Effortlessly Linearize y(x)= a(1-e-bx) with Expert Help

[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:

time(min) - %Rec
0 - 0
1 - 45
3 - 72
5 - 80
9 - 88
12 - 91.8
15 - 92​

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 $y=a(1- e^{-bx})$ 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 $abe^{-7.5b}$.