Question

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 \(v(t)=1000 e^{0.065 t}\)

Short answer

If so, what is the steady-state value? Answer: No, the given investment function does not have a steady state, as the value of the investment increases indefinitely over time.

Step-by-step solution

Identify the function and variable

The function representing the value of the investment is given by \(v(t)=1000 e^{0.065 t}\), with \(t\) as the time in years.

Determine the limit as time approaches infinity

We need to find the limit of the function as time approaches infinity: \(\lim_{t\to\infty} v(t) = \lim_{t\to\infty} 1000 e^{0.065 t}\).

Apply properties of limits and exponential functions

Since the exponential function \(e^{0.065 t}\) grows without bound as \(t\) goes to infinity, the limit does not exist. Therefore, there is no steady-state value for this investment as the value keeps increasing indefinitely.

Conclusion

For the given investment function \(v(t)=1000 e^{0.065 t}\), no steady state exists, as the value of the investment keeps increasing indefinitely over time.

Key concepts

Limits to Infinity

In calculus, the concept of limits is fundamental for analyzing the behavior of functions as they approach a particular point or infinity. When investigating the long-term behavior of time-varying systems, engineers and scientists often look at the limits to infinity. This involves finding the value that a function approaches as the independent variable (often time, t) goes to infinity.

For instance, if we are considering the growth of an investment over time, we might use limits to infinity to determine whether the investment's value will settle at a certain amount, known as a steady state, or if it will continue to increase or decrease without bound. Unfortunately, when a limit to infinity does not result in a finite number, the system does not reach a steady state. In our example of the function v(t)=1000e^{0.065t}, it was concluded that the steady-state value does not exist since the value of the investment grows without bound as time marches on to infinity.

Exponential Functions

Exponential functions are powerful mathematical expressions involving constants raised to a variable exponent. In the context of time-varying systems, exponential functions like v(t)=1000e^{0.065t} often describe scenarios where growth or decay happens at a rate that is proportional to the current value.

The base of the exponential function, in this case, 'e', is an important mathematical constant approximately equal to 2.71828, known as Euler's number. It appears frequently in the natural descriptions of growth and decay processes, such as population growth, radioactive decay, and, as in our textbook problem, financial investments.

The key characteristic of exponential growth is that the rate of increase becomes faster over time. This is reflected in the investment example, where the value of the investment increases more rapidly as time passes. This feature is what leads to the absence of a limit, and thus no steady-state value when time approaches infinity.

Time-Varying Systems

Time-varying systems are dynamic systems whose behavior changes over time. They are encountered in various fields, such as physics, engineering, biology, and economics. The behavior of time-varying systems can often be modeled using mathematical functions which encapsulate how system variables evolve with time.

In the example of the investment value v(t), the exponential function encapsulates a financial model where the value increases continually. In a time-varying system like this, reaching a stable value (steady state) is significant because it implies a predictable, constant outcome after a long period. However, not all systems reach such an equilibrium; some grow indefinitely or oscillate without settling down. Understanding whether a steady state exists and determining the steady-state value are critical for long-term planning and analysis in such scenarios. In the given problem, it is clear that no steady state is achieved, meaning that anyone invested in this particular financial product should expect its value to keep rising as time goes on.