What is the name and utility of the half-moon symbol on top of the last note?
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The symbol over the note (which can also be below a note, upside down) is called a fermata and nowadays it has become a standard indication of a pause*. Meaning that the duration of the note or a rest associated with the fermata will be longer than it's supposed to be. say for example, if it's above a quarter note then the duration of the quarter note could be as long as a half note (i.e. twice the duration) Basically the music takes a pause here in the sense that it does not move forward for a short while and lingers on a note. How long it should linger is subjective.
How long this pause should be, although a matter of personal discretion, should still not be exaggerated. If the composer truly wanted a long pause he could've simply notated lunga pausa
In the example in the photo the regular beat is abandoned altogether and the fermata takes up the duration of thrice its nominal value. (i.e. the notes are sustained up till the rest)
*note that in the baroque era the fermata sign was used to merely show an important structural point, e.g. the end of a section in the music.