos previamente dos funciones, una de una variable y otra de dos \ variables y posteriormente las representamos.\ \>", "Text", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[BoxData[ RowBox[{ RowBox[{"f", "[", "x_", "]"}], ":=", RowBox[{ RowBox[{"Sin", "[", "x", "]"}], "+", RowBox[{ RowBox[{"x", "^", "2"}], "/", "2"}]}]}]], "Input", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[BoxData[ RowBox[{ RowBox[{"g", "[", RowBox[{"x_", ",", "y_"}], "]"}], ":=", RowBox[{ RowBox[{"x", "^", "2"}], "+", RowBox[{"y", "*", "x"}], "-", RowBox[{"2", " ", "y"}]}]}]], "Input", PageBreakWithin->Automatic, PageBreakBelow->Automatic, 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Por otro lado una gr\[AAcute]n cantidad de funciones no vienen \ representadas de forma expl\[IAcute]cita e incluso es imposible despejar una \ de las variables en funci\[OAcute]n del resto. Por ello se necesita otra \ serie de comandos que nos permitan representar esas funciones. Las otras dos \ formas en que mas habitualmente se presentan las funciones son la forma param\ \[EAcute]trica y la forma impl\[IAcute]cita. A continuaci\[OAcute]n veremos \ como se representan las funciones cuando viene definidas de esta manera.\ \>", "Text", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}] }, Open ]], Cell[CellGroupData[{ Cell["8.2.- ParametricPlot, ParametricPlot3D", "Section", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell["\<\ Estos comandos permiten representar funciones que vienen definidas a trav\ \[EAcute]s de par\[AAcute]metros. Una funci\[OAcute]n de dos variables se \ representar\[IAcute]a de la siguiente manera: x = x(t) y = y(t)\tA medida que var\[IAcute]a el parametro se obtienen puntos de la \ curva. Para el caso de tres dimensiones a trav\[EAcute]s de expresiones param\ \[EAcute]tricas se pueden representar tanto curvas como superficies. Una \ curva se representa de la siguiente manera: x = x(t) y = y(t)\tVariando t se obtiene una familia simplemente infinita de puntos ( \ una curva ) z = z(t) Para representar una superficie se necesitan dos par\[AAcute]metros, de forma \ que se obtiene una familia doblemente infinita de puntos que generan la \ superficie. La superficie se reptresenta de la siguiente manera: x=x( u,v) y=y( u,v) z=z( u,v) Vista esta introducci\[OAcute]n podemos presentar la forma de utilizar los \ comandos ParametricPlot y ParametricPlot3D. 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3}, {-3, 3}}, PlotRangePadding->{ Scaled[0.02], Scaled[0.02], Scaled[0.02]}, ViewPoint->{2.0481815215954167`, -2.055532422433216, 1.7406145222078655`}, ViewVertical->{0.1746051650878924, 0.03427837564802309, 0.9840416806657952}]], "Output", CellChangeTimes->{3.505715655229435*^9, 3.5057158730888104`*^9, 3.50634072865625*^9}] }, Open ]], Cell[CellGroupData[{ Cell["Ejercicios de la secci\[OAcute]n 8", "Subtitle", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell["\<\ - Dibujar la funci\[OAcute]n sen(x)/x as\[IAcute] como la funci\[OAcute]n \ sen(x) en el intervalo (-10,10) - Dibujar la funci\[OAcute]n x*y - Dibujar la funci\[OAcute]n param\[EAcute]trica\tx = 4 Cos(-11t / 4)+7 Cos(t) \t\t\t\t\t y = 4 Sin(-11t / 4)+7 Sin(t)\t\tDesde 0 a 8Pi - Dibujar la funcion \tx= Cos(u)Sin(v) \t\t\ty= Cos(u)Cos(v) \t\t\tz=v - Dibujar la funci\[OAcute]n : 4x^2+y^2=1\ \>", "Text", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}] }, Open ]], Cell[CellGroupData[{ Cell["9.- Resoluci\[OAcute]n de Ecuaciones", "Subtitle", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, CellChangeTimes->{3.494068518640625*^9}, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " dispone de un amplio grupo de sentencias que permiten resolver \ ecuaciones. Dependiendo del tipo de ecuaci\[OAcute]n que se desee resolver y \ las variables y par\[AAcute]metros que tenga ser\[AAcute] conveniente \ utilizar uno u otro de los comandos que a continuaci\[OAcute]n se exponen.\n", StyleBox["Roots[ecuacion,variable]", FontWeight->"Bold"], " permite obtener las raices de una ecuaci\[OAcute]n polin\[OAcute]mica en \ la variable que se indica en la expresi\[OAcute]n. Este comando funcionar\ \[AAcute] correctamente siempre que se pueda obtener de forma exacta la raiz \ del polinomio." }], "Text", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, CellChangeTimes->{{3.5057158221513104`*^9, 3.505715835791935*^9}}, TextAlignment->Left, TextJustification->1, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Roots", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"x", "+", "2"}], ")"}], "^", "3"}], "*", RowBox[{"(", RowBox[{"x", "-", "3"}], ")"}]}], "==", "0"}], ",", "x"}], "]"}]], "Input", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[BoxData[ RowBox[{ RowBox[{"x", "\[Equal]", "3"}], "||", RowBox[{"x", "\[Equal]", RowBox[{"-", "2"}]}], "||", RowBox[{"x", "\[Equal]", RowBox[{"-", "2"}]}], "||", RowBox[{"x", "\[Equal]", RowBox[{"-", "2"}]}]}]], "Output", CellChangeTimes->{3.5057156610575604`*^9, 3.5057158731200604`*^9, 3.506340728703125*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Roots", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"x", "+", "y", "-", "1"}], ")"}], "*", RowBox[{"(", RowBox[{"x", "-", "2"}], ")"}]}], "==", "0"}], ",", "x"}], "]"}]], "Input", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[BoxData[ RowBox[{ RowBox[{"x", "\[Equal]", "2"}], "||", RowBox[{"x", "\[Equal]", RowBox[{"1", "-", "y"}]}]}]], "Output", CellChangeTimes->{3.5057156619325604`*^9, 3.5057158731513104`*^9, 3.506340728734375*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Roots", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"x", "+", "y", "-", "1"}], ")"}], "*", RowBox[{"(", RowBox[{"x", "-", "2"}], ")"}]}], "==", "0"}], ",", "y"}], "]"}]], "Input", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[BoxData[ RowBox[{"y", "\[Equal]", RowBox[{"1", "-", "x"}]}]], "Output", CellChangeTimes->{3.5057156623700604`*^9, 3.505715873166935*^9, 3.506340728765625*^9}] }, Open ]], Cell["\<\ La expresi\[OAcute]n Roots[ ] devuelve todas las raices de la \ ecuaci\[OAcute]n.En caso de haber mas de una soluci\[OAcute]n relaciona todas \ ellas mediante operadores l\[OAcute]gicos ( | | significa OR , && significa \ AND ). En caso que no se pueda obtener el valor exacto de la raiz se utilizar\ \[AGrave] el comando NRoots[ ] que proporciona una soluci\[OAcute]n \ aproximada. \ \>", "Text", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, TextAlignment->Left, TextJustification->1, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Roots", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{"x", "^", "7"}], "+", "x", "+", "4"}], "==", "0"}], ",", "x"}], "]"}]], "Input", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[BoxData[ RowBox[{ RowBox[{"x", "\[Equal]", RowBox[{"Root", "[", RowBox[{ RowBox[{ RowBox[{"4", "+", "#1", "+", SuperscriptBox["#1", "7"]}], "&"}], ",", "1"}], "]"}]}], "||", RowBox[{"x", "\[Equal]", RowBox[{"Root", "[", RowBox[{ RowBox[{ RowBox[{"4", "+", "#1", "+", 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"\[Equal]", RowBox[{"1.1302715848445464`", "\[VeryThinSpace]", "-", RowBox[{"0.5663488814623097`", " ", "\[ImaginaryI]"}]}]}], "||", RowBox[{"x", "\[Equal]", RowBox[{"1.1302715848445464`", "\[VeryThinSpace]", "+", RowBox[{"0.5663488814623097`", " ", "\[ImaginaryI]"}]}]}]}]], "Output", CellChangeTimes->{3.505715664073185*^9, 3.5057158732138104`*^9, 3.50634072884375*^9}] }, Open ]], Cell[TextData[{ "En este \[UAcute]ltimo ejemplo se puede ver como intentando obtener las \ raices con el comando Roots no se llega a una soluci\[OAcute]n por lo que hay \ que buscar la soluci\[OAcute]n aproximada.\nPara un caso mas general en que \ se desea resolver una ecuaci\[OAcute]n cualquiera se utiliza el comando ", StyleBox["Solve[ ]", FontWeight->"Bold"], " que funciona de la misma manera que el comando Roots[ ]" }], "Text", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, CellChangeTimes->{{3.5057158043700604`*^9, 3.505715806323185*^9}}, TextAlignment->Left, TextJustification->1, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}, CellTags->"Solve"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Solve", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"x", "+", "2"}], ")"}], "^", "3"}], "*", RowBox[{"(", RowBox[{"x", "-", "3"}], ")"}]}], "==", "0"}], ",", "x"}], "]"}]], "Input", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"x", "\[Rule]", RowBox[{"-", "2"}]}], "}"}], ",", RowBox[{"{", RowBox[{"x", "\[Rule]", RowBox[{"-", "2"}]}], "}"}], ",", RowBox[{"{", RowBox[{"x", "\[Rule]", RowBox[{"-", "2"}]}], "}"}], ",", RowBox[{"{", RowBox[{"x", "\[Rule]", "3"}], "}"}]}], "}"}]], "Output", CellChangeTimes->{3.505715665541935*^9, 3.5057158732450604`*^9, 3.506340728875*^9}] }, Open ]], Cell["\<\ Los resultados se presentan en una lista en que se asignan a x un valor con \ el signo x->. Cuando no se pueden obtener los valores exactos la soluci\[OAcute]n queda en \ funci\[OAcute]n del comando Root y se puede obtener la soluci\[OAcute]n \ aproximada aplicando el comando N[ ]\ \>", "Text", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, TextAlignment->Left, TextJustification->1, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Solve", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{"x", "^", "7"}], "+", "x", "+", "4"}], "==", "0"}], ",", "x"}], "]"}]], "Input", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"x", "\[Rule]", RowBox[{"Root", "[", RowBox[{ RowBox[{ RowBox[{"4", "+", "#1", "+", SuperscriptBox["#1", "7"]}], "&"}], ",", "1"}], "]"}]}], "}"}], ",", RowBox[{"{", RowBox[{"x", "\[Rule]", RowBox[{"Root", "[", RowBox[{ RowBox[{ RowBox[{"4", "+", "#1", "+", SuperscriptBox["#1", "7"]}], "&"}], ",", "2"}], "]"}]}], "}"}], ",", RowBox[{"{", RowBox[{"x", "\[Rule]", RowBox[{"Root", "[", RowBox[{ RowBox[{ RowBox[{"4", "+", "#1", "+", SuperscriptBox["#1", "7"]}], "&"}], ",", "3"}], "]"}]}], "}"}], ",", RowBox[{"{", RowBox[{"x", "\[Rule]", RowBox[{"Root", "[", RowBox[{ RowBox[{ RowBox[{"4", "+", "#1", "+", SuperscriptBox["#1", "7"]}], "&"}], ",", "4"}], "]"}]}], "}"}], ",", RowBox[{"{", RowBox[{"x", "\[Rule]", RowBox[{"Root", "[", RowBox[{ RowBox[{ RowBox[{"4", "+", "#1", "+", SuperscriptBox["#1", "7"]}], "&"}], ",", "5"}], "]"}]}], "}"}], ",", RowBox[{"{", RowBox[{"x", "\[Rule]", RowBox[{"Root", "[", RowBox[{ RowBox[{ RowBox[{"4", "+", "#1", "+", SuperscriptBox["#1", "7"]}], "&"}], ",", "6"}], "]"}]}], "}"}], ",", RowBox[{"{", RowBox[{"x", "\[Rule]", RowBox[{"Root", "[", RowBox[{ RowBox[{ RowBox[{"4", "+", "#1", "+", SuperscriptBox["#1", "7"]}], "&"}], ",", "7"}], "]"}]}], "}"}]}], "}"}]], "Output", CellChangeTimes->{3.505715666635685*^9, 3.505715873260685*^9, 3.506340728921875*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"N", "[", "%", "]"}]], "Input", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"x", "\[Rule]", RowBox[{"-", "1.1607618283269017`"}]}], "}"}], ",", RowBox[{"{", RowBox[{"x", "\[Rule]", RowBox[{ RowBox[{"-", "0.776477729610295`"}], "-", RowBox[{"0.8995904913612532`", " ", "\[ImaginaryI]"}]}]}], "}"}], ",", RowBox[{"{", RowBox[{"x", "\[Rule]", RowBox[{ RowBox[{"-", "0.776477729610295`"}], "+", RowBox[{"0.8995904913612532`", " ", "\[ImaginaryI]"}]}]}], "}"}], ",", RowBox[{"{", RowBox[{"x", "\[Rule]", RowBox[{"0.22658705892920028`", "\[VeryThinSpace]", "-", RowBox[{"1.2146781955444665`", " ", "\[ImaginaryI]"}]}]}], "}"}], ",", RowBox[{"{", RowBox[{"x", "\[Rule]", RowBox[{"0.22658705892920028`", "\[VeryThinSpace]", "+", RowBox[{"1.2146781955444665`", " ", "\[ImaginaryI]"}]}]}], "}"}], ",", RowBox[{"{", RowBox[{"x", "\[Rule]", RowBox[{"1.1302715848445464`", "\[VeryThinSpace]", "-", RowBox[{"0.5663488814623097`", " ", "\[ImaginaryI]"}]}]}], "}"}], ",", RowBox[{"{", RowBox[{"x", "\[Rule]", RowBox[{"1.1302715848445464`", "\[VeryThinSpace]", "+", RowBox[{"0.5663488814623097`", " ", "\[ImaginaryI]"}]}]}], "}"}]}], "}"}]], "Output", CellChangeTimes->{3.505715667198185*^9, 3.505715873291935*^9, 3.50634072896875*^9}] }, Open ]], Cell["\<\ Este Comando permite resolver ecuaciones que no sean polin\[OAcute]micas como \ por ejemplo:\ \>", "Text", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Solve", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{"Sin", "[", "x", "]"}], " ", RowBox[{"Cos", "[", "x", "]"}]}], "==", "0"}], ",", "x"}], "]"}]], "Input", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[BoxData[ RowBox[{ StyleBox[ RowBox[{"Solve", "::", "ifun"}], "MessageName"], RowBox[{ ":", " "}], "\<\"Inverse functions are being used by \ \[NoBreak]\\!\\(Solve\\)\[NoBreak], so some solutions may not be found; use \ Reduce for complete solution information. \\!\\(\\*ButtonBox[\\\"\ \[RightSkeleton]\\\", ButtonStyle->\\\"Link\\\", ButtonFrame->None, \ ButtonData:>\\\"paclet:ref/Solve\\\", ButtonNote -> \\\"Solve::ifun\\\"]\\)\"\ \>"}]], "Message", "MSG", CellChangeTimes->{3.5057156684325604`*^9, 3.505715873354435*^9, 3.5063407290625*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"x", "\[Rule]", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"x", "\[Rule]", RowBox[{"-", FractionBox["\[Pi]", "2"]}]}], "}"}], ",", RowBox[{"{", RowBox[{"x", "\[Rule]", FractionBox["\[Pi]", "2"]}], "}"}]}], "}"}]], "Output", CellChangeTimes->{3.5057156684325604`*^9, 3.505715873354435*^9, 3.5063407290625*^9}] }, Open ]], Cell["\<\ En este caso se obtienen algunas soluciones triviales y devuelve un mensaje \ que indica la posible existencia de mas soluciones\ \>", "Text", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[TextData[{ "Este comando Solve permite tambi\[EAcute]n resolver sistemas de ecuaciones \ que se representan en una lista o bien unidas a trav\[EAcute]s del \ s\[IAcute]mbolo && (And). Ejemplos de este tipo se ver\[AAcute]n en las pr\ \[AAcute]cticas correspondientes al Algebra.\nEn muchas ocasiones se emplean \ par\[AAcute]metros que pueden ir variando seg\[UAcute]n se desee y que \ mejoran o empeoran el comportamiento de un sistema. Por ello es muy \ interesante, poder resolver ecuaciones en las que se encuentren \ par\[AAcute]metros. Para ello el comando mas indicado en el paquete ", StyleBox["Mathematica", FontSlant->"Italic"], " es ", StyleBox["Reduce[ ]", FontWeight->"Bold"], ". La forma de presentarlo es id\[EAcute]ntica a los anteriores. Veamos alg\ \[UAcute]n ejemplo:" }], "Text", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, CellChangeTimes->{{3.5057156830575604`*^9, 3.5057156858700604`*^9}}, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[BoxData[ RowBox[{"Clear", "[", RowBox[{"a", ",", "x"}], "]"}]], "Input", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Reduce", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{"3", " ", "a", " ", RowBox[{"x", "^", "3"}]}], "-", RowBox[{"2", "x"}], "+", "3"}], "==", "0"}], ",", "x"}], "]"}]], "Input", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[BoxData[ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"a", "\[Equal]", "0"}], "&&", RowBox[{"x", "\[Equal]", FractionBox["3", "2"]}]}], ")"}], "||", RowBox[{"(", RowBox[{ RowBox[{"a", "\[NotEqual]", "0"}], "&&", RowBox[{"(", RowBox[{ RowBox[{"x", "\[Equal]", RowBox[{"Root", "[", RowBox[{ RowBox[{ RowBox[{"3", "-", RowBox[{"2", " ", "#1"}], "+", RowBox[{"3", " ", "a", " ", SuperscriptBox["#1", "3"]}]}], "&"}], ",", "1"}], "]"}]}], "||", RowBox[{"x", "\[Equal]", RowBox[{"Root", "[", RowBox[{ RowBox[{ RowBox[{"3", "-", RowBox[{"2", " ", "#1"}], "+", RowBox[{"3", " ", "a", " ", SuperscriptBox["#1", "3"]}]}], "&"}], ",", "2"}], "]"}]}], "||", RowBox[{"x", "\[Equal]", RowBox[{"Root", "[", RowBox[{ RowBox[{ RowBox[{"3", "-", RowBox[{"2", " ", "#1"}], "+", RowBox[{"3", " ", "a", " ", SuperscriptBox["#1", "3"]}]}], "&"}], ",", "3"}], "]"}]}]}], ")"}]}], ")"}]}]], "Output", CellChangeTimes->{3.505715670666935*^9, 3.505715873385685*^9, 3.50634072915625*^9}] }, Open ]], Cell["\<\ En este ejemplo se ve claramente el funcionamiento del comando. Si a vale \ cero logicamente se trata de una ecuaci\[OAcute]n en que solo hay una raiz. \ Si a es distinto de cero se obtienen tres raices distintas dado que estamos \ estudiando un polinomio de grado tres. Este comando tambi\[EAcute]n se puede emplear cuando se trabaja con un \ sistema de ecuaciones como en el caso anterior, definiendo todos ellos entre \ llaves o con el s\[IAcute]mbolo &&. Ej:\ \>", "Text", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[BoxData[ RowBox[{"Clear", "[", RowBox[{"a", ",", "r", ",", "s"}], "]"}]], "Input", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"2", ",", "a"}], "}"}], ",", RowBox[{"{", RowBox[{"3", ",", "1"}], "}"}]}], "}"}], ".", RowBox[{"{", RowBox[{"r", ",", "s"}], "}"}]}], " ", "==", RowBox[{"{", RowBox[{"0", ",", "1"}], "}"}]}]], "Input", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[BoxData[ RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"2", " ", "r"}], "+", RowBox[{"a", " ", "s"}]}], ",", RowBox[{ RowBox[{"3", " ", "r"}], "+", "s"}]}], "}"}], "\[Equal]", RowBox[{"{", RowBox[{"0", ",", "1"}], "}"}]}]], "Output", CellChangeTimes->{3.5057156724325604`*^9, 3.505715873416935*^9, 3.506340729203125*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Reduce", "[", RowBox[{"%", ",", RowBox[{"{", RowBox[{"r", ",", "s"}], "}"}]}], "]"}]], "Input", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{ RowBox[{"-", "2"}], "+", RowBox[{"3", " ", "a"}]}], "\[NotEqual]", "0"}], "&&", RowBox[{"r", "\[Equal]", FractionBox["a", RowBox[{ RowBox[{"-", "2"}], "+", RowBox[{"3", " ", "a"}]}]]}], "&&", RowBox[{"s", "\[Equal]", RowBox[{"1", "-", RowBox[{"3", " ", "r"}]}]}]}]], "Output", CellChangeTimes->{3.505715672760685*^9, 3.505715873448185*^9, 3.506340729296875*^9}] }, Open ]], Cell[TextData[{ "Resuelve el sistema matricial considerando el valor del par\[AAcute]metro ", StyleBox["a", FontWeight->"Bold"], ". L\[OAcute]gicamente el valor del determinante de la matriz tiene que ser \ no nulo ya que el rango debe ser dos para que exista soluci\[OAcute]n.\n\ Cuando no se pueden obtener soluciones mediante los comandos que se han \ explicado se debe pasar a los m\[EAcute]todos iterativos para la obtenci\ \[OAcute]n de soluciones de la ecuaci\[OAcute]n.\nPara ello se utiliza el \ comando ", StyleBox["FindRoot[ecuacion,{x,sol_aprox}]", FontWeight->"Bold"], " en el que se indica la ecuaci\[OAcute]n que se desea resolver y un valor \ cercano a la soluci\[OAcute]n de la ecuaci\[OAcute]n. Debido a esto se \ recomienda anteriormente dibujar la funci\[OAcute]n para tener una idea \ aproximada de la posici\[OAcute]n de la soluci\[OAcute]n. Ejemplo: Calcular \ las soluciones de la ecuaci\[OAcute]n x Sin[x]-1/2= 0" }], "Text", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, CellChangeTimes->{{3.5057156939013104`*^9, 3.5057157407138104`*^9}}, TextAlignment->Left, TextJustification->1, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Plot", "[", RowBox[{ RowBox[{ RowBox[{"x", " ", RowBox[{"Sin", "[", "x", "]"}]}], "-", RowBox[{"1", "/", "2"}]}], ",", RowBox[{"{", RowBox[{"x", ",", "0", ",", "4"}], "}"}]}], "]"}]], "Input", PageBreakWithin->Automatic, PageBreakBelow->Automatic, GroupPageBreakWithin->Automatic, ImageSize->{150, 150}, RenderingOptions->{"ImageCacheDepth"->"DeepestScreen"}], Cell[BoxData[ GraphicsBox[{{}, {}, {Hue[0.67, 0.6, 0.6], LineBox[CompressedData[" 1:eJwd1nk8FP//APDZdYTFIqVEkkiSdFLh9Y5UQuUOJRUVH8lV0qUcRbkqoXLm KFSOUESzuZL7rJwrIXLbWXLt/ub7m3/m8Xw85vGeeb/mdbzlz142daRiGHaE 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