
aufgeben
to give up
To stop trying or continuing something because one does not expect success.
Obwohl das Rennen sehr anstrengend war, hat sie nicht aufgegeben.
Although the race was very exhausting, she did not give up.
to give up / submit
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to give up
To stop trying or continuing something because one does not expect success.
Obwohl das Rennen sehr anstrengend war, hat sie nicht aufgegeben.
Although the race was very exhausting, she did not give up.

to send / submit
To hand something over for processing or dispatch, for example a letter, a package, or an order.
Ich muss nach der Arbeit noch schnell ein Paket bei der Post aufgeben.
After work, I still have to quickly send a package at the post office.

to assign
To give someone a task, especially homework, to complete.
Der Mathematiklehrer hat uns über das Wochenende viele Übungen aufgegeben.
The math teacher assigned us many exercises over the weekend.

to give up
To give up or stop pursuing something one owns or holds on to, such as a plan, a hope, or a position.
Nach langer Suche mussten sie die Hoffnung aufgeben, die Katze wiederzufinden.
After a long search, they had to give up hope of finding the cat again.
'Aufgeben' (to give up / submit) is a separable verb. The prefix 'auf' comes at the end in a main clause. In subordinate clauses and in the infinitive with 'zu', the verb stays together.
Ich gebe diesen Plan nicht auf.
I am not giving up this plan.
The verb is made up of the prefix 'auf' and the verb 'geben' (to give). The basic idea of giving something away from oneself or letting go of something can be recognized in many meanings.
Er musste seinen Traum vom Profisport aufgeben.
He had to give up his dream of professional sports.
In the meaning 'to surrender' or 'to stop continuing', 'aufgeben' (to give up) can also be used without an accusative object. This does not apply to the other meanings.
Kurz vor dem Ziel war er so erschöpft, dass er aufgab.
Shortly before the finish line, he was so exhausted that he gave up.
In the context of 'handing over', 'aufgeben' (to send / submit) is often combined with things that one hands in or checks in at an institution. Typical phrases are: ein Paket aufgeben (to send a package), einen Brief aufgeben (to mail a letter), Gepäck aufgeben (to check in luggage), or eine Bestellung aufgeben (to place an order).
Am Flughafen müssen Sie Ihr Gepäck am Schalter aufgeben.
At the airport, you have to check in your luggage at the counter.
When a teacher assigns tasks, the structure 'jemandem (Dativ) etwas (Akkusativ) aufgeben' (to assign someone something) is used. This meaning is common almost only in school or university contexts.
Der Professor hat uns eine schwere Aufgabe aufgegeben.
The professor assigned us a difficult task.
to result / surrender
Both verbs can mean that someone ends their resistance. Someone who aufgibt stops an attempt or fight, without surrender necessarily being implied; someone who sich ergibt explicitly capitulates to the opposing side. Use „sich ergeben“ when surrender is the decisive point.

Nach mehreren erfolglosen Versuchen gab der Läufer das Rennen auf.
After several unsuccessful attempts, the runner gave up on the race.
Als die Polizei das Gebäude umstellte, ergab sich der bewaffnete Mann.
When the police surrounded the building, the armed man surrendered.