Problem 54 - Essential Calculus Early Transcendentals

Question

Show that if \(a \neq 0\) and \(b \neq 0\) , then there exist numbers \(\alpha\) and \(\beta\) such that \(a e^{x}+b e^{-x}\) equals either \(\alpha \sinh (x+\beta)\) or \(\alpha \cosh (x+\beta)\) . In other words, almost every function of the form \(f(x)=a e^{x}+b e^{-x}\) is a shifted and stretched hyperbolic sine or cosine function.

Step-by-Step Solution

Answer

\(f(x) = a e^{x} + b e^{-x}\) can be expressed as \(\alpha \cosh(x + \beta)\) for certain \(\alpha\) and \(\beta\).

Step 1: Recall Hyperbolic Definitions

Recall the definitions of hyperbolic sine and cosine: \[ \sinh(x) = \frac{e^x - e^{-x}}{2} \] \[ \cosh(x) = \frac{e^x + e^{-x}}{2} \] These definitions will help you express the linear combination of exponential terms in terms of hyperbolic functions.

Step 2: Match with Hyperbolic Function Formulas

Consider expressing \(f(x) = a e^{x} + b e^{-x}\) in terms of \(\cosh(x)\) and \(\sinh(x)\):Notice that: \[ a e^{x} + b e^{-x} = \frac{a+b}{2}(e^x + e^{-x}) + \frac{a-b}{2}(e^x - e^{-x}) \] This can be rewritten using hyperbolic functions as: \[ a e^{x} + b e^{-x} = \left(\frac{a+b}{2}\right) \cosh(x) + \left(\frac{a-b}{2}\right) \sinh(x) \]

Step 3: Introduce Parameters \(\alpha\) and \(\beta\)

To express this as either \(\alpha \cosh(x + \beta)\) or \(\alpha \sinh(x + \beta)\), recognize that a linear combination of \(\sinh(x)\) and \(\cosh(x)\) can be expressed as a single hyperbolic function:By trigonometric identities, any expression \( C \cosh(x) + D \sinh(x) \) can take the form of \( E \cosh(x + F) \) where the parameters \(E\) and \(F\) involve \(C\) and \(D\). Use: \[ \alpha = \sqrt{ \left(\frac{a+b}{2}\right)^2 + \left(\frac{a-b}{2}\right)^2 } \]\[ \beta = \tanh^{-1}\left(\frac{a-b}{a+b}\right) \] This will account for the scaling and shifting in \(\cosh(x + \beta)\).

Step 4: Conclude with General Form

With \(\alpha\) and \(\beta\) defined, \(f(x) = a e^{x} + b e^{-x}\) can be written as a hyperbolic function of the form:\[ f(x) = \alpha \cosh(x + \beta) \] This shows that, for non-zero \(a\) and \(b\), the function is indeed a shifted and stretched hyperbolic cosine function. You can similarly explore that it can take the form of a \(\sinh\) function with appropriately chosen parameters.

Key Concepts

Hyperbolic Sine

The hyperbolic sine function, denoted as \(\sinh(x)\), is one of the fundamental hyperbolic functions. It is defined using exponential functions as follows:\[ \sinh(x) = \frac{e^x - e^{-x}}{2} \]. This function can be thought of as a version of the sine function, but it operates in a different, "hyperbolic" context. Unlike the traditional sine function, \(\sinh(x)\) is not bounded between -1 and 1. Instead, as the value of \(x\) increases, \(\sinh(x)\) approaches infinity.
Let's break down why this function behaves as it does. The key lies in its definition: it uses exponential growth \((e^x)\) and decay \((e^{-x})\) to create a unique pattern.

• When \(x\) is positive, \(e^x\) grows swiftly, making \(\sinh(x)\) positive.
• When \(x\) is negative, \(e^x\) is small, but \(e^{-x}\) takes a larger value, making \(\sinh(x)\) negative.

This trait is what gives the hyperbolic sine its characteristic "S" shape, with it hugging the x-axis closely near the origin before rising sharply.
Understanding \(\sinh(x)\) is crucial for comprehending more complex expressions that involve hyperbolic functions, such as the shifted and stretched variants in the related exercise.

Hyperbolic Cosine

The hyperbolic cosine, denoted as \(\cosh(x)\), complements the hyperbolic sine function. Defined as \[ \cosh(x) = \frac{e^x + e^{-x}}{2} \], it involves both exponential growth and decay terms.
Unlike \(\sinh(x)\), the hyperbolic cosine function is always positive and includes no imaginary component. Its graph resembles a U-shape, symmetrically situated with respect to the y-axis. At \(x = 0\), \(\cosh(x)\) starts at 1, and as \(x\) moves towards either positive or negative infinity, \(\cosh(x)\) increases exponentially.

• The symmetry of \(\cosh(x)\) is due to the way it incorporates \(e^{-x}\), offering a balance that the \(\sinh\) function does not.
• This function has real-world applications, such as modeling the shape of a hanging cable or chain, known as a catenary.

In the context of the exercise, \(f(x)\), which is written as a sum of \(e^x\) and \(e^{-x}\), can be easily expressed using \(\cosh(x)\). The ability to represent these exponential expressions using \(\cosh(x)\) and \(\sinh(x)\) demonstrates the strength of hyperbolic functions in mathematical modeling.

Exponential Functions

Exponential functions are the building blocks of both hyperbolic sine and cosine functions. The general form for exponential functions is \(e^x\), where \(e\) is approximately equal to 2.718. This function is defined for all real numbers and plays a significant role in calculus and complex analysis due to its properties.
Here are some defining aspects of exponential functions:

• Exponential Growth: As \(x\) increases, \(e^x\) grows very rapidly.
• Exponential Decay: Similarly, as \(x\) decreases, \(e^{-x}\) goes towards zero.

In the context of hyperbolic functions, these two exponential behaviors are combined. \(e^x - e^{-x}\) defines the hyperbolic sine \(\sinh(x)\), reflecting the net growth, while \(e^x + e^{-x}\) forms \(\cosh(x)\), reflecting a balanced contribution of both growth and decay.
Understanding how these exponential functions combine allows us to recompute any expression involving \(a e^x + b e^{-x}\) into either a \(\sinh(x)\) or \(\cosh(x)\) form, giving rise to the utility of hyperbolic functions in solving complex mathematical problems.