If a function f represents a system that varies in time, the e...

Question

If a function f represents a system that varies in time, the existence of lim ${$\displaystyle\[\lim$_{t$\rightarrow\]\infty$}{f(t)}}$ means that the system reaches a steady state (or equilibrium). For the following systems, determine whether a steady state exists and give the steady-state value.

The population of a colony of squirrels is given by $p (t) = \frac{1500}{3 + 2 e^{- 0.1 t}}$ .

The population of a colony of squirrels is given by $p (t) = \frac{1500}{3 + 2 e^{- 0.1 t}}$ .

Accepted Answer

Welcome back everyone. The concentration of a pollutant in a reservoir modeled by the function P of T equals 800 divided by four plus five E to the negative 0.1 T decreases over time determine whether a steady state exists and provide its value. As T approaches infinity. A says the steady state exists and its value is 160 B says it exists and its value is 200 C, it exists and its value is 800. While D says the steady state does not exist now to determine if a steady state concentration exists. As T approaches infinity, we can evaluate the limit of the function P of T as T approaches infinity. In other words, the limit as T approaches infinity of P of T is going to be equal to the limit as T approaches infinity of 800 divided by four plus five E to the negative 0.1 T I know if we evaluate this and get a result, if the limit exists, that means the steady state exists. And the value of the limit is going to be the value for our steady state. Now, how can we go ahead and solve here? How can we find this limit as T approaches infinity? Well as T approaches infinity, meaning as T gets bigger, the term five E, let's write that here. That's right. Now. Before I continue, as T approaches infinity, the term five E to the negative 0.1 T, OK is going to approach zero. OK? Because the exponent negative 0.1 T becomes large and very negative making E to the negative 0.1 T very close to zero. OK? Let me just fix that here. So that means then that our limit is going to be equal to 800 divided by four plus zero, which is 800 divided by four, which equals 200. Therefore, since the limit exists, a steady state exists and its value is 200 B is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Key Concepts

Limit of a Function

Steady State (Equilibrium)

Exponential Decay

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