Steady states If a function f represents a system that...

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 if a steady state exists and give the steady-state value. The population of a culture of tumor cells is given by $p(t)=\frac{3500 t}{t+1}.$

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 if a steady state exists and give the steady-state value.
The population of a culture of tumor cells is given by $p(t)=\frac{3500 t}{t+1}.$

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Asymptotic Analysis

Asymptotic analysis involves studying the behavior of functions as the input parameter approaches a particular limit, usually infinity. It is key for understanding how a system behaves in its long-term, steady state, and it provides insight into the trend and bounding behavior of the system functions regardless of short-term fluctuations.

Limit of a Function

The limit of a function as its independent variable approaches a particular value (often infinity) is a fundamental concept in calculus. This limit describes the value that the function approaches as the input grows unbounded, and it is used to determine the long-term behavior or asymptotic value of the system being modeled.

Steady State

A steady state in a dynamic system is a condition where the system variables no longer change over time, indicating that the system has reached an equilibrium. This means that after sufficient time, the behavior of the system stabilizes, and any further changes become negligible or zero, allowing for predictions based on this long-term behavior.

Please give Ace some feedback