Karajan Beginner is a free iPhone App, very handy for piano students. It’s an easy to use music and ear training application for the iPhone and iPod Touch which provides lessons for learning to recognize intervals, chords, scales, pitch and tempo with detailed statistics. The free version is limited to one level and features 5 built-in instruments and 4 different play modes.
For more features including a multi-touch keyboard upgrade to the Karajan for $14.99 (see video). I don’t see much point to the BPM exercise unless you really don’t want to have to deal with metronomes. But nevertheless, I wish I had something like this years ago!