This slur exercise is not just useful by itself but can demonstrate an important point. The intervals are derived from the opening of the Hindemith trombone sonata. Whenever we confront awkward articulated intervals, it can be very helpful to slur them as a practice technique. If the embouchure can cleanly navigate these leaps while slurring, chances are the tongued version will sound much better!
Here’s a pdf of these slurs.
Achieving a solid sound and attack in the lower register takes work. You can’t force it. Personally, I (and many people) pivot the mouthpiece a bit in search of the best placement for lower notes. I believe all of this pivoting serves the goal of allowing the lower lip to set in the right place for best vibration.
Also, lower notes call for a more air stream to allow good resonance. Some people like to think of ‘warmer’ air such as that you would use to fog up a mirror.
Here’s a pdf of Octave Leaps 
Sometimes long notes wobble a bit. The tone gets a little quiver.
There is a simple mental trick I use that helps with this. I imagine my sustained note as moving forward from my bell through the space in front of me.
It’s almost as if my sound is a column of light moving forward from the bell. In fact, I sometimes like to visualize an entire phrase as a single, unbroken column of light that changes color for the different notes of the phrase.
In lessons, I sometimes use a hand motion where I begin with my hand close to the bell and then, as the student sustains the note, I move my hand slowly away from the bell, giving them a visual image of forward motion to the sound. This often helps.
Playing any note without a sense of forward motion is often a source of trouble. Not only is the note less musically satisfying, the tone is often less resonant as well.
In that way, you can almost imagine a little, nearly imperceptible crescendo as you sustain the note.
When buzzing, it is nice to get visual feedback of the air-in-motion. A pinwheel is good for this. So is a piece of tissue paper suspended in front of the mouthpiece.
Think of this analogy: If you were to drink from a stream, it is better to drink from flowing water…
..than it is to drink from stagnant water!
Just as you wouldn’t drink from stagnant water, don’t subject your audience to stagnant notes.
So often, I see students decide that a passage is difficult. Once they have placed that label on the music, it can become a self-fulfilling prophecy!
We see that batch of notes, we think, “Oh man, that’s difficult.” and we tense up accordingly.
This reminds me of that famous experiment with Pavlov and the dogs.
The bell is the stimulus, the salivation is the response.
Do we have similar learned responses?
As you listened to that music, perhaps you envisioned yourself getting ready to play the solo. Did it evoke a learned response?
Students studying for psychology tests will sometimes encapsulate this Pavlovian experiment as the “ding-slurp” theory. Thus, the name of our monster!
So, if we have developed a learned response that hinders us, we’ll have to learn a new response to replace the old one.
Here’s one simple example. When a younger student is struggling to play higher notes, they often label a note as ‘really high’ and proceed to freak out. Stepping away from the music, I simply play a gliss from 6th to 1st and back down, asking them to imitate me. Back and forth we go, moving to higher partials. When they hit the partial on which they can’t gliss to 1st position, I suggest that they just gliss to 3rd instead. In each case, I play and ask them to imitate.
Usually, they end up playing higher than they thought they could! Then, I tell them what high note they just reached and even write it down on the page.
Sometimes, their eyes grow wide with the thought, “Whoa, I just played *that* note?”
Now they are looking at the printed note (stimulus) and realizing that they just played with far more ease than they thought possible (new learned response).
It’s a good beginning.
Pause to consider what monsters you create as you practice your music.
Your mind automatically fixates on the things that didn’t go well. That’s human nature, I guess.
However, if you only criticize yourself for the bad moments, without giving equal praise for the ‘shining moments,’ practicing and performing can become a pretty dark affair.
Don’t live in denial, vaguely thinking that everything is just wonderful. But I have found that acknowledging the good moments is a more effective ‘springboard’ for improvement.
How to get started with this?
Here’s a simple trick:
1. Play a passage twice.
2. Ask yourself which one was better. Which one had more ‘shining moments?’
See if you can replicate that success.
Always building, always finding something positive in your playing is just as valid as finding all the negatives.