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Calculus function

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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 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.

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

Can you please explain it step by step?
asked Sep 25, 2014 in CALCULUS by anonymous

3 Answers

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(a)

P(t) is learning curve , it is varying with respect to time

image represents rate  at which performance improves.

From the above P increases most  rapidly when psychologists learn in smaller interval of time.

mathematically ,  numerator in higher only when the denominator is less .

If t increases image will be less in value , which means the performance decreased . 

answered Sep 25, 2014 by friend Mentor
0 votes

(b)

M is the maximum capable level of performance.

k is constant 

P is reasonable model for learning .

image

In the begining stage of learning ,psychologists need to learn more    (image is maximum ) 

After infinite point of time , psychologists no need to learn more compare to previous  

  (image decresed to 0 ) 

 
answered Sep 25, 2014 by friend Mentor
0 votes

(c)

The differential equation is image

image

So the solution for differential equation is image

answered Sep 25, 2014 by friend Mentor
edited Sep 25, 2014 by bradely

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