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#1 2006-10-05 10:48:48

shudan54
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Registered: 2006-10-05
Posts: 4

Can I derive a hyperbolic formula using 2 points and an asymptote?

I'm trying to derive a learning curve formula, using the x-axis for elapsed time and the y-axis for the percent of acquired knowledge/skill.  Assuming the curve begins at 0,0 and the estimate of learning reaches 95% after n units of time, what would the formula be for this linear function?  (The asymptote would have a y value of 1.)  I'd really appreciate your help because I've forgotten if this is even enough information to write the formula.  I want to assume that the learner never actually reaches 100% but keeps getting closer to it over time.

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#2 2006-10-05 12:15:30

John E. Franklin
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Registered: 2005-08-29
Posts: 3,588

Re: Can I derive a hyperbolic formula using 2 points and an asymptote?

How do you know it is a hyperbola and not an exponential decay toward 100%?

Last edited by John E. Franklin (2006-10-05 12:15:56)


igloo myrtilles fourmis

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#3 2006-10-06 07:25:10

shudan54
Member
Registered: 2006-10-05
Posts: 4

Re: Can I derive a hyperbolic formula using 2 points and an asymptote?

Ah, well, I don't - know that it's a hyperbola.  What I'm really after is an approximation (key word) of productivity, given that learning "appears" to be complete after a certain duration.  And I'll assume that, say, 95% learning equates to "trained" and that more learning happens over time, but resulting in a negligible increase in productivity.  I've looked at write-ups for learning curves but they all seem to be based on tracking the number of units completed (say, per day) and this problem is more subjective, a qualitative measurement rather than a quantitative one.


Make sense?  It's just something that got stuck in my craw and I can't seem to let go of it.  I'd really appreciate it if you can put me in the ballpark.

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#4 2006-10-06 07:30:17

polylog
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Registered: 2006-09-28
Posts: 162

Re: Can I derive a hyperbolic formula using 2 points and an asymptote?

As John said, an exponential approaching 100% might work nicely.

It has an asymptote at y = 1.

Such a curve would be:

Where k would be a rate constant.

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#5 2006-10-09 10:48:55

shudan54
Member
Registered: 2006-10-05
Posts: 4

Re: Can I derive a hyperbolic formula using 2 points and an asymptote?

OK - thanks.  I feel like I'm getting somewhere, except I'm not familiar or have forgotten the notation.  I don't know what e & t represent, and I don't know what you mean by "rate constant."  Sorry.

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#6 2006-10-09 11:23:59

polylog
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Registered: 2006-09-28
Posts: 162

Re: Can I derive a hyperbolic formula using 2 points and an asymptote?

t is the independent variable, which would be time in this case, on the horizontal axis.

e is the base of the natural logarithm, and is equal to about 2.718281828

e^t is just the standard exponential function that is used to model all kinds of exponential growth in science.


here is an example of such a curve (the top one in this picture):

http://cognitrn.psych.indiana.edu/busey/WWWPubs/PsychRev/Image2.gif

k, which I called a rate constant, is a number that determines how sharply the curve rises at the beginning.

If k = 100 for example, it will rise very sharply, if k = 1, the rise is moderate.

I would suggest plotting a few of these with graph plotting software, like Winplot (which is free and easy to use).

Last edited by polylog (2006-10-09 11:25:03)

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#7 2006-10-11 04:39:48

shudan54
Member
Registered: 2006-10-05
Posts: 4

Re: Can I derive a hyperbolic formula using 2 points and an asymptote?

Ah - PERFECT!  That's EXACTLY what I was after and it works GREAT!  Thank you very much!

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#8 2006-10-11 13:08:16

polylog
Member
Registered: 2006-09-28
Posts: 162

Re: Can I derive a hyperbolic formula using 2 points and an asymptote?

Cool. smile

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