Exercises 13—15 consider an elementary model of the learning process: Although human learning is an extremely complicated process, it is possible to build models of certain simple types of memorization. For example, consider a person presented with a list to be studied. The subject is given periodic quizzes to determine exactly how much of the list has been memorized. (The lists are usually things like nonsense syllables, randomly generated three-digit numbers, or entries from tables of integrals.) If we let L(t) be the fraction of the list learned at time t, where L = 0 corresponds to knowing nothing and L = 1 corresponds to knowing the entire list, then we can form a simple model of this type of learning based on the assumption:
• The rate dL/dt is proportional to the fraction of the list left to be learned.
Since L = 1 corresponds to knowing the entire list, the model is
where k is the constant of proportionality.
14. Suppose two students memorize lists according to the model
(a) If one of the students knows one-half of the list at time t = 0 and the other knows none of the list, which student is learning more rapidly at this instant?
(b) Will the student who starts out knowing none of the list ever catch up to the student who starts out knowing one-half of the list?
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Differential Equations
- What is your predicted average population growth rate from this method, in terms of lambda AND % per year? Does that predict an increasing or decreasing population, on average? Year (t) Nests at time t Nests at t + 1 N(t+1)/N(t) 1983 746 780 1.046 1984 780 702 0.900 1985 702 744 1.060 1986 744 729 0.980 1987 729 843 1.156 1988 843 905 1.074 1989 905 992 1.096 1990 992 1180 1.190 1991 1180 1275 1.081 1992 1275 1241 0.973 1993 1241 1566 1.262 1994 1566 1930 1.232 1995 1930 1915 0.992 1996 1915 2219 1.159 1997 2219 3482 1.569 1998 3482 3365 0.966 1999 3365 5834 1.734 2000 5834 4927 0.845 2001 4927 5525 1.121 2002 5525 7601 1.376 2003 7601 6446 0.848 2004 6446 9258 1.436 2005 9258 10899 1.177 2006 10899 average 1.142arrow_forwardUse the Laplace transform to solve the initial-value problem, ?′′ + 4? = 1 − ?(? − 1); ?(0) = 0, ?′(0) = −1arrow_forwardUse Laplace transform to solve the given initial value linear systemsarrow_forward
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