Problem 57 - Calculus Early Transcendentals: Pearson New International Edition

Question

Show that \(\cosh x\) is an even function.

Step-by-Step Solution

Answer

\(\cosh x\) is even because \(\cosh(-x) = \cosh(x)\).

Step 1: Understand the Definition of Hyperbolic Cosine

The hyperbolic cosine function, denoted as \(\cosh x\), is defined as \(\cosh x = \frac{e^x + e^{-x}}{2}\). It's important to start with this formula in order to show it's an even function.

Step 2: Find the Expression for \(\cosh(-x)\)

To prove \(\cosh x\) is an even function, calculate \(\cosh(-x)\) using the definition. Substitute \(-x\) into the formula: \(\cosh(-x) = \frac{e^{-x} + e^{x}}{2}\).

Step 3: Simplify \(\cosh(-x)\)

Notice that \(\cosh(-x) = \frac{e^{-x} + e^x}{2}\) is the same as \(\cosh(x) = \frac{e^x + e^{-x}}{2}\). Since addition is commutative, these are indeed equal.

Step 4: Conclude the Even Function Property

Since \(\cosh(-x) = \cosh(x)\) for all \(x\), this satisfies the condition for even functions, which states \(f(-x) = f(x)\). Hence, \(\cosh x\) is an even function.

Key Concepts

Hyperbolic Cosine

The hyperbolic cosine function is a special type of mathematical function. It's named for its resemblance to the cosine function found in trigonometry, but it relates to the hyperbola, not the circle. The notation for hyperbolic cosine is \( \cosh x \). It is expressed with the formula:

• \( \cosh x = \frac{e^x + e^{-x}}{2} \)

Here, \( e \) represents the base of the natural logarithm, also known as Euler's number, which is approximately 2.71828. The formula combines the exponential function \( e^x \) and its inverse \( e^{-x} \).

Using this formula revolves around understanding how exponential growth and decay can combine to form a balanced curve. The resulting graph of \( \cosh x \) starts at 1 when \( x = 0 \) and rises exponentially in both negative and positive directions as \( x \) increases, forming a U-shaped curve.

Despite its roots in hyperbolic functions, \( \cosh x \) appears in disciplines beyond just pure mathematics. It is used in physics for calculating the shape of a hanging cable, known as a catenary, or in engineering models where certain types of heat dissipation or wave equations are solved.

Function Symmetry

Symmetry in functions is all about balance and mirroring. A function can be classified as even if replacing the variable \( x \) with \( -x \) does not change the value of the function. Essentially, a function is even if:

• \( f(-x) = f(x) \) for every \( x \).

One of the key characteristic of even functions is their symmetric graph, which is mirrored along the y-axis. This symmetry is akin to how a perfectly folded piece of paper aligns along the fold. Understanding this property helps in solving various mathematical challenges, including complex equations and integral calculations.

For \( \cosh x \), when we compute \( \cosh(-x) \), we find:

• \( \cosh(-x) = \frac{e^{-x} + e^x}{2} = \cosh x \)

Since \( \cosh(-x) \) equals \( \cosh x \), we can confidently conclude that \( \cosh x \) is an even function. Recognizing symmetry helps simplify understanding of the function's behavior across both sides of the y-axis.

Cosh x Property

The properties of \( \cosh x \) extend beyond just being classified as an even function. Due to its definition through exponentials, several interesting characteristics come into play.

1. Exponential Growth: As \( x \) grows in the positive or negative direction, \( \cosh x \) experiences exponential growth. This results in the U-shaped graph that spans upwards as \( x \) moves away from zero.

2. Behavior at Zero: At \( x = 0 \), the function evaluates to 1, as the exponentials cancel each other out:

• \( \cosh 0 = \frac{e^0 + e^{-0}}{2} = \frac{1 + 1}{2} = 1 \)

This makes \( \cosh x \) a smooth starting point for understanding behavior at or around the origin.

3. Not Limited by Length: Unlike its trigonometric cousin cosine, \( \cosh x \)'s values are not restricted to the width of a circle. This makes it suitable for modeling scenarios where distances or dimensions are virtually unlimited, such as very high or low energy states in physics.

These properties aid in visualizing how \( \cosh x \) can be utilized in real-world scenarios, providing a reliable tool for a range of scientific and mathematical applications.