(* 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[ 37467, 982] NotebookOptionsPosition[ 18594, 610] NotebookOutlinePosition[ 35235, 910] CellTagsIndexPosition[ 35192, 907] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["\<\ Tarea 8\.aa: Operadores L\[OAcute]gicos y Relacionales. Ciclos y Estructuras de Control. M\ \[EAcute]todos de resoluci\[OAcute]n de una Ecuaci\[OAcute]n No Lineal.\ \>", "Title", CellChangeTimes->{{3.42734278509206*^9, 3.4273428050291452`*^9}, 3.428126571703125*^9, {3.458980892578125*^9, 3.45898096840625*^9}, { 3.459666071703125*^9, 3.45966608446875*^9}, 3.4596661445*^9, { 3.45966645709375*^9, 3.459666512296875*^9}, {3.459667067609375*^9, 3.459667072453125*^9}}, TextAlignment->Center, TextJustification->0], Cell[CellGroupData[{ Cell["\<\ Ejercicio 1\.ba.- Utilizando el comando Table, representar sucesivamente las gr\[AAcute]ficas \ de sin(nx) para n desde 1 hasta 5. Repetir el ejercicio con el comando \ Animate.\ \>", "Subsection", CellChangeTimes->{ 3.4280497007994742`*^9, 3.428134779234375*^9, {3.459671224796875*^9, 3.4596712326875*^9}}], Cell[BoxData["\[IndentingNewLine]"], "Input", CellChangeTimes->{3.472217528296875*^9}] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Ejercicio 2\.ba.- Siendo Sn la suma de los n primeros n\[UAcute]meros naturales. \ \[DownQuestion]C\[UAcute]al es el mayor valor de n tal que Sn<11?. En este \ caso, \[DownQuestion]Cuanto es exactamente la suma Sn?. Repetir el ejercicio de forma que Sn<10^4.\ \>", "Subsection", CellChangeTimes->{ 3.4280497092369742`*^9, 3.4281348065625*^9, {3.459754381734375*^9, 3.459754384046875*^9}}], Cell[BoxData["\[IndentingNewLine]"], "Input", CellChangeTimes->{3.472217538046875*^9}] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "Ejercicio 3\.ba.-\n", "Utilizando la funci\[OAcute]n Prime, crear una lista con los 20 primeros n\ \[UAcute]meros primos. Hallar todos los n\[UAcute]meros primos menores que \ 1000 guard\[AAcute]ndolos en variables de la forma ", Cell[BoxData[ FormBox[ SubscriptBox["a", "i"], TraditionalForm]]], "." }], "Subsection", CellChangeTimes->{{3.4280497616275992`*^9, 3.4280497617838492`*^9}, 3.428134856828125*^9, {3.4597543951875*^9, 3.459754397265625*^9}}], Cell[BoxData["\[IndentingNewLine]"], "Input", CellChangeTimes->{3.47221754478125*^9}] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "Ejercicio 4\.ba.-\n", "Comprobar gr\[AAcute]ficamente que la funci\[OAcute]n f(x) = x- 0.2 sen(x) \ - 0,5 tiene una \[UAcute]nica raiz real en el intervalo [0.5 , 1]. Aplicar \ los m\[EAcute]todos de la bisecci\[OAcute]n, y Newton-Raphson, contando en \ cada caso el n\[UAcute]mero de iteraciones que se han realizado , para \ determinar la raiz con una precisi\[OAcute]n de ", Cell[BoxData[ FormBox[ SuperscriptBox["10", RowBox[{"-", "4"}]], TraditionalForm]]], ". Establecer un n\.ba m\[AAcute]ximo de iteraciones a realizar igual a 50." }], "Subsection", CellChangeTimes->{ 3.42813486221875*^9, {3.459754405234375*^9, 3.4597544071875*^9}}], Cell[BoxData["\[IndentingNewLine]"], "Input", CellChangeTimes->{3.472217548765625*^9}], Cell["\ta) Bisecci\[OAcute]n", "Text"], Cell[BoxData["\[IndentingNewLine]"], "Input", CellChangeTimes->{3.472217559640625*^9}], Cell["\tb) Newton-Raphson", "Text"], Cell[BoxData["\[IndentingNewLine]"], "Input", CellChangeTimes->{3.47221756415625*^9}] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "Ejercicio 5\.ba.-\n", "Representar en una misma gr\[AAcute]fica las funciones f(x) = 2 cos(x) y \ g(x) = ", Cell[BoxData[ FormBox[ SuperscriptBox["e", "x"], TraditionalForm]]], " para obtener aproximaciones iniciales de las raices de 2cos(x) - ", Cell[BoxData[ FormBox[ SuperscriptBox["e", "x"], TraditionalForm]]], "= 0. \[DownQuestion]Cu\[AAcute]ntas ra\[IAcute]ces tiene?. \ \[DownQuestion]Cu\[AAcute]ntas son positivas?. Determinar las raices con una \ precisi\[OAcute]n de ", Cell[BoxData[ FormBox[ SuperscriptBox["10", RowBox[{"-", "8"}]], TraditionalForm]]], " utilizando el m\[EAcute]todo de Newton-Raphson ." }], "Subsection", CellChangeTimes->{ 3.428134866296875*^9, {3.459754417265625*^9, 3.459754418796875*^9}}], Cell[BoxData["\[IndentingNewLine]"], "Input", CellChangeTimes->{3.472217571671875*^9}] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "Ejercicio 6\.ba.-\n", "Comprobar de forma gr\[AAcute]fica que la ecuaci\[OAcute]n .\n\t\t4 sen(x) \ = 1+x\ntiene tres raices reales : ", Cell[BoxData[ FormBox[ SubscriptBox["r", "1"], TraditionalForm]]], "<", Cell[BoxData[ FormBox[ SubscriptBox["r", "2"], TraditionalForm]]], "<", Cell[BoxData[ FormBox[ SubscriptBox["r", "3"], TraditionalForm]]], " . A continuaci\[OAcute]n determinar con una precisi\[OAcute]n de ", Cell[BoxData[ FormBox[ SuperscriptBox["10", RowBox[{"-", "6"}]], TraditionalForm]]], ",\n\ta) ", Cell[BoxData[ FormBox[ SubscriptBox["r", "1"], TraditionalForm]]], " utilizando el mdo de la secante.\n\tb) ", Cell[BoxData[ FormBox[ SubscriptBox["r", "2"], TraditionalForm]]], " \" \" \" \" \" bisecci\[OAcute]n.\n\tc) ", Cell[BoxData[ FormBox[ SubscriptBox["r", "3"], TraditionalForm]]], " \" \" \" \" Newton." }], "Subsection", CellChangeTimes->{ 3.4281348698125*^9, {3.45975442978125*^9, 3.45975443153125*^9}}], Cell[BoxData["\[IndentingNewLine]"], "Input", CellChangeTimes->{3.472217576234375*^9}], Cell["\ta) M\[EAcute]todo de la secante:", "Text"], Cell[BoxData["\[IndentingNewLine]"], "Input", CellChangeTimes->{3.47221758403125*^9}], Cell["\tb) M\[EAcute]todo de la bisecci\[OAcute]n", "Text"], Cell[BoxData["\[IndentingNewLine]"], "Input", CellChangeTimes->{3.4722175876875*^9}], Cell["\tc) M\[EAcute]todo de Newton-Raphson", "Text"], Cell[BoxData["\[IndentingNewLine]"], "Input", CellChangeTimes->{3.472217591796875*^9}] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "Ejercicio 7\.ba.-\n", "Encontrar una raiz aproximada de la ecuaci\[OAcute]n :\n\t\t", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SuperscriptBox["x", "3"], "-", "x", "-", "1"}], " ", "=", " ", "0"}], TraditionalForm]]], "\nen el intervalo [1 , 2] con una precisi\[OAcute]n de ", Cell[BoxData[ FormBox[ SuperscriptBox["10", RowBox[{"-", "10"}]], TraditionalForm]]], " primero por el m\[EAcute]todo de Newton y luego por el de la secante. \ Contar el n\.ba de iteraciones que se han realizado en ambos casos para \ comprobar cu\[AAcute]l de los dos m\[EAcute]todos tiene una convergencia m\ \[AAcute]s r\[AAcute]pida. " }], "Subsection", CellChangeTimes->{ 3.428134875234375*^9, {3.459754446390625*^9, 3.459754449078125*^9}}], Cell[BoxData["\[IndentingNewLine]"], "Input", CellChangeTimes->{3.472217597734375*^9}], Cell["\ta) M\[EAcute]todo de Newton", "Text"], Cell[BoxData["\[IndentingNewLine]"], "Input", CellChangeTimes->{3.472217600671875*^9}], Cell["\tb) M\[EAcute]todo de la secante", "Text"], Cell[BoxData["\[IndentingNewLine]"], "Input", CellChangeTimes->{3.4722176035*^9}] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "Ejercicio 8\.ba.-\n", "Hacer una representaci\[OAcute]n gr\[AAcute]fica de las funciones f(x) = \ 2cosx y g(x) = ", Cell[BoxData[ FormBox[ SuperscriptBox["e", "x"], TraditionalForm]]], " para obtener estimaciones iniciales de las raices de la ec. 2cosx - ", Cell[BoxData[ FormBox[ SuperscriptBox["e", "x"], TraditionalForm]]], "=0. \[DownQuestion]Cu\[AAcute]ntas raices tiene ? \ \[DownQuestion]Cu\[AAcute]ntas son positivas?. Determinar las raices \ positivas con una precisi\[OAcute]n \[VerticalSeparator]f(r)\ \[VerticalSeparator] < ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["10", RowBox[{"-", "3"}]], " "}], TraditionalForm]]], "utilizando el m\[EAcute]todo de la secante y el de Newton-Raphson." }], "Subsection", CellChangeTimes->{ 3.42813487921875*^9, {3.459754459671875*^9, 3.45975446125*^9}}], Cell[BoxData["\[IndentingNewLine]"], "Input", CellChangeTimes->{3.47221760759375*^9}], Cell["Infinitas raices negativas. Una raiz positiva.", "Text"], Cell["a) Con el m\[EAcute]todo de Newton:", "Text"], Cell[BoxData["\[IndentingNewLine]"], "Input", CellChangeTimes->{3.472217619546875*^9}], Cell["\<\ b) Con el m\[EAcute]todo de la secante:\ \>", "Text"], Cell[BoxData["\[IndentingNewLine]"], "Input", CellChangeTimes->{3.472217621734375*^9}] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "Ejercicio 9\.ba.-\n", "Resolver con un error relativo inferior al 0.05% la ec. f(x) = ", Cell[BoxData[ FormBox[ SuperscriptBox["e", RowBox[{"-", "x"}]], TraditionalForm]]], "-x = 0, mediante el m\[EAcute]todo de la secante tomando como valores \ iniciales ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SubscriptBox["x", RowBox[{"-", "1"}]], "=", "0"}], ",", " ", RowBox[{ RowBox[{"y", " ", SubscriptBox["x", "0"]}], "=", " ", "1"}], ","}], TraditionalForm]]], " y sabiendo que la soluci\[OAcute]n exacta es 0.56714329. Trabajar con \ redondeo a seis cifras significativas." }], "Subsection", CellChangeTimes->{ 3.42813488321875*^9, {3.459754469171875*^9, 3.459754470796875*^9}}], Cell[BoxData["\[IndentingNewLine]"], "Input", CellChangeTimes->{3.472217625921875*^9}] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "Ejercicio 10\.ba.-\n", "Sabiendo que ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["e", "x"], " ", "=", RowBox[{"1", "+", "x", "+", FractionBox[ SuperscriptBox["x", "2"], RowBox[{"2", "!"}]]}]}], TraditionalForm]]], "+", Cell[BoxData[ FormBox[ FractionBox[ SuperscriptBox["x", "3"], RowBox[{"3", "!"}]], TraditionalForm]]], "+.... . Encontrar el polinomio que nos da, para x=5, una valor aproximado \ de ", Cell[BoxData[ FormBox[ SuperscriptBox["e", RowBox[{"5", " "}]], TraditionalForm]]], "con un error menor que", Cell[BoxData[ FormBox[ RowBox[{" ", SuperscriptBox["10", RowBox[{"-", "4"}]]}], TraditionalForm]]], ". Utilizar como criterio de parada que el valor absoluto del sumando a a\ \[NTilde]adir sea menor que ", Cell[BoxData[ FormBox[ SuperscriptBox["10", RowBox[{"-", "4"}]], TraditionalForm]]], ". El n\.ba de operaciones se puede optimizar teniendo en cuenta que el \ nuevo sumando que hay que a\[NTilde]adir es igual al \[UAcute]ltimo que se ha \ a\[NTilde]adido dividido por n. Por ejemplo : ", Cell[BoxData[ FormBox[ FractionBox[ SuperscriptBox["x", "3"], RowBox[{"3", "!"}]], TraditionalForm]]], " = (", Cell[BoxData[ FormBox[ FractionBox[ SuperscriptBox["x", "2"], RowBox[{"2", "!"}]], TraditionalForm]]], ")*", Cell[BoxData[ FormBox[ FractionBox["x", "3"], TraditionalForm]]], ", es decir , term = term/n y a continuaci\[OAcute]n : suma = suma +term" }], "Subsection", CellChangeTimes->{ 3.4280497268150992`*^9, {3.42813488653125*^9, 3.42813488765625*^9}, { 3.45975448325*^9, 3.45975448496875*^9}}], Cell[TextData[{ "1\.aa Forma: Definiendo el polinomio en s(x), el t\[EAcute]rmino a a\ \[NTilde]adir en funci\[OAcute]n de x como ", Cell[BoxData[ FormBox[ FractionBox[ SuperscriptBox["x", "n"], RowBox[{"n", "!"}]], TraditionalForm]]], ", y evaluando este t\[EAcute]rmino en 5. para comparar con ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["10", RowBox[{"-", "4"}]], "."}], TraditionalForm]]] }], "Text", CellChangeTimes->{{3.459754495765625*^9, 3.4597544968125*^9}}], Cell[BoxData["\[IndentingNewLine]"], "Input", CellChangeTimes->{3.4722176318125*^9}], Cell[TextData[{ "2\.aa Forma: Definiendo, igual que antes, el polinomio en s(x), el t\ \[EAcute]rmino a a\[NTilde]adir en funci\[OAcute]n de x como ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"el", " ", "anterior"}], ",", " ", RowBox[{ FractionBox[ SuperscriptBox["x", RowBox[{"n", "-", "1"}]], RowBox[{ RowBox[{"(", RowBox[{"n", "-", "1"}], ")"}], "!"}]], ".", FractionBox["x", "n"]}]}], TraditionalForm]]], ", es decir, term[x_]=term[x].", Cell[BoxData[ FormBox[ FractionBox["x", "n"], TraditionalForm]]], ", y evaluando este t\[EAcute]rmino en 5. para comparar con ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["10", RowBox[{"-", "4"}]], "."}], TraditionalForm]]] }], "Text", CellChangeTimes->{{3.45975450184375*^9, 3.459754502796875*^9}}], Cell[BoxData["\[IndentingNewLine]"], "Input", CellChangeTimes->{3.472217635859375*^9}], Cell[TextData[{ "3\.aa Forma: Sin definir el polinomio como funci\[OAcute]n de x, sino \ calculando directamente el valor n\[UAcute]merico en 5. y lo mismo para el t\ \[EAcute]rmino a a\[NTilde]adir calculando el valor num\[EAcute]rico de la \ forma \[OAcute]ptima, es decir como term=term*5/n, para comparar con ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["10", RowBox[{"-", "4"}]], "."}], TraditionalForm]]] }], "Text", CellChangeTimes->{{3.459754508703125*^9, 3.459754509703125*^9}}], Cell[BoxData["\[IndentingNewLine]"], "Input", CellChangeTimes->{3.47221763790625*^9}] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "Ejercicio 11\.ba.-\n", StyleBox[" La sucesi\[OAcute]n ", FontColor->GrayLevel[0]], Cell[BoxData[ FormBox[ RowBox[{"{", SubscriptBox["a", "n"]}], TraditionalForm]]], "}", StyleBox[" de los n\[UAcute]meros de Fibonacci verifica que ", FontColor->GrayLevel[0]], Cell[BoxData[ FormBox[ SubscriptBox["a", "n"], TraditionalForm]], FontColor->GrayLevel[0]], StyleBox[" = ", FontColor->GrayLevel[0]], Cell[BoxData[ FormBox[ SubscriptBox["a", RowBox[{"n", "-", "1"}]], TraditionalForm]], FontColor->GrayLevel[0]], StyleBox["+", FontColor->GrayLevel[0]], Cell[BoxData[ FormBox[ SubscriptBox["a", RowBox[{"n", "-", "2"}]], TraditionalForm]], FontColor->GrayLevel[0]], StyleBox[", siendo a0 = a1 = 1. Crear una lista que contenga a los 20 \ primeros t\[EAcute]rminos de la sucesi\[OAcute]n. Calcular los \ t\[EAcute]rminos de la sucesi\[OAcute]n que sean menores que 1000.\nCon el \ comando StringForm conseguimos que dentro de un bucle se imprima el valor del \ sub\[IAcute]ndice.(", FontColor->GrayLevel[0]], Cell[BoxData[ RowBox[{"Print", "[", RowBox[{"StringForm", "[", RowBox[{"\<\"\\!\\(x\\_``\\) = ``, \\!\\(y\\_``\\) = ``\"\>", ",", "1", ",", RowBox[{"a", "^", "2"}], ",", "1", ",", RowBox[{"b", "^", "2"}]}], "]"}], "]"}]]], StyleBox[")", FontColor->GrayLevel[0]] }], "Subsection", CellChangeTimes->{{3.4280497363307242`*^9, 3.4280497370182242`*^9}, 3.428134839171875*^9, 3.428134892953125*^9, {3.45975451903125*^9, 3.459754520171875*^9}}], Cell[BoxData["\[IndentingNewLine]"], "Input", CellChangeTimes->{3.47221764665625*^9}] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "Ejercicio 11\.ba.-\n", "Los polinomios de Laguerre son una familia de polinomios que verifican la \ ecuaci\[OAcute]n diferencial de Laguerre. Entre sus propiedades m\[AAcute]s \ importantes se puede citar que son ortogonales en [-1,1] , esto es,\n", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"], RowBox[{ RowBox[{ SubscriptBox["L", "n"], "(", "x", ")"}], RowBox[{ SubscriptBox["L", "m"], "(", "x", ")"}], SuperscriptBox["e", RowBox[{"-", "x"}]], RowBox[{"\[DifferentialD]", "x"}]}]}], "=", RowBox[{ RowBox[{"(", RowBox[{"n", "!"}], ")"}], SubscriptBox["\[Delta]", "nm"]}]}], TraditionalForm]]], " siendo ", Cell[BoxData[ FormBox[ SubscriptBox["\[Delta]", "nm"], TraditionalForm]]], " la Delta de Kronecker.\nOtra propiedad interesante es que verifican la f\ \[OAcute]rmula de recursi\[OAcute]n:\n", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SubscriptBox["L", RowBox[{"n", "+", "1"}]], "(", "x", ")"}], " ", "="}], TraditionalForm]]], Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"2", "n"}], "+", "1", "-", "x"}], ")"}], RowBox[{ SubscriptBox["L", "n"], "(", "x", 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Representarlos en el intervalo [0,10], en tres l\[IAcute]neas de cuatro gr\ \[AAcute]ficos.\nDefinir una funci\[OAcute]n de dos variables, m y n, que sea \ igual a ", Cell[BoxData[ FormBox[ RowBox[{ SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"], RowBox[{ RowBox[{ SubscriptBox["L", "n"], "(", "x", ")"}], RowBox[{ SubscriptBox["L", "m"], "(", "x", ")"}], SuperscriptBox["e", RowBox[{"-", "x"}]], RowBox[{"\[DifferentialD]", "x"}]}]}], TraditionalForm]]], " para comprobar la ortogonalidad." }], "Subsection", CellChangeTimes->{{3.4280497363307242`*^9, 3.4280497370182242`*^9}, 3.428134839171875*^9, 3.428134892953125*^9, 3.43619967125*^9, 3.43624980384375*^9, {3.459754548953125*^9, 3.459754550359375*^9}}], Cell[BoxData["\[IndentingNewLine]"], "Input", CellChangeTimes->{3.47221766203125*^9}] }, Open ]] }, Open ]] }, WindowToolbars->"EditBar", WindowSize->{1672, 933}, WindowMargins->{{0, Automatic}, {Automatic, 0}}, DockedCells->FEPrivate`If[ FEPrivate`SameQ[FEPrivate`$ProductIDName, 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