Problem 54 - Essential Calculus Early Transcendentals

Question

Show that the curve \(y=\sqrt{x^{2}+4 x}\) has two slant asymptotes: \(y=x+2\) and \(y=-x-2 .\) Use this fact to help sketch the curve.

Step-by-Step Solution

Answer

The curve \(y=\sqrt{x^2+4x}\) has slant asymptotes \(y=x+2\) and \(y=-x-2\).

Step 1: Understand the Curve Equation

The curve is given as \(y = \sqrt{x^2 + 4x}\). To find slant asymptotes, we should analyze the behavior of the function as \(x\) approaches infinity or negative infinity.

Step 2: Factor Inside the Square Root

Let's rewrite \(x^2 + 4x\) as \((x+2)^2 - 4\). This decomposition will help in simplifying expressions later. This turns the function into \(y = \sqrt{(x+2)^2 - 4}\).

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

For large \(x\), the expression \((x+2)^2 - 4\) simplifies approximately to \((x+2)^2\). Thus, \(y \approx \sqrt{(x+2)^2} = |x+2|\). For large positive \(x\), \(|x+2| = x+2\). Hence, one asymptote is \(y = x + 2\).

Step 4: Analyze Behavior as \(x \to -\infty\)

Similarly, for large negative \(x\), \(|x+2| = -(x+2)\). So \(y \approx -x-2\). Therefore, the other asymptote is \(y = -x - 2\).

Step 5: Sketch the Curve

The curve approaches the line \(y = x + 2\) as \(x \to \infty\) and the line \(y = -x - 2\) as \(x \to -\infty\). To sketch the curve, note that it will lie above \(y = 0\) for all real \(x\), and it will transition smoothly between the two asymptotes as \(x\) ranges from \(-\infty\) to \(\infty\).

Key Concepts

Asymptotic Behavior

Asymptotic behavior gives us a glimpse into how a function behaves as the input becomes very large or very small. In this case, the function is given by \( y = \sqrt{x^2 + 4x} \). Identifying asymptotes helps understand the general direction in which the curve heads.

• A slant asymptote typically occurs when, for a rational function, the degree of the polynomial in the numerator is one more than the degree of the polynomial in the denominator.
• For the given equation, directionality as \( x \to \infty \) helps discover the slant asymptotes: \( y = x + 2 \) as \( x \to \infty \) and \( y = -x - 2 \) as \( x \to -\infty \).
• This indicates that as \( x \) becomes very large (positively or negatively), the curve will closely follow these linear paths.

Recognizing such asymptotic behavior is useful in sketching the curve, providing an understanding of the trend as opposed to the precise path.

Curve Sketching

When sketching curves, understanding their asymptotic behavior is fundamental. It helps define the visual boundary and overall direction. The curve described by \( y = \sqrt{x^2 + 4x} \) serves as a valuable case.

• First, identify the vertical and horizontal tends. This is crucial as they assist in predicting behaviors across the plane.
• Since the function includes a square root, the results must be non-negative. Therefore, the graph will remain above the \( y = 0 \) line.
• The transition between slant asymptotes \( y = x + 2 \) and \( y = -x - 2 \) requires a smooth path. The curve will arc smoothly, starting above the more negative asymptote and changing direction as it heads toward the more positive one.

Remember, as part of curve sketching, the function's roots and any sharp turning points or inflections must be acknowledged, even as they might not be explicitly tackled with this function.

Limit Analysis

Limit analysis is the mathematical approach to understanding the behavior of a function as the input approaches a certain value. In this context, we're interested in the values as \( x \to \infty \) or \( x \to -\infty \).

• By expressing \( x^2 + 4x \) as \((x+2)^2 - 4\), and looking at the limit as \( x \to \infty \), the equation simplifies to \( y \approx x + 2 \).
• Similarly, analyzing as \( x \to -\infty \) results in \( y \approx -x - 2 \). This simplification helps identify asymptotic trends.
• Such limits are instrumental in pinpointing where the function is heading and gauging the function's end behavior.

Performing a limit analysis rests at the core of not only understanding slant asymptotes but also anticipating how these mathematical behaviors will translate onto the graph. Recognizing these key trends is crucial for solving complex mathematical tasks effectively.