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"with a positive mood," 1947, apparently from the musical noun upbeat (1869), referring to the beat of a bar at which the conductor's baton is in a raised position; the "optimistic" sense apparently for no other reason than that it sounds like a happy word (the musical upbeat is no more inherently "positive" than any other beat). Expression on the upbeat "improving, getting better" is recorded from 1934.
Very intensively; very diligently; very competently: and they're really dancing up a storm (1953+)