help with this linear regression

cicleriano

Hello! how I linearize this function?

y(x)= a(1-e^-bx)

a and b are constants

algebrat

Unfortunately, it seems you do not understand the problem completely, as it only make sense if you offer some points at which to approximate the function. For starters...

cicleriano

sorry!
In the flotation laboratory was determined following table of values:


A mathematical model representing these results is R(t) = R_max(1-e^-kt). Linearize the function and determine the parameters R_max and k.

HallsofIvy

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 [itex]y=a(1- e^{-bx})[/itex] is [itex]y'= abe^{-bx}[/itex] and at x= 0 that is [itex]ab[/itex]. So your linear approximation, around x= 0, is the line through (0, 0) with slope ab.

But the derivative at x= 7.5 is [itex]abe^{-7.5b}[/itex] so the linearization would be the line through [itex](7.5, a(1- e^{-7.5b}))[/itex] with slope [itex]abe^{-7.5b}[/itex].