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Cell[TextData[{
 "Tema : 6\nIntroducci\[OAcute]n al c\[AAcute]lculo diferencial e Integral \
con ",
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Cell["Introducci\[OAcute]n", "Section 1",
 CellChangeTimes->{{3.458981002109375*^9, 3.45898101496875*^9}}],

Cell[TextData[{
 "Dentro del C\[AAcute]lculo Diferencial e Integral , ",
 StyleBox["Mathematica",
  FontSlant->"Italic"],
 " incorpora un peque\[NTilde]o n\[UAcute]mero de funciones de potencia \
excepcional. As\[IAcute], es posible calcular complejas derivadas, derivadas \
parciales, o l\[IAcute]mites. Del mismo modo podremos calcular complicadas \
integrales, definidas o indefinidas , de forma exacta o aproximada. Tambi\
\[EAcute]n podremos obtener desarrollos en serie de funciones de una o varias \
variables y resolver tambi\[EAcute]n de forma exacta o aproximada ecuaciones \
diferenciales ordinarias o en derivadas parciales. ",
 StyleBox["Mathematica",
  FontSlant->"Italic"],
 " nos permite tambi\[EAcute]n resolver problemas de C\[AAcute]lculo \
Operacional obteniendo de forma sencilla tanto las transformadas directas \
como inversas de Laplace o Fourier.\nA continuaci\[OAcute]n veremos \
\[UAcute]nicamente cuestiones sencillas relativas a l\[IAcute]mites, derivaci\
\[OAcute]n e integraci\[OAcute]n."
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Cell[CellGroupData[{

Cell["  El comando Limit", "Section 1",
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Cell[TextData[{
 "El comando Limit permite obtener el l\[IAcute]mite de una sucesi\[OAcute]n \
num\[EAcute]rica o funcional, o de funci\[OAcute]n cuando la variable tiende \
a un determinado valor. Su sintaxis es la siguiente:\n\tLimit[expr, ",
 Cell[BoxData[
  FormBox[
   RowBox[{"x", "\[Rule]", 
    SubscriptBox["x", "0"]}], TraditionalForm]]],
 "]\nEn el caso de funciones se pueden obtener tambi\[EAcute]n los \
l\[IAcute]mites por la derecha y por la izquierda utilizando la \
opci\[OAcute]n Directi\[OAcute]n. El l\[IAcute]mite por la izquierda \
asignando el valor 1:\n\tLimit[expr, ",
 Cell[BoxData[
  FormBox[
   RowBox[{"x", "\[Rule]", 
    SubscriptBox["x", "0"]}], TraditionalForm]]],
 ", Direction \[Rule]1]\nY el l\[IAcute]mite por la derecha asignando el \
valor -1\n\tLimit[expr, ",
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  FormBox[
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    SubscriptBox["x", "0"]}], TraditionalForm]]],
 ", Direction \[Rule] - 1]"
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Cell["\<\
Ejemplo 1
Calcular el l\[IAcute]mite de f(x) = sinx/x , cuando x tiende a 0. Obtener \
tambi\[EAcute]n los l\[IAcute]mites por la derecha y por la izquierda\
\>", "Subsubsection",
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Cell[TextData[{
 "Ejemplo 2\nCalcular el l\[IAcute]mite de la sucesi\[OAcute]n : ",
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  FormBox[
   SubscriptBox["a", "n"], TraditionalForm]]],
 "=1/",
 Cell[BoxData[
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 " cuando n tiende a ",
 "\[Infinity]."
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Cell[BoxData[
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Cell["Derivaci\[OAcute]n  de funciones", "Section 1",
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Cell["\<\
\tEl operador D, nos permite obtener la derivada de funciones de una variable \
o las derivadas parciales en el caso de funciones de varias variables. Por \
ejemplo la derivada de sinx .\
\>", "Text",
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Cell[TextData[{
 "Ejemplo 3\nObtener la primera derivada de f(x) = ",
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   SuperscriptBox["x", "n"], TraditionalForm]]]
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Cell["\<\
Podemos obtener tambi\[EAcute]n las derivadas de \[OAcute]rdenes superiores. \
Para obtener la derivas de orden n, la sintaxis es :
\tD[f[x],{x,n}]\
\>", "Text",
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Cell[CellGroupData[{

Cell[TextData[{
 "Ejemplo 4\nObtener la  derivada tercera de f(x) = ",
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   SuperscriptBox["x", "n"], TraditionalForm]]]
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Cell[BoxData[
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Cell["\<\
Otra forma sencilla de derivar una funci\[OAcute]n de una variable es \
semejante a la que se utiliza en la pizarra, utilizando el acento que est\
\[AAcute] debajo del signo de interrogaci\[OAcute]n en el teclado. Por \
ejemplo para obtener la derivada segunda de f(x) = sinx+cosx,\
\>", "Text",
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Cell[CellGroupData[{

Cell[BoxData[
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Cell["\<\
Podemos utilizar tambi\[EAcute]n el operador D para obtener derivadas \
parciales. Por ejemplo, en la siguiente entrada, se calcula la primera \
derivada parcial con respecto a x de sinx+x*cosy, \
\>", "Text",
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Cell["\<\
Y en esta otra entrada la derivada parcial con respecto a x e y :\
\>", "Text",
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[TextData[{
 "Podemos utilizar tambi\[EAcute]n el s\[IAcute]mbolo de la paleta Basic Math \
Inputs, ",
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   RowBox[{
    SubscriptBox["\[PartialD]", 
     RowBox[{"\[Placeholder]", ",", "\[Placeholder]"}]], "\[Placeholder]"}], 
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 " , para realizar el mismo c\[AAcute]lculo."
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[CellGroupData[{

Cell["  Integraci\[OAcute]n de funciones.", "Section 1",
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Cell[TextData[{
 StyleBox["Mathematica",
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 " nos permite evaluar de forma sencilla una gran cantidad de integrales, \
indefinidas, definidas y m\[UAcute]ltiples. Para ello se utiliza el comando \
Integrate o bien los s\[IAcute]mbolos de la paleta BasicMathInputs, ",
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 " y ",
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 " .\nEn el caso de la integral indefinida, ",
 StyleBox["Mathematica",
  FontSlant->"Italic"],
 " da como resultado una funci\[OAcute]n cuya derivada es el dato. Se pueden \
obtener muchas m\[AAcute]s que se diferenciar\[AAcute]n de la obtenida en una \
constante. Por ejemplo, la integral indefinida de ",
 Cell[BoxData[
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 " se obtiene en la siguiente entrada:"
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Cell[BoxData[
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Cell[CellGroupData[{

Cell[TextData[{
 "Ejemplo 5\nCalcular ",
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      RowBox[{
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 " y comprobar que su derivada es la funci\[OAcute]n subintegral."
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Cell["\<\
La evaluaci\[OAcute]n de integrales puede ser mucho m\[AAcute]s complicada \
que la de derivadas en cuyo caso existe un porocedimiento sistem\[AAcute]tico \
que siempre nos permite obtener la derivada. En ocasiones el resultado que se \
obtendr\[AAcute] al evaluar una integral depender\[AAcute] de funciones \
complejas como las integrales de Fressnel o las funciones Zeta o Gamma y en \
otros casos ser\[AAcute] imposible obtener el resultado.\
\>", "Text",
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Cell[TextData[{
 "Para calcular integrales definidas podemos utilizar el comando Integrate o \
el s\[IAcute]mbolo de la paleta ",
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 ". As\[IAcute], por ejemplo, para calcular ",
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 " , podemos hacer:"
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Cell["\<\
Podemos evaluar tambi\[EAcute]n integrales m\[UAcute]ltiples. As\[IAcute], \
para integrar la funci\[OAcute]n f(x,y) = x+y, en la regi\[OAcute]n : { (x,y) \
/ 0 \[LessEqual]y\[LessEqual]x , 0\[LessEqual]x\[LessEqual]1 }, haremos :\
\>", "Text",
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Cell[CellGroupData[{

Cell[BoxData[
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   RowBox[{"{", 
    RowBox[{"y", ",", "0", ",", "x"}], "}"}]}], "]"}]], "Input",
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Cell[CellGroupData[{

Cell["\<\
Ejemplo 6
Hallar el volumen del tetraedro limitado por los planos y=0, z=0, x=0 y el \
plano y-x+z=1.\
\>", "Subsubsection",
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Cell["\<\
Mediante el comando RegionPlot3D (consultar la ayuda relativa a este comando) \
representamos la regi\[OAcute]n cuyo volumen queremos calcular.\
\>", "Text",
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}, Open  ]],

Cell["\<\
En primer lugar consideramos la proyecci\[OAcute]n del tetraedro en el plano \
z que obtenemos haciendo z = 0 en la ecuaci\[OAcute]n del plano. La siguiente \
entrada representa esta regi\[OAcute]n mediante el comando RegionPlot \
(consultar la ayuda relativa a este comando)\
\>", "Text",
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A continuaci\[OAcute]n, para calcular el volumen  planteamos la integral \
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(* NotebookSignature SwDUWvlNSfQbiAKfw7Nhj7yV *)