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Differential equation- find max slope

  1. Oct 11, 2010 #1
    1. The problem statement, all variables and given/known data

    Another model for a growth function for a limited pupulation is given by the Gompertz function, which is a solution of the differential equation

    dP/dt = c*ln(K/P)*P

    where 'c' is a constant and 'K' is the carrying capacity.



    At what value of P does P grow fastest?

    2. Relevant equations

    c = .05
    K=1000
    P_0 = 500 (initial condition)

    P(t) = 1000/e^(e^(-.05t-.3665)) (this is the specific solution)

    3. The attempt at a solution

    I think it is asking to find what the max value of dP/dt. However I think to do this, I need to find the derivative of dP/dt which is d^2P/dt^2 and set it equal to zero. This will let me find t when the slope of P is at its max and then I plug t back into P. Is this correct?

    If it is, I am in for one hell of a derivative...

    Thanks!
     
  2. jcsd
  3. Oct 11, 2010 #2

    HallsofIvy

    User Avatar
    Science Advisor

    Why solve for P? You were asked for a value of P for which P grows fastest which mean its derivative is a maximum. You can find a maximum of a function by setting its derivative to 0. Here, the function is dP/dt so differentiate that:
    [tex]\frac{d}{dt}\frac{dP}{dt}= \frac{d^2P}{dt^2}[/tex][tex]= C\frac{d(P(ln(K)- ln(P))}{dt}= 0[/tex]
     
  4. Oct 11, 2010 #3
    Ok. Solving that derivative I get t = -7.33 and in the end P has a max slope at P= 367.879. These are right.

    Thanks!
     
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