CrashOveride
Active Member
Logistic growth equation:
dP/dt = kP(1 - RP)
k, R are constants.
Show by differentiation that P = I / [RI + (1 - RI)e<sup>-kt</sup>] , where I is the inital population (a constant), is a soluution of the logistic equation.
And also, just something i want to clear up:
I'll use an arbitary question:
For a body falling under gravity in air, the rate of change of velocity is given by dV/dt = -k(V - P) where P and k are constants. This has as a solution: V = P + Ae<sup>-kt</sup>
If the rate of change is given by dV/dt = -0.02(V - 490)
part (a) tells us to derive an equation for V, remembering that initally V = 0. But why should the velocity be zero inititally ? What if a scenario came about where a body was already falling and when timing began, initially it had some z velocity ? Here the velocity initally would not be 0. And as the question doesn't say anything the body was inititally at rest when it was dropped etc. etc. how can we assume this, other than "remembering" it, as was asked of us. "Remembering" would imply that we should already know, just in case we forgot. Somebody correct me if i am brazenly wrong here or anywhere =p
part (b) went on to ask for the velocity after 10 seconds. Now, suppose it had asked us for the velocity after 10min. Or both. What 't' values would i be substituting here.
dP/dt = kP(1 - RP)
k, R are constants.
Show by differentiation that P = I / [RI + (1 - RI)e<sup>-kt</sup>] , where I is the inital population (a constant), is a soluution of the logistic equation.
And also, just something i want to clear up:
I'll use an arbitary question:
For a body falling under gravity in air, the rate of change of velocity is given by dV/dt = -k(V - P) where P and k are constants. This has as a solution: V = P + Ae<sup>-kt</sup>
If the rate of change is given by dV/dt = -0.02(V - 490)
part (a) tells us to derive an equation for V, remembering that initally V = 0. But why should the velocity be zero inititally ? What if a scenario came about where a body was already falling and when timing began, initially it had some z velocity ? Here the velocity initally would not be 0. And as the question doesn't say anything the body was inititally at rest when it was dropped etc. etc. how can we assume this, other than "remembering" it, as was asked of us. "Remembering" would imply that we should already know, just in case we forgot. Somebody correct me if i am brazenly wrong here or anywhere =p
part (b) went on to ask for the velocity after 10 seconds. Now, suppose it had asked us for the velocity after 10min. Or both. What 't' values would i be substituting here.
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