Calculus function

Question

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?

Answer 1

(a)

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

 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  will be less in value , which means the performance decreased . 

Answer 2

(b)

M is the maximum capable level of performance.

k is constant 

P is reasonable model for learning .

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

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

  ( decresed to 0 ) 

 

Answer 3

(c)

The differential equation is 

So the solution for differential equation is