Low Range
1. Low notes require a large amount of slow-moving air. If you use air that is too fast, you will overblow the note.
2. When you descend into the low register your embouchure changes shape; your jaw should drop and your chin might move forward or even to the side a little. Every trombonist is unique in what should happen with regard to this change to accommodate the low register. As you descend and your embouchure changes shape, the instrument must be freely responsive to the changing shape of your embouchure. In other words, allow your trombone to move as a reaction to the needs of your embouchure – not the other way around.
These are the two most common problems I see with students: sometimes a student will confuse blowing fast air with blowing lots of slow-moving air and overblow the note; sometimes a student will hold the instrument too rigidly, not allowing its angle to be freely responsive to the inevitable changing shape of the embouchure.
I am not opposed to changing the embouchure for various registers so long as the change can be made while playing. If you are compelled to remove the mouthpiece for every shift in order to get into the low range then you have some more work to do. Practice connected scales and arpeggios until you can move into the low range smoothly.
I recommend the following books for practicing in the low range:
Blazhevich – 70 Studies for BB flat Tuba, Book 1
Blume/Fink – 36 Studies for Trombone with F Attachment
Bordogni – 43 Bel Canto Studies for Tuba
Bordogni – Melodious Etudes, book 1 (down one octave)
Snedecor – Low Etudes for Tuba
Vining – Daily Routines for the Student Trombone Player (this book features a routine devoted to learning F-attachment valve skills; great for those who have just purchased their first F-attachment instrument)
Rangsongs explores both high and low ranges. Here is a sound file of me playing the low D-flat 2 Rangesong. I will include the etude so you can follow along as you listen:
Rangesongs is a terrific resource for improving low range:
Rangesongs develops your high and low ranges by using target notes in a musical context to encourage you to move your air properly. For the high range, Rangesongs offers sixty-five songs designed to improve your range and endurance systematically by using two strategies: the phrases within each song are specifically composed to build melodically to a designated target note, providing a compelling musical incentive to support the tone with the proper air speed; and the songs proceed chromatically to reach a high F for gradual improvement. For the low range, Rangesongs offers forty songs to extend the range downward in the same fashion to a pedal B-flat.
Trombonetools 
