turning point / inflection point
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turning point
A point in time at which a decisive change occurs in a development or course of events, fundamentally changing its direction.
Die Erfindung des Internets war ein Wendepunkt in der Geschichte der Kommunikation.
The invention of the internet was a turning point in the history of communication.
inflection point
In mathematics, a point on the graph of a function at which the curvature behavior changes.
Um den Wendepunkt zu finden, müssen wir die zweite Ableitung der Funktion null setzen.
To find the inflection point, we have to set the second derivative of the function equal to zero.
'Wendepunkt' is made up of 'Wende' (turn, change) and 'Punkt' (point). So it literally describes a point at which a decisive turn takes place.
Seine Entscheidung, das Land zu verlassen, war ein Wendepunkt in seinem Leben.
His decision to leave the country was a turning point in his life.
To name the area in which the change takes place, the preposition 'in' is used with the dative. Typical patterns are 'ein Wendepunkt im Leben' (a turning point in life), 'in der Geschichte' (in history), or 'in einer Karriere' (in a career).
Die Schlacht von Stalingrad war ein entscheidender Wendepunkt im Zweiten Weltkrieg.
The Battle of Stalingrad was a decisive turning point in the Second World War.
To emphasize the importance of the turning point, adjectives such as 'entscheidend' (decisive), 'historisch' (historical), 'persönlich' (personal), or 'wichtig' (important) are often used.
Das war ein persönlicher Wendepunkt für mich.
That was a personal turning point for me.
You can immediately recognize the mathematical meaning from words like 'Funktion' (function), 'Graph' (graph), 'Kurve' (curve), or 'Ableitung' (derivative). Outside this context, the general meaning is almost always intended.
Der Lehrer erklärte uns, wie man den Wendepunkt einer Funktion berechnet.
The teacher explained to us how to calculate the inflection point of a function.
milestone
Both words refer to something decisive in a development. A Meilenstein is an important advance and can be called that even without a fundamental change of direction; a Wendepunkt, by contrast, explicitly involves a change in direction. Choose Wendepunkt when the course of events is fundamentally different afterward.
Die neue Maschine wurde nach dem schweren Unfall zum Wendepunkt in der Entwicklung.
After the serious accident, the new machine became a turning point in the development process.
Der erfolgreiche Test war ein wichtiger Meilenstein auf dem Weg zur neuen Maschine.
The successful test was an important milestone on the way to the new machine.
