Problem 32 - Calculus: Graphical, Numerical, Algebraic

Question

In Exercises \(27-34,\) (a) find the vertical asymptotes of the graph of \(f(x) .(\) b) Describe the behavior of \(f(x)\) to the left and right of each vertical asymptote. $$f(x)=\sec x$$

Step-by-Step Solution

Answer

The vertical asymptotes of the secant function are at \(x = (2n+1)\pi/2\), where n is any integer. On the intervals \((-2n\pi -\pi/2, -2n\pi +\pi/2)\) and \((2n\pi +\pi/2, 2n\pi +3\pi/2)\), the secant function approaches positive infinity, and on the intervals \((-2n\pi +\pi/2, -2n\pi +3\pi/2)\) and \((2n\pi -\pi/2, 2n\pi +\pi/2)\), the secant function approaches negative infinity.

Step 1: Identify the secant function

The secant function, denoted as \(\sec x\), is the reciprocal of the cosine function. It is defined as \(\sec x = 1/ \cos x\).

Step 2: Determine the vertical asymptotes

The vertical asymptotes of the secant function are located where the cosine function equals to zero. This occurs at \(x = (2n+1)\frac{\pi}{2}\), where n is any integer.

Step 3: Describe the behavior

On the intervals \((-2n\pi -\pi/2, -2n\pi +\pi/2)\) and \((2n\pi +\pi/2, 2n\pi +3\pi/2)\), the secant function increases without bound (i.e., approaches positive infinity). On the intervals \((-2n\pi +\pi/2, -2n\pi +3\pi/2)\) and \((2n\pi -\pi/2, 2n\pi +\pi/2)\), the secant function decreases without bound (i.e., approaches negative infinity). This is the behavior to the left and right of each vertical asymptote.

Key Concepts

Secant Function

The secant function is one of the primary trigonometric functions, and it's expressed as \( \sec x \). It is the reciprocal of the cosine function, meaning \( \sec x = \frac{1}{\cos x} \). Because it relies on the cosine function, its domain excludes values where the cosine of \( x \) equals zero. These values cause the secant function to be undefined, leading to vertical asymptotes. Understanding the secant function helps in analyzing trigonometric graphs and their behaviors.

Reciprocal Function

A reciprocal function is one where a given function is flipped over its identity line. For the secant function, this involves the cosine function. If you recall, when one takes the reciprocal of a number, you essentially flip the fraction. The reciprocal of a non-zero number \( a \) is \( \frac{1}{a} \).
This concept of reciprocal is key to understanding the secant function, as it directly influences where this function is undefined. Since division by zero is not allowed, wherever \( \cos x = 0 \), the reciprocal function \( \sec x \) becomes undefined, resulting in vertical asymptotes.

Trigonometric Functions

Trigonometric functions are fundamental in mathematics, originating from the relationships within triangles. They are commonly used to describe oscillations, wave patterns, and periodic phenomena. The primary trigonometric functions include sine, cosine, and tangent, with their reciprocal functions being cosecant, secant, and cotangent.
The secant function, being the reciprocal of cosine, shares its periodic nature but differs in domain and behavior due to its asymptotic properties. Grasping trigonometric functions is essential for studying full behaviors of waves and periodic functions in various fields such as physics and engineering.

Asymptotic Behavior

Asymptotic behavior describes how functions behave as they get closer to their undefined points or infinity. For the secant function, this behavior is observed around vertical asymptotes, occurring where the cosine function is zero. This is particularly important for understanding the graph of \( \sec x \) as it approaches infinity.

• To the left and right of a vertical asymptote, \( \sec x \) either increases without bound (approaches \( +\infty \)) or decreases without bound (approaches \( -\infty \)).
• The vertical asymptotes occur at \( x = (2n+1)\frac{\pi}{2} \), where \( n \) is any integer.

This knowledge of asymptotic behavior is crucial for interpreting the complete graph and nature of the secant function.