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Micron Document

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<title>Formelsammlung Analysis</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading">Formelsammlung Analysis</h1>

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<div class="mw-heading mw-heading2"><h2 id="Folgen_und_Reihen"><a href="Folge_(Mathematik)" title="Folge (Mathematik)">Folgen</a> und <a href="Reihe_(Mathematik)" title="Reihe (Mathematik)">Reihen</a></h2></div>

<div class="mw-heading mw-heading3"><h3 id="Arithmetische_und_geometrische_Folgen">Arithmetische und geometrische Folgen</h3></div>

<dl><dt><a href="Arithmetische_Folge" title="Arithmetische Folge">Arithmetische Folge</a></dt>

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<annotation encoding="application/x-tex">{\displaystyle a_{n+1}-a_{n}=d\quad \mathrm {f{\ddot {u}}r\;alle\;} n}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cc8587c4875904c8699577776c511bc6376c5357.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:25.717ex; height:2.509ex;" alt="{\displaystyle a_{n+1}-a_{n}=d\quad \mathrm {f{\ddot {u}}r\;alle\;} n}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle a_{n}={\tfrac {1}{2}}(a_{n-1}+a_{n+1})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/76ab8a1955fb59b735dbec94bba7038648fb7758.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:20.952ex; height:3.509ex;" alt="{\displaystyle a_{n}={\tfrac {1}{2}}(a_{n-1}+a_{n+1})}" loading="lazy"></dd>

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \,a_{n}=a_{1}+(n-1)d}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f80c6721bb16394e8728fca734af9ef93bf814.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.481ex; height:2.843ex;" alt="{\displaystyle \,a_{n}=a_{1}+(n-1)d}" loading="lazy"></dd>

<dt><a href="Geometrische_Folge" title="Geometrische Folge">Geometrische Folge</a></dt>

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<mover>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {a_{n+1}}{a_{n}}}=q\quad \mathrm {f{\ddot {u}}r\;alle\;} n,q\in \mathbb {R} \setminus \{0\}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fc5826aa2038626e6c9c132bf73b498b4bdff78e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:33.423ex; height:5.176ex;" alt="{\displaystyle {\frac {a_{n+1}}{a_{n}}}=q\quad \mathrm {f{\ddot {u}}r\;alle\;} n,q\in \mathbb {R} \setminus \{0\}}" loading="lazy"></dd>

<msub>

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<msqrt>

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<annotation encoding="application/x-tex">{\displaystyle a_{n}={\sqrt {a_{n-1}\cdot a_{n+1}}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2e8404dd0ef89523923a4dd690b26891b5b7381b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:18.259ex; height:3.009ex;" alt="{\displaystyle a_{n}={\sqrt {a_{n-1}\cdot a_{n+1}}}}" loading="lazy"></dd>

<msub>

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<annotation encoding="application/x-tex">{\displaystyle a_{n}=a_{1}\cdot q^{n-1}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d661c6435667c7a4e56549f178a4cdf91f270dd4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.908ex; height:3.009ex;" alt="{\displaystyle a_{n}=a_{1}\cdot q^{n-1}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Grenzwerte:_Definition_(Folgen)"><a href="Grenzwert_(Folge)" title="Grenzwert (Folge)">Grenzwerte</a>: Definition (Folgen)</h3></div>

<ul><li>Die Folge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a_{n})}">

<msub>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle (a_{n})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/18bc33c7c35d82b00f88d3a9103ed4738cde41f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.258ex; height:2.843ex;" alt="{\displaystyle (a_{n})}" loading="lazy"> heißt <a href="Nullfolge" title="Nullfolge">Nullfolge</a>, wenn es zu jedem <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon >0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \epsilon >0}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/568095ad3924314374a5ab68fae17343661f2a71.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.205ex; height:2.176ex;" alt="{\displaystyle \epsilon >0}" loading="lazy"> eine Nummer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{0}}">

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<annotation encoding="application/x-tex">{\displaystyle n_{0}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/63584d203ecb012a7bcb90f422408bbfe4018956.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.449ex; height:2.009ex;" alt="{\displaystyle n_{0}}" loading="lazy"> gibt, sodass für alle Folgeglieder mit höherer Nummer, also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n>n_{0}}">

<msub>

</mrow>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle n>n_{0}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/deb934b76829498dd1264524bb7a7a1981aed31e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.942ex; height:2.176ex;" alt="{\displaystyle n>n_{0}}" loading="lazy"> gilt:</li></ul>

</mrow>

<msub>

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</mrow>

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<annotation encoding="application/x-tex">{\displaystyle \,|a_{n}|<\epsilon }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c2ba04cbc5b4bcdfb1d550820b12c08e36f6620a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.172ex; height:2.843ex;" alt="{\displaystyle \,|a_{n}|<\epsilon }" loading="lazy"></dd></dl></dd></dl>

<ul><li>Eine Folge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a_{n})}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (a_{n})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/18bc33c7c35d82b00f88d3a9103ed4738cde41f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.258ex; height:2.843ex;" alt="{\displaystyle (a_{n})}" loading="lazy"> hat den Grenzwert a, wenn die Folge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a_{n}-a)}">

<msub>

</mrow>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle (a_{n}-a)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c7d9b43d724bc3f29a82b5b271911196bf25defa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.328ex; height:2.843ex;" alt="{\displaystyle (a_{n}-a)}" loading="lazy"> den Grenzwert 0 hat.</li>

<li>Folgen ohne Grenzwert heißen divergent.</li>

<li>Eine Folge heißt beschränkt, wenn es eine Zahl <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K>0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K>0}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f89502f5e80eaec4b6f5b1f822aebd0c50d0a303.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.327ex; height:2.176ex;" alt="{\displaystyle K>0}" loading="lazy"> gibt, sodass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |f_{n}|<K}">

</mrow>

<msub>

</mrow>

</msub>

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle |f_{n}|<K}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0a097c9e8487763295b9805baaa63b7836008ecf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.816ex; height:2.843ex;" alt="{\displaystyle |f_{n}|<K}" loading="lazy"> für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\in \mathbb {N} }">

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<annotation encoding="application/x-tex">{\displaystyle n\in \mathbb {N} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d059936e77a2d707e9ee0a1d9575a1d693ce5d0b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.913ex; height:2.176ex;" alt="{\displaystyle n\in \mathbb {N} }" loading="lazy"> gilt.</li></ul>

<div class="mw-heading mw-heading3"><h3 id="Grenzwertsätze_(Folgen)">Grenzwertsätze (Folgen)</h3></div>

Hat die Folge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a_{n})}">

<msub>

</mrow>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (a_{n})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/18bc33c7c35d82b00f88d3a9103ed4738cde41f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.258ex; height:2.843ex;" alt="{\displaystyle (a_{n})}" loading="lazy"> den Grenzwert a, die Folge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (b_{n})}">

<msub>

</mrow>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle (b_{n})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5ca3872ee8054312f21ff267bc6831745e42b9c9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.025ex; height:2.843ex;" alt="{\displaystyle (b_{n})}" loading="lazy"> den Grenzwert b, so gilt:

</mrow>

</munder>

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</mrow>

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</mrow>

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<annotation encoding="application/x-tex">{\displaystyle \lim _{n\to \infty }(a_{n}\pm b_{n})=a\pm b}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/41ff933bc34591361aa9aa7da157d978ba3a655d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:21.753ex; height:3.843ex;" alt="{\displaystyle \lim _{n\to \infty }(a_{n}\pm b_{n})=a\pm b}" loading="lazy"></li>

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle \lim _{n\to \infty }(a_{n}\cdot b_{n})=a\cdot b}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a9a0c0d33ae3e9dae9dc428065e6436d58e0ff2d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.43ex; height:3.843ex;" alt="{\displaystyle \lim _{n\to \infty }(a_{n}\cdot b_{n})=a\cdot b}" loading="lazy"></li>

</mrow>

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<mfrac>

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</mrow>

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<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle \lim _{n\to \infty }{\frac {a_{n}}{b_{n}}}={\frac {a}{b}}\qquad b\not =0}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8d58277f7bc5b223dea7896a357a604eecc181c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:23.012ex; height:5.176ex;" alt="{\displaystyle \lim _{n\to \infty }{\frac {a_{n}}{b_{n}}}={\frac {a}{b}}\qquad b\not =0}" loading="lazy"></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Funktionen_(formale_Eigenschaften)">Funktionen (formale Eigenschaften)</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Grenzwerte_von_Funktionen"><a href="Grenzwert_(Funktion)" title="Grenzwert (Funktion)">Grenzwerte von Funktionen</a></h3></div>

<ul><li>[Definition, Eigenschaften, Grenzwertsätze analog]</li></ul>

<div class="mw-heading mw-heading4"><h4 id="Regel_von_de_L’Hospital"><a href="Regel_von_de_L%E2%80%99Hospital" title="Regel von de L’Hospital">Regel von de L’Hospital</a></h4></div>

Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)={\frac {u(x)}{v(x)}}.}">

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle f(x)={\frac {u(x)}{v(x)}}.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/030df6f28b50032f79b249fb72af6bf06a5f313d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:13.468ex; height:6.509ex;" alt="{\displaystyle f(x)={\frac {u(x)}{v(x)}}.}" loading="lazy">

Voraussetzungen:

<ul><li>Es gibt eine Stelle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy">, sodass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u(a)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u(a)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f34d1cfcb353a646f89533529e4d97cd1cd72686.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.369ex; height:2.843ex;" alt="{\displaystyle u(a)}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v(a)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v(a)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aee7d41de7724f3437da67556cc1ea586b51e011.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.167ex; height:2.843ex;" alt="{\displaystyle v(a)}" loading="lazy"> entweder Null sind oder bestimmt divergieren</li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy"> sind in einer Umgebung von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a}</annotation>

</semantics>

<li>Der Grenzwert <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{x\to a}{\frac {u'(x)}{v'(x)}}}">

</mrow>

</munder>

<mfrac>

<mrow>

<msup>

</msup>

</mrow>

<mrow>

<msup>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lim _{x\to a}{\frac {u'(x)}{v'(x)}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/531d87d3c99785f7373a673d831cee69ff36dd6b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:9.83ex; height:6.509ex;" alt="{\displaystyle \lim _{x\to a}{\frac {u'(x)}{v'(x)}}}" loading="lazy"> existiert.</li></ul>

Dann gilt:

</mrow>

</munder>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</munder>

<mfrac>

<mrow>

<msup>

</msup>

</mrow>

<mrow>

<msup>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lim _{x\to a}{\frac {u(x)}{v(x)}}=\lim _{x\to a}{\frac {u'(x)}{v'(x)}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c30d048845be88738202f028b4e5bed6244ceb86.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:22.073ex; height:6.509ex;" alt="{\displaystyle \lim _{x\to a}{\frac {u(x)}{v(x)}}=\lim _{x\to a}{\frac {u'(x)}{v'(x)}}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading4"><h4 id="Einseitige_Grenzwerte">Einseitige Grenzwerte</h4></div>

Die Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon X\to \mathbb {R} \;}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\colon X\to \mathbb {R} \;}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b283aac1eb91af7ecb0ef5f7bba8a03ba02841db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.23ex; height:2.509ex;" alt="{\displaystyle f\colon X\to \mathbb {R} \;}" loading="lazy"> hat für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\to p+}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\to p+}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9454231471d04f12ffe2bd1cc2041901203c6f58.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.921ex; height:2.343ex;" alt="{\displaystyle x\to p+}" loading="lazy"> den Limes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L\;}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L\;}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4eaa3f97b12fdbe439b649fe24b7d8e882ce85d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.228ex; height:2.176ex;" alt="{\displaystyle L\;}" loading="lazy">, wenn es zu jedem (noch so kleinen) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon >0\;}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varepsilon >0\;}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bafdf74a7967ecd0cae8b4eb1da348b9790d6289.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.99ex; height:2.176ex;" alt="{\displaystyle \varepsilon >0\;}" loading="lazy"> ein <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta >0\;}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \delta >0\;}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/508ccebac7b4100b722d32d66245db959d8b86c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.955ex; height:2.343ex;" alt="{\displaystyle \delta >0\;}" loading="lazy"> gibt, sodass für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\;}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\;}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/19d9db5c9a6544bf5e20f4a51ac44e7cdfee0099.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.975ex; height:1.676ex;" alt="{\displaystyle x\;}" loading="lazy">-Werte aus dem Definitionsbereich <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\;}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X\;}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/acb11a982da4633651f3b90f5d569e31a31bb31b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.625ex; height:2.176ex;" alt="{\displaystyle X\;}" loading="lazy"> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\;}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\;}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1754a429c1438384f5271fe6065a5aee70952e8b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.924ex; height:2.509ex;" alt="{\displaystyle f\;}" loading="lazy">, die der Bedingung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<x-p<\delta \;}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0<x-p<\delta \;}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3e988860b9ae413be92b75c44cf462171d0a1bea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.393ex; height:2.676ex;" alt="{\displaystyle 0<x-p<\delta \;}" loading="lazy"> genügen, auch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |f(x)-L|<\varepsilon \;}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |f(x)-L|<\varepsilon \;}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dd3a76935e29e5a17518201bf79d06fd8f968a90.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.962ex; height:2.843ex;" alt="{\displaystyle |f(x)-L|<\varepsilon \;}" loading="lazy"> gilt.

<dl><dd>In diesem Falle nennt man den Grenzwert <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{x\to p+}f(x):=L}">

</mrow>

</munder>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lim _{x\to p+}f(x):=L}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c30cb99a700482e862976f6a6d931abc7226ca93.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:14.822ex; height:4.176ex;" alt="{\displaystyle \lim _{x\to p+}f(x):=L}" loading="lazy"> konvergent.</dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Stetigkeit"><a href="Stetige_Funktion" title="Stetige Funktion">Stetigkeit</a></h3></div>

Eine Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> heißt an einer Stelle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/86f21d0e31751534cd6584264ecf864a6aa792cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}" loading="lazy"> stetig, wenn der Grenzwert von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> gegen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/86f21d0e31751534cd6584264ecf864a6aa792cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}" loading="lazy"> existiert und mit dem Funktionswert <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x_{0})}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x_{0})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/27cf1dbaefc6038a22779fb2943aff758a592a3a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.472ex; height:2.843ex;" alt="{\displaystyle f(x_{0})}" loading="lazy"> übereinstimmt

<msub>

</mrow>

</msub>

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mrow>

</munder>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</munder>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x_{0})=\lim _{h\to 0}f(x_{0}+h)=\lim _{h\to 0}f(x_{0}-h)=\lim _{x\to x_{0}}f(x)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6fdd8009cfea7a9753cdce656ea84c6c961aeea3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:50.827ex; height:4.176ex;" alt="{\displaystyle f(x_{0})=\lim _{h\to 0}f(x_{0}+h)=\lim _{h\to 0}f(x_{0}-h)=\lim _{x\to x_{0}}f(x)}" loading="lazy"></dd></dl>

<ul><li>Epsilon-Delta-Kriterium:<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon D\to \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\colon D\to \mathbb {R} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/229e3a78dd4be03159691c3d4c84a9095784d26e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.529ex; height:2.509ex;" alt="{\displaystyle f\colon D\to \mathbb {R} }" loading="lazy"> ist stetig in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}\in D}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{0}\in D}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b107b130543b591ac3037f4bc152da314ec51586.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.149ex; height:2.509ex;" alt="{\displaystyle x_{0}\in D}" loading="lazy">, wenn zu jedem <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon >0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varepsilon >0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e04ec3670b50384a3ce48aca42e7cc5131a06b12.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.344ex; height:2.176ex;" alt="{\displaystyle \varepsilon >0}" loading="lazy"> ein <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta >0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \delta >0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/595d5cea06fdcaf2642caf549eda2cfc537958a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.31ex; height:2.343ex;" alt="{\displaystyle \delta >0}" loading="lazy"> existiert, so dass für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in D}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in D}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aaf3e2c3607ccf1c1d7ee6620b44ef9d9e2e1f6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.095ex; height:2.176ex;" alt="{\displaystyle x\in D}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |x-x_{0}|<\delta }">

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |x-x_{0}|<\delta }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b2f902c12a7f7e5be35218a7d6f5cde43e036d86.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.995ex; height:2.843ex;" alt="{\displaystyle |x-x_{0}|<\delta }" loading="lazy"> gilt: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |f(x)-f(x_{0})|<\varepsilon }">

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |f(x)-f(x_{0})|<\varepsilon }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/59c85be121472f6c05895e45cbbdcdbe0b2d4e6e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.205ex; height:2.843ex;" alt="{\displaystyle |f(x)-f(x_{0})|<\varepsilon }" loading="lazy">.</li>

<li>Folgenkriterium: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon D\to \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\colon D\to \mathbb {R} }</annotation>

</semantics>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{0}\in D}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b107b130543b591ac3037f4bc152da314ec51586.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.149ex; height:2.509ex;" alt="{\displaystyle x_{0}\in D}" loading="lazy">, wenn für jede Folge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{k})_{k\in \mathbb {N} }}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (x_{k})_{k\in \mathbb {N} }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5721f623481f88d169056cadc4212b9aee6c4a0f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.599ex; height:2.843ex;" alt="{\displaystyle (x_{k})_{k\in \mathbb {N} }}" loading="lazy"> mit Elementen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{k}\in D}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{k}\in D}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b09ce92224e1ae9c87779d49e3a709beff18e504.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.183ex; height:2.509ex;" alt="{\displaystyle x_{k}\in D}" loading="lazy">, die gegen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/86f21d0e31751534cd6584264ecf864a6aa792cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}" loading="lazy"> konvergiert, auch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x_{k})}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x_{k})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/20d432970858c21563481c08f9c2bf50dffbeb06.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.506ex; height:2.843ex;" alt="{\displaystyle f(x_{k})}" loading="lazy"> gegen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x_{0})}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x_{0})}</annotation>

</semantics>

<div class="mw-heading mw-heading3"><h3 id="Grundlegendes">Grundlegendes</h3></div>

<dl><dt><a href="Zwischenwertsatz" title="Zwischenwertsatz">Zwischenwertsatz</a></dt>

<dd>Eine im Intervall <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [a,b]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [a,b]}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9c4b788fc5c637e26ee98b45f89a5c08c85f7935.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.555ex; height:2.843ex;" alt="{\displaystyle [a,b]}" loading="lazy"> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a<b}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a<b}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/91a7698e4c7401bb321f97888b872b583a9e4642.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.326ex; height:2.176ex;" alt="{\displaystyle a<b}" loading="lazy">) stetige Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> nimmt jeden Funktionswert zwischen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(a)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(a)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/368cb4b81ba5754d7a354a4ce49c2f1084bdaace.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.318ex; height:2.843ex;" alt="{\displaystyle f(a)}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(b)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(b)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/adbef9aaf500c30a3f107e5d7efb7a8627d3bba9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.085ex; height:2.843ex;" alt="{\displaystyle f(b)}" loading="lazy"> mindestens einmal an.</dd></dl>

Spezialfall: <a href="Nullstellensatz_von_Bolzano" class="mw-redirect" title="Nullstellensatz von Bolzano">Nullstellensatz</a>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/535ea7fc4134a31cbe2251d9d3511374bc41be9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}" loading="lazy"> stetige Funktion, bei der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(a)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(a)}</annotation>

</semantics>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(b)}</annotation>

</semantics>

<dt><a href="Extremwertsatz" class="mw-redirect" title="Extremwertsatz">Extremwertsatz</a></dt>

<dd>Eine in einem Intervall stetige Funktion hat dort stets einen größten und einen kleinsten Funktionswert.</dd>

<dt><a href="Mittelwertsatz_der_Differentialrechnung" title="Mittelwertsatz der Differentialrechnung">Mittelwertsatz</a></dt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f:[a,b]\to \mathbb {R} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b592d102ccd1ba134d401c5b3ea177baaba3ffac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.063ex; height:2.843ex;" alt="{\displaystyle f:[a,b]\to \mathbb {R} }" loading="lazy"> auf dem abgeschlossenen Intervall <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [a,b]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [a,b]}</annotation>

</semantics>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a<b}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/91a7698e4c7401bb321f97888b872b583a9e4642.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.326ex; height:2.176ex;" alt="{\displaystyle a<b}" loading="lazy">) stetig und differenzierbar. Dann gibt es mindestens ein <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}\in (a,b)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{0}\in (a,b)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c4d03f71523a733bbcdbd4ef0601099717cd1b75.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.295ex; height:2.843ex;" alt="{\displaystyle x_{0}\in (a,b)}" loading="lazy">, so dass

<msup>

</msup>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<mfrac>

<mrow>

<mrow>

</mrow>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f'\left(x_{0}\right)={\frac {f\left(b\right)-f\left(a\right)}{b-a}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/407c245fa5f48f7a4872ec7dbc8330261d3b72b6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:22.538ex; height:6.009ex;" alt="{\displaystyle f'\left(x_{0}\right)={\frac {f\left(b\right)-f\left(a\right)}{b-a}}}" loading="lazy"></dd></dl></dd>

<div class="mw-heading mw-heading2"><h2 id="Differentialrechnung"><a href="Differentialrechnung" title="Differentialrechnung">Differentialrechnung</a></h2></div>

<div class="mw-heading mw-heading3"><h3 id="Differenzierbarkeit:_Definitionen">Differenzierbarkeit: Definitionen</h3></div>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> ist genau dann differenzierbar an einer Stelle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation>

</semantics>

<msub>

</mrow>

</msub>

</mrow>

</munder>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lim _{x\rightarrow x_{0}}{\frac {f(x)-f(x_{0})}{x-x_{0}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1544ea3d274ef767d1ab121674f453639c9234d9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:18.308ex; height:6.009ex;" alt="{\displaystyle \lim _{x\rightarrow x_{0}}{\frac {f(x)-f(x_{0})}{x-x_{0}}}}" loading="lazy"></dd></dl>

existiert. Diesen Grenzwert bezeichnet man als die Ableitung von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> an der Stelle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation>

</semantics>

<div class="mw-heading mw-heading3"><h3 id="Geometrisches:_Tangenten">Geometrisches: Tangenten</h3></div>

<dl><dt><a href="Tangentengleichung" class="mw-redirect" title="Tangentengleichung">Tangentengleichung</a> zu <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> im Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(x_{0}|f(x_{0}))}">

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(x_{0}|f(x_{0}))}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fe0935f383bb121286230b8ad00a079348b12fd6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.057ex; height:2.843ex;" alt="{\displaystyle P(x_{0}|f(x_{0}))}" loading="lazy"></dt>

<msup>

</msup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y=f'(x_{0})\!\,(x-x_{0})+f(x_{0})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/51b63201a4adc25e468e26d5ddeaf89f167f43ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.128ex; height:3.009ex;" alt="{\displaystyle y=f'(x_{0})\!\,(x-x_{0})+f(x_{0})}" loading="lazy"></dd>

<dt>Normale (Senkrechte)</dt>

<mfrac>

<mrow>

</mrow>

<mrow>

<msup>

</msup>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y={\frac {-1}{f'(x_{0})}}(x-x_{0})+f(x_{0})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5d57dd6a1caa5d686fe0e4e4597e8e96190eefc4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:27.964ex; height:6.009ex;" alt="{\displaystyle y={\frac {-1}{f'(x_{0})}}(x-x_{0})+f(x_{0})}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Ableitungsregeln">Ableitungsregeln</h3></div>

<dl><dt><a href="Konstante_Funktion" title="Konstante Funktion">Konstante Funktion</a></dt>

<msup>

<mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left(a\right)'=0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1050a10bfbb7449309462816d32ecbd7dea6164b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.985ex; height:3.176ex;" alt="{\displaystyle \left(a\right)'=0}" loading="lazy"></dd>

<dt><a href="Faktorregel" title="Faktorregel">Faktorregel</a></dt>

<msup>

</msup>

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (a\cdot f)'=a\cdot f'}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3e308728043f480d59a3f9bb7109c8d9cba954f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.694ex; height:3.009ex;" alt="{\displaystyle (a\cdot f)'=a\cdot f'}" loading="lazy"></dd>

<dt><a href="Summenregel" title="Summenregel">Summenregel</a></dt>

<msup>

<mrow>

<mrow>

</mrow>

</mrow>

</msup>

<msup>

</msup>

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left(g\pm h\right)'=g'\pm h'}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b46620201b8360c490f5041f327d8825e14d616c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.555ex; height:3.176ex;" alt="{\displaystyle \left(g\pm h\right)'=g'\pm h'}" loading="lazy"></dd>

<dt><a href="Produktregel" title="Produktregel">Produktregel</a></dt>

<msup>

</msup>

<msup>

</msup>

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (g\cdot h)'=g'\cdot h+g\cdot h'}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/05cff9657c544b74ab128545a57aaef16431f575.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.207ex; height:3.009ex;" alt="{\displaystyle (g\cdot h)'=g'\cdot h+g\cdot h'}" loading="lazy"></dd>

<dt><a href="Quotientenregel" title="Quotientenregel">Quotientenregel</a></dt>

<msup>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</msup>

<mfrac>

<mrow>

<msup>

</msup>

<msup>

</msup>

</mrow>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left({\frac {g}{h}}\right)'={\frac {g'\cdot h-g\cdot h'}{h^{2}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f14cbf031987cefc5420324fafa426decf4cd314.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:22.05ex; height:5.843ex;" alt="{\displaystyle \left({\frac {g}{h}}\right)'={\frac {g'\cdot h-g\cdot h'}{h^{2}}}}" loading="lazy"></dd>

<dt><a href="Potenzregel" title="Potenzregel">Potenzregel</a></dt>

<msup>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left(x^{n}\right)'=nx^{n-1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8302b879b6e60565e573d710bea60c69436f1ba9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.184ex; height:3.176ex;" alt="{\displaystyle \left(x^{n}\right)'=nx^{n-1}}" loading="lazy"></dd>

<dt><a href="Kettenregel" title="Kettenregel">Kettenregel</a></dt>

<msup>

</msup>

<msup>

</msup>

<msup>

</msup>

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (g\circ h)'(x)=(g(h(x)))'=g'(h(x))\cdot h'(x)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4eb45ef8447352cbe95eeca6c9e9cb12d0c271f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.309ex; height:3.009ex;" alt="{\displaystyle (g\circ h)'(x)=(g(h(x)))'=g'(h(x))\cdot h'(x)}" loading="lazy"></dd>

<dt>Ableitung der Potenzfunktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=g(x)^{h(x)}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)=g(x)^{h(x)}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6a31e7f794f6d3b03d1bb9507e3e5e1e9dc1dd41.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.17ex; height:3.343ex;" alt="{\displaystyle f(x)=g(x)^{h(x)}}" loading="lazy"></dt>

<msup>

</msup>

<mrow>

<mrow>

<msup>

</msup>

<mfrac>

<mrow>

<msup>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f'(x)=\left(h'(x)\ln(g(x))+h(x){\frac {g'(x)}{g(x)}}\right)g(x)^{h(x)}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9d799cde87539ee10ed139cbb1ddb9361e87d3be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:46.354ex; height:6.509ex;" alt="{\displaystyle f'(x)=\left(h'(x)\ln(g(x))+h(x){\frac {g'(x)}{g(x)}}\right)g(x)^{h(x)}}" loading="lazy">.</dd>

<dt><a href="Produktregel#Leibniz-Regel_für_dividierte_Differenzen" title="Produktregel">Leibnizsche Regel</a></dt>

<dd>Die Ableitung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">-ter Ordnung für ein Produkt aus zwei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">-fach differenzierbaren Funktionen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

</semantics>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</munderover>

<mrow>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (fg)^{(n)}=\sum _{k=0}^{n}{n \choose k}f^{(k)}g^{(n-k)}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d6096866646c0c3c0e4ebe0cd563076a39a89b92.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:27.798ex; height:7.009ex;" alt="{\displaystyle (fg)^{(n)}=\sum _{k=0}^{n}{n \choose k}f^{(k)}g^{(n-k)}}" loading="lazy">.</dd>

<dd>Die hier auftretenden Ausdrücke der Form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tbinom {n}{k}}}">

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tbinom {n}{k}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/206415d3742167e319b2e52c2ca7563b799abad7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.116ex; height:3.176ex;" alt="{\displaystyle {\tbinom {n}{k}}}" loading="lazy"> sind <a href="Binomialkoeffizient" title="Binomialkoeffizient">Binomialkoeffizienten</a>.</dd>

<dt><a href="Formel_von_Fa%C3%A0_di_Bruno" title="Formel von Faà di Bruno">Formel von Faà di Bruno</a></dt>

<dd>Diese Formel ermöglicht die geschlossene Darstellung der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">-ten Ableitung der Komposition zweier <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">-fach differenzierbarer Funktionen. Sie verallgemeinert die Kettenregel auf höhere Ableitungen.</dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Ableitungen_wichtiger_Funktionen">Ableitungen wichtiger Funktionen</h3></div>

siehe <a href="Tabelle_von_Ableitungs-_und_Stammfunktionen" title="Tabelle von Ableitungs- und Stammfunktionen">Tabelle von Ableitungs- und Stammfunktionen</a>

<div class="mw-heading mw-heading3"><h3 id="Geometrische_Anwendungen:_Eigenschaften_von_Kurven_(Kurvendiskussion)">Geometrische Anwendungen: Eigenschaften von Kurven (<a href="Kurvendiskussion" title="Kurvendiskussion">Kurvendiskussion</a>)</h3></div>

Betrachtet wird <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon x\mapsto f(x)}">

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<annotation encoding="application/x-tex">{\displaystyle f\colon x\mapsto f(x)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/118a044796fbaca30af1e8c4ea60f7a2ca68e95a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.674ex; height:2.843ex;" alt="{\displaystyle f\colon x\mapsto f(x)}" loading="lazy">

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<th>Untersuchungsaspekt

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<th>Kriterium

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<td style="background:lightblue"><a href="Nullstelle" title="Nullstelle">Nullstelle</a>

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<td style="background:lightyellow"><a href="Extremwert" title="Extremwert">Extremwert</a>

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<annotation encoding="application/x-tex">{\displaystyle f'(x_{E})=0\quad {\text{und}}\quad f''(x_{E})\neq 0}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/177154fe7fb368f28835594cd70191daeb7f74f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.761ex; height:3.009ex;" alt="{\displaystyle f'(x_{E})=0\quad {\text{und}}\quad f''(x_{E})\neq 0}" loading="lazy">

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<td style="background:lightyellow"><a href="Lokales_Minimum" class="mw-redirect" title="Lokales Minimum">Minimum</a>

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<annotation encoding="application/x-tex">{\displaystyle f'(x_{E})=0\quad {\text{und}}\quad f''(x_{E})>0}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a80c7a5a56a569eb72db41adfaf998b5ddfb7dba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.761ex; height:3.009ex;" alt="{\displaystyle f'(x_{E})=0\quad {\text{und}}\quad f''(x_{E})>0}" loading="lazy">

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<td style="background:lightyellow"><a href="Lokales_Maximum" class="mw-redirect" title="Lokales Maximum">Maximum</a>

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<annotation encoding="application/x-tex">{\displaystyle f'(x_{E})=0\quad {\text{und}}\quad f''(x_{E})<0}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/823ff30a17ca50046a5b0e2a808eeac13dfd98c4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.761ex; height:3.009ex;" alt="{\displaystyle f'(x_{E})=0\quad {\text{und}}\quad f''(x_{E})<0}" loading="lazy">

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<td style="background:lightblue"><a href="Wendepunkt" title="Wendepunkt">Wendepunkt</a>

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<annotation encoding="application/x-tex">{\displaystyle f''(x_{W})=0\quad {\text{und}}\quad f'''(x_{W})\neq 0}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9e2224ef71b5ff946cc0c4d1d574b18b822274e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.599ex; height:3.009ex;" alt="{\displaystyle f''(x_{W})=0\quad {\text{und}}\quad f'''(x_{W})\neq 0}" loading="lazy">

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<td style="background:lightblue"><a href="Sattelpunkt" title="Sattelpunkt">Sattelpunkt</a>

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<annotation encoding="application/x-tex">{\displaystyle f'(x_{W})=0\quad {\text{und}}\quad f''(x_{W})=0\quad {\text{und}}\quad f'''(x_{W})\neq 0}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4d12f159dd60921abe9356d08df6709392f5068c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:52.481ex; height:3.009ex;" alt="{\displaystyle f'(x_{W})=0\quad {\text{und}}\quad f''(x_{W})=0\quad {\text{und}}\quad f'''(x_{W})\neq 0}" loading="lazy">

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<td style="background:lightyellow">Verhalten im Unendlichen

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<annotation encoding="application/x-tex">{\displaystyle \lim _{x\to \infty }f(x)\quad {\text{und}}\quad \lim _{x\to -\infty }f(x)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d68d0ceb82f8f608b6e8274fcb418fe2755ee315.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:28.251ex; height:4.009ex;" alt="{\displaystyle \lim _{x\to \infty }f(x)\quad {\text{und}}\quad \lim _{x\to -\infty }f(x)}" loading="lazy">

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<td style="background:lightblue">Symmetrie

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<td style="background:lightblue">Achsensymmetrie zur Koordinatenachse („gerade“)

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<annotation encoding="application/x-tex">{\displaystyle f(x)=f(-x)\,}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/42e4dd12a0ba400ed63dc508d4a5a9569b85b14e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.129ex; height:2.843ex;" alt="{\displaystyle f(x)=f(-x)\,}" loading="lazy">

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<td style="background:lightblue">Punktsymmetrie zum Koordinatenursprung („ungerade“)

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<annotation encoding="application/x-tex">{\displaystyle -f(x)=f(-x)\,}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9863c870b4baefdfb29631f04b168eacad6a3203.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.937ex; height:2.843ex;" alt="{\displaystyle -f(x)=f(-x)\,}" loading="lazy">

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<td style="background:lightyellow"><a href="Reelle_monotone_Funktion" class="mw-redirect" title="Reelle monotone Funktion">Monotonie</a>

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<td style="background:lightyellow">monoton steigend bzw. streng monoton steigend

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<annotation encoding="application/x-tex">{\displaystyle f'(x)\geq 0\quad {\text{bzw.}}\quad f'(x)>0\,}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cc8a35f116d6490d3afc2c0745a02056726986dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.492ex; height:3.009ex;" alt="{\displaystyle f'(x)\geq 0\quad {\text{bzw.}}\quad f'(x)>0\,}" loading="lazy">

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<td style="background:lightyellow">monoton fallend bzw. streng monoton fallend

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<annotation encoding="application/x-tex">{\displaystyle f'(x)\leq 0\quad {\text{und}}\quad f'(x)<0}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c435567f391b3af86efd2ff28fdfb484eaab8208.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.333ex; height:3.009ex;" alt="{\displaystyle f'(x)\leq 0\quad {\text{und}}\quad f'(x)<0}" loading="lazy">

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<td style="background:lightblue">Krümmung

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<td style="background:lightblue">Linkskrümmung/Konvexbogen (nach oben offen)

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<annotation encoding="application/x-tex">{\displaystyle f''(x)\geq 0\,}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d9c7238968305c40de32c21a31a827dce4a416ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.245ex; height:3.009ex;" alt="{\displaystyle f''(x)\geq 0\,}" loading="lazy">

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<td style="background:lightblue">Rechtskrümmung/Konkavbogen (nach unten offen)

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<annotation encoding="application/x-tex">{\displaystyle f''(x)\leq 0\,}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/edd64f68b8b89b1b57a4d63c059a70d32982c785.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.245ex; height:3.009ex;" alt="{\displaystyle f''(x)\leq 0\,}" loading="lazy">

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<td style="background:lightyellow"><a href="Periodizit%C3%A4t" title="Periodizität">Periodizität</a>

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<annotation encoding="application/x-tex">{\displaystyle f(x+p)=f(x)\,}</annotation>

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<div class="mw-heading mw-heading4"><h4 id="Gebrochenrationale_Funktionen"><a href="Gebrochenrationale_Funktion" class="mw-redirect" title="Gebrochenrationale Funktion">Gebrochenrationale Funktionen</a></h4></div>

Funktionsterm:

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<annotation encoding="application/x-tex">{\displaystyle f(x)={\frac {a_{z}x^{z}+a_{z-1}x^{z-1}+\cdots +a_{1}x+a_{0}}{b_{n}x^{n}+b_{n-1}x^{n-1}+\cdots +b_{1}x+b_{0}}}={\frac {P_{z}(x)}{Q_{n}(x)}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8d77cfdf6554c77d943fc4ac28bdbb7bcad7763d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:51.792ex; height:6.509ex;" alt="{\displaystyle f(x)={\frac {a_{z}x^{z}+a_{z-1}x^{z-1}+\cdots +a_{1}x+a_{0}}{b_{n}x^{n}+b_{n-1}x^{n-1}+\cdots +b_{1}x+b_{0}}}={\frac {P_{z}(x)}{Q_{n}(x)}}}" loading="lazy"></dd></dl>

<ul><li>Einteilung

<ul><li>Ist das Nennerpolynom <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{n}}">

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<annotation encoding="application/x-tex">{\displaystyle Q_{n}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/503d0af3998f76cd4eaf8b3cc5e8834e254cb71b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.057ex; height:2.509ex;" alt="{\displaystyle Q_{n}}" loading="lazy"> vom Grad 0 (also n = 0 und b0 ≠ 0) und ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{z}}">

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<annotation encoding="application/x-tex">{\displaystyle P_{z}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/313d4fff47414070eb90eb707e6f32355dc71625.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.494ex; height:2.509ex;" alt="{\displaystyle P_{z}}" loading="lazy"> nicht das <a href="Nullpolynom" class="mw-redirect" title="Nullpolynom">Nullpolynom</a>, so spricht man von einer <a href="Ganzrationale_Funktion" title="Ganzrationale Funktion">ganzrationalen</a> oder einer <a href="Polynom" title="Polynom">Polynomfunktion</a>.</li>

<li>Ist n > 0 , so handelt es sich um eine gebrochenrationale Funktion.</li>

<li>Ist n > 0 und z ≥ n, so handelt es sich um eine unecht gebrochenrationale Funktion. Sie kann mittels <a href="Polynomdivision" title="Polynomdivision">Polynomdivision</a> in eine ganzrationale Funktion und eine echt gebrochenrationale Funktion aufgeteilt werden.</li></ul></li>

<li>Definitionsbereich

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {D} =\mathbb {R} \setminus \lbrace x_{0}\mid Q_{n}(x_{0})=0\rbrace }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8ca1a9915f2329a887ea465d05b4fbe68b0b6b2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.806ex; height:2.843ex;" alt="{\displaystyle \mathbb {D} =\mathbb {R} \setminus \lbrace x_{0}\mid Q_{n}(x_{0})=0\rbrace }" loading="lazy"></li></ul></li>

<li>Asymptotisches Verhalten: Für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\to \infty }">

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<annotation encoding="application/x-tex">{\displaystyle x\to \infty }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/eda2caf97ec29f30d5f0c0cd7135393361efc020.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.268ex; height:1.843ex;" alt="{\displaystyle x\to \infty }" loading="lazy"> strebt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}">

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<annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation>

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<ul><li>[falls <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z>n}">

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<annotation encoding="application/x-tex">{\displaystyle z>n}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a85753258723bc494c1ae3327950fe4b8e577439.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.581ex; height:1.843ex;" alt="{\displaystyle z>n}" loading="lazy">] gegen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {sgn}(a_{z})\cdot \operatorname {sgn}(b_{n})\cdot \infty }">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {sgn}(a_{z})\cdot \operatorname {sgn}(b_{n})\cdot \infty }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6c13a24048d7f2daa4dd9b3dace8429d704f3d81.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.491ex; height:2.843ex;" alt="{\displaystyle \operatorname {sgn} (a_{z})\cdot \operatorname {sgn} (b_{n})\cdot \infty }" loading="lazy">, wobei sgn die <a href="Vorzeichenfunktion" title="Vorzeichenfunktion">Vorzeichenfunktion</a> bezeichnet.</li>

<li>[falls <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=n}">

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<annotation encoding="application/x-tex">{\displaystyle z=n}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e57eb18158a74260ef03b71ed63ad5d67d645eea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.581ex; height:1.676ex;" alt="{\displaystyle z=n}" loading="lazy">] gegen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {a_{z}}{b_{n}}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\tfrac {a_{z}}{b_{n}}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3d9db6ea1d62490f8a3f919544cdf38878575010.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:2.506ex; height:3.843ex;" alt="{\displaystyle {\tfrac {a_{z}}{b_{n}}}}" loading="lazy"></li>

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<annotation encoding="application/x-tex">{\displaystyle z<n}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/76eb7197e1e13599a9ffbddafed86f38ace3dcbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.581ex; height:1.843ex;" alt="{\displaystyle z<n}" loading="lazy">] gegen 0 (die x-Achse)</li></ul></li>

<li>Symmetrie

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<annotation encoding="application/x-tex">{\displaystyle P_{z}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/313d4fff47414070eb90eb707e6f32355dc71625.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.494ex; height:2.509ex;" alt="{\displaystyle P_{z}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{n}}">

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<annotation encoding="application/x-tex">{\displaystyle Q_{n}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/503d0af3998f76cd4eaf8b3cc5e8834e254cb71b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.057ex; height:2.509ex;" alt="{\displaystyle Q_{n}}" loading="lazy"> beide gerade oder beide ungerade, so ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P_{z}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/313d4fff47414070eb90eb707e6f32355dc71625.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.494ex; height:2.509ex;" alt="{\displaystyle P_{z}}" loading="lazy"> gerade und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{n}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Q_{n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/503d0af3998f76cd4eaf8b3cc5e8834e254cb71b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.057ex; height:2.509ex;" alt="{\displaystyle Q_{n}}" loading="lazy"> ungerade, so ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> ungerade (punktsymmetrisch zum Ursprung); Gleiches gilt, wenn <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{z}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P_{z}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/313d4fff47414070eb90eb707e6f32355dc71625.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.494ex; height:2.509ex;" alt="{\displaystyle P_{z}}" loading="lazy"> ungerade und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{n}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Q_{n}}</annotation>

</semantics>

<li>Polstellen: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{p}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{p}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0bec554743fa797409a83ad8d00b4d35e110a50a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.389ex; height:2.343ex;" alt="{\displaystyle x_{p}}" loading="lazy"> heißt <a href="Polstelle" title="Polstelle">Polstelle</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Q_{n}(x_{p})=0\quad {\text{und}}\quad P_{z}(x_{p})\neq 0.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4d7e84da43f6486cbda46940f4cd920340d25bbb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:31.638ex; height:3.009ex;" alt="{\displaystyle Q_{n}(x_{p})=0\quad {\text{und}}\quad P_{z}(x_{p})\neq 0.}" loading="lazy"></li></ul></li>

<li>Asymptoten: Mittels Polynomdivision von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> durch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy"> erhält man <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=g\cdot q+r}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p=g\cdot q+r}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f73e461becbdad76276809e03fb098fb9272d9a2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:12.111ex; height:2.343ex;" alt="{\displaystyle p=g\cdot q+r}" loading="lazy"> mit Polynomen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" loading="lazy">, wobei der Grad von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" loading="lazy"> kleiner als der von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy"> ist. Das asymptotische Verhalten von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f={\tfrac {p}{q}}=g+{\tfrac {r}{q}}}">

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f={\tfrac {p}{q}}=g+{\tfrac {r}{q}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1388d5d56ed339d14feb91ff3bfa6ffc6648174b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:14.687ex; height:3.676ex;" alt="{\displaystyle f={\tfrac {p}{q}}=g+{\tfrac {r}{q}}}" loading="lazy"> ist damit durch die ganzrationale Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

</semantics>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [z<n]\,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/17195fdf6e90a2bb05929456876c67052e0f9ff1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.262ex; height:2.843ex;" alt="{\displaystyle [z<n]\,}" loading="lazy"> x-Achse ist Asymptote: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x)=0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g(x)=0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2c9bb64a9a8234edd8fb1c622dddfab478719776.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.516ex; height:2.843ex;" alt="{\displaystyle g(x)=0}" loading="lazy"></li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [z=n]\,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fad8ba51530b70457ae8b25e30623cc433d40943.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.262ex; height:2.843ex;" alt="{\displaystyle [z=n]\,}" loading="lazy"> waagerechte Asymptote: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x)={\frac {a_{z}}{b_{n}}}}">

<mfrac>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g(x)={\frac {a_{z}}{b_{n}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cbd43e352af55b422bc011c5b50c3acf66404a8d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:10.421ex; height:5.176ex;" alt="{\displaystyle g(x)={\frac {a_{z}}{b_{n}}}}" loading="lazy"></li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [z=n+1]\,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fd7a9034512af49dba989a4c68ba240c9447c72a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.265ex; height:2.843ex;" alt="{\displaystyle [z=n+1]\,}" loading="lazy"> schräge Asymptote: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x)=mx+c\,;m\neq 0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g(x)=mx+c\,;m\neq 0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1256c4c6caf3c013d0be227b98aa722a05bdfd6e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.293ex; height:2.843ex;" alt="{\displaystyle g(x)=mx+c\,;m\neq 0}" loading="lazy"></li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [z>n+1]\,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3768bc2bc96258e13be6815eee94eb2987bd1076.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.265ex; height:2.843ex;" alt="{\displaystyle [z>n+1]\,}" loading="lazy"> ganzrationale <a href="N%C3%A4herungsfunktion" class="mw-redirect" title="Näherungsfunktion">Näherungsfunktion</a></li></ul></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Integralrechnung"><a href="Integralrechnung" title="Integralrechnung">Integralrechnung</a></h2></div>

<div class="mw-heading mw-heading3"><h3 id="Flächenberechnung">Flächenberechnung</h3></div>

Der <a href="Fl%C3%A4cheninhalt" title="Flächeninhalt">Flächeninhalt</a> zwischen der x-Achse und dem Graphen der Funktion f(x) im Intervall von a bis b ist

</mrow>

</mrow>

</msubsup>

</mrow>

<mtext>falls </mtext>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int _{a}^{b}f(x)\mathrm {d} x,\qquad {\text{falls }}f(x)\geq 0\forall x\in [a,b]}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b506c88774f4ab310172759590a01006def54570.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:39.869ex; height:6.343ex;" alt="{\displaystyle \int _{a}^{b}f(x)\mathrm {d} x,\qquad {\text{falls }}f(x)\geq 0\forall x\in [a,b]}" loading="lazy"></li>

</mrow>

</mrow>

</msubsup>

</mrow>

<mtext>falls </mtext>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -\int _{a}^{b}f(x)\mathrm {d} x,\qquad {\text{falls }}f(x)\leq 0\forall x\in [a,b]}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b3e4e09ec4cf8deb90f91d91a7eff037087637e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:42.064ex; height:6.343ex;" alt="{\displaystyle -\int _{a}^{b}f(x)\mathrm {d} x,\qquad {\text{falls }}f(x)\leq 0\forall x\in [a,b]}" loading="lazy"></li>

<li>Andernfalls ist das Intervall durch Bestimmung der Nullstellen in solche Teilintervalle zu zerlegen.</li></ul>

<div class="mw-heading mw-heading3"><h3 id="Eigenschaften_des_bestimmten_Integrals">Eigenschaften des bestimmten Integrals</h3></div>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int _{a}^{b}f(x)\mathrm {d} x=-\int _{b}^{a}f(x)\mathrm {d} x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f067fc07d92fa0c13467bb1d7b3ef00b0af4c204.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:27.115ex; height:6.343ex;" alt="{\displaystyle \int _{a}^{b}f(x)\mathrm {d} x=-\int _{b}^{a}f(x)\mathrm {d} x}" loading="lazy"></dd>

</mrow>

</mrow>

</msubsup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int _{a}^{a}f(x)\mathrm {d} x=0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5f38387b6488ba85ae4597e7591c5bd9ca27b557.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.254ex; height:5.843ex;" alt="{\displaystyle \int _{a}^{a}f(x)\mathrm {d} x=0}" loading="lazy"></dd>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int _{a}^{c}f(x)\mathrm {d} x=\int _{a}^{b}f(x)\mathrm {d} x+\int _{b}^{c}f(x)\mathrm {d} x,\qquad a<b<c}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7f4401b46ba6313f2f031363db4da451a0f1082f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:53.548ex; height:6.343ex;" alt="{\displaystyle \int _{a}^{c}f(x)\mathrm {d} x=\int _{a}^{b}f(x)\mathrm {d} x+\int _{b}^{c}f(x)\mathrm {d} x,\qquad a<b<c}" loading="lazy"></dd>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int _{a}^{b}k\cdot f(x)\mathrm {d} x=k\cdot \int _{a}^{b}f(x)\mathrm {d} x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e9209cbcbe03a3574923fe09f7e26cc5cc021ede.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:30.536ex; height:6.343ex;" alt="{\displaystyle \int _{a}^{b}k\cdot f(x)\mathrm {d} x=k\cdot \int _{a}^{b}f(x)\mathrm {d} x}" loading="lazy"></dd>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int _{a}^{b}(f(x)+g(x))\,\mathrm {d} x=\int _{a}^{b}f(x)\mathrm {d} x+\int _{a}^{b}g(x)\mathrm {d} x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3dab3201bd12662e528c81bf2fe3a1cd19078d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:47.167ex; height:6.343ex;" alt="{\displaystyle \int _{a}^{b}(f(x)+g(x))\,\mathrm {d} x=\int _{a}^{b}f(x)\mathrm {d} x+\int _{a}^{b}g(x)\mathrm {d} x}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Integralfunktion_und_Hauptsatz_der_Differential-_und_Integralrechnung">Integralfunktion und <a href="Fundamentalsatz_der_Analysis" title="Fundamentalsatz der Analysis">Hauptsatz der Differential- und Integralrechnung</a></h3></div>

<dl><dt>Integralfunktion</dt>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{a}(x)=\int \limits _{a}^{x}f(t)\mathrm {d} t}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b73c07d3551908f15777952e2ca23b8628acfea3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:17.474ex; height:8.676ex;" alt="{\displaystyle F_{a}(x)=\int \limits _{a}^{x}f(t)\mathrm {d} t}" loading="lazy"></dd>

<dt>Hauptsatz der Infinitesimalrechnung</dt>

<msub>

</mrow>

</msub>

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{a}(x)'=f(x)\,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a7f54b7bad6f9009caf0ea99dec612341b68c4ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.323ex; height:3.009ex;" alt="{\displaystyle F_{a}(x)'=f(x)\,}" loading="lazy"></dd>

<dt><a href="Stammfunktion" title="Stammfunktion">Stammfunktion</a></dt>

<dd>Jede Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"> heißt Stammfunktion von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F'(x)=f(x)\,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/92a91d207cdb58e0f70d76379092ce34db6f5993.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.542ex; height:3.009ex;" alt="{\displaystyle F'(x)=f(x)\,}" loading="lazy"></dd></dl></dd>

<dd>Dies bezeichnet der Ausdruck <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int f(x)\mathrm {d} x}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int f(x)\mathrm {d} x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/07dcdf22244ef6077217b1916828efe4882cb01a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:9.621ex; height:5.676ex;" alt="{\displaystyle \int f(x)\mathrm {d} x}" loading="lazy"></dd>

<dt>Integration</dt>

<dd>Ist F irgendeine Stammfunktion von f, so gilt

</mrow>

</mrow>

</msubsup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int _{a}^{b}f(x)\mathrm {d} x=F(b)-F(a)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dab18d926a8d405db54b8a1a1ffae71795a85d09.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:26.095ex; height:6.343ex;" alt="{\displaystyle \int _{a}^{b}f(x)\mathrm {d} x=F(b)-F(a)}" loading="lazy"></dd></dl></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Spezielle_Stammfunktionen">Spezielle Stammfunktionen</h3></div>

Die Stammfunktionen von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=x^{n}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)=x^{n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8d09a7e3b534fe963d72c17a19e5bdd8b09eebc1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.064ex; height:2.843ex;" alt="{\displaystyle f(x)=x^{n}}" loading="lazy"> sind

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F(x)={\frac {x^{n+1}}{n+1}}+c,\qquad n\not =-1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2a0a9fc8a5aefa7b3f3847bf007d442fe93feca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:31.202ex; height:5.843ex;" alt="{\displaystyle F(x)={\frac {x^{n+1}}{n+1}}+c,\qquad n\not =-1}" loading="lazy"></dd></dl>

Alles Weitere siehe <a href="Tabelle_von_Ableitungs-_und_Stammfunktionen" title="Tabelle von Ableitungs- und Stammfunktionen">Tabelle von Ableitungs- und Stammfunktionen</a>

<div class="mw-heading mw-heading3"><h3 id="Integrationsmethoden">Integrationsmethoden</h3></div>

<a href="Partielle_Integration" title="Partielle Integration">Produkt-, Teil- oder partielle Integration</a>

<ul><li>unbestimmt

<msup>

</msup>

</mrow>

<msup>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int f(x)g'(x)\mathrm {d} x=f(x)\cdot g(x)-\int f'(x)\cdot g(x)\mathrm {d} x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/27c60f5ae080415b4f49fe5bc675279d89591774.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:47.134ex; height:5.676ex;" alt="{\displaystyle \int f(x)g'(x)\mathrm {d} x=f(x)\cdot g(x)-\int f'(x)\cdot g(x)\mathrm {d} x}" loading="lazy"></dd>

</mrow>

<msup>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int f(x)\cdot g(x)\mathrm {d} x=f(x)\cdot G(x)-\int f'(x)\cdot G(x)\mathrm {d} x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bc175f3885d5a41ccd48b44e971632997e8623bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:49.548ex; height:5.676ex;" alt="{\displaystyle \int f(x)\cdot g(x)\mathrm {d} x=f(x)\cdot G(x)-\int f'(x)\cdot G(x)\mathrm {d} x}" loading="lazy"></dd></dl></li>

<li>bestimmt

</mrow>

</mrow>

</msubsup>

<msup>

</msup>

</mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

<msup>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int _{a}^{b}f(x)\cdot g'(x)\mathrm {d} x=[f(x)\cdot g(x)]_{a}^{b}-\int _{a}^{b}f'(x)\cdot g(x)\mathrm {d} x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d740fbf93fba8f74edf2079e84b13caa8f896e53.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:53.625ex; height:6.343ex;" alt="{\displaystyle \int _{a}^{b}f(x)\cdot g'(x)\mathrm {d} x=[f(x)\cdot g(x)]_{a}^{b}-\int _{a}^{b}f'(x)\cdot g(x)\mathrm {d} x}" loading="lazy"></dd></dl></li></ul>

<a href="Integration_durch_Substitution" title="Integration durch Substitution">Integration durch Substitution</a>

<ul><li>unbestimmt

</mrow>

<msup>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int f(x)\mathrm {d} x=\int f(\varphi (t))\varphi '(t)\mathrm {d} t}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9c0a097bea5ea8078be32a7c51313185679354b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:29.543ex; height:5.676ex;" alt="{\displaystyle \int f(x)\mathrm {d} x=\int f(\varphi (t))\varphi '(t)\mathrm {d} t}" loading="lazy"></dd></dl></li>

<li>bestimmt

</mrow>

</mrow>

</msubsup>

<msup>

</msup>

</mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int _{a}^{b}f(\varphi (t))\cdot \varphi '(t)\mathrm {d} t=\int _{\varphi (a)}^{\varphi (b)}f(x)\mathrm {d} x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9206510146d2553f287d00c16ec8800bdd81c183.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:35.992ex; height:6.676ex;" alt="{\displaystyle \int _{a}^{b}f(\varphi (t))\cdot \varphi '(t)\mathrm {d} t=\int _{\varphi (a)}^{\varphi (b)}f(x)\mathrm {d} x}" loading="lazy"></dd></dl></li>

<li>Spezialfall: lineare Substitution

</mrow>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int f(mx+n)\mathrm {d} x={\frac {1}{m}}F(mx+n)+C,\qquad m\neq 0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/785505be52cf3f9e5ccfd50bfc37b44ab99313ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:49.613ex; height:5.676ex;" alt="{\displaystyle \int f(mx+n)\mathrm {d} x={\frac {1}{m}}F(mx+n)+C,\qquad m\neq 0}" loading="lazy"></dd>

</mrow>

</mrow>

</msubsup>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int _{a}^{b}f(mx+n)\mathrm {d} x={\frac {1}{m}}\lbrack F(mx+n)\rbrack _{a}^{b},\qquad m\neq 0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4dbccfcd946c7a5a5d3ffc0fede28a0b6534c0ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:48.61ex; height:6.343ex;" alt="{\displaystyle \int _{a}^{b}f(mx+n)\mathrm {d} x={\frac {1}{m}}\lbrack F(mx+n)\rbrack _{a}^{b},\qquad m\neq 0}" loading="lazy"></dd></dl></li>

<li>Spezialfall: logarithmische Integration

<mfrac>

<mrow>

<msup>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int {\frac {f'(x)}{f(x)}}\mathrm {d} x=\ln |f(x)|+C,\qquad f(x)\neq 0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b902b88ca9a04919e80cee6ff090f29e4c2558fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:41.284ex; height:6.509ex;" alt="{\displaystyle \int {\frac {f'(x)}{f(x)}}\mathrm {d} x=\ln |f(x)|+C,\qquad f(x)\neq 0}" loading="lazy"></dd></dl></li></ul>

<div class="mw-heading mw-heading3"><h3 id="Angewandtes">Angewandtes</h3></div>

<div class="mw-heading mw-heading4"><h4 id="Volumenbestimmung">Volumenbestimmung</h4></div>

<ul><li>Volumen des Körpers bei Rotation der Fläche zwischen dem Graphen von f und der x-Achse im Intervall [a,b]

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi \cdot \int _{a}^{b}f^{2}(x)\mathrm {d} x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/022009f046c817d086732f73721f69055d47fa62.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:14.936ex; height:6.343ex;" alt="{\displaystyle \pi \cdot \int _{a}^{b}f^{2}(x)\mathrm {d} x}" loading="lazy"></dd></dl></li>

<li>Volumen des Körpers bei Rotation der Fläche zwischen dem Graphen der umkehrbaren Funktion f und der y-Achse im Intervall [a,b]

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi \cdot \int _{f(a)}^{f(b)}(f^{-1}(y))^{2}\mathrm {d} y}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f4daeed02252d7073463581781d972d2354a11cc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:20.526ex; height:6.676ex;" alt="{\displaystyle \pi \cdot \int _{f(a)}^{f(b)}(f^{-1}(y))^{2}\mathrm {d} y}" loading="lazy"></dd></dl></li>

<li>Volumen des Körpers, der bei y-Rotation der Fläche, welche durch den Graphen der Funktion f im Intervall [a,b], der x-Achse und den beiden Geraden <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=a}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x=a}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aaae23950e96a955ab5b07015a168fd931d4d82b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.658ex; height:1.676ex;" alt="{\displaystyle x=a}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=b}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x=b}</annotation>

</semantics>

</mrow>

</mrow>

</msubsup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2\pi \cdot \int _{a}^{b}(x\cdot f(x))\mathrm {d} x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/69a38385328aba902aee0ac851360a4955c0871c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:19.433ex; height:6.343ex;" alt="{\displaystyle 2\pi \cdot \int _{a}^{b}(x\cdot f(x))\mathrm {d} x}" loading="lazy"></dd></dl></li></ul>

<div class="mw-heading mw-heading5"><h5 id="Guldinsche_Regeln"><a href="Guldinsche_Regeln" class="mw-redirect" title="Guldinsche Regeln">Guldinsche Regeln</a></h5></div>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> Oberflächeninhalt</dd>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"> Volumen</dd>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy"> Länge der erzeugenden Linie (Profillinie)</dd>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> Flächeninhalt der erzeugenden Fläche</dd>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"> Radius des Schwerpunktkreises</dd>

<dt>Erste Regel</dt>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M=L\cdot 2\pi R}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bed22cd849f559833673807bb4cfcdfd47560327.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.061ex; height:2.176ex;" alt="{\displaystyle M=L\cdot 2\pi R}" loading="lazy"></dd></dl>

Ausgedrückt in Abhängigkeit von der Funktion f(x) der erzeugenden Linie ergibt sich dies als:

<ul><li>bei Rotation um die x-Achse

</mrow>

</mrow>

</msubsup>

<msqrt>

<msup>

<mrow>

<mrow>

<msup>

</msup>

</mrow>

</mrow>

</mrow>

</msup>

</msqrt>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M=2\pi \int _{a}^{b}f(x){\sqrt {1+\left[f'(x)\right]^{2}}}\mathrm {d} x.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/812752f64957bfffad25c9c5dffb7a9f4ee05c11.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:33.716ex; height:6.343ex;" alt="{\displaystyle M=2\pi \int _{a}^{b}f(x){\sqrt {1+\left[f'(x)\right]^{2}}}\mathrm {d} x.}" loading="lazy"></dd></dl></li>

<li>bei Rotation um die y-Achse

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

<msqrt>

<msup>

<mrow>

<msup>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</msup>

</mrow>

</mrow>

</msup>

</msqrt>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M=2\pi \int _{\min(f(a),f(b))}^{\max(f(a),f(b))}f^{-1}(y){\sqrt {1+\left[\left(f^{-1}(y)\right)'\right]^{2}}}\mathrm {d} y.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/60dbd3e3c04d90026dc884f4e2fe34a9693b389b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:50.964ex; height:6.676ex;" alt="{\displaystyle M=2\pi \int _{\min(f(a),f(b))}^{\max(f(a),f(b))}f^{-1}(y){\sqrt {1+\left[\left(f^{-1}(y)\right)'\right]^{2}}}\mathrm {d} y.}" loading="lazy"></dd></dl></li></ul>

<dl><dt>Zweite Regel</dt>

<dd>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V=A\cdot 2\pi R.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/72f0af2fa209e8f9f517c251dc416a588cdac8e7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.213ex; height:2.176ex;" alt="{\displaystyle V=A\cdot 2\pi R.}" loading="lazy"></dd></dl></dd></dl>

Im Fall der Rotation um die x-Achse einer Fläche zwischen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/202945cce41ecebb6f643f31d119c514bec7a074.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}" loading="lazy">, der x-Achse und den Grenzen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=a}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x=a}</annotation>

</semantics>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x=b}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/229dd3f0f42d375fa257bdc1389f92f7b225c415.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.426ex; height:2.176ex;" alt="{\displaystyle x=b}" loading="lazy"> ergibt sich das Volumen ausgedrückt durch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/202945cce41ecebb6f643f31d119c514bec7a074.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R}</annotation>

</semantics>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V=A\cdot 2\pi {\tfrac {1}{A}}\int _{A}y\mathrm {d} A=\pi \cdot \int _{a}^{b}(f(x))^{2}\mathrm {d} x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d73e0157a18747100b55056b36ebd234efb433cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:40.008ex; height:6.343ex;" alt="{\displaystyle V=A\cdot 2\pi {\tfrac {1}{A}}\int _{A}y\mathrm {d} A=\pi \cdot \int _{a}^{b}(f(x))^{2}\mathrm {d} x}" loading="lazy"></dd></dl>

mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y={\tfrac {f(x)}{2}}}">

<mfrac>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y={\tfrac {f(x)}{2}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1da39cf92691f556c01724faceedc76ed8e7ad56.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:8.214ex; height:4.176ex;" alt="{\displaystyle y={\tfrac {f(x)}{2}}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} A=f(x)\mathrm {d} x.}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} A=f(x)\mathrm {d} x.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/83e9c6b4673bd57b16c6454efcec3a63019c40fe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.821ex; height:2.843ex;" alt="{\displaystyle \mathrm {d} A=f(x)\mathrm {d} x.}" loading="lazy">

<div class="mw-heading mw-heading4"><h4 id="Weiteres">Weiteres</h4></div>

<ul><li>Ist f auf [a,b] stetig, so heißt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {m}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\bar {m}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/69328c84c2cc565cd6aade2135b40be6f520f5f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:2.009ex;" alt="{\displaystyle {\bar {m}}}" loading="lazy"> der <a href="Mittelwert" title="Mittelwert">Mittelwert</a> der Funktionswerte von f auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [a,b]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [a,b]}</annotation>

</semantics>

<mover>

</mover>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\bar {m}}={\frac {1}{b-a}}\cdot \int _{a}^{b}f(x)\mathrm {d} x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4a785b44c1baf91a5a95a8fed02ea5418e34b3e7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:23.551ex; height:6.343ex;" alt="{\displaystyle {\bar {m}}={\frac {1}{b-a}}\cdot \int _{a}^{b}f(x)\mathrm {d} x}" loading="lazy"></dd></dl></li>

<li>Länge des Bogens der differenzierbaren Funktion f im Intervall [a,b]:

</mrow>

</mrow>

</msubsup>

<msqrt>

<msup>

</msup>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L=\int _{a}^{b}{\sqrt {1+[f'(x)]^{2}}}\mathrm {d} x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/567dddae40338fb66b3d9285c3f7eca54dc6c40c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:24.911ex; height:6.343ex;" alt="{\displaystyle L=\int _{a}^{b}{\sqrt {1+[f'(x)]^{2}}}\mathrm {d} x}" loading="lazy"></dd></dl></li></ul>

<div class="mw-heading mw-heading3"><h3 id="Näherungsweises_Berechnen_von_Integralen:_Numerische_Integration">Näherungsweises Berechnen von Integralen: <a href="Numerische_Integration" title="Numerische Integration">Numerische Integration</a></h3></div>

<ul><li>Zerlegungssummen

</mrow>

</mrow>

</msubsup>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int _{a}^{b}f(x)\mathrm {d} x\approx hf(x_{1})+hf(x_{2})+\cdots +hf(x_{n})\qquad {\text{mit }}h={\frac {b-a}{n}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/75ffa2d49f0941254df494f982269b9fed19c72b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:64.823ex; height:6.343ex;" alt="{\displaystyle \int _{a}^{b}f(x)\mathrm {d} x\approx hf(x_{1})+hf(x_{2})+\cdots +hf(x_{n})\qquad {\text{mit }}h={\frac {b-a}{n}}}" loading="lazy"></dd></dl></li>

<li><a href="Keplersche_Fassregel" class="mw-redirect" title="Keplersche Fassregel">Keplersche Fassregel</a>

</mrow>

</mrow>

</msubsup>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mrow>

<mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int _{a}^{b}f(x)\mathrm {d} x\approx {\frac {1}{6}}\cdot \left(f(a)+4\cdot f\left({\frac {a+b}{2}}\right)+f(b)\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/20cc3b79c3db0fd9c8d6985d6cea3356cea73c24.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:48.942ex; height:6.509ex;" alt="{\displaystyle \int _{a}^{b}f(x)\mathrm {d} x\approx {\frac {1}{6}}\cdot \left(f(a)+4\cdot f\left({\frac {a+b}{2}}\right)+f(b)\right)}" loading="lazy"></dd></dl></li>

<li><a href="Trapezregel" title="Trapezregel">Trapezregel</a>

<ul><li>Sehnentrapez</li></ul>

</mrow>

</mrow>

</msubsup>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int _{a}^{b}f(x)\mathrm {d} x\approx {\frac {f(b)+f(a)}{2}}\cdot (b-a)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0ed0410670adb851d55661ab2b591a4ccdcd13ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:34.563ex; height:6.343ex;" alt="{\displaystyle \int _{a}^{b}f(x)\mathrm {d} x\approx {\frac {f(b)+f(a)}{2}}\cdot (b-a)}" loading="lazy"></dd>

</mrow>

</mrow>

</msubsup>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int _{a}^{b}f(x)\mathrm {d} x\approx {\frac {b-a}{2n}}\left(f(x_{0})+2f(x_{1})+\cdots +2f(x_{n-1}+f(x_{n})\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/520af81f0efbefaf1f02f4c13b270c20935ef940.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:61.849ex; height:6.343ex;" alt="{\displaystyle \int _{a}^{b}f(x)\mathrm {d} x\approx {\frac {b-a}{2n}}\left(f(x_{0})+2f(x_{1})+\cdots +2f(x_{n-1}+f(x_{n})\right)}" loading="lazy"></dd></dl></dd></dl>

<ul><li>Tangententrapez</li></ul>

</mrow>

</mrow>

</msubsup>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int _{a}^{b}f(x)\mathrm {d} x\approx {\frac {b-a}{2}}\cdot {\frac {b-a}{2}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0741fa68e231e2632e5f787804578a3b95ad4a44.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:27.414ex; height:6.343ex;" alt="{\displaystyle \int _{a}^{b}f(x)\mathrm {d} x\approx {\frac {b-a}{2}}\cdot {\frac {b-a}{2}}}" loading="lazy"></dd></dl></dd></dl></li>

<li><a href="Simpsonregel" title="Simpsonregel">Simpsonregel</a>

</mrow>

</mrow>

</msubsup>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mrow>

<mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int _{a}^{b}f(x)\mathrm {d} x\approx {\frac {b-a}{6}}\cdot \left(f(a)+4f\left({\frac {a+b}{2}}\right)+f(b)\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6ec5c927729059e393b01aeb355130fa15adc057.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:51.168ex; height:6.509ex;" alt="{\displaystyle \int _{a}^{b}f(x)\mathrm {d} x\approx {\frac {b-a}{6}}\cdot \left(f(a)+4f\left({\frac {a+b}{2}}\right)+f(b)\right)}" loading="lazy"></dd>

</mrow>

</mrow>

</msubsup>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int _{a}^{b}f(x)\mathrm {d} x\approx {\frac {b-a}{6n}}\cdot \left(f(x_{0})+4f(x_{1})+2f(x_{2})+4f(x_{3})+2f(x_{4})+\cdots f(x_{n})\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f0fcf5b640602a2f22d50dffca6e4a1033e4ee15.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:78.277ex; height:6.343ex;" alt="{\displaystyle \int _{a}^{b}f(x)\mathrm {d} x\approx {\frac {b-a}{6n}}\cdot \left(f(x_{0})+4f(x_{1})+2f(x_{2})+4f(x_{3})+2f(x_{4})+\cdots f(x_{n})\right)}" loading="lazy"></dd></dl></dd></dl></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Quellen">Quellen</h2></div>

<ul><li><a href="Konrad_K%C3%B6nigsberger" title="Konrad Königsberger">Konrad Königsberger</a>: Analysis 1. Springer-Verlag, Berlin u. a., 2004, ISBN 3-540-41282-4.</li></ul></div><div><div class="zim-footer">

Dieser Artikel wurde von <a class="external text" title="Zuletzt bearbeitet am 2026-01-08" href="https://de.wikipedia.org/wiki/?title=Formelsammlung_Analysis&oldid=263137066">Wikipedia</a> herausgegeben. Der Text ist unter <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.de">Creative Commons Attribution-Share Alike 4.0</a> verfügbar, sofern nicht anders angegeben. Für die Mediendateien können zusätzliche Bedingungen gelten.

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