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Problem 13 Medium Difficulty

Psychologists interested in learning theory study learning curves. A learning curve is the graph of a function $P(t),$ the performance of someone learning a skill as a function of the
training time $t .$ The derivative $d P / d t$ represents the rate at which performance improves.
(a) When do you think $P$ increases most rapidly? What happens to $d P / d t$ as $t$ increases? Explain.

(b) If $M$ is the maximum level of performance of which the learner is capable, explain why the differential equation
$$\frac{d P}{d t}=k(M-P)$$
k a positive constant is a reasonable model for learning.

(c) Make a rough sketch of a possible solution of this differential equation.

Answer

Psychologists would like to study learning curves, or how the performance of a person learning a new activity depends with time, $P(t)$. What would this look like? Well for one, we know that if you haven't spent any time $(t=0)$, there is no performance we can measure, so $P(0) =0$. We also know that there are limitations to performance, so there has to be a maximum: we'll call it $M$ (for Maximum, of course). So we go from $0$ to $M$ as time passes by. What point is the change in performance, $\frac{dP}{dt}$, the largest? In other words, when are you improving the fastest? We can all agree that it's not at the maximum, since you can't improve any more, so it has to be before that. I think it's sensible to agree that we learn the most at the point we start, since we didn't have any prior performance. Thus $P$ increases the most at $t=0$, and when $t$ increases, $\frac{dP}{dt}$ (the change in learning) goes to $0$, since we reach a maximum, $M$.
Thus, a sensible differential equation should be:
$$\frac{dP}{dt} = k(M-P)$$

An increase in performance makes it so that the change in performance gets smaller and smaller until we reach the maximum, $M$, where the change in performance will be zero.

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Video Transcript

all right, So let's hear some scientists and psychologists meant to understand burning cruise, and they want to measure the change of performance with respect to time in this case will measure time these days and performance. I was a bit of natural concept start to measure it, but I guess that doesn't benefit such. So let's think about when that learning could wind. The performance changes the most. And of course, you start. You have zero performance, of course, because you haven't performed at all. And as time goes on, we all know that it looks Mike gets used from better and better and better. And so, of course, we reach capacity because we can only improve so much. And that's anything like that. Some maximum. It doesn't actually lower there, but you get the night goes on forever, and there's a value here and performance returned last month about him that's supposed to be, and her performance usually looks like this. So we started zero. He end up at a maximum, so if you look whether that changes the most, it's reasonable to say that zero it does because you have no performance and I suddenly have some sort Your change in performances degree is there. And what happens as time goes on to the change in performance. We see that eventually we hit a peak, right? We had a flat. So where are changing performance doesn't change. And so we call this So if there's a maximum level of performance than he would have a change in performance that looks something like this. So you're changing performance. We respect the time or the day spent performing would be the fun Constant right now doesn't matter times maximum Highness from this. And so the reason why this is a good choice out differential equations because the more you perform, the smaller you want smaller, you're changing performance to be. We know that because we couldn't. And, of course, the eventually when you hit the maximum, you're when P is equal to end. Then your you want your changing performance to be zero, and that's what we have a flat in here. So this is a good way to represents our performance. And so this is a nice graph. We know that this is a good solution for the problem that we haven't had