Question

In each exercise, a function \(f(t)\) is given. In Exercises 28 and 29, the symbol \(\llbracket u \rrbracket\) denotes the greatest integer function, \(\llbracket u \rrbracket=n\) when \(n \leq u

Short answer

If so, find the values for \(M\) and \(a\). Answer: Yes, the function \(f(t) = \cosh 2t\) is continuous on the interval \(0 \leq t < \infty\) and exponentially bounded with constants \(M = 1\) and \(a = 2\).

Step-by-step solution

Analyze the continuity of \(f(t)\)

To check whether \(f(t)\) is continuous on \(0 \leq t < \infty\), we first need to recall the definition of the cosine hyperbolic function: $$ \cosh x = \frac{e^x + e^{-x}}{2} $$ Now, we can rewrite \(f(t)\) as: $$ f(t) = \cosh 2t = \frac{e^{2t} + e^{-2t}}{2} $$ Since the exponential functions are continuous on their domain, the cosine hyperbolic function is also continuous on its domain, which is \(-\infty < t < \infty\). Therefore, \(f(t)\) is continuous on \(0 \leq t<\infty\).

Analyze if \(f(t)\) is exponentially bounded

To check if \(f(t)\) is exponentially bounded, we need to find constants \(M\) and \(a\) such that \(|f(t)| \leq M e^{at}\) for all \(0 \leq t < \infty\). Considering the definition of \(f(t) = \cosh 2t\) and that the hyperbolic cosine is always non-negative, we have: $$ f(t) = \frac{e^{2t} + e^{-2t}}{2} \leq \frac{e^{2t} + e^{2t}}{2} = e^{2t} $$ So, we can choose \(M = 1\) and \(a = 2\). Thus, \(f(t)\) is exponentially bounded, and we have \(|f(t)| \leq e^{2t}\) for \(0 \leq t<\infty\). In conclusion, the function \(f(t) = \cosh 2t\) is continuous on \([0,\infty)\) and is exponentially bounded with constants \(M = 1\) and \(a = 2\).

Key concepts

Greatest Integer Function

The greatest integer function, denoted by \(\llbracket u \rrbracket\), is a way to round down any real number \(u\) to the nearest integer less than or equal to \(u\). This function can be seen as a "step function," moving down to the previous whole number.

• If \(u = 3.7\), then \(\llbracket u \rrbracket = 3\).
• If \(u = -2.3\), then \(\llbracket u \rrbracket = -3\).
• If \(u\) is already an integer, \(\llbracket u \rrbracket = u\).

These characteristics yield step-like graphs with jumps at integer values. Each jump results in a discontinuity. Understanding this step function helps in identifying how functions behave at these integer boundaries, important for piecewise continuity analysis.

Continuity

Continuity in a function means that the graph of the function is unbroken and you can sketch it without lifting your pencil. In mathematical terms, a function \(f(x)\) is continuous at a point \(x = c\) if:

• \(f(c)\) is defined.
• The limit of \(f(x)\) as \(x\) approaches \(c\) from both sides exists.
• The limit of \(f(x)\) as \(x\) approaches \(c\) is equal to \(f(c)\).

For the whole domain (such as \([0, \infty) \)), the function must be continuous at every point within that domain. In the context of the original problem, \(f(t) = \cosh 2t\) is continuous because hyperbolic functions (like exponential functions) are inherently smooth across their entire domain. There are no breaks, jumps, or holes, satisfying all conditions for continuity.

Exponential Bounds

Exponential bounds are crucial in determining if a function's growth is manageable over an infinite interval. We say a function \(f(t)\) is exponentially bounded if there exist constants \(M\) and \(a\) such that:\[|f(t)| \leq M e^{at} \]for all relevant \(t\), typically \(0 \leq t < \infty\). This defines a ceiling on how rapidly \(f(t)\) can grow relative to the exponential function \(e^{at}\). In the original problem, \(f(t) = \cosh 2t\) is demonstrated to be uniformly bounded by another exponential function, indicating controlled growth. By identifying \(M = 1\) and \(a = 2\), one can ensure that \(f(t)\)'s values stay below \(e^{2t}\) as \(t\) increases, keeping the behavior in check and easily predictable.

Hyperbolic Functions

Hyperbolic functions, such as sine hyperbolic \(\sinh\) and cosine hyperbolic \(\cosh\), are analogs of the standard trigonometric functions but for hyperbolas. These are particularly useful in various areas of mathematics, including differential equations and complex analysis. Defined as:\[\cosh x = \frac{e^x + e^{-x}}{2} \\sinh x = \frac{e^x - e^{-x}}{2}\]These functions share symmetric properties and identities similar to their trigonometric counterparts. For example, \(\cosh 2t\) in the original problem forms a smooth, continuous curve. Unlike trigonometric functions which oscillate, hyperbolic functions showcase exponential growth. Understanding these functions helps solve differential equations involving exponential terms, as they naturally model rapid growth in systems and processes.