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

Question

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 ?

Answer 1

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 $700$).

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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).