Problem 68 - Calculus: Early Transcendentals

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

Step-by-Step Solution

Answer

If so, what is the steady-state value? Answer: Yes, a steady state exists for the population of squirrels. The steady-state value is 500 squirrels.

Step 1: Identify the function

We are given the function \(p(t) = \frac{1500}{3 + 2e^{-0.1t}}\). We want to determine the steady-state value, which is the limit as \(t\) approaches infinity.

Step 2: Find the limit as t approaches infinity

We'll find the limit \(\lim_{t\to\infty} p(t) = \lim_{t\to\infty} \frac{1500}{3 + 2e^{-0.1t}}.\)

Step 3: Observe the exponential term

Notice that as \(t\) approaches infinity, the exponential term \(e^{-0.1t}\) approaches zero. This is because the exponential function decreases rapidly as the exponent becomes large and negative.

Step 4: Substitute the limit of the exponential term

Knowing that the exponential term approaches zero as \(t\) approaches infinity, we can substitute the limit: \(\lim_{t\to\infty} \frac{1500}{3 + 2e^{-0.1t}} = \frac{1500}{3 + 2 \cdot 0}.\)

Step 5: Compute the steady-state value

Now, we can simplify and calculate the steady-state value: \(\frac{1500}{3 + 2 \cdot 0} = \frac{1500}{3} = 500.\) So, the steady-state value exists, and it is 500 squirrels.

Key Concepts

Limit as t approaches infinity

In mathematics, the concept of a limit is essential for understanding the behavior of functions as the input approaches a certain value. When we talk about the "limit as \(t\) approaches infinity," we mean observing what happens to a function \(f(t)\) as time \(t\) becomes very large. This concept helps us understand if a system stabilizes over time, often described as reaching a "steady state."

When dealing with dynamic systems, calculating limits can tell us about long-term behavior.- If the limit is a finite number, the system stabilizes, reaching an equilibrium state.- On the other hand, if the limit is infinite or undefined, the system does not settle into an equilibrium as time goes forward.

In the squirrel population example, we calculated the limit of the function representing the population as \(t\) goes to infinity. This allowed us to determine if the population size reaches a stable, long-term value.

Exponential Decay

Exponential decay describes functions that decrease rapidly over time, often in physical, natural, or economic systems. It is characterized by an exponential function with a negative exponent. In our exercise, the term \(e^{-0.1t}\) denotes exponential decay. As \(t\) becomes very large, the expression \[e^{-0.1t} = \frac{1}{e^{0.1t}}\]shrinks towards zero.

This occurs because multiplying a number slightly larger than zero by successively smaller fractions (like \(\frac{1}{e^{0.1t}}\)), results in smaller and smaller products. The rapid decline makes such terms virtually negligible in the context of calculating limits for long-time behavior. Thus, over time, the decay factor becomes insignificant, leaving us with a simplified expression that aids in recognizing steady states.

Equilibrium in dynamic systems

In dynamic systems, equilibrium refers to a stable state where the system's behavior becomes predictable and constant. Reaching equilibrium means the system doesn't change significantly as time moves on. For many processes, this is the point where inputs and outputs balance. Dynamic systems can include biological populations, chemical reactions, or even economic models. Analyzing equilibrium involves understanding how these systems evolve over time and predicting their long-term behavior. - If the system reaches a steady state, it has found equilibrium. For our squirrel population, we concluded a steady state existed at 500 squirrels, indicating equilibrium in the system. - Reaching this state can imply that resources are balanced, or growth is matched by factors like mortality or resource limits. Understanding equilibrium helps in predicting outcomes and planning strategies in various fields of study.