# Thread: differential equation world problems

1. ## differential equation world problems

A learning curve is the graph of a function P(t), the performance of someone learning a skill as a function of training time t. The derivative dP/dt represents the rate at which performance improves.

a) When do you think P increases most rapidly? What happens to dP/dt as t increases? Explain.
b) If M is the maximum level of performance of which the learner is capable, explain why the differential equation dP / dt = k(M-P), k a positive constant is a reasonable model for learning.

a) answer is that it increases most rapidly in the beginning then begins to decline. can anyone tell me why this is so?

b)?

2. 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 $\displaystyle \frac{dP}{dt} \alpha (M-P)$ i.e. the rate of learning is proportional to (the student's potential - how much the student already knows.)

Therefore $\displaystyle \frac{dP}{dt} = k(M-P)$ 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, $\displaystyle \frac{dP}{dt}=0$ which makes perfect sense since at that point, the rate of learning is at it's highest point, and cannot increase any further.

3. Originally Posted by xfyz
A learning curve is the graph of a function P(t), the performance of someone learning a skill as a function of training time t. The derivative dP/dt represents the rate at which performance improves.

a) When do you think P increases most rapidly? What happens to dP/dt as t increases? Explain.

a) answer is that it increases most rapidly in the beginning then begins to decline. can anyone tell me why this is so?
This is not quite the kind of problem that has an analytical answer as much as it has an intuitive answer. (That is to say, the answer may turn out to be wrong after the data are taken, but it makes the most sense of any of the other possibilities, so we take it to be true as an assumption.)

Consider the fact that P(0) = 0. So to start out the amount of learning per unit time is the largest because we are starting from nothing and there is the most to learn. This corresponds to any teaching technique I have heard of: one starts with the simpler and most basic definitions, which most people typically find easy to comprehend. So at the beginning the learning is easiest and hence most rapid.

-Dan