To "linearize" a fuction means to approxiate it by a linear function and that can only be done accurately in a limited range. One of the things we should learn in basic Calculus is that the tangent line to a graph gives the best linear approximation to the function in a neighborhood of the given point.
The difficulty is that you can't have a linear function that accurately approximates a function for all x and here you are not saying where you want it approximated. In the list you give, x varies from 0 to 15. It would be easiest to linearize at x= 0 but I would be inclined to use the midpoint x= 7.5.
The derivative of y=a(1- e^{-bx}) is y'= abe^{-bx} and at x= 0 that is ab. So your linear approximation, around x= 0, is the line through (0, 0) with slope ab.
But the derivative at x= 7.5 is abe^{-7.5b} so the linearization would be the line through (7.5, a(1- e^{-7.5b})) with slope abe^{-7.5b}.