This is a fabulous tone exercise requiring great embouchure control, from my friend and mentor Elizabeth Koch. Start very softly, make a huge crescendo, slur up the octave seamlessly, glide across the semitone and diminuendo beautifully!
Here’s an article I wrote about how we breathe. Hopefully you’ll find it useful.
Taffanel once said the breath is the soul of the flute. Worthwhile keeping at the forefront of your mind.