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I am working on a tutoring system. When a student first starts we set the mastery threshold at A and they also have to have done at least a questions. But most students won't achieve that standard - after a student has done b questions we will lower the threshold to B. Finally we set a lower threshold, at c questions where the student only has to achieve a mastery of C. The thresholds remains at C past c questions.

My (poor) attempt at drawing this in paint is below. I need to extrapolate good values between the points. I'd like the curve to be smooth (smooth higher order derivatives are nice, but not necessary) and monotonically non-decreasing.

Bonus points I would like a gradient of 0 at the point (c, C) so that it smoothly joins with the straight line. But this isn't strictly necessary if it will greatly complicate the function.

I know that we should be able to fit a cubic equation to those three points, but I don't believe this would always guarantee the monotonically decreasing property, nor would we be able to get a gradient of 0 at the end

enter image description here

Casebash
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  • Can the function be defined piecewise? – MattyZ Mar 15 '14 at 04:09
  • Would a hyperbolic function of the form $f(x)=y_0+\frac{m}{x-x_0}$ due? Three points {(a,A), (b,B), (c,C)} should determine the parameters $x_0,y_0,m$. – David H Mar 15 '14 at 04:12
  • @Bitrex: Sure. It would be nice if it weren't just smooth in terms of gradient - but had continuous higher order derivatives too, but I won't say that that is strictly necessary – Casebash Mar 15 '14 at 04:27
  • @DavidH: That's a nice solution for the simple problem. Now if only there was a way to get the gradient to 0 at the end (in some cases the gradient might be heading towards minus infinity instead of 0) – Casebash Mar 15 '14 at 04:32
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    Maybe put point A at the point of inflection of $C_1\mathrm{sech}(C_2 x)$, for some constants c1 and c2. Connect A and B with the sech, and then join a parabola from B to C? – MattyZ Mar 15 '14 at 06:09

1 Answers1

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If I understand it correctly you want a smooth function $f\colon [a,+\infty)\to \mathbb R$ such that $f(a) = A$, $f(b)=B$ and $f(x)=C$ for all $x\ge c$. Moreover you want the function to be decreasing (we need $C<B<A$).

Since $f$ is required to be constant on some interval, the function cannot be analytical, hence it must be defined piecewise.

A possibility is: $$ f(x) = \begin{cases} C+(A-C)\exp\left(\beta\frac{x-a}{c-x}\right) & \text{if $x<c$} \\ C & \text{if $x\ge c$} \end{cases} $$ with $\beta = \frac{c-b}{b-a} \log\frac{B-C}{A-C}$.

Emanuele Paolini
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