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

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Key Concepts

Steady State (Equilibrium)

A steady state, or equilibrium, is reached in a dynamic system when its state variables settle to a constant value as time approaches infinity. This concept is critical in analyzing long-term behavior because it indicates whether the system's output eventually stops changing or continues to evolve indefinitely.

Limit of a Function at Infinity

The limit of a function as time approaches infinity is used to determine the long-term behavior of systems. By evaluating lim(t ? ?) f(t), one can decide if the function converges to a finite value, implying a steady state, or diverges, indicating ongoing growth or decay.

Exponential Growth Behavior

Exponential growth occurs when a function's rate of change is proportional to its current value, leading the function to increase rapidly over time. When the exponent is positive, the value of the function grows without bounds as time increases, which is a key factor in determining the existence of a steady state.

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