Problem 53 - Essential Calculus Early Transcendentals

Question

Show that the curve \(y=x-\tan ^{-1} x\) has two slant asymptotes: \(y=x+\pi / 2\) and \(y=x-\pi / 2 .\) Use this fact to help sketch the curve.

Step-by-Step Solution

Answer

The curve has slant asymptotes at \( y = x + \frac{\pi}{2} \) and \( y = x - \frac{\pi}{2} \).

Step 1: Understand the Concept of Slant Asymptotes

A slant asymptote is a line that a curve approaches as the independent variable (often denoted as \( x \)) approaches positive or negative infinity. For the given curve \( y = x - \tan^{-1}x \), we must analyze the behavior of \( y \) as \( x \to \infty \) and \( x \to -\infty \).

Step 2: Analyze Behavior of \( y \) as \( x \to \infty \)

As \( x \to \infty \), the inverse tangent function \( \tan^{-1}x \to \frac{\pi}{2} \). Therefore, the expression for \( y \) becomes \( y = x - \tan^{-1}x \approx x - \frac{\pi}{2} \). This means the slant asymptote for large positive \( x \) is \( y = x - \frac{\pi}{2} \).

Step 3: Analyze Behavior of \( y \) as \( x \to -\infty \)

As \( x \to -\infty \), the inverse tangent function \( \tan^{-1}x \to -\frac{\pi}{2} \). Therefore, the expression for \( y \) becomes \( y = x - \tan^{-1}x \approx x + \frac{\pi}{2} \). This means the slant asymptote for large negative \( x \) is \( y = x + \frac{\pi}{2} \).

Step 4: Confirm the Slant Asymptotes

Both behaviors confirmed that as \( x \to \infty \), the curve approaches \( y = x - \frac{\pi}{2} \), and as \( x \to -\infty \), the curve approaches \( y = x + \frac{\pi}{2} \). Thus, these are indeed the slant asymptotes of the curve.

Step 5: Sketch the Curve With Asymptotes

To sketch the curve, draw the lines for the asymptotes \( y = x + \frac{\pi}{2} \) and \( y = x - \frac{\pi}{2} \). Sketch the behavior of \( y = x - \tan^{-1}x \) such that it approaches these lines asymptotically, i.e., getting closer to these lines as \( x \to \pm\infty \), without ever touching them.

Key Concepts

Slant Asymptotes

Slant asymptotes occur when the graph of a function approaches a straight line, which isn't necessarily horizontal or vertical, as the input value, typically denoted as \( x \), heads to infinity or minus infinity. They are often observed in rational functions where the degree of the polynomial in the numerator is exactly one higher than that in the denominator. However, any function can have slant asymptotes if certain conditions are met at infinity.
For the function \( y = x - \tan^{-1}x \), slant asymptotes are determined by examining the behavior as \( x \to \infty \) and \( x \to -\infty \). As \( x \to \infty \), the inverse tangent \( \tan^{-1}x \) approaches \( \pi/2 \), leading the function toward the line \( y = x - \pi/2 \). As \( x \to -\infty \), \( \tan^{-1}x \) approaches \( -\pi/2 \), steering the function towards \( y = x + \pi/2 \). Thus, these lines are the slant asymptotes for the given function.

Inverse Trigonometric Functions

Inverse trigonometric functions, like \( \tan^{-1}x \), allow us to determine angles given trigonometric ratios. They are the inverse functions of the trigonometric ratios: sine, cosine, and tangent.
The function \( \tan^{-1}x \), specifically, transforms a numeric input back into an angle, effectively yielding the angle whose tangent is \( x \). The range of \( \tan^{-1}x \) is usually restricted between \( -\pi/2 \) and \( \pi/2 \) to ensure it is a true function—single-valued for any given input. Understanding how \( \tan^{-1}x \) behaves as \( x \to \pm \infty \) is crucial for identifying asymptotic behaviors, particularly in this context of curve analysis as it controls how the values approach their limits.

Curve Sketching

Curve sketching involves depicting a graph's behavior through the understanding of its equation without actually plotting innumerable points. The key steps to effective curve sketching include identifying intercepts, recognizing symmetries, plotting critical points like maxima or minima, and understanding long-term behavior such as asymptotes.
For the curve given by \( y = x - \tan^{-1}x \), sketching involves drawing the slant asymptotes, \( y = x + \pi/2 \) and \( y = x - \pi/2 \), as references. These lines help us understand how the curve behaves far off the y-axis. From there, by plotting a few strategic points and knowing it approaches but never crosses these asymptotes, a rough accurate depiction can be formed. This process provides a snapshot of the curve's overall trajectory and behavior.

Limits at Infinity

Limits at infinity are crucial for grasping how functions behave as the input grows indefinitely large or small. In calculus, evaluating limits at infinity helps in determining horizontal or slant asymptotes and understanding a function's end-behavior.
The expression \( y = x - \tan^{-1}x \) prompts the analysis of limits. As \( x \to \infty \), \( \tan^{-1}x \to \pi/2 \) leading to a limit of \( x - \pi/2 \). Conversely, as \( x \to -\infty \), \( \tan^{-1}x \to -\pi/2 \), resulting in \( x + \pi/2 \). These results contribute directly to identifying the slant asymptotes discussed earlier. Understanding these limits allows us to predict and visualize how the function will behave along the infinite horizon.