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 $\lim f(t)$ 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 value of an investment in dollars is given by $v(t)=1000 e^{0.065 t}$

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 value of an investment in dollars is given by $v(t)=1000 e^{0.065 t}$

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Key Concepts

Steady State

A steady state refers to the condition of a system where its behavior becomes unchanging over time. In mathematical modeling, this is observed when a function reaches a constant value as time tends to infinity, indicating that the impacts within the system balance out and the system ceases to evolve further.

Limit of a Function

The limit of a function, particularly as time approaches infinity, is a fundamental concept in calculus used to determine the long-term behavior of a system. When assessing if a steady state exists, one examines the limit of the system’s function; if a finite value is reached, the steady state is that value, otherwise the system may not reach equilibrium.

Exponential Growth

Exponential growth describes a process in which a quantity increases at a rate proportional to its current value, typically represented by functions of the form f(t) = a * e^(kt). In these models, when k is positive, the function tends to infinity as time increases, suggesting that in a system characterized by exponential growth, a steady state is not reached since the growth continues indefinitely.

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