Problem 29 - Essential Calculus Early Transcendentals

Question

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

Step-by-Step Solution

Answer

Sketch the curve with points at the origin, symmetry, and vertical asymptotes at \(-\pi/2\) and \(\pi/2\).

Step 1: Identify the Function

The function we are working with is given by the equation \(y = x \tan x\). This is a product of \(x\) and \(\tan x\).

Step 2: Determine the Domain of the Function

The given interval for \(x\) is \(-\pi/2 < x < \pi/2\). Since \(\tan x\) becomes undefined at \(x = -\pi/2\) and \(x = \pi/2\), this matches the domain we need to consider for sketching the curve.

Step 3: Consider Symmetry and Intercepts

Determine if the function is even, odd, or neither. Since \(y = x \tan x\) involves a product of \(x\) (odd) and a function (\(\tan x\)) with an odd domain, \(y\) is an odd function. Therefore, it is symmetric with respect to the origin. Since the function is undefined at the endpoints, analyze the intercepts by checking where \(x \tan x = 0\): this occurs at \(x = 0\), giving us the intercept \((0,0)\).

Step 4: Consider Asymptotic Behavior

Since \(\tan x\) approaches \(+\infty\) as \(x\) approaches \(\pi/2\) from the left and \(-\infty\) as \(x\) approaches \(-\pi/2\) from the right, the function \(y = x \tan x\) will also exhibit this asymptotic behavior on these intervals. Hence, there are vertical asymptotes at \(x = \pi/2\) and \(x = -\pi/2\).

Step 5: Analyze Critical Points and Derivatives

Take the derivative of the function \(y = x \tan x\): \(y' = \tan x + x \sec^2 x\). Setting the derivative to zero gives \(\tan x + x \sec^2 x = 0\). Solve this equation for critical points. This involves numerical approximation or graphical methods, as analytical solutions may be complex.

Step 6: Sketch the Curve with Information

Combine all gathered information to sketch the curve: identify symmetry about the origin, plot the intercept at \((0,0)\), and mark the vertical asymptotes at \(x = -\pi/2\) and \(x = \pi/2\). Indicate the tendency of the function to rise and fall sharply as \(x\) approaches these asymptotes.

Key Concepts

Derivative Analysis

The role of derivatives in calculus is crucial for understanding how a function behaves at various points. For the function \(y = x \tan x\), we can better understand its rate of change by analyzing its derivative, \(y' = \tan x + x \sec^2 x\). Calculating the derivative involves using both the product rule and the knowledge of the derivatives of trigonometric functions.

When we set \(y' = 0\), it helps us find the critical points of the function. Critical points are where the slope of the tangent to the curve is zero, indicating a potential maximum, minimum, or point of inflection. Solving \(\tan x + x \sec^2 x = 0\) can be challenging analytically, so numerical or graphical methods are often used to approximate such points.

• The critical points help us understand where the function might change its increasing or decreasing behavior.
• Knowing the signs of the derivative helps determine intervals where the function is increasing or decreasing.

By combining the analysis of derivatives with other techniques, we gain a comprehensive understanding of the function's behavior.

Asymptotic Behavior

Asymptotic behavior describes how a function behaves as it approaches a certain value, often leading to unbounded behavior. For the function \(y = x \tan x\), the asymptotic behavior is important near the points where \(\tan x\) becomes undefined—specifically, at \(x = -\pi/2\) and \(x = \pi/2\).

• As \(x\) approaches \(\pi/2\) from the left, \(\tan x\) tends towards \(+\infty\), pushing \(y = x \tan x\) towards \(+\infty\) as well.
• Conversely, as \(x\) approaches \(-\pi/2\) from the right, \(\tan x\) heads towards \(-\infty\), making the function \(y = x \tan x\) drop to \(-\infty\).

This behavior indicates vertical asymptotes at those points, where the graph will become infinitely large in magnitude. Recognizing these vertical asymptotes is crucial in producing an accurate curve sketch.

Function Symmetry

Function symmetry is a key component in understanding how a function behaves across its domain. It involves checking whether a function is even, odd, or neither. For the function \(y = x \tan x\), identifying its symmetry can simplify the process of sketching the graph.

• An even function is symmetric about the y-axis, satisfying \(f(x) = f(-x)\).
• An odd function is symmetric about the origin, meaning \(f(-x) = -f(x)\).

In this case, the function \(y = x \tan x\) is a product of two odd functions, \(x\) and \(\tan x\). Thus, it satisfies the condition for being an odd function.

This symmetry about the origin simplifies analysis as it requires studying only half of the curve, then reflecting that behavior through the origin to understand the entire function. This symmetry helps in better visualizing the function's behavior and facilitates a more efficient approach to curve sketching.