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Help with this linear regression

  1. Jun 18, 2012 #1
    Hello! how I linearize this function?

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

    a and b are constants
     
  2. jcsd
  3. Jun 18, 2012 #2
    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...
     
  4. Jun 18, 2012 #3
    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) = Rmax(1-e^-kt). Linearize the function and determine the parameters Rmax and k.
     
  5. Jun 18, 2012 #4

    HallsofIvy

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    Staff Emeritus
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    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].
     
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