Problem 64 - 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 bacteria culture is given by \(p(t)=\frac{2500}{t+1}\)

Step-by-Step Solution

Answer

If so, what is the steady-state value? Answer: Yes, a steady state exists and the steady-state value is 0.

Step 1: Identify the given function

The given function representing the population of the bacteria culture over time is: \(p(t) = \frac{2500}{t+1}\).

Step 2: Determine the limit as t goes to infinity

To check for the existence of a steady state, we need to compute the limit of the function \(p(t)\) as \(t\) approaches infinity: \(\lim_{t \rightarrow \infty} \frac{2500}{t+1}\).

Step 3: Compute the limit using algebraic methods

As \(t\) approaches infinity, we can observe that the denominator will become infinitely large. We can rewrite the function to help us compute the limit: \(p(t) = \frac{2500}{t+1} = \frac{1}{\frac{1}{2500}(t+1)}\) Now, as \(t\) approaches infinity, we can see that the term \(\frac{1}{2500}(t+1)\) also approaches infinity. This means that the fraction \({\frac{1}{\frac{1}{2500}(t+1)}}\) approaches zero as the denominator gets infinitely large. Therefore, the limit is: \(\lim_{t \rightarrow \infty} p(t) = \lim_{t \rightarrow \infty} \frac{1}{\frac{1}{2500}(t+1)} = 0\).

Step 4: Determine if a steady state exists and state the value

Since the limit of the given function exists and is equal to 0 as \(t\) approaches infinity, this indicates that the system reaches a steady state. Thus, a steady state exists and the steady-state value is 0.

Key Concepts

Understanding Steady State

The concept of a steady state is crucial when studying systems that evolve over time. It's essentially a condition where the system no longer changes, even though time keeps moving forward. Think of it like a calm sea where the waves have settled after a storm. Mathematically, when we say a system reaches a steady state, we mean that the function representing it approaches a constant value as time progresses infinitely. In our bacteria population example, determining the steady state involves calculating

• the limit of the function as time approaches infinity.
• If this limit is a finite number, the system stabilizes.

For the bacteria population function, \[p(t) = \frac{2500}{t+1}\] we evaluated its limit as time (\(t\)increases without bound, which turned out to be zero. This means that eventually, the bacteria population will maintain a steady state at 0, meaning the population declines over time and eventually disappears. Understanding this concept equips us with a tool to predict long-term behavior in various dynamic systems.

Exploring Bacteria Population Models

Bacteria populations can be fascinating to study as they display rapid changes over short time periods. In mathematical modeling, these populations are often depicted by functions that illustrate growth or decline over time. Our particular function, \[p(t) = \frac{2500}{t+1}\], is a model used to represent how a bacteria population changes. Factors influencing the population dynamics include resources, environmental conditions, and intrinsic biological processes. For this model:

• The numerator (2500) represents an initial factor affecting population size.
• The denominator (\(t+1\)suggests a process where maximum population starts decreasing as time passes.

So, as time increases, the population decreases towards zero, indicating resource limitation or environmental changes that can't sustain a larger population. Such functions help us not only predict when a steady state occurs but also understand biological interactions in ecosystems.

Equilibrium in Dynamic Systems

Equilibrium in dynamic systems refers to a state where a system experiences no net change as time progresses. It is the point where forces acting upon the system are balanced. This concept is prevalent across various disciplines such as physics, chemistry, and ecology, including our study of bacteria populations. In evaluating equilibrium states:

• We assess whether the changes within a system stabilize over time.
• Mathematically, it implies the limit of the function modeling the system becomes a constant.

In our bacteria population example, reaching an equilibrium means that the limit \(lim _{t \rightarrow \infty} p(t)\)=0, indicating no further change in population size past a certain point. In biological terms, this often indicates a depletion of necessary resources or a balance with environmental pressures. Understanding equilibrium can offer vital insights into how to manage and preserve natural populations or control microbial growth.