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Notebook[{
Cell["Repaso de las funciones elementales b\[AAcute]sicas", "Title"],

Cell[TextData[{
  "Usaremos las posibilidades gr\[AAcute]ficas y de c\[AAcute]lculo de ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " para repasar las funciones elementales. En este cuaderno debes usar la \
versi\[OAcute]n 5.0 de ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " o superior porque algunos comandos no funcionan en versiones anteriores. \
"
}], "Text"],

Cell[CellGroupData[{

Cell["Logaritmos", "Section"],

Cell[TextData[{
  "Te recuerdo que, dado un n\[UAcute]mero positivo ",
  Cell[BoxData[
      \(TraditionalForm\`a > 0\)]],
  ",  ",
  Cell[BoxData[
      \(TraditionalForm\`a \[NotEqual] 1\)]],
  ",  y un n\[UAcute]mero real ",
  Cell[BoxData[
      \(TraditionalForm\`x > 0\)]],
  ", se define el logaritmo en base ",
  StyleBox["a",
    FontSlant->"Italic"],
  " de ",
  StyleBox["x",
    FontSlant->"Italic"],
  " como el \[UAcute]nico n\[UAcute]mero real ",
  StyleBox["y",
    FontSlant->"Italic"],
  " tal que ",
  Cell[BoxData[
      \(TraditionalForm\`a\^y = x\)]],
  ". Los logaritmos en base E se llaman logaritmos naturales (o neperianos). \
"
}], "Text"],

Cell[BoxData[{
    \(Log[10, 10. ]\  (*\ 
      Log[a, x]\ es\ el\ logaritmo\ de\ x\ en\ base\ a\ *) \), "\n", 
    \(Log[10. ]\  (*\ 
      Log[x]\ es\ el\ logaritmo\ natural\ de\ x\ *) \), "\n", 
    \(Log[E, 10. ]\)}], "Input"],

Cell[BoxData[
    \(\(\(Plot[Log[2, x], {x,  .1, 9}]\)\(;\)\(\ \)\( (*\ 
      Log[a, x]\ es\ creciente\ si\ a > 1\ *) \)\)\)], "Input"],

Cell[BoxData[
    \(\(\(\(Plot[Log[ .5, x], {x,  .1, 9}]\) \)\(;\)\(\ \)\( (*\ 
      Log[a, x]\ es\ decreciente\ si\ a < 1\ *) \)\)\)], "Input"],

Cell[BoxData[
    \(\(Limit[Log[2, x], x \[Rule] \(+\[Infinity]\), 
      Assumptions \[Rule] a > 1]\  (*\ 
      el\ l\[IAcute]mite\ en\  + 
          \[Infinity]\ de\ la\ funci\[OAcute]n\ logaritmo\ de\ base\ a > 
        1\ es\  + \[Infinity]\ *) \)\)], "Input"],

Cell[BoxData[
    \(\(Limit[Log[a, x], x \[Rule] 0, Assumptions \[Rule] a > 1]\  (*\ 
      el\ l\[IAcute]mite\ en\ 0\ de\ la\ funci\[OAcute]n\ logaritmo\ de\ base
          \ a > 1\ es\  - \[Infinity]\ *) \)\)], "Input"],

Cell[BoxData[
    \(\(Limit[Log[a, x], x \[Rule] \(+\[Infinity]\), 
      Assumptions \[Rule] 0 < a < 1]\  (*\ 
      el\ l\[IAcute]mite\ en\  + 
          \[Infinity]\ de\ la\ funci\[OAcute]n\ logaritmo\ de\ base\ a < 
        1\ es\  - \[Infinity]\ *) \)\)], "Input"],

Cell[BoxData[
    \(\(Limit[Log[a, x], x \[Rule] 0, Assumptions \[Rule] 0 < a < 1]\n (*\ 
      el\ l\[IAcute]mite\ en\ 0\ de\ la\ funci\[OAcute]n\ logaritmo\ de\ base
          \ a < 1\ es\  + \[Infinity]\ *) \)\)], "Input"],

Cell[TextData[{
  "Podemos comprobar con ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " la propiedad pricipal de los logaritmos: el logaritmo de un producto de n\
\[UAcute]meros reales positivos es igual a la suma de los logaritmos de los \
factores."
}], "Text"],

Cell[BoxData[
    \(\(\(Simplify[Log[x*y] \[Equal] Log[x] + Log[y], 
      x > 0\  && \ y > 0]\)\(\ \)\( (*\ 
      esta\ es\ la\ propiedad\ principal\ de\ los\ logaritmos\ *) \)\)\)], \
"Input"],

Cell["\<\
Es f\[AAcute]cil relacionar los logaritmnos en cualquier base con los \
logaritmos neperianos.\
\>", "Text"],

Cell[BoxData[
    \(\(\(Simplify[Log[a, x] \[Equal] Log[x]/Log[a], 
      x > 0\  && \ a > 0]\)\(\ \)\( (*\ 
      es\ suficiente\ conocer\ los\ logaritmos\ naturales\ *) \)\)\)], "Input"],

Cell["\<\
En esta asignatura trabajaremos siempre con logaritmos naturales o \
neperianos.\
\>", "Text"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Exponenciales", "Section"],

Cell[TextData[{
  "La funci\[OAcute]n exponencial de base ",
  StyleBox["a",
    FontSlant->"Italic"],
  " ",
  Cell[BoxData[
      \(TraditionalForm\`\((a > 0, \ a \[NotEqual] \ 1)\)\)]],
  " es la funci\[OAcute]n inversa de la funci\[OAcute]n logaritmo de base ",
  StyleBox["a",
    FontSlant->"Italic"],
  ". La exponencal de base E se llama exponencial natural y es la inversa de \
la funci\[OAcute]n logaritmo natural. "
}], "Text"],

Cell[BoxData[
    \(\(\(2^\((3.14)\) (*\ 
      a^x\ representa\ el\ valor\ en\ x\ de\ la\ funci\[OAcute]n\ exponencial\
\ de\ base\ a\ *) \[IndentingNewLine]
    Exp[3.14]\)\(\ \)\( (*\ 
      Exp[x]\ es\ la\ funci\[OAcute]n\ exponencial\ natural\ *) \)\)\)], \
"Input"],

Cell[BoxData[
    \(\(\(Plot[\((1.8)\)^x, {x, \(-5\), 5}]; 
    \) (*\ la\ funci\[OAcute]n\ exponencial\ de\ base\ a > 
        1\ es\ creciente\ *) \)\)], "Input"],

Cell[BoxData[
    \(\(\(Plot[\((0.5)\)^x, {x, \(-5\), 5}]; 
    \) (*\ la\ funci\[OAcute]n\ exponencial\ de\ base\ a < 
        1\ es\ decreciente\ *) \)\)], "Input"],

Cell[BoxData[
    \(\(Limit[a^x, x \[Rule] \(+Infinity\), Assumptions \[Rule] a > 1] (*\ 
      la\ funci\[OAcute]n\ exponencial\ de\ base\ a > 
        1\ tiene\ l\[IAcute]mite\  + \[Infinity]\ en\  + \[Infinity]\ *) 
      \)\)], "Input"],

Cell[BoxData[
    \(\(Limit[a^x, x \[Rule] \(-Infinity\), Assumptions \[Rule] a > 1] (*\ 
      la\ funci\[OAcute]n\ exponencial\ de\ base\ a > 
        1\ tiene\ l\[IAcute]mite\ 0\ en\  - \[Infinity]\ *) \)\)], "Input"],

Cell[BoxData[
    \(\(Limit[a^x, x \[Rule] \(+Infinity\), Assumptions \[Rule] 0 < a < 1] (*
      \ la\ funci\[OAcute]n\ exponencial\ de\ base\ a < 
        1\ tiene\ l\[IAcute]mite\ 0\ en\  + \[Infinity]\ *) \)\)], "Input"],

Cell[BoxData[
    \(\(Limit[a^x, x \[Rule] \(-Infinity\), Assumptions \[Rule] 0 < a < 1] (*
      \ la\ funci\[OAcute]n\ exponencial\ de\ base\ a < 
        1\ tiene\ l\[IAcute]mite\  + \[Infinity]\ en\  - \[Infinity]\ *) 
      \)\)], "Input"],

Cell[TextData[{
  "Podemos comprobar con ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " la propiedad principal de la exponencial: la exponencial de una suma es \
igual al producto de las exponenciales de los sumandos."
}], "Text"],

Cell[BoxData[
    \(\(\(Simplify[Exp[x + y] \[Equal] Exp[x]*Exp[y]]\)\( (*\ 
      propiedad\ principal\ de\ las\ exponenciales\ *) \)\)\)], "Input"],

Cell["\<\
No debes olvidar que la exponencial de un n\[UAcute]mero real es siempre un n\
\[UAcute]mero positivo.\
\>", "Text"],

Cell[BoxData[
    \(\(\(Simplify[Exp[x] > 0, x \[Element] Reals]\)\( (*\ 
      la\ exponencial\ es\ siempre\ positiva\ *) \)\)\)], "Input"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Potencias de base y exponente real", "Section"],

Cell[TextData[{
  "Dados ",
  Cell[BoxData[
      \(TraditionalForm\`x > 0\)]],
  " e ",
  Cell[BoxData[
      \(TraditionalForm\`y \[Element] \[DoubleStruckCapitalR]\)]],
  " se define  ",
  Cell[BoxData[
      \(TraditionalForm\`x\^y = \[ExponentialE]\^\(y\ log\ x\)\)]],
  ". La funci\[OAcute]n potencia de exponente real ",
  StyleBox["b",
    FontSlant->"Italic"],
  " es la funci\[OAcute]n que a cada n\[UAcute]mero real ",
  Cell[BoxData[
      \(TraditionalForm\`x > 0\)]],
  " hace corresponder ",
  Cell[BoxData[
      \(TraditionalForm\`x\^b = \[ExponentialE]\^\(b\ log\ x\)\)]],
  ". "
}], "Text"],

Cell[BoxData[
    \(\(\(Plot[x^\(( .36)\), {x, 0, 5}]\ ; 
    \)\n (*\ la\ funci\[OAcute]n\ potencia\ de\ exponente\ b > 
        0\ es\ creciente\ *) \)\)], "Input"],

Cell[BoxData[
    \(\(\(Plot[x^\((\(-0.36\))\), {x, 0.01, 5}]\)\(\ \)\(;\)\( (*\ 
      la\ funci\[OAcute]n\ potencia\ de\ exponente\ b < 
        0\ es\ decreciente\ *) \)\)\)], "Input"],

Cell[TextData[{
  "Podemos comprobar con ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " la propiedad principal de las potencias reales: la potencia de un \
producto de n\[UAcute]meros positivos es igual al producto de las potencias \
de los factores."
}], "Text"],

Cell[BoxData[
    \(Simplify[\((x*y)\)^b \[Equal] \((x^b)\)*\((y^b)\), x > 0\  && \ y > 0]
      \)], "Input"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Escala de infinitos", "Section"],

Cell[BoxData[
    \(Limit[\((1 + 10^\((\(-5\))\))\)\^x\/x\^100000, 
      x \[Rule] \(+\[Infinity]\)]\)], "Input"],

Cell[BoxData[
    \(\(Limit[a\^x\/x\^b, x \[Rule] \(+\[Infinity]\), 
      Assumptions \[Rule] a > 1\  && \ b > 0]\n (*\ 
      una\ exponencial\ de\ base\ a > 1, \ 
      crece\ m\[AAcute]s\ r\[AAcute]pidamente\  - \ aunque\ a > 
        1\ est\[EAcute]\ muy\ cerca\ de\ 1\  - \ \ 
          que\ cualquier\ potencia\ de\ exponente\ b\  - \ 
          por\ grande\ que\ pueda\ ser\ b\  - \ *) \)\)], "Input"],

Cell[BoxData[
    \(Limit[x\^ .0000001\/\(\((Log[x])\)\(\ \)\)\^100, 
      x \[Rule] \(+Infinity\)]\)], "Input"],

Cell[BoxData[
    \(\(\(Limit[x\^b\/\((Log[x])\)\^c, x \[Rule] \(+\[Infinity]\), 
      Assumptions \[Rule] b > 0\  && \ c > 0]\)\(\ \)\( (*\ 
      una\ potencia\ de\ exponente\ positivo, \ b > 0, \ 
      crece\ m\[AAcute]s\ r\[AAcute]pidamente\  - \ 
        por\ peque\[NTilde]o\ que\ sea\ b\  - \ 
        que\ cualquier\ potencia\ del\ logaritmo\  - \ 
        por\ grande\ que\ sea\ c\ *) \)\)\)], "Input"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Funciones trigonom\[EAcute]tricas", "Section"],

Cell[BoxData[
    \(\(Plot[Sin[x], {x, \(-2\) \[Pi], 2  \[Pi]}, 
        Ticks \[Rule] {\((Range[9] - 5)\)\ \[Pi]/
                2, {1, \(-1\)}}];\)\)], "Input"],

Cell[BoxData[
    \(Simplify[Sin[x + 2  \[Pi]]]\)], "Input"],

Cell[BoxData[
    \(\(Plot[Cos[x], {x, \(-2\) \[Pi], 2  \[Pi]}, 
        Ticks \[Rule] {\((Range[9] - 5)\)\ \[Pi]/
                2, {1, \(-1\)}}];\)\)], "Input"],

Cell[BoxData[
    \(Simplify[Sin[x + \[Pi]/2]]\)], "Input"],

Cell[BoxData[
    \(Simplify[Cos[x]^2 + Sin[x]^2]\)], "Input"],

Cell[BoxData[{
    \(Simplify[Cos[\(-x\)]]\), 
    \(Simplify[Sin[\(-x\)]] (*\ 
      la\ funci\[OAcute]n\ coseno\ es\ par\ y\ el\ seno\ es\ impar\ *) \)}], 
  "Input"],

Cell[BoxData[
    \(\(\(TrigExpand[Cos[x + y]]\)\( (*\ 
      f\[OAcute]rmula\ de\ adici\[OAcute]n\ para\ el\ coseno\ *) \)\)\)], \
"Input"],

Cell[BoxData[
    \(\(\(TrigExpand[Sin[x + y]]\)\(\ \)\( (*\ 
      f\[OAcute]rmula\ de\ adici\[OAcute]n\ para\ el\ seno\ *) \)\)\)], \
"Input"],

Cell[BoxData[
    \(\(Plot[Tan[x], {x, \(-\[Pi]\)/2 +  .5, \[Pi]/2 -  .5}, 
        AspectRatio \[Rule] Automatic];\)\)], "Input"],

Cell[BoxData[
    \(Simplify[Tan[x + \[Pi]] \[Equal] Tan[x]]\)], "Input"]
}, Open  ]],

Cell[CellGroupData[{

Cell["El permanente problema de los grados y los radianes", "Section"],

Cell[TextData[StyleBox[
"Las funciones trigonom\[EAcute]tricas trabajan todas por defecto en \
radianes.",
  FontWeight->"Bold"]], "Text"],

Cell[BoxData[{
    \(Sin[45. ] (*\ 
      esto\ no\ es\ el\ seno\ de\ 45\ grados\ sino\ de\ 45\ radianes\ *) \), 
    \(Sin[\[Pi]/4] (*\ 
      esto\ es\ el\ seno\ de\ \[Pi]/4\ radianes\  = \ 45\ grados\ *) \)}], 
  "Input"],

Cell[BoxData[{
    \(Sin[90. ] (*\ 
      esto\ no\ es\ el\ seno\ de\ 90\ grados\ sino\ de\ 90\ radianes\ *) \), 
    \(Sin[\[Pi]/2] (*\ 
      esto\ es\ el\ seno\ de\ \[Pi]/2\ radianes\  = \ 90\ grados\ *) \)}], 
  "Input"],

Cell[TextData[{
  "Para pasar de grados a radianes se multiplica por ",
  Cell[BoxData[
      \(TraditionalForm\`\[Pi]\/180\)]],
  " esa constante se llama \"Degree\"."
}], "Text"],

Cell[BoxData[{
    \(Sin[90\ Degree]\  (*\ seno\ de\ 90\ grados\ *) \), 
    \(Sin[45\ Degree] (*\ seno\ de\ 45\ grados\ *) \)}], "Input"],

Cell[BoxData[
    \({N[Degree], N[\[Pi]/180]}\)], "Input"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Las funciones arcotangente, arcoseno, arcocoseno", "Section"],

Cell["\<\
Estas funciones est\[AAcute]n definidas en los apuntes de teor\[IAcute]a. Lo \
que sigue es un complemento de lo all\[IAcute] expuesto.\
\>", "Text"],

Cell[BoxData[
    \(\(Plot[ArcSin[x], {x, \(-1\), 1}];\)\)], "Input"],

Cell[BoxData[
    \(\(\(Simplify[ArcSin[\(-x\)]]\)\( (*\ 
      el\ arcoseno\ es\ una\ funci\[OAcute]n\ impar\ *) \)\)\)], "Input"],

Cell[BoxData[
    \(Simplify[Sin[ArcSin[x]]]\)], "Input"],

Cell[TextData[{
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " no puede simplificar la siguiente expresi\[OAcute]n."
}], "Text"],

Cell[BoxData[
    \(Simplify[ArcSin[Sin[x]]]\)], "Input"],

Cell["En general, el ArcSin[Sin[x]] no es igual a x.", "Text"],

Cell[BoxData[
    \(ArcSin[Sin[3. ]]\)], "Input"],

Cell["\<\
ArcSin[Sin[x]] es igual a x solamente cuando x est\[AAcute] en el intervalo \
[-Pi/2,Pi/2] que es donde toma valores la funci\[OAcute]n arcoseno.\
\>", "Text"],

Cell[BoxData[
    \(Simplify[ArcSin[Sin[x]], \(-Pi\)/2 \[LessEqual] x \[LessEqual] Pi/2]
      \)], "Input"],

Cell["Podemos comprobar este resultado gr\[AAcute]ficamente.", "Text"],

Cell[BoxData[
    \(\(\(Plot[ArcSin[Sin[x]], {x, \(-\[Pi]\)/2, \[Pi]/2}]; 
    \) (*\ la\ funci\[OAcute]n\ arcoseno\ es\ la\ inversa\ de\ la\ 
        funci\[OAcute]n\ seno\ restringuida\ al\ 
        intervalo\ [\(-\[Pi]\)/2, \[Pi]/2]\ *) \)\)], "Input"],

Cell[BoxData[
    \(\(\(Plot[ArcSin[Sin[x]], {x, \(-2\) \[Pi], 2  \[Pi]}]; 
    \) (*\ en\ este\ intervalo\ ArcSin[Sin[x]]\ No\ es\ igual\ a\ x\ *) 
      \)\)], "Input"],

Cell[BoxData[
    \(\(Plot[ArcCos[x], {x, \(-1\), 1}];\)\)], "Input"],

Cell[BoxData[
    \(FullSimplify[ArcSin[x] + ArcCos[x]]\)], "Input"],

Cell[BoxData[
    \(Simplify[Cos[ArcCos[x]]]\)], "Input"],

Cell[TextData[{
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " no puede simplificar la siguiente expresi\[OAcute]n."
}], "Text"],

Cell[BoxData[
    \(Simplify[ArcCos[Cos[x]]]\)], "Input"],

Cell["En general, el ArcCos[Cos[x]] no es igual a x.", "Text"],

Cell[BoxData[
    \(ArcCos[Cos[4. ]]\)], "Input"],

Cell["\<\
ArcCos[Cos[x]] es igual a x solamente cuando x est\[AAcute] en el intervalo \
[0,Pi] que es donde toma valores la funci\[OAcute]n arcocoseno.\
\>", "Text"],

Cell[BoxData[
    \(Simplify[ArcCos[Cos[x]], 0 \[LessEqual] x \[LessEqual] Pi]\)], "Input"],

Cell["Podemos comprobar este resultado gr\[AAcute]ficamente.", "Text"],

Cell[BoxData[
    \(\(\(Plot[ArcCos[Cos[x]], {x, 0, \[Pi]}, AspectRatio -> 1]; 
    \) (*\ la\ funci\[OAcute]n\ arcocoseno\ es\ la\ inversa\ de\ la\ 
        restricci\[OAcute]n\ de\ la\ funci\[OAcute]n\ coseno\ al\ 
        intervalo\ [0, \[Pi]]\ *) \)\)], "Input"],

Cell["Aqu\[IAcute] puedes ver la gr\[AAcute]fica de la funci\[OAcute]n \
arcotangente", "Text"],

Cell[BoxData[
    \(\(Plot[{\[Pi]/2, ArcTan[x], \(-\[Pi]\)/2}, {x, \(-2\), 2}]; \)\)], 
  "Input"],

Cell[TextData[{
  "Recuerda que la funci\[OAcute]n arcotangente est\[AAcute] acotada y toma \
valores en el intervalo ]-Pi/2,Pi/2[. Puedes comprobarlo con ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  "."
}], "Text"],

Cell[BoxData[
    \(\(Simplify[\(-\[Pi]\)/2 < ArcTan[x] < \[Pi]/2, x \[Element] Reals] (*\ 
      esto\ no\ funciona\ en\ la\ versi\[OAcute]n\ 3.0\ *) \)\)], "Input"],

Cell[BoxData[{
    \(Limit[ArcTan[x], x \[Rule] \(+\[Infinity]\)]\), 
    \(Limit[ArcTan[x], x \[Rule] \(-\[Infinity]\)]\)}], "Input"],

Cell[BoxData[
    \(Simplify[Tan[ArcTan[x]]]\)], "Input"],

Cell[TextData[{
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " no puede simplificar la siguiente expresi\[OAcute]n."
}], "Text"],

Cell[BoxData[
    \(Simplify[ArcTan[Tan[x]]]\)], "Input"],

Cell["La raz\[OAcute]n es que no siempre ArcTan[Tan[x]] es igual a x.", "Text"],

Cell[BoxData[
    \(ArcTan[Tan[3.5]]\)], "Input"],

Cell["\<\
ArcTan[Tan[x]] es igual a x solamente cuando x est\[AAcute] en el \
intervalodonde toma valores la funci\[OAcute]n arcotangente, esto es en  \
]-Pi/2,Pi/2[.\
\>", "Text"],

Cell[BoxData[
    \(Simplify[ArcTan[Tan[x]], \(-Pi\)/2 < x < Pi/2]\)], "Input"],

Cell["Podemos comprobar este resultado gr\[AAcute]ficamente.", "Text"],

Cell[BoxData[
    \(\(\(Plot[ArcTan[Tan[x]], {x, \(-\[Pi]\)/2, \[Pi]/2}]; 
    \) (*\(\ la\ funci\[OAcute]n\ arcotangente\ es\ la\ inversa\ de\ la\ 
            funci\[OAcute]n\ tangente\ restringida\ al\ intervalo\n\t\t\t\ ]
          \) - \[Pi]/2, \[Pi]/2  [\ *) \)\)], "Input"],

Cell[BoxData[
    \(\(\(Plot[ArcTan[Tan[x]], {x, \(-2\) \[Pi], 2  \[Pi]}]; 
    \) (*\(\ la\ funci\[OAcute]n\ arcotangente\ es\ la\ inversa\ de\ la\ 
            funci\[OAcute]n\ tangente\ restringida\ al\ intervalo\n\t\t\t\ ]
          \) - \[Pi]/2, \[Pi]/\(2[\ \)*) \)\)], "Input"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Funciones hiperb\[OAcute]licas", "Section"],

Cell[BoxData[
    \(\(Plot[Sinh[x], {x, \(-2\), 2}];\)\)], "Input"],

Cell[BoxData[
    \(\(Plot[Cosh[x], {x, \(-2\), 2}];\)\)], "Input"],

Cell[BoxData[
    \(TrigToExp[Sinh[x]]\)], "Input"],

Cell[BoxData[
    \(TrigToExp[Cosh[x]]\)], "Input"],

Cell[BoxData[
    \(Plot[ArcSinh[x], {x, \(-2\), 2}]\)], "Input"],

Cell[BoxData[
    \(Plot[ArcCosh[x], {x, 0, 2}]\)], "Input"],

Cell[BoxData[
    \(Simplify[Sinh[ArcSinh[x]]]\)], "Input"],

Cell[BoxData[
    \(Simplify[Cosh[ArcCosh[x]]]\)], "Input"],

Cell[BoxData[
    \({Simplify[ArcCosh[Cosh[x]]], 
      Simplify[ArcCosh[Cosh[x]], x > 0]}\)], "Input"],

Cell[BoxData[
    \({TrigToExp[ArcSinh[x]], TrigToExp[ArcCosh[x]]}\)], "Input"]
}, Open  ]]
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End of Mathematica Notebook file.
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