quiz_11B - learning decreases.) Thus dP dt = k ( M-P ( t ))...

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Math 213 — Quiz 11 Name Solution 1. 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. If M is the maximum level of performance of which the learner is capable, it is reasonable to assume that dP/dt is proportional to M - P ( t ). (At first, learning is rapid. Then, as performance increases and approaches its maximal value, the rate of
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Unformatted text preview: learning decreases.) Thus dP dt = k ( M-P ( t )) where k is a positive constant. Solve this linear differential equation. We first rewrite the equation as dP dt + kP ( t ) = kM, which is a linear equation. The integration factor is I ( t ) = exp( Z k dt ) = e kt . So the solution is given by P ( t ) = 1 I ( t ) Z kMI ( t ) dt = e-kt ( Me kt + C ) = M + Ce-kt for some constant C ....
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This note was uploaded on 05/09/2008 for the course MATH 2130 taught by Professor Dorais during the Spring '08 term at Cornell.

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