(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 7.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 38278, 1005] NotebookOptionsPosition[ 19519, 651] NotebookOutlinePosition[ 36255, 951] CellTagsIndexPosition[ 36212, 948] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["\<\ TAREA 6\.aa: C\[AAcute]lculo diferencial e integral\ \>", "Title", CellChangeTimes->{ 3.427778666641778*^9, {3.467914451796875*^9, 3.467914464328125*^9}}], Cell[TextData[{ "Ejercicio 1\.ba. Se considera la funci\[OAcute]n : f(x)=", Cell[BoxData[ RowBox[{ FractionBox["1", RowBox[{"1", "-", RowBox[{"Exp", "[", RowBox[{"x", "/", RowBox[{"(", RowBox[{"1", "-", "x"}], ")"}]}], "]"}]}]], "."}]], CellChangeTimes->{{3.46764812421875*^9, 3.467648192765625*^9}, { 3.467648231078125*^9, 3.46764830290625*^9}}], " Estudiar la existencia de los l\[IAcute]mites en x=0 y en x=1." }], "Text", CellChangeTimes->{{3.467648359625*^9, 3.467648410015625*^9}, { 3.467914380828125*^9, 3.46791441475*^9}, {3.467914475125*^9, 3.46791448109375*^9}}], Cell[BoxData["\[IndentingNewLine]"], "Input", CellChangeTimes->{3.472305536234375*^9}], Cell[TextData[{ "Ejercicio 2 \.ba.- Calcular los siguientes l\[IAcute]mites:\n\t1) ", Cell[BoxData[ FormBox[ UnderscriptBox["lim", RowBox[{"x", "\[RightArrow]", "\[Infinity]"}]], TraditionalForm]]], " ", Cell[BoxData[ FormBox[ SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"sen", "(", FractionBox["1", "x"], ")"}], "+", RowBox[{"cos", "(", FractionBox["1", "x"], ")"}]}], 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RowBox[{ SuperscriptBox["x", "a"], RowBox[{"sin", "(", RowBox[{"1", "/", "x"}], ")"}]}], RowBox[{"x", "\[NotEqual]", "0"}]}, {"0", RowBox[{"x", "=", "0"}]} }]}], TraditionalForm]]] }], "Text", CellChangeTimes->{{3.46769056125*^9, 3.4676905839375*^9}, { 3.46769065909375*^9, 3.467690853796875*^9}, {3.4679148240625*^9, 3.4679148280625*^9}}], Cell[BoxData["\[IndentingNewLine]"], "Input", CellChangeTimes->{3.4723055590625*^9}], Cell[TextData[{ "Ejercicio 4\.ba.- Estudiar la derivabilidad en su campo de \ definici\[OAcute]n de las siguientes funciones :\nf(x) = ", Cell[BoxData[ FormBox[ SuperscriptBox["e", RowBox[{"-", RowBox[{"|", "x", "|"}]}]], TraditionalForm]]], " \nf(x) = ", Cell[BoxData[ FormBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"|", "x", "|"}], ")"}], RowBox[{"1", "/", "2"}]], TraditionalForm]]], "\nf(x) = ", Cell[BoxData[ FormBox[ RowBox[{"\[Piecewise]", "\[NoBreak]", GridBox[{ { SuperscriptBox["e", FractionBox[ RowBox[{"-", "1"}], RowBox[{"1", "-", SuperscriptBox["x", "2"]}]]], RowBox[{"|", "x", "|", RowBox[{"<", "1"}]}]}, {"9", RowBox[{"|", "x", "|", RowBox[{"\[GreaterEqual]", " ", "1"}]}]} }]}], TraditionalForm]]] }], "Text", CellChangeTimes->{{3.46769098178125*^9, 3.46769124471875*^9}, { 3.467914831859375*^9, 3.467914836140625*^9}}], Cell[BoxData["\[IndentingNewLine]"], "Input", CellChangeTimes->{3.472305587359375*^9}], Cell[TextData[{ "Ejercicio 5\.ba.- Calcular las derivadas de las siguientes funciones :\n\t1\ \.ba.- arctg", Cell[BoxData[ FormBox[ SqrtBox[ RowBox[{"x", "-", "1"}]], TraditionalForm]]], "-arcsin", Cell[BoxData[ FormBox[ SqrtBox[ FractionBox[ RowBox[{"x", "-", "1"}], "x"]], TraditionalForm]]], "\n\t2\.ba.- ", Cell[BoxData[ FormBox[ RowBox[{"log", "(", FractionBox[ RowBox[{ SqrtBox[ RowBox[{"1", "+", "x"}]], "+", SqrtBox[ RowBox[{"1", "-", "x"}]]}], RowBox[{ SqrtBox[ RowBox[{"1", "+", "x"}]], "-", SqrtBox[ RowBox[{"1", "-", "x"}]]}]], ")"}], TraditionalForm]]], "\n\t3\.ba.- ", Cell[BoxData[ FormBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"arctgx", "+", "arcsinx"}], ")"}], "n"], TraditionalForm]]], "\n\t4\.ba.- ", Cell[BoxData[ FormBox[ RadicalBox[ FractionBox[ RowBox[{"1", "+", "tghx"}], RowBox[{"1", "-", "tghx"}]], "4"], TraditionalForm]]], "\n\t5\.ba.- ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SuperscriptBox["cos", "n"], "(", "x", ")"}], RowBox[{"cos", "(", "nx", ")"}]}], TraditionalForm]]], " , n\[Element]", "\[DoubleStruckCapitalN]" }], "Text", CellChangeTimes->{{3.467691265171875*^9, 3.4676915789375*^9}, { 3.467691843625*^9, 3.467691843625*^9}, {3.46791484*^9, 3.4679148444375*^9}, {3.46791490053125*^9, 3.467914904390625*^9}}], Cell["1 \.ba.- ", "Text", CellChangeTimes->{{3.46867625053125*^9, 3.4686762535*^9}}], Cell[BoxData["\[IndentingNewLine]"], "Input", CellChangeTimes->{3.4723055933125*^9}], Cell["2 \.ba.- ", "Text", CellChangeTimes->{{3.46867625053125*^9, 3.4686762535*^9}, 3.4718458373125*^9} ], Cell[BoxData["\[IndentingNewLine]"], "Input", CellChangeTimes->{3.472305595609375*^9}], Cell["3 \.ba.- ", "Text", CellChangeTimes->{{3.46867625053125*^9, 3.4686762535*^9}, 3.4718458373125*^9, 3.471845994875*^9}], Cell[BoxData["\[IndentingNewLine]"], "Input", CellChangeTimes->{3.472305597671875*^9}], Cell["4 \.ba.- ", "Text", CellChangeTimes->{{3.46867625053125*^9, 3.4686762535*^9}, 3.4718458373125*^9, 3.471846126390625*^9}], Cell[BoxData["\[IndentingNewLine]"], "Input", CellChangeTimes->{3.472305600234375*^9}], Cell["5 \.ba.- ", "Text", CellChangeTimes->{{3.46867625053125*^9, 3.4686762535*^9}, 3.4718458373125*^9, 3.471846216234375*^9}], Cell[BoxData["\[IndentingNewLine]"], "Input", CellChangeTimes->{3.47230560309375*^9}], Cell[TextData[{ "Ejercicio 6\.ba.- Calcular los siguientes l\[IAcute]mites mediante la regla \ de L' Hopital:\n\t1) ", Cell[BoxData[ FormBox[ UnderscriptBox["lim", RowBox[{"x", "\[RightArrow]", RowBox[{"\[Pi]", "/", "2"}]}]], TraditionalForm]]], "(\[Pi]-2x)tgx\n\t", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"2", ")"}], RowBox[{ UnderscriptBox["lim", RowBox[{"x", "\[RightArrow]", "0"}]], FractionBox[ RowBox[{ RowBox[{"log", "(", RowBox[{"1", "+", "x"}], ")"}], "-", "x"}], RowBox[{"1", "-", SqrtBox[ RowBox[{"1", "-", " ", SuperscriptBox["x", "2"]}]]}]]}]}], TraditionalForm]]], "\n\t3) ", Cell[BoxData[ FormBox[ RowBox[{ UnderscriptBox["lim", RowBox[{"x", "\[RightArrow]", "0"}]], RowBox[{"(", RowBox[{ FractionBox["1", RowBox[{ SuperscriptBox["sen", "2"], "x"}]], "-", FractionBox["1", RowBox[{"1", "-", "cosx"}]]}], ")"}]}], TraditionalForm]]], "\n\t4) ", Cell[BoxData[ FormBox[ UnderscriptBox["lim", RowBox[{"x", "\[RightArrow]", "\[Infinity]"}]], TraditionalForm]]], " ", Cell[BoxData[ FormBox[ SuperscriptBox[ RowBox[{"(", RowBox[{ FractionBox["2", "\[Pi]"], "arctgx"}], ")"}], "x"], TraditionalForm]]], "\t" }], "Text", CellChangeTimes->{{3.467912697203125*^9, 3.46791269959375*^9}, { 3.4679128036875*^9, 3.467913196296875*^9}, {3.46791324946875*^9, 3.467913266609375*^9}, {3.467914848796875*^9, 3.4679148536875*^9}, 3.46791491203125*^9}], Cell["\<\ Con el comando Limit podemos calcular directamente el valor del \ l\[IAcute]mite. En primer lugar aplicaremos el comando Limit a la funci\ \[OAcute]n dada para pasar despu\[EAcute]s a aplicar L' Hopital\ \>", "Text", CellChangeTimes->{{3.472283136933854*^9, 3.4722832136697025`*^9}}], Cell["1)", "Text", CellChangeTimes->{{3.4722837243658094`*^9, 3.472283725412698*^9}}], Cell[BoxData["\[IndentingNewLine]"], "Input", CellChangeTimes->{3.472305609125*^9}], Cell["2)", "Text", CellChangeTimes->{{3.4722837243658094`*^9, 3.472283725412698*^9}, { 3.472283851836191*^9, 3.4722838519455676`*^9}}], Cell[BoxData["\[IndentingNewLine]"], "Input", CellChangeTimes->{3.472305612984375*^9}], Cell["3)", "Text", CellChangeTimes->{{3.4722837243658094`*^9, 3.472283725412698*^9}, { 3.472283851836191*^9, 3.4722838519455676`*^9}, {3.472283942462351*^9, 3.472283943524865*^9}}], Cell[BoxData["\[IndentingNewLine]"], "Input", CellChangeTimes->{3.4723056170625*^9}], Cell["\<\ 4)\ \>", "Text", CellChangeTimes->{ 3.46791491378125*^9, {3.4722828105213375`*^9, 3.4722828116932344`*^9}}], Cell[BoxData["\[IndentingNewLine]"], "Input", CellChangeTimes->{3.47230563003125*^9}], Cell[TextData[{ "Ejercicio 7\.ba.- Un canal abierto de fondo horizontal y cuyas paredes \ tienen una inclinaci\[OAcute]n de 45\.ba ha de tener una secci\[OAcute]n de \ 12 ", Cell[BoxData[ FormBox[ SuperscriptBox["m", "2"], TraditionalForm]]], ". Determinar las dimensiones de la secci\[OAcute]n que hacen m\[IAcute]nimo \ el per\[IAcute]metro que se encuentra en contacto con el fluido. " }], "Text", CellChangeTimes->{{3.467913349234375*^9, 3.467913486875*^9}, { 3.467914857703125*^9, 3.467914862109375*^9}}], Cell[BoxData["\[IndentingNewLine]"], "Input", CellChangeTimes->{3.47230565375*^9}], Cell[TextData[{ "Ejercicio 8\.ba.- Determinar un polinomio P(x) tal que P(0)=P(1)=0 y ", Cell[BoxData[ FormBox[ RowBox[{ SubsuperscriptBox["\[Integral]", "0", "1"], RowBox[{ RowBox[{"P", "(", "x", ")"}], RowBox[{"\[DifferentialD]", "x"}]}]}], TraditionalForm]]], "=1" }], "Text", CellChangeTimes->{{3.467913593171875*^9, 3.467913650046875*^9}, { 3.467914865671875*^9, 3.467914870375*^9}}], Cell[BoxData["\[IndentingNewLine]"], "Input", CellChangeTimes->{3.472305660421875*^9}], Cell[TextData[{ "Ejercicio 9\.ba.- Calcular las siguientes integrales :\n\t1) ", Cell[BoxData[ FormBox[ RowBox[{ SubsuperscriptBox["\[Integral]", "0", "2"], RowBox[{ FractionBox[ SuperscriptBox["x", "2"], SuperscriptBox[ RowBox[{"(", RowBox[{"1", "+", " ", SuperscriptBox["x", "2"]}], ")"}], RowBox[{"3", "/", "2"}]]], RowBox[{"\[DifferentialD]", "x"}]}]}], TraditionalForm]]], "\n\t2)", Cell[BoxData[ FormBox[ RowBox[{ SubsuperscriptBox["\[Integral]", RowBox[{"-", "3"}], "3"], RowBox[{"|", RowBox[{ RowBox[{"x", "(", RowBox[{"x", "-", "1"}], ")"}], RowBox[{"(", RowBox[{"x", "+", "1"}], ")"}], RowBox[{"(", RowBox[{"x", "-", "2"}], ")"}]}], "|", RowBox[{"\[DifferentialD]", "x"}]}]}], TraditionalForm]]] }], "Text", CellChangeTimes->{{3.467913748703125*^9, 3.467913940625*^9}, { 3.46791487375*^9, 3.467914877140625*^9}}], Cell[BoxData["\[IndentingNewLine]"], "Input", CellChangeTimes->{3.472305666328125*^9}], Cell[TextData[{ "Ejercicio 10\.ba.- Calcular el volumen de la figura engendrada por la \ curva:\n\t\t ", Cell[BoxData[ FormBox[ RowBox[{"y", "=", FractionBox[ SuperscriptBox["x", "2"], "2"]}], TraditionalForm]]], "\ncon x ", "\[Element][1,2]" }], "Text", CellChangeTimes->{{3.46791407478125*^9, 3.467914253578125*^9}, { 3.467914880296875*^9, 3.467914884890625*^9}, {3.4721480065*^9, 3.472148011390625*^9}, {3.472148058140625*^9, 3.4721480610625*^9}, { 3.47214817775*^9, 3.4721482203125*^9}, {3.47214830340625*^9, 3.472148317359375*^9}, {3.472148445015625*^9, 3.4721485074375*^9}}], Cell[BoxData["\[IndentingNewLine]"], "Input", CellChangeTimes->{3.472305669859375*^9}], Cell[TextData[{ "Ejercicio 11 \.ba.- Calcular el \[AAcute]rea de la regi\[OAcute]n del plano \ comprendida entre la hip\[EAcute]rbola de ecuaci\[OAcute]n :\n\t", Cell[BoxData[ FormBox[ RowBox[{"\t", RowBox[{ FractionBox[ SuperscriptBox["x", "2"], SuperscriptBox["a", "2"]], "-"}]}], TraditionalForm]]], Cell[BoxData[ FormBox[ FractionBox[ SuperscriptBox["y", "2"], SuperscriptBox["b", "2"]], TraditionalForm]]], "=1\ny la cuerda de ecuaci\[OAcute]n x=h, con h>a." }], "Text", CellChangeTimes->{{3.467914947609375*^9, 3.467915039109375*^9}, { 3.47214728365625*^9, 3.47214728365625*^9}}], Cell["\<\ En este caso no podemos hacer la representaci\[OAcute]n gr\[AAcute]fica ya \ que la curva viene dada en funci\[OAcute]n de un par\[AAcute]metro : a y la \ cuerda en funci\[OAcute]n de otro : h. Sin embargo podemos tomar dos valores \ concretos para tener una idea de la representaci\[OAcute]n gr\[AAcute]fica \ del \[AAcute]rea pedida, por ejemplo a=2, b=4 y h=4. Posteriormente se \ resuelve de forma general.\ \>", "Text", CellChangeTimes->{{3.472105347875*^9, 3.47210546378125*^9}, { 3.47214733296875*^9, 3.472147342296875*^9}, {3.4723057075*^9, 3.472305719328125*^9}}], Cell[BoxData["\[IndentingNewLine]"], "Input", CellChangeTimes->{3.47230568221875*^9}], Cell[TextData[{ "Ejercicio 12 \.ba.- Calcular el \[AAcute]rea de la regi\[OAcute]n del plano \ acotada por la curva de ecuaci\[OAcute]n :\n\t\t", Cell[BoxData[ FormBox[ RowBox[{"y", "=", RowBox[{ SuperscriptBox["x", "3"], "-", "x"}]}], TraditionalForm]]], " \ny su tangente en el punto de abscisa x=1." }], "Text", CellChangeTimes->{{3.46791506575*^9, 3.467915142390625*^9}}], Cell[BoxData["\[IndentingNewLine]"], "Input", CellChangeTimes->{3.472305735171875*^9}], Cell[TextData[{ "Ejercicio 13 \.ba.- Calcular el \[AAcute]rea de la regi\[OAcute]n del plano \ comprendida entre el eje OX y cada una de las siguientes curvas de \ ecuaciones, representando gr\[AAcute]ficamente las curvas previamente :\n\ta) \ ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["y", "2"], " ", "=", " ", RowBox[{ RowBox[{"4", SuperscriptBox["x", "2"]}], "-", SuperscriptBox["x", "4"], " "}]}], TraditionalForm]]], " , y \[GreaterEqual] 0\n\tb) y= sen(2x) , 0\[LessEqual] x \ \[LessEqual]4", "\[Pi] ." }], "Text", CellChangeTimes->{{3.4679151575625*^9, 3.46791532846875*^9}, { 3.47210424284375*^9, 3.47210425903125*^9}}], Cell["a)", "Text", CellChangeTimes->{{3.4723057563125*^9, 3.472305757453125*^9}}], Cell[BoxData["\[IndentingNewLine]"], "Input", CellChangeTimes->{3.47230574846875*^9}], Cell["b)", "Text", CellChangeTimes->{{3.472104605203125*^9, 3.472104606109375*^9}}], Cell[BoxData["\[IndentingNewLine]"], "Input", CellChangeTimes->{3.47230577459375*^9}], Cell[TextData[{ "Ejercicio 14 \.ba.- Calcular la longitud de las siguientes curvas planas :\n\ \t1) El arco de la par\[AAcute]bola y = ", Cell[BoxData[ FormBox[ SuperscriptBox["x", "2"], TraditionalForm]]], " comprendido entre los puntos (0,0) y (1,1)\n\t2) El arco de la cicloide de \ ecuaciones param\[EAcute]tricas : ", Cell[BoxData[ FormBox[ RowBox[{"\[Piecewise]", GridBox[{ { RowBox[{ RowBox[{"x", "(", "t", ")"}], " ", "=", " ", RowBox[{"r", "(", RowBox[{"t", "-", "sent"}], ")"}]}]}, { RowBox[{ RowBox[{"y", "(", "t", ")"}], " ", "=", " ", RowBox[{"r", "(", RowBox[{"1", "-", "cost"}], ")"}]}]} }, GridBoxAlignment->{ "Columns" -> {{Left}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxItemSize->{ "Columns" -> {{Automatic}}, "ColumnsIndexed" -> {}, "Rows" -> {{1.}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.84]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}]}], TraditionalForm]]], " , t\[Element][0, 2\[Pi])" }], "Text", CellChangeTimes->{{3.4679153635*^9, 3.467915528046875*^9}, { 3.472103734796875*^9, 3.4721037395*^9}, {3.472103852265625*^9, 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