🎉 Announcing Numerade's $26M Series A, led by IDG Capital!Read how Numerade will revolutionize STEM Learning University of North Texas ### Problem 15 Medium Difficulty # Psychologists interested in learning theory study learning curves. A learning curve is the graph of a function$ P(t) $, the performance of someone learning a skill as a function of the training time$ t. $The derivative$ 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 $increase? Explain.(b) If$ M $is the maximum level of performance of which the learner is capable, explain why the differential equation$ \frac {dP}{dt} = k(M - P)  K $a positive constant ### Answer ## (a)$P$increases most rapidly at the beginning, since there are usually many simple, easily-learned sub-skills associated with learning a skill. As$t$increases, we would expect$d P / d t$to remain positive, but decrease. This is because as time progresses, the only points left to learn are the more difficult ones.(b)$\frac{d P}{d t}=k(M-P)$is always positive, so the level of performance$P$is increasing.As$P$gets close to$M, \frac{d P}{d t}$gets close to$0 ;\$ that is, the performance levels off, as explained in part (a)(c) See graph

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Differential Equations

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