Problem 18 - Calculus - AP Edition

Question

Show that \(y=3-x^{2}\) is a curvilinear asymptote of the graph of \(f(x)=\left(2+3 x-x^{3}\right) / x .\) Sketch the graph of \(y=\) \(f(x)\) showing this asymptotic behavior.

Step-by-Step Solution

Answer

The function \( f(x) \) has the curvilinear asymptote \( y = 3 - x^2 \).

Step 1: Determine the Type of Asymptote

Our goal is to show that the function \( f(x) = \frac{2 + 3x - x^3}{x} \) has the curvilinear asymptote \( y = 3 - x^2 \). A curvilinear asymptote refers to a function \( y = g(x) \) that \( f(x) \) approaches as \( |x| \to \infty \).We will therefore investigate the behavior of \( f(x) \) for large \( |x| \), and check if \( f(x) \approx g(x) \), where \( g(x) = 3 - x^2 \).

Step 2: Simplify the Function

First, simplify the given function:\[ f(x) = \frac{2 + 3x - x^3}{x} = \frac{2}{x} + 3 - x^2 \]As \( x \rightarrow \pm \infty \), we note that \( \frac{2}{x} \to 0 \). Thus, the dominant terms in \( f(x) \) as \( |x| \) becomes large are:\[ f(x) \approx 3 - x^2 \]indicating that \( y = 3 - x^2 \) is the curvilinear asymptote.

Step 3: Confirm Asymptotic Behavior

Next, check that as \( x \to \infty \) and \( x \to -\infty \), the discrepancy between \( f(x) \) and \( g(x) = 3-x^2 \) tends towards zero:\[ f(x) - (3 - x^2) = \frac{2}{x} \]As \( x \to \pm \infty \), the term \( \frac{2}{x} \to 0 \). Therefore, \( f(x) \) approaches \( g(x) = 3 - x^2 \), confirming \( y = 3 - x^2 \) as the curvilinear asymptote.

Step 4: Sketch the Graph

To sketch the graph, highlight the curvilinear asymptote \( y = 3 - x^2 \) and the behavior of \( f(x) \):1. Draw the parabola \( 3 - x^2 \), a downward-opening parabola shifted up 3 units.2. Show \( f(x) \) approaching this parabola from above as \( x \to \infty \) and below as \( x \to -\infty \), due to the diminishing term \( \frac{2}{x} \).

Key Concepts

Function Behavior at Infinity

When analyzing the behavior of functions, understanding their behavior at infinity is crucial. As the variable \(x\) becomes very large or very small (approaching positive or negative infinity), it's essential to observe how the function \(f(x)\) behaves. To determine if a function has a curvilinear asymptote like \(y = 3 - x^2\), one must investigate how \(f(x)\) behaves for large \(|x|\). This involves simplifying the function to better see how its components behave as \(x\) moves away from zero.

For example, in the function \(f(x) = \frac{2 + 3x - x^3}{x}\), by simplifying we find that it approximates \(3 - x^2\) as \(\frac{2}{x} \to 0\). This simplification allows us to deduce that \(f(x)\) behaves like \(3 - x^2\) when \(x\) is very large or very small, showing the curvilinear asymptote behavior.

Graph Sketching

Graph sketching becomes much easier when we understand the asymptotic behavior of a function. The goal is to visually represent important characteristics of the function including asymptotes and general trends.

For the function \(f(x) = \frac{2 + 3x - x^3}{x}\), sketching begins with drawing its asymptotic behavior. Start by plotting the parabola \(y = 3 - x^2\), which opens downwards and is shifted up 3 units. This parabola is predominant in showing where \(f(x)\) approaches as \(|x|\) grows. Then, illustrate how \(f(x)\) approaches this parabola from above as \(x\) increases and from below as \(x\) decreases due to the small term \(\frac{2}{x}\) which vanishes for large \(|x|\).

By incorporating these asymptotic details, we can create a clearer sketch that highlights how the function behaves across its domain.

Asymptotic Analysis

Asymptotic analysis is a mathematical approach used to describe the limiting behavior of a function. When we consider the asymptotes of a function, we're essentially looking at what the function "looks like" as it approaches infinity.

In the context of our function \(f(x) = \frac{2 + 3x - x^3}{x}\), the asymptotic analysis involves breaking down the function and seeing which terms dominate as \(x\) becomes large. By simplifying \(f(x)\) to \(\frac{2}{x} + 3 - x^2\), it becomes evident that the term \(3 - x^2\) will determine the behavior of \(f(x)\) at infinity, as \(\frac{2}{x}\) becomes negligible.

This analysis is crucial in determining that \(y = 3 - x^2\) acts as a curvilinear asymptote, effectively shaping our understanding of the function's behavior in extreme scenarios.