Question

Steady states If a function \(f\) represents a system that varies in time, the existence of \(\lim f(t)\) 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 \(p(t)=\frac{2500}{t+1}\)

Short answer

Answer: Yes, the bacteria population reaches a steady state. The steady-state value is 0.

Step-by-step solution

Identify the function

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

Evaluate the limit as t approaches infinity

To find the steady-state value, we need to evaluate the limit of the population function as \(t\) goes to infinity: $$ \lim_{t \to \infty} p(t) = \lim_{t \to \infty} \frac{2500}{t+1} $$

Use properties of limits

We can simplify the expression by dividing both the numerator and denominator by \(t\): $$ \lim_{t \to \infty} \frac{2500/t}{1+(1/t)} $$

Evaluate the limit

Now we can evaluate the limit directly: $$ \lim_{t \to \infty} \frac{2500/t}{1+(1/t)} = \frac{\lim_{t \to \infty} 2500/t}{\lim_{t \to \infty} (1+(1/t))} $$ Both limits in the numerator and denominator go to 0: $$ \frac{0}{1+0} = 0 $$

Interpret the result

Since the limit exists and is equal to 0, we can conclude that the population of the bacteria culture reaches a steady state. The steady-state value of the population is 0.

Key concepts

Steady State

In mathematical and scientific contexts, the concept of a "steady state" describes a situation where a system settles into an unchanging state over time. This means that despite fluctuations or changes at the beginning, the system will eventually stabilize. Whether it's a chemical reaction or the cooling of coffee, reaching a steady state implies reaching an equilibrium.
For functions and systems described mathematically, if the limit of the function exists as time goes to infinity, it implies that the system eventually becomes constant. This end point of function indicates that no further significant changes occur, providing valuable insight into the future behavior of the system.
In the context of our exercise, the function's behavior over time is explored to find out if a steady state is achieved. Determining this helps predict long-term trends and equilibrium points of various processes, such as population dynamics, temperature regulation, and more.

Bacteria Population Modeling

Bacteria population modeling involves using mathematical functions to predict the growth or decline of bacterial colonies over time. These models are crucial for understanding behaviors like growth rates in favorable conditions and decline or stabilization in constrained environments.
The given exercise provides us with a function, \(p(t) = \frac{2500}{t+1}\), where \(p(t)\) represents the population at time \(t\). Here, the numerator \(2500\) represents the maximum possible population when time is at its initial moment, and \(t+1\) reflects the time-dependent growth factor.
As time \(t\) grows larger, the denominator (\(t+1\)) increases and the fraction \(\frac{2500}{t+1}\) becomes smaller. This implies that over time, the rate at which the bacteria population can grow diminishes. Ultimately, modeling population in this way lets scientists simulate real-world conditions, such as resource constraints or growth limits, extrapolating detailed insights into bacterial behavior.

Evaluating Limits

Evaluating limits is a fundamental process in calculus, especially to study the behavior of functions as the independent variable approaches a particular point, often infinity. By examining limits, we aim to understand the eventual value or trend of a function, making it a powerful tool for predicting long-term behaviors in various mathematical models.
In the exercise, we evaluated the limit \(\lim_{t \to \infty} \frac{2500}{t+1}\) to reveal the system's steady state. We achieve this by transforming the function into \(\frac{2500/t}{1+1/t}\), where both the numerator and denominator simplify as \(t\) grows towards infinity. With this simplification, both terms approach zero, resulting in \(\frac{0}{1+0} = 0\).

• Concluding with a result of zero indicates that the bacteria population will eventually reach a steady state, which in this case is extinction or stabilization at zero population.
• This showcases how limits serve as critical indicators of trends and potential outcomes, helping students and professionals alike to grasp complex changes over time.

Understanding and applying limits thus provides clear insights into various dynamic phenomena, making them indispensable in scientific and mathematical explorations.