Problem 65 - Calculus: Early Transcendentals

Question

Asymptotes Use analytical methods and/or a graphing utility to identify the vertical asymptotes (if any) of the following functions. $$f(x)=\frac{1}{\sqrt{x} \sec x}$$

Step-by-Step Solution

Answer

Question: Determine the vertical asymptotes of the function $$f(x)=\frac{1}{\sqrt{x} \sec x}$$. Answer: The function has no vertical asymptotes.

Step 1: Find the denominator and simplify

The denominator of the given function is $$\sqrt{x}\sec x$$. Before continuing, let's simplify this by rewriting it as: $$\sqrt{x}\frac{1}{\cos x}$$. Now our denominator is $$\frac{\sqrt{x}}{\cos x}$$.

Step 2: Determine values of x for which the denominator is zero

Our goal now is to find the x values that will make the denominator zero. In order for the denominator to equal zero, we know that either $$\sqrt{x}$$ or $$1/\cos x$$ must be zero. But, note that \(\sqrt{x}\) is never equal to zero for any positive value of x. Therefore, we will focus on finding values when $$1/\cos x=0$$. However, $$1/\cos x$$ (secant function) does not have any real root. Therefore, the function has no vertical asymptotes.

Key Concepts

Asymptotes

In the study of calculus, asymptotes are of significant importance. They represent lines that a graph approaches as the inputs (or 'x' values) either increase without bound or decrease without bound—essentially as they head towards infinity or negative infinity.

There are three types of asymptotes—horizontal, vertical, and oblique (or slant). For this article, we'll focus on vertical asymptotes, which occur at certain values of 'x' where a function is undefined and the graph of the function shoots up towards positive or negative infinity.

Mathematically, to find vertical asymptotes, we look for values of 'x' that make the denominator of a fraction equal to zero since division by zero is undefined. In the expression \( \frac{1}{{\sqrt{x} \sec x}} \), to find vertical asymptotes, we must find the 'x' values for which the denominator—\(\sqrt{x}\sec x\)—becomes zero. It's a crucial concept because it shows us the limitations and behavior of functions at certain critical points.

Graphing Utility

Graphing utilities are vital tools in mathematics, specifically for visualizing the behavior of functions. With them, we can easily identify features like intercepts, extrema, and importantly, asymptotes.

In an educational setting, a graphing utility takes the abstraction out of mathematical functions and gives students a more intuitive understanding of the subject. When asked to find the vertical asymptotes of a function like \(f(x)=\frac{1}{{\sqrt{x} \sec x}}\), a graphing utility can help by enabling us to plot the function and see visually where the graph approaches infinity.

Using a graphing utility, we reinforce the analytical method by verifying that there are no vertical asymptotes, as suggested by the algebraic investigation. It greatly aids in learning as it provides a visual confirmation of the mathematical conclusions reached.

Secant Function

The secant function, denoted as \(\sec x\), is a trigonometric function that is the reciprocal of the cosine function. Specifically, it's defined as \(\sec x = \frac{1}{\cos x}\). This function is undefined when the cosine of 'x' is zero since division by zero is not possible.

In terms of graphing, the secant function has its own distinct features and behavior. For instance, it has vertical asymptotes at locations where \(\cos x = 0\). In the given function \(f(x)=\frac{1}{{\sqrt{x} \sec x}}\), though it might initially seem that vertical asymptotes would occur at these points, we must consider the domain of the square root function as well, which only allows for non-negative values of 'x'.

Understanding when and where the secant function will have vertical asymptotes contributes to a deeper grasp of how trigonometric functions behave within more complex functions, as in the given exercise.