Zentraler Differenzenquotient Beispiel Essay

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Der Differenzenquotient ist ein Begriff aus der Mathematik. Er beschreibt das Verhältnis der Veränderung einer Größe zu der Veränderung einer anderen, wobei die erste Größe von der zweiten abhängt. In der Analysis verwendet man Differenzenquotienten, um die Ableitung einer Funktion zu definieren. In der numerischen Mathematik werden sie zum Lösen von Differentialgleichungen und für die näherungsweise Bestimmung der Ableitung einer Funktion (Numerische Differentiation) benutzt.

Definition[Bearbeiten | Quelltext bearbeiten]

Ist eine reellwertige Funktion, die im Bereich definiert ist, und ist , so nennt man den Quotienten

Differenzenquotient von im Intervall.

Schreibt man und , dann ergibt sich die alternative Schreibweise

.

Setzt man , also , so erhält man die Schreibweise

.

Geometrisch entspricht der Differenzenquotient der Steigung der Sekante des Graphen von durch die Punkte und . Für bzw. wird aus der Sekante eine Tangente an der Stelle .

Differentialrechnung[Bearbeiten | Quelltext bearbeiten]

Differenzenquotienten bilden zusammen mit dem Grenzwertbegriff die theoretische Grundlage der Differentialrechnung. Den Grenzwert des Differenzenquotienten für bezeichnet man als Differentialquotienten oder Ableitung der Funktion an der Stelle (kurz: ), sofern dieser Grenzwert existiert. Das Berechnen dieses Grenzwerts nennt man Ableiten oder Differenzieren. Die Tabelle zeigt die Ableitungen einiger Funktionen. Dabei stimmt der Differenzenquotient jeweils nur für .

Numerische Mathematik[Bearbeiten | Quelltext bearbeiten]

Bei differenzierbaren Funktionen kann der Differenzenquotient als Näherung für die lokale Ableitung benutzt werden. In der Finite-Differenzen-Methode wird diese Eigenschaft zur Lösung von Differentialgleichungen benutzt. Ebenso wird dies für die numerische Differentiation von Funktionen verwendet.

Dabei ist der Differenzenquotient nicht auf die erste Ableitung beschränkt. Es existieren Differenzenquotienten für höhere sowie partielle Ableitungen.

Beispiel[Bearbeiten | Quelltext bearbeiten]

Es sei .

Der Graph von ist eine Normalparabel. Wollen wir die Ableitung z. B. in der Nähe der Stelle ungefähr berechnen, so wählen wir für einen kleinen Wert, z. B. 0,001. Das ergibt als Differenzenquotienten im Intervall den Wert . Dieser ist die Sekantensteigung des Funktionsgraphen im Intervall und eine Näherung der Steigung der Tangente an der Stelle .

Varianten[Bearbeiten | Quelltext bearbeiten]

In der Praxis werden verschiedene Varianten des Differenzenquotienten verwendet, die sich in der Definition von unterscheiden, etwa um die Genauigkeit bei der Bestimmung des lokalen Wachstums, z. B. der Sekantensteigung eines Graphen, zu verbessern oder um an den Randstellen einer Funktion deren Sekantensteigung „rückwärts“ in Richtung des Inneren ihres Definitionsbereichs zu ermitteln.

Vorwärtsdifferenzenquotient[Bearbeiten | Quelltext bearbeiten]

Der oben definierte Ausdruck

wird auch Vorwärtsdifferenzenquotient genannt, weil zur Bestimmung des ersten Funktionswertes, der zur Bildung von notwendig ist, von aus nach rechts, also „vorwärts“ gegangen wird.

Rückwärtsdifferenzenquotient[Bearbeiten | Quelltext bearbeiten]

Analog bezeichnet man den Ausdruck

als Rückwärtsdifferenzenquotienten, da zur Differenzbildung von aus nach links, also „rückwärts“ gegangen wird, um den zweiten Funktionswert zu erhalten.

Zentraler Differenzenquotient[Bearbeiten | Quelltext bearbeiten]

Gebräuchlich ist auch der zentrale Differenzenquotient, den man z. B. durch Mittelwertbildung des Vorwärtsdifferenzen- und Rückwärtsdifferenzenquotienten erhält. Er ist durch

gegeben. Bei ihm liegen die zur Differenzbildung verwendeten Stellen symmetrisch um den -Wert, für den die Ableitung angenähert werden soll.

Im Gegensatz zu den beiden vorherigen Differenzenquotienten, deren Fehlerterme beim Annähern der ersten Ableitung an der Stelle nur von der Klasse sind, falls die Funktion zweimal differenzierbar ist, liegt der Fehler des zentralen Differenzenquotienten in , falls die Funktion zusätzlich dreifach differenzierbar in ist. Zur -Notation siehe Landau-Symbole.

Höhere Differenzenquotienten[Bearbeiten | Quelltext bearbeiten]

Ebenso wie die erste Ableitung durch Differenzenquotienten angenähert werden kann, gilt dies auch für höhere Ableitungen, die über Differenzenquotienten höherer Ordnung approximierbar sind.

Die Herleitung der höheren Differenzenquotienten kann man durch eine rekursive Entwicklungsvorschrift darstellen:

Für die zweite Ableitung kann zum Beispiel der Zusammenhang

Veranschaulichung des Differenzenquotienten: Er entspricht der Steigung der blauen Geraden

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