Question

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

Short answer

Answer: The vertical asymptotes of the function $$f(x) = \frac{1}{\sqrt{x}sec(x)}$$ occur at $$x = \frac{\pi}{2}, \frac{3\pi}{2}, \frac{5\pi}{2}, \ldots$$

Step-by-step solution

Determine the Domain of the Square Root of x

The square root of x is only defined for non-negative values of x (i.e., x ≥ 0). So, we'll consider only non-negative values of x for our analysis.

Determine the Domain of the Secant of x

The secant function, $$sec(x)$$, is equal to $$\frac{1}{cos(x)}$$. It is undefined when cos(x) = 0. The values of x for which cos(x) = 0 are $$x = \frac{\pi}{2} + k\pi$$, where k is an integer. Since we're considering non-negative values of x, we can write all the undefined values of the secant function as: $$x = \frac{\pi}{2}, \frac{3\pi}{2}, \frac{5\pi}{2}, \ldots$$

Find the Vertical Asymptotes

Now, we have the domains of the square root of x and the secant of x. To find the vertical asymptotes, we'll combine the undefined values of both functions in the denominator. In this case, those values are the ones when the secant function is undefined, which we found in step 2. The vertical asymptotes of the given function are as follows: $$x = \frac{\pi}{2}, \frac{3\pi}{2}, \frac{5\pi}{2}, \ldots$$

Key concepts

Analytical Methods in Calculus

The application of analytical methods in calculus is integral to identifying the behavior of functions, including their limits and asymptotes. Through analysis, we study functions by looking at their algebraic structure, using limits to understand their behavior as they approach certain points. Analytical methods involve setting conditions, testing hypotheses, and deducing properties of functions. For instance, to find vertical asymptotes of a function, we investigate where the function's denominator equals zero—a point where the function is undefined but the limit approaches infinity.

In the exercise, analytical methods are used to combine the domains of the square root and secant functions. Identifying where the secant function, which corresponds to \(1/ \cos(x)\), is undefined requires analyzing the cosine function's properties. From this, we learn that vertical asymptotes occur at points where the cosine function equals zero, applying the concept that vertical asymptotes are typically at inputs where the function becomes unbounded.

Domain of a Function

The domain of a function defines the set of all possible inputs (x-values) for which the function is valid, or the outputs (y-values) are defined. A function's domain is crucial to understanding its behavior since it tells us where the function can and cannot compute values. For example, the square root function is only defined for non-negative x-values—as square roots of negative numbers are not real. Similarly, certain trigonometric functions have restricted domains due to their periodic properties.

When analyzing the function \(f(x)=\frac{1}{\sqrt{x} \sec x}\), we consider the domains of both the square root and secant functions. The exclusion of negative numbers for the square root and the points where cosine is zero for the secant determine where the composite function is defined or undefined, hence influencing its domain.

Secant Function

The secant function, denoted as \(\sec(x)\), is a trigonometric function and the reciprocal of the cosine function, represented by \(\sec(x) = \frac{1}{\cos(x)}\). Unlike the more commonly known sine and cosine functions, the secant function has a graph that consists of alternating curves with undefined points—these correspond to the angles where the cosine equals zero.

Characteristics of the Secant Function

• Periodic with a period of \(2\pi\)
• Has no maximum or minimum values
• Has vertical asymptotes at \(x = \frac{\pi}{2} + k\pi\), with \(\bi{k\)) being any integer

In the context of the exercise, we focus on the undefined values of the secant function, specifically where \(\cos(x) = 0\), to determine the vertical asymptotes of \(f(x)\).

Vertical Asymptote Identification

Vertical asymptotes are lines to which a function's value approaches infinity (or negative infinity) as the input gets closer to a specific value. Identifying vertical asymptotes is crucial when analyzing the behavior of functions, particularly around their discontinuities. Asymptotes indicate that there's a kind of 'boundary' that the function will approach but never cross or reach.

To identify vertical asymptotes analytically, we look for values where the function becomes undefined—typically where the denominator of a rational function reaches zero while the numerator remains non-zero. In trigonometric functions, like the secant function, vertical asymptotes also arise at angles where the function is inherently undefined. The exercise demonstrates this by showing that the vertical asymptotes of \(f(x)=\frac{1}{\sqrt{x} \sec x}\) occur at values of x where \(\sec(x)\) is undefined—those being integer multiples of \(\pi\) away from \(\frac{\pi}{2}\).