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 culture of tumor cells is given by $p (t) = \frac{3500 t}{t + 1}$ .

The population of a culture of tumor cells is given by $p (t) = \frac{3500 t}{t + 1}$ .

Accepted Answer

Welcome back, everyone. In this problem, the revenue of a company over time can be modeled by the function R of T equals 5,000T divided by 2 T + 5. Determine whether a steady state exists and provide its value as T approaches infinity. A says the steady state exists and its value is 0. B, it exists and its value is 2500. C, it exists and is 5000, and the DA says it does not exist. Now, to determine if a steady state exists for the revenue function R of T as T approaches infinity, we can evaluate the limit of R of T as T approaches infinity. In other words, Let's find out if the limit, or sorry, let's try to find a value for the limit as T approaches infinity of R of T. That means it's going to be the limit as T approaches infinity of 5000 T divided by 2 T. Plus 5. Now, to simplify, we can divide every term in the fraction by T, the highest power of T in the denominator, OK. So now we can rewrite this as the limit as T approaches infinity of 5000 T divided by T all divided by 2 T divided by T plus 5 divided by T. Now let me make some space on the right here, OK, because no, that means we can when we divide through by T. Is going to be equal to the limit as T approaches infinity of 5000 T divided by T equals 12 T divided by T equals 2, and we're gonna be left with 5 divided by T. Now we can take the limit, and that means this is going to be equal to 5000 divided by 2 plus 5 divided by infinity. Now 5 divided by infinity as T approaches infinity, the quotient 5 divided by infinity is going to approach 0, OK? So we can rewrite this as 5000 divided by 2 + 0. Actually, we'll write it here, OK. And 2 + 0 equals 2, 5000 divided by 2 equals 2500. Thus, a steady state exists for the revenue function as T approaches infinity and its value is 2500. Therefore, B is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

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