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

Short answer

Answer: The investment function does not have a steady-state value because the limit of the function as \(t\) approaches infinity is infinity.

Step-by-step solution

Identify the function

The system is represented by the function \(v(t)=1000 e^{0.065 t}\). We need to determine the steady-state value of this function, which means we need to find the limit as \(t\) approaches infinity.

Calculate the limit

Determine the limit of the function as \(t\) approaches infinity: \(\lim_{t \rightarrow \infty} v(t) = \lim_{t \rightarrow \infty} 1000 e^{0.065 t}\)

Analyze the exponent

The exponent in the function is \(0.065t\). As \(t\) approaches infinity, the exponent will also go to infinity (\(0.065 * \infty = \infty\)).

Analyze the exponential term

Since the exponent goes to infinity, then the exponential function \(e^{0.065t}\) will go to infinity as well, because \(e^x\) is a continuous and strictly increasing function in the domain of \((-\infty,\infty)\). Therefore, \(e^{\infty}=\infty\).

Determine the limit

Since the exponential term goes to infinity, the whole function will go to infinity: \(\lim_{t \rightarrow \infty} 1000 e^{0.065 t} = 1000 \cdot \infty = \infty\)

Conclusion

The system does not reach a steady state, because the limit of the function as \(t\) approaches infinity is infinity.

Key concepts

Limits at Infinity

When we talk about limits at infinity in calculus, we are interested in understanding the behavior of a function as the input value grows without bound. Think of it as following a road that stretches out endlessly; you want to see where it ultimately leads. For a function like v(t) = 1000e^{0.065t}, we looked at where it goes as t heads off towards infinity. In our exercise, we found that as t gets larger and larger, v(t) grows without limitation, heading towards infinity. This means that the value of the investment increases continuously over time and does not settle at a finite number, indicating no steady state.

Exponential Growth

Exponential growth is characterized by an increase that becomes more rapid proportional to the growing total number or size. In the context of the investment function v(t) = 1000e^{0.065t}, the base of the exponential, e (Euler's number), is a clear indicator of this type of growth. Here's the fascinating part: the rate of growth (0.065) is constant, but the amount it grows by over equal time intervals increases. For instance, the investment's value does not just add a fixed amount each year; it multiplies by a larger and larger amount due to the compound interest effect in this case. This relentless multiplication means that, theoretically, the investment grows indefinitely over time.

Equilibrium in Calculus

In calculus, when we identify an equilibrium or a steady state, we're pinpointing a stable condition where a function doesn't change as time goes on. It's like a sailboat reaching a point where the forces of the wind and the resistance of the water balance perfectly, so the boat sails smoothly without accelerating. For the investment function in our exercise, we sought the steady state by looking for the limit of v(t) as t approaches infinity. Unfortunately, the exponential nature of the function meant that the investment value continued to rise indefinitely, never achieving a steady state. This is crucial for financial models, where recognizing such unbounded growth is essential for predicting long-term trends.

Continuous Functions

Continuous functions are the mathematical equivalent of a smooth, uninterrupted line with no breaks, jumps, or gaps. You can think of it as drawing a line without lifting your pencil off the paper. For a function to be continuous, at every point within its domain, the function's value (limit) as we approach that point from either direction must be equal to the function's value at that point. The investment function we discussed, v(t) = 1000e^{0.065t}, is a prime example of a continuous function. Since exponentials never hit a sudden break or sharp turn, you can always know where an investment stands at any moment in time, which is helpful for predicting future values. Continuous functions like this allow for thorough and precise mathematical analysis, which underpins concepts like derivatives and integrals.