Ch. 2 - Limits

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 2 - Limits

Problem 2.5.64

Chapter 2, Problem 2.5.64

If a function f represents a system that varies in time, the existence of lim $\lim_{t \to \infty} f (t)$ 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 $p (t) = \frac{2500}{t + 1}$ .

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Identify the function given: $ p(t) = \frac{2500}{t+1} $.

Understand that we need to find $ \lim_{t \to \infty} p(t) $ to determine if a steady state exists.

Set up the limit: $ \lim_{t \to \infty} \frac{2500}{t+1} $.

Recognize that as $ t \to \infty $, the denominator $ t+1 $ becomes very large, making the fraction approach zero.

Conclude that the limit is zero, indicating the system reaches a steady state at a population of 0.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Limits

In calculus, a limit describes the behavior of a function as its input approaches a certain value. Specifically, the limit as t approaches infinity examines how the function behaves as time progresses indefinitely. Understanding limits is crucial for determining the long-term behavior of dynamic systems, such as whether they stabilize or diverge.

Steady State

A steady state in a system occurs when the variables of interest no longer change over time, indicating that the system has reached equilibrium. Mathematically, this is represented by the existence of a limit as time approaches infinity. In the context of the given function, identifying the steady-state value involves evaluating the limit of the population function as time increases.

Population Dynamics

Population dynamics is a field of study that examines how populations change over time due to various factors such as birth, death, and resource availability. The function provided, p(t) = 2500/(t+1), models the growth of a bacterial culture, illustrating how the population evolves and approaches a maximum capacity as time progresses. Understanding these dynamics is essential for analyzing the behavior of biological systems.

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