Problem 49 - Essential Calculus Early Transcendentals

Question

\(49-52=\) The line \(y=m x+b\) is called a slant asymptote if \(f(x)-(m x+b) \rightarrow 0\) as \(x \rightarrow \infty\) or \(x \rightarrow-\infty\) because the vertical distance between the curve \(y=f(x)\) and the line \(y=m x+b\) approaches 0 as \(x\) becomes large. Find an equation of the slant asymptote of the function and use it to help sketch the graph. [ For rational functions, a slant asymptote occurs when the degree of the numerator is one more than the degree of the denominator. To find it, use long division to write $$f(x)=m x+b+R(x) / Q(x) ]$$ $$y=\frac{x^{2}}{x-1}$$

Step-by-Step Solution

Answer

The slant asymptote is \( y = x + 1 \).

Step 1: Determine the Degree of Numerator and Denominator

The given function is \( y = \frac{x^2}{x-1} \). The degree of the numerator \( x^2 \) is 2, and the degree of the denominator \( x-1 \) is 1. Since the degree of the numerator is one more than the degree of the denominator, a slant asymptote exists.

Step 2: Perform Polynomial Long Division

Divide \( x^2 \) by \( x-1 \) using polynomial long division:1. Divide the first term of the numerator \( x^2 \) by the first term of the denominator \( x \), resulting in \( x \).2. Multiply \( x \) by \( x - 1 \), giving \( x^2 - x \).3. Subtract \( x^2 - x \) from \( x^2 \), leaving \( x \).4. Bring down \(x\) and repeat: divide \( x \) by \( x \) to get 1.5. Multiply 1 by \( x - 1 \), giving \( x - 1 \).6. Subtract \( x - 1 \) from \( x \), resulting in a remainder of \( 1 \). The division gives: \( \frac{x^2}{x-1} = x + 1 + \frac{1}{x-1} \).

Step 3: Identify the Slant Asymptote Equation

The slant asymptote is the linear part of the polynomial long division without the remainder. From \( \frac{x^2}{x-1} = x + 1 + \frac{1}{x-1} \), the slant asymptote is given by \( y = x + 1 \).

Step 4: Sketch the Graph with the Slant Asymptote

Draw the graph of \( y = \frac{x^2}{x-1} \). As \( x \rightarrow \pm \infty \), the graph approaches the slant asymptote \( y = x + 1 \). Include this slant asymptote as a dashed line on your graph to demonstrate the behavior of the function for large values of \( x \).

Key Concepts

Polynomial Long Division

Polynomial long division is a method used to divide polynomials, much like long division is used with numbers. It's particularly useful when dealing with rational functions and finding slant asymptotes. Here's how it works:

• Write down the dividend (the numerator of the rational function) and the divisor (the denominator of the rational function) in standard form.
• Start by dividing the first term of the numerator by the first term of the denominator. This gives the initial term of the quotient.
• Multiply the entire divisor by this term and subtract the result from the original numerator. This subtraction often yields a new polynomial, which acts as the new numerator.
• Repeat the process with the new numerator, dividing, multiplying, and subtracting until you can't go any further, i.e., the degree of the remainder is less than the degree of the divisor.

The resulting quotient is the polynomial part, and if you're analyzing rational functions, it's often used to help identify features like slant asymptotes. In the given example, we divided \( x^2 \) by \( x-1 \) and found \( x + 1 \) as the result, with a remainder of \(1\). The quotient \( x + 1 \) represents the slant asymptote.

Rational Functions

Rational functions are expressions of the form \( \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials. These functions can be quite complex but have certain identifiable characteristics:

• The domain of rational functions excludes values that make the denominator zero.
• Zeros and poles, or undefined points, often indicate where the function changes behavior.
• Rational functions can have vertical, horizontal, or slant asymptotes depending on the degrees of the numerator and denominator.

When examining a rational function like \( y = \frac{x^2}{x-1} \), we observe that the degree of the numerator (2) is one more than the degree of the denominator (1). This indicates the presence of a slant asymptote, which can be found using polynomial long division. The resulting quotient, \( x + 1 \), highlights the path the function will approach as \( x \) reaches very large or very small values.

Graphing Asymptotes

Understanding how to graph asymptotes is crucial when analyzing the behavior of rational functions. Asymptotes are lines that the graph of the function approaches but never quite reaches. They provide significant insight into the long-term behavior of the function. Here's how you can identify and graph them:

• Vertical asymptotes occur where the denominator of the rational function is zero (i.e., undefined parts of the function). They are usually represented as vertical dashed lines.
• Horizontal asymptotes are found when the degree of the denominator is equal to the degree of the numerator. For slant asymptotes, however, the degree of the numerator is one more than that of the denominator.
• To graph a slant asymptote, use the linear equation obtained from polynomial long division. Draw this line as a dashed line across your graph, indicating that as \( x \) approaches infinity (positive or negative), the function gets closer to this line.

In our example, the slant asymptote is \( y = x + 1 \). When plotting the graph of \( y = \frac{x^2}{x-1} \), draw this line to visually show the tendency of the graph to align along it, as \( x \) becomes very large or very small.