Question

Steady states If a function f represents a system that varies in time, the existence of \(\lim _{t \rightarrow \infty} f(t)\) 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 amount of drug (in milligrams) in the blood after an IV tube is inserted is given by \(m(t)=200\left(1-2^{-t}\right)\).

Short answer

Answer: Yes, a steady-state exists for this function, and the steady-state value is 200 milligrams.

Step-by-step solution

Find the limit as t approaches infinity

To find the limit of \(m(t)\) as \(t\) approaches infinity, we need to analyze the function and see how it behaves as \(t\) increases without bound: \(\lim _{t \rightarrow \infty} m(t) = \lim _{t \rightarrow \infty} 200\left(1-2^{-t}\right)\)

Analyze the exponential term

In this step, we will analyze the behavior of the exponential term as \(t\) approaches infinity: \(\lim _{t \rightarrow \infty} 2^{-t}\) As \(t\) goes to infinity, \(2^{-t}\) goes to zero (since any positive number raised to a negative infinite power tends to zero). So, \(\lim _{t \rightarrow \infty} 2^{-t} = 0\)

Find the limit of the function

Now that we know the behavior of the exponential term, we can find the limit of the function: \(\lim _{t \rightarrow \infty} m(t) = \lim _{t \rightarrow \infty} 200\left(1-2^{-t}\right) = 200(1-0) = 200\)

Determine if a steady-state exists

Since the limit of the function \(m(t)\) exists and is equal to \(200\), a steady-state exists for this system.

Determine the steady-state value

As the limit of the function is equal to \(200\), the steady-state value for this system is 200 milligrams.

Key concepts

Limits at Infinity

Understanding limits at infinity is crucial when analyzing the long-term behavior of functions, which is often required in fields such as engineering, physics, and economics. A limit at infinity investigates what value a function approaches as the variable goes to infinity. For instance, in the given exercise, the limit of the function m(t) as t approaches infinity is evaluated to determine if a steady state exists for the amount of a drug in the bloodstream over time.

When you come across a term like 2^-t, as t becomes very large, the value diminishes and gets infinitesimally close to zero. This behavior impacts the entire function, as the subtracted part within the parentheses becomes negligible, effectively turning the function's value into just 200 in this case. In calculus, this process of finding limits as the variable grows indefinitely large is not only about finding the final value but also confirming the existence of a predictable outcome or 'destination' for the function's value.

Exponential Decay

Exponential decay refers to the process by which an amount decreases at a rate proportional to its current value, which results in its value decreasing exponentially over time. Mathematically, exponential decay functions commonly have a base raised to a negative variable exponent, such as 2^-t seen in the exercise.

In our context, m(t)=200(1-2^-t) represents the amount of drug in the blood, where the term 2^-t signifies the rate at which the drug's presence is dwindling. As time t becomes large, the exponential component approaches zero, indicating that the amount of drug is stabilizing and no further significant decay is occurring. This concept is pivotal to fields like pharmacokinetics, nuclear physics, and resource depletion, where understanding the rate of decay is vital for making predictions and decisions.

Equilibrium in Systems

The concept of equilibrium in systems often comes into play in science and engineering when a system reaches a condition where its state no longer changes with time. The equilibrium connotes balance and is synonymous with the term 'steady state' in certain contexts.

In the solution process, evaluating the existence of a limit as t goes to infinity serves to check for the system's equilibrium. If such a limit exists and is finite, as with the 200 milligrams of the drug in the blood system in the given exercise, the system is said to have reached equilibrium. The importance of equilibrium lies in its implication of stability and predictability, making it a vital concept for disciplines ranging from chemistry to economics, where systems' behavior over time determines critical outcomes and strategies.

Steady-State Value

The steady-state value of a system is the value that a function approaches as the independent variable, typically time, approaches infinity. In the context of the exercise, the steady-state value represents the amount of the drug that remains constant in the blood after a prolonged period of time, assuming no more of the drug is introduced and the rates of metabolism and excretion are constant.

As confirmed in the step-by-step solution, the steady-state value for this drug dosage is 200 milligrams. Clinically, recognizing and calculating the steady-state value is imperative in designing effective dosing regimens. More broadly, in any dynamic system where performance stabilizes over time, understanding the steady-state value informs decisions on how to manage or alter the system's conditions for the desired outcome.