# Learning Curve

Mastering a new area or a skill always takes some time. In this section we try to model the learning process using a differential equation.

First of all, we introduce a measurable learning function $$L\left( t \right).$$ This function, for example, may describe the current labour productivity of an employee. Let $${L_{\max }}$$ be the maximum available value of $$L\left( t \right).$$ In many cases the following thumb rule is valid: the learning speed is proportional to the volume of remaining (unlearned) material. Mathematically, this can be represented by the equation:

$\frac{{dL}}{{dt}} = k\left( {{L_{\max }} - L} \right),$

where $$k$$ is a coefficient of proportionality. The given differential equation is a separable equation, so it can be easily solved in general form:

$\frac{{dL}}{{dt}} = k\left( {{L_{\max }} - L} \right),\;\; \Rightarrow \frac{{dL}}{{{L_{\max }} - L}} = kdt,\;\; \Rightarrow \int {\frac{{dL}}{{{L_{\max }} - L}}} = \int {kdt} ,\;\; \Rightarrow - \int {\frac{{d\left( {{L_{\max }} - L} \right)}}{{{L_{\max }} - L}}} = \int {kdt} ,\;\; \Rightarrow - \ln \left( {{L_{\max }} - L} \right) = kt + \ln C,\;\; \Rightarrow \ln \left( {{L_{\max }} - L} \right) = - kt + \ln C,\;\; \Rightarrow \ln \left( {{L_{\max }} - L} \right) = \ln {e^{ - kt}} + \ln C.$

After eliminating the logarithms, we obtain the general solution in the form:

${L_{\max }} - L = C{e^{ - kt}}.$

The constant $$C$$ can be found from the initial condition: $$L\left( {t = 0} \right) = M.$$ Hence, $$C = {L_{\max }} - M.$$ As a result, the learning curve is described by the formula

$L\left( t \right) = {L_{\max }} - \left( {{L_{\max }} - M} \right){e^{ - kt}}.$

The parameter $$M$$ is the last expression means the initial level of knowledge or skills. In the simplest case, we can suppose that $$M = 0.$$ The other parameter $$k$$ controls how fast the curve rises. View of the learning curves at different values of $$M$$ and $$k$$ is shown in Figure $$1$$ and $$2,$$ respectively.

As it can be seen, the learning level $$L$$ in all cases increases in the beginning of the process, and then the learning rate slows down as the level $$L$$ approaches the maximum value $${L_{\max }}.$$

See solved problems on Page 2.