Question

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 whether a steady state exists and give the steady-state value. The amplitude of an oscillator is given by \(a(t)=2\left(\frac{t+\sin t}{t}\right)\)

Short answer

If so, what is the steady-state value? Answer: No, a steady state does not exist for the given amplitude function as time approaches infinity.

Step-by-step solution

Identify the function

The function given for the amplitude of an oscillator is \(a(t)=2\left(\frac{t+\sin t}{t}\right)\), where \(t\) represents time.

Calculate the limit as t approaches infinity

Our goal is to find \(\lim _{t \rightarrow \infty} a(t)=\lim _{t \rightarrow \infty} 2\left(\frac{t+\sin t}{t}\right)\).

Apply L'Hopital's Rule

The limit is of type \(\frac{\infty}{\infty}\) which requires a use of L'Hopital's Rule. L'Hopital's Rule states that if \(\lim_{x \rightarrow c}\frac{f(x)}{g(x)}\) is of indeterminate form \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\), then \(\lim_{x \rightarrow c}\frac{f(x)}{g(x)}\) = \(\lim_{x \rightarrow c}\frac{f'(x)}{g'(x)}\) if the limit on the right-hand side exists. Let us calculate the derivatives of the numerator and the denominator of the fraction: \(\frac{d(t+\sin t)}{dt} = 1 + \cos t\) \(\frac{d(t)}{dt} = 1\) Now apply L'Hopital's Rule: \(\lim _{t \rightarrow \infty} \frac{t+\sin t}{t} = \lim _{t \rightarrow \infty} \frac{1 + \cos t}{1}\)

Evaluate the limit

Now, let's evaluate the limit: \(\lim _{t \rightarrow \infty} \frac{1 + \cos t}{1}\) Since \(\cos t\) oscillates between -1 and 1, the limit does not exist. Therefore, no steady state exists for the given function, and no steady-state value can be determined.

Key concepts

Limit of a Function

Understanding the concept of limits is fundamental in calculus, as it helps us approach the behavior of functions as they head towards a particular point or as time goes to infinity. In our exercise, to find out whether a system reaches a steady state, we look at the limit of the function as time approaches infinity, written mathematically as \(\lim _{t \rightarrow \infty} f(t)\).

If this limit results in a finite number, the system is said to reach a steady state. However, if the limit is infinite, oscillates without a pattern, or doesn't approach a specific value, the system does not settle into a steady state. In the given exercise, the amplitude of an oscillator does not reach a steady state as time goes to infinity due to the oscillatory nature of the \(\cos t\) term within the function.

L'Hopital's Rule

When confronting indeterminate forms such as \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\), L'Hopital's Rule provides an invaluable tool. It allows us to differentiate the numerator and the denominator of the fraction separately and then take the limit.

In the exercise, applying L'Hopital's Rule is crucial since the original limit presents an indeterminate form \(\frac{\infty}{\infty}\). Despite the application of the rule, the limit did not provide a steady state value due to the persistent oscillation of the \(\cos t\) function which doesn't settle to a finite value as time goes to infinity.

System Equilibrium

System equilibrium, or steady state, is a condition where the system's variables do not change as time passes, implying that there's a balance in the system's forces or factors. In calculus, we say that a system reaches equilibrium if the limit of the system's function as time approaches infinity exists and is finite.

The absence of a steady state, as seen in our exercise, indicates that the system will keep changing over time, in this case due to the sinusoidal component of the amplitude function. The concept of equilibrium is essential in many fields, including physics, economics, and engineering, as it signifies stability within the system being studied.

Calculus in System Analysis

Calculus is an invaluable tool in system analysis, allowing us to model and predict the behavior of dynamic systems over time. Through differentiation and integration, we can understand rates of change, accumulation, and the impact of these changes on the system as a whole.

In the context of our problem, calculus enables us to examine the behavior of the amplitude of an oscillator over time. It helps us identify whether the system will eventually stabilize or continue to exhibit dynamic behavior. The non-existence of a limit when applied to our oscillator's amplitude indicates that calculus, while powerful, can also inform us when a system refuses to stabilize, highlighting the importance of analyzing systems through a calculus lens.