Problem 32 - Essential Calculus Early Transcendentals

Question

Use the guidelines of this section to sketch the curve. $$y=\sec x+\tan x, \quad 0< x <\pi / 2$$

Step-by-Step Solution

Answer

The curve increases from near 1 at \(x=0\) to infinity as \(x\) approaches \(\frac{\pi}{2}\) with a vertical asymptote there.

Step 1: Understand the Functions Involved

The function we're dealing with is the sum of the secant and tangent functions, \(y = \sec x + \tan x\). In trigonometry, \(\sec x\) is the reciprocal of \(\cos x\) and \(\tan x\) is \(\frac{{\sin x}}{{\cos x}}\). Both functions have a vertical asymptote at \(x = \frac{\pi}{2}\), which causes them to go to infinity as \(x\) approaches \(\frac{\pi}{2}\) from the left.

Step 2: Identify the Domain and Behavior

The given domain is \(0 < x < \frac{\pi}{2}\). Both \(\sec x\) and \(\tan x\) are undefined at \(x = \frac{\pi}{2}\), causing a vertical asymptote there. As \(x\) approaches \(\frac{\pi}{2}\) from the left, both \(\sec x\) and \(\tan x\) go to positive infinity.

Step 3: Consider the Combined Function

The function is \(y = \sec x + \tan x\). For small values of \(x\) close to 0, \(\sec x\) is approximately 1 and \(\tan x\) is approximately \(x\). As \(x\) increases toward \(\frac{\pi}{2}\), both \(\sec x\) and \(\tan x\) increase rapidly, causing \(y\) to also increase rapidly.

Step 4: Calculate Values at Key Points

Calculate a few values to understand the behavior:- At \(x = \frac{\pi}{6}\), \(\sec(\frac{\pi}{6}) = \frac{2}{\sqrt{3}}\) and \(\tan(\frac{\pi}{6}) = \frac{1}{\sqrt{3}}\).- At \(x = \frac{\pi}{4}\), \(\sec(\frac{\pi}{4}) = \sqrt{2}\) and \(\tan(\frac{\pi}{4}) = 1\).Use these values to sketch a rough plot of the curve behavior.

Step 5: Sketch the Curve

Based on the calculated points and asymptote, sketch the curve starting near (0, 1) and increasing through the calculated values towards infinity as \(x\) approaches \(\frac{\pi}{2}\). Ensure the plot shows the increase and indicates the vertical asymptote.

Key Concepts

Secant Function

The secant function, denoted by \( \sec x \), is one of the six fundamental trigonometric functions. It is defined as the reciprocal of the cosine function. In mathematical terms, this means \( \sec x = \frac{1}{\cos x} \). When plotting the secant function, it is important to remember a few key characteristics:

• The secant function is undefined whenever the cosine function is zero, which occurs at odd multiples of \( \frac{\pi}{2} \).
• Due to this, the secant function has vertical asymptotes at those points.
• The function reaches its maximum and minimum values when the cosine is at its maximum and minimum, respectively.

Because \( \sec x \) tends to infinity as \( x \) approaches its vertical asymptote, it is crucial to consider this behavior when sketching curves that involve \( \sec x \). Particularly, for the range \( 0 < x < \frac{\pi}{2} \), \( \sec x \) increases rapidly as \( x \) nears \( \frac{\pi}{2} \).

Tangent Function

The tangent function, represented by \( \tan x \), is another basic trigonometric function. It's defined as the ratio of the sine and cosine functions, so \( \tan x = \frac{\sin x}{\cos x} \). Here are some important aspects to consider when dealing with the tangent function:

• It is periodic with a period of \( \pi \).
• The tangent function is undefined where the cosine is zero, similar to the secant function, leading to vertical asymptotes at those points.
• The function traverses from negative to positive infinity across each period without reaching a maximum or minimum bound.

For the specified range \( 0 < x < \frac{\pi}{2} \), the tangent function starts from zero and increases towards infinity as it approaches the asymptote at \( x = \frac{\pi}{2} \). This behavior has a significant impact on any combined functions involving \( \tan x \). Being aware of these properties is essential for accurate curve sketching.

Curve Sketching

Curve sketching involves drawing a graph based on the behavior of a function. When sketching the curve for combined functions like \( y = \sec x + \tan x \), it is important to incorporate the characteristics of each individual function as well as their collective behavior:

• Identify and mark vertical asymptotes and note any undefined points in the domain.
• Understand how each component function behaves on its own. As noted earlier, both \( \sec x \) and \( \tan x \) increase rapidly towards \( \frac{\pi}{2} \).
• Combine these behaviors by calculating the sum at specific key points to determine how the combined function behaves.

For example, for \( y = \sec x + \tan x \), both parts add together and push the curve upwards rapidly as \( x \) nears \( \frac{\pi}{2} \). Identifying points of interest, such as \( x = \frac{\pi}{4} \), helps in understanding how the graph unfolds. Here, sketching starts near \( (0,1) \) and moves upwards through calculated values, ultimately showing an increase to infinity, accurately reflecting both functions' contributions.