Hmm...interesting question.

a) Part a might be very easy, but I'm not seeing a mathematical way to answer it. Hopefully someone else will come along and help you out.

b) the key here is to note that if M is the M is the maximum level of performance that the learner can reach, it follows that the learner can't 'learn' beyond that point. (Sorry for the pathetic use of English ).

So i.e. the rate of learning is proportional to (the student's potential - how much the student already knows.)

Therefore where k is a positive constant.

Also note think about what the maximum of P is? It's M - since that is it's maximum level of 'learning'. So when P=M, which makes perfect sense since at that point, the rate of learning is at it's highest point, and cannot increase any further.