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Cell[CellGroupData[{
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  "Materialien zur Vorlesung ",
  StyleBox["Elementare Analysis",
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  ", Wintersemester 2003 / 4"
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  "\n\nWie kann der Fl\[ADoubleDot]cheninhalt eines Kreises berechnet werden? \
Das war eine Fragestellung zu Beginn dieser Vorlesung.\nDie Kreisgleichung f\
\[UDoubleDot]r einen Kreis mit Radius Eins und dem Mittelpunkt  (0, 0)  \
lautet\n\t\t",
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  "  die Funktion  ",
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  "  sehen Sie im linken Bild:\n"
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Cell[TextData[{
  "Es entspricht unserem anschaulichen Verst\[ADoubleDot]ndnis, die Fl\
\[ADoubleDot]che \"unter\" dem Graphen von  ",
  StyleBox["f",
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  "  (rechtes Bild) als Halbkreis zu interpretieren und dieser \
Fl\[ADoubleDot]che den Wert  ",
  Cell[BoxData[
      \(TraditionalForm\`\[Pi]\/2\)]],
  "  zuzuordnen. Diese geometrischen Vorstellungen f\[UDoubleDot]hren zu den \
Begriffen Integration und Integral einer Funktion."
}], "Text"],

Cell[TextData[{
  StyleBox["Definition 6.1:\n",
    FontFamily->"Arial",
    FontVariations->{"Underline"->True}],
  StyleBox["INTEGRAL einer positiven, stetigen Funktion",
    FontFamily->"Arial"],
  StyleBox["\n",
    FontVariations->{"Underline"->True}],
  "Es seien  ",
  Cell[BoxData[
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  " und  ",
  Cell[BoxData[
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  "  stetig. Es gelte  ",
  Cell[BoxData[
      \(TraditionalForm\`f(x) \[GreaterEqual] 0\)]],
  "  f\[UDoubleDot]r  ",
  Cell[BoxData[
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  ". Der Fl\[ADoubleDot]cheninhalt der durch die ",
  StyleBox["x",
    FontSlant->"Italic"],
  "-Achse, die Geraden  ",
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Eine andere M\[ODoubleDot]glichkeit Integrale zu definieren, besteht darin \
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  "Zum Kennenlernen einiger Eigenschaften des oben definierten Integrals \
seien auf  ",
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  "  die Funktionen  ",
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  "  betrachtet, wobei  ",
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      \(TraditionalForm\`\(f\_1\)(x) := x + 1\ , \ \(f\_2\)(x) := 
        2 \((x + 1)\)\)]],
  " ,  also  ",
  Cell[BoxData[
      \(TraditionalForm\`f\_2 = 2\[CenterDot]f\_1\)]],
  "  sei.\n"
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{407.875, 269.438}} -> {-4.41892, 6.94876, 0.0229524, 0.0229524}, {{230.813, \
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Cell[TextData[{
  "Die Graphen motivieren die Beziehung\n\n\t\t ",
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  " ,\n\t\t \nwas auch bei nicht so einfachen Funktionen gilt.\n"
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F
0 g
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Mistroke
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Mistroke
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Mistroke
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Mistroke
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Mfstroke
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.12 .33879 L
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.06757 .23684 L
F
0 g
.06757 .36842 m
.08358 .40959 L
.09174 .4274 L
.10046 .44208 L
.10285 .44513 L
.1051 .44761 L
.1073 .44962 L
.10932 .45112 L
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.22343 .26929 L
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.225 .26943 L
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.22661 .26973 L
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Mistroke
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.38643 .35656 L
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Mistroke
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.47471 .24637 L
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.69093 .40179 L
.6932 .40272 L
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Mistroke
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.84567 .30332 L
.84668 .3033 L
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.84857 .30324 L
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.8508 .3031 L
Mistroke
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.86285 .30067 L
.865 .29986 L
.86905 .29797 L
.87813 .29195 L
.87838 .29175 L
Mfstroke
.06757 .23684 m
.08358 .28494 L
.09174 .30566 L
.10046 .32304 L
.1051 .32985 L
.10932 .33443 L
.11136 .33607 L
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Mfstroke
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F
0 g
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Mistroke
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Mistroke
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Mistroke
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Mistroke
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Mfstroke
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Mistroke
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  "Wenn auch hier das Integral blo\[SZ] ein Ma\[SZ] f\[UDoubleDot]r den Fl\
\[ADoubleDot]cheninhalt sein sollte, so lie\[SZ]e sich wieder eine positive \
Zahl f\[UDoubleDot]r ein solches Integral  \[ContourIntegral]  zuordnen. Man \
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  "-Achse, den Geraden  ",
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    FontVariations->{"Underline"->True}],
  StyleBox["Beispiel 6.1:",
    FontFamily->"Arial",
    FontVariations->{"Underline"->True}],
  StyleBox["\n",
    FontVariations->{"Underline"->True}],
  "Beachten Sie, dass ein solches Integral sich z.B. aus unendlich vielen \
\"Fl\[ADoubleDot]chenst\[UDoubleDot]cken\" zusammensetzen kann. Es seien  ",
  StyleBox["a",
    FontSlant->"Italic"],
  " = 0  und  ",
  StyleBox["b",
    FontSlant->"Italic"],
  "  geeignet und  ",
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  StyleBox["Hauptsatz der Differentialrechnung und Integralrechnung, Teil 1",
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  "\nEs sei ein beliebiges  ",
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  " . Die Behauptung l\[ADoubleDot]\[SZ]t sich nun mit diesen Bezeichnungen \
so formulieren:\n\t\t\t",
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  "\n\t\t\t\nDurch geschicktes Umformen wird diese Gleichung beweisbar sein:\n\
\nEs gilt\t \t",
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  "\t\n\t\t\t\t\t\t   \t\nnach obigem Beispiel 6.2. \nAlso gilt\n \t\t",
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  "\nWie oben errechnet sich\n\t\t",
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  "\nNun werde der Einfachheit halber der Spezialfall angenommen, dass es f\
\[UDoubleDot]r  ",
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Cell[TextData[{
  "Die hellgraue, krummlinig berandete Fl\[ADoubleDot]che hat also den \
gleichen Fl\[ADoubleDot]cheninhalt wie der untere Halbkreis, also  ",
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}], "Text"],

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  StyleBox[" Substitutionsregel:",
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  "\n\n\t\t",
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  "\nF\[UDoubleDot]r  ",
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  " > 0  sei die Funktion  ",
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  " .\nF\[UDoubleDot]r eine Basis  \[Alpha] ",
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  "  ist der ",
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entsteht der Graph der Umkehrfunktion des  ln . Diese Funktion hei\[SZ]t ",
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  ".\n\nDeshalb werden jetzt einige Eigenschaften von Umkehrfunktionen \
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  StyleBox["UMKEHRFUNKTION f\[UDoubleDot]r eine streng monotone, stetige \
Funktion",
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  "Es sei  ",
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Funktion  ",
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  "  bezeichnet.\nWeiterhin gilt\n\t\t",
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  " !!!)\n\nUmkehrfunktionen f\[UDoubleDot]r streng monoton fallende \
Funktionen lassen sich analog definieren.\n\n"
}], "Text"],

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Cell[TextData[{
  "Bei diese Grafik ist zu beachten, dass der Wertebereich von  tan  ganz  \
\[DoubleStruckCapitalR]  ist und daher der Definitionsbereich von arctan auch \
 \[DoubleStruckCapitalR]  ist!\n\nWeiterhin gilt:\n\t\t",
  Cell[BoxData[
      FormBox[
        RowBox[{
          RowBox[{\(arctan' \((t)\)\), "=", 
            FractionBox[
              StyleBox["1",
                FontSize->11], 
              RowBox[{
                StyleBox["1",
                  FontSize->11], " ", "+", 
                StyleBox[" ",
                  FontSize->11], 
                SuperscriptBox[
                  StyleBox["t",
                    FontSize->11], "2"]}]]}], " ", ",", 
          " ", \(t \[Element] \[DoubleStruckCapitalR]\), " ", ",", 
          "\[IndentingNewLine]", 
          RowBox[{\(arccot' \((t)\)\), "=", 
            FractionBox[
              RowBox[{"-", 
                StyleBox["1",
                  FontSize->11]}], 
              RowBox[{
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                StyleBox[" ",
                  FontSize->11], 
                SuperscriptBox[
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          " ", \(t \[Element] \(\(\[DoubleStruckCapitalR]\)\(\ \)\(.\)\)\)}], 
        TraditionalForm]]],
  "\n\n"
}], "Text"],

Cell[TextData[{
  "Beweis f\[UDoubleDot]r die Ableitung von  arctan:\nZur Erinnerung die \
Formel f\[UDoubleDot]r die Ableitung einer Umkehrfunktion:\n\n\t\t",
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            FractionBox[
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              RowBox[{
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                StyleBox[
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                        RowBox[{"-", 
                          StyleBox["1",
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                  FontSize->10]}]]}]}], TraditionalForm]]],
  " ,\n\t\t\n\t\t      ",
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          FractionBox[
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            RowBox[{
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                  FontSize->10], 
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                  FontSize->10], 
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                FontSize->10], 
              StyleBox[" ",
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              StyleBox[
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                  RowBox[{"  ", 
                    RowBox[{"-", 
                      StyleBox["1",
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  " .",
  "\n"
}], "Text"],

Cell[TextData[{
  "Zuerst die Ableitung von  tan. Mit der Quotientenregel folgt:\n\n\t",
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                      FontSize->10], 
                    StyleBox["cos",
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