# Could you help me to find a model for this curve?

I am very bad in mathematics and I'm not able to find by myself the model corresponding to this kind of curve.

I wish to have a quick growth at the beginning, then it should increase slowly for a long time and then a quick growth until the end value (here at 10).

This curve represents the basic learning curve: You learn a lot of things at the beginning then you have a time when you don't improve a lot (you are in fact memorizing what you've learnt) and then you become better and better.

Could someone give me a function / model to match this curve ?

Damien

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You may try something like :

tanh(x/150)*4+(arctan((x-700)/100)+PI/2)*2


with this result :

The $\tanh$ gives the growing part at the left and the $\arctan$ the inflection (at $700$).

Using $\tanh$ alone seems better :

tanh(x/120)*5+(tanh((x-670)/150)+1)*2.5


Some 'tuning' may be useful.

Other functions of this kind are sigmoids (see this picture).

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Thanks, it's what I was looking for :) – MARTIN Damien Dec 15 '12 at 12:02
@MARTINDamien: You are welcome! – Raymond Manzoni Dec 15 '12 at 12:42