Question

Let \(f\) be piecewise continuous on \([0, \infty)\). If \(\lim _{t \rightarrow \infty} f(t) e^{-a t}=0\) for some constant \(\alpha\), prove that \(f\) is of exponential order.

Short answer

Question: Prove that the given function \(f(t)\) on the interval \([0, \infty)\) is of exponential order using the given limit condition: \(\lim_{t \rightarrow \infty} f(t) e^{-at} = 0\). Answer: The given function \(f(t)\) is of exponential order because we can find constants \(M > 0\) and \(\beta > 0\) such that \(|f(t)| \leq M e^{\beta t}\) for all \(t \geq 0\). We can choose \(M = \max \{\epsilon e^{aT}, \sup_{t \in [0, T]} |f(t)|\}\) and \(\beta = a\), where \(T > 0\) and \(\epsilon > 0\) are values such that \(|f(t)e^{-at}|< \epsilon\) when \(t \geq T\). This satisfies the definition of exponential order, proving that \(f\) has exponential order.

Step-by-step solution

Examine the given condition

We know that \(\lim_{t \rightarrow \infty} f(t) e^{-at} = 0\). By definition of limit, for any \(\epsilon > 0\), there exists \(T > 0\) such that for all \(t \geq T\), we have \(|f(t)e^{-at}|< \epsilon\).

Find an upper bound for |f(t)|

For \(t \geq T\), we can write $$ |f(t)| = |f(t) e^{-at} e^{at}| \leq \epsilon e^{at} $$ Where, in the last inequality, we used the fact that \(|f(t) e^{-at}| < \epsilon\) when \(t \geq T\).

Define constants M and β

Choose \(M = \max \{\epsilon e^{aT}, \sup_{t \in [0, T]} |f(t)|\}\) and \(\beta = a\). Now, we show that these constants satisfy the definition of exponential order.

Prove the function is of exponential order

For \(t \in [0, T]\), we have $$ |f(t)| \leq \sup_{t \in [0, T]} |f(t)| \leq M $$ For \(t \geq T\), we have $$ |f(t)| \leq \epsilon e^{aT} \leq M $$ In both cases, we have \(|f(t)| \leq Me^{\beta t}\). Therefore, by definition, \(f\) is of exponential order.

Key concepts

Exponential Order

Exponential order is a term used to describe the growth behavior of functions, specifically how fast or slow a function grows as its input becomes very large. A function \( f(t) \) is said to be of exponential order \( a \) if there exist positive constants \( M \) and \( \beta \) such that for large values of \( t \), the inequality \( |f(t)| \leq Me^{\beta t} \) holds.

This means that beyond a certain point, the function grows at a rate that does not exceed that of an exponential function \( e^{\beta t} \).
This property is crucial in understanding the long-term behavior of solutions to differential equations.

Piecewise Continuous Functions

A piecewise continuous function is a function that is continuous on specific intervals within its domain, except at a finite number of points where it may have jumps or discontinuities.

For our function \( f(t) \) defined on \([0, \infty)\), this means that \( f(t) \) is continuous within each subinterval, though it might have isolated points where it jumps.

• These functions are often used in mathematical modeling where systems have operations that vary at distinct points.
• They are particularly common in applications involving switching mechanisms or where conditions change at set times.

Most importantly, piecewise continuity allows piecewise-defined functions to be analyzed using various calculus and differential equation techniques, essential for solving real-world problems.

Limit Definition

Limits are a fundamental concept in calculus that describe the behavior of functions as inputs approach a given point - often infinity in discussing infinite intervals.

In our exercise, we are considering the limit \( \lim_{t \rightarrow \infty} f(t) e^{-at} = 0 \). This implies that as \( t \) becomes very large, the product \( f(t)e^{-at} \) approaches zero. This behavior underlies the conclusion that \( f(t) \) is of exponential order.

• Given any small number \( \epsilon > 0 \), there exists some point \( T \) beyond which the absolute value \(|f(t)e^{-at}|\) will remain less than \( \epsilon \), showcasing the 'tending to zero' behavior.

This notion of limits is instrumental in determining long-term properties of functions, especially in the field of differential equations.

Upper Bound

An upper bound is a value that a function does not exceed as it grows infinitely large. In terms of proving exponential order, finding an appropriate upper bound is key.

In our context, \(|f(t)| \leq Me^{\beta t}\) serves as an upper bound for \( f(t) \). This means that beyond a point, the function's magnitude is limited by an exponential curve.

• The constants \( M \) and \( \beta \) are chosen such that they reflect the maximum manageable size of \( f(t) \) as \( t \) increases.
• This assures that even if \( f(t) \) is increasing, its growth is controllable, mirrored by exponential terms.

Such bounds help in simplifying complex functions and are particularly useful for mathematicians and engineers in predicting the long-term behavior of systems described by differential equations.