Grafisches Ableiten Beispiel Essay

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Rima ix de becquer analysis essay

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Rima analysis de ix essay becquer

Ableitungsfunktion f'(x)

Die Ableitungsfunktion zu einer gegebenen Funktion f ist – vereinfacht ausgedrückt – die Tangentensteigungsfunktion. Das bedeutet: An jeder Stelle x hat die Steigung der Tangente an den Graphen von f einen bestimmten Wert, welcher der Ableitung der Funktion entspricht. Nun kannst du quasi mit einem Lineal den Graphen von f "abfahren". Das Lineal soll dabei immer tangential an den Graphen anliegen. Wenn du dann die Steigung des Lineals Punkt für Punkt in ein Diagramm überträgst, erhältst du ein Bild der Ableitungsfunktion. Dabei musst du die Funktionsgleichung selbst nicht kennen, sondern nur den Graphen.

Wie leitet man graphisch ab?

Beim graphischen Ableiten gehst du aber nicht Punkt für Punkt vor. Vielmehr nutzt du einige wichtige Zusammenhänge zwischen der Funktion f und ihrer Ableitungsfunktion f ′:

  • Für Bereiche von f mit positiver Steigung sind auch die y-Werte von f ′ positiv. Sie liegen oberhalb der x-Achse.
  • Für alle Punkte von f mit negativer Steigung hat auch die Ableitung f ′ negative y-Werte. Diese Punkte von f ′ liegen unterhalb der x-Achse.
  • Für alle Extrempunkte von f, also Punkte mit der Steigung null und waagerechter Tangente, hat die Ableitung f ′ den y-Wert null. Diese Punkte liegen auf der x-Achse. Der Graph der Ableitungsfunktion hat hier eine Nullstelle.
  • Dort, wo der Graph von f einen Wendepunkt hat, besitzt der Graph von f ′ einen Extrempunkt. Für einen RL-Wechsel des Graphen von f ist dieser Extrempunkt ein Tiefpunkt. Für einen LR-Wechsel des Graphen von f ist es ein Hochpunkt.

Beispiel zum graphischen Ableiten

Gegeben ist die Funktion f(x) = 0,5x³ - 3,6x² + 6,96x – 2,24 und die Ableitungsfunktion f'(x) = 1,5x² - 7,2x² + 6,96. In der Abbildung siehst du: Dort, wo der Graph der Funktion f(x) ansteigt (jeweils in roter Farbe), sind die Funktionswerte der Ableitungsfunktion f'(x) positiv (jeweils blau). Diese Werte sind jedoch negativ (violet), wo der Graph von f(x) abfällt (hellgrün).

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