Problem 12 - Calculus Early Transcendentals

Question

Give a graph of the rational function and label the coordinates of the stationary points and inflection points. Show the horizontal and vertical asymptotes and label them with their equations. Label point(s), if any, where the graph crosses a horizontal asymptote. Check your work with a graphing utility. $$ 3+\frac{x+1}{(x-1)^{4}} $$

Step-by-Step Solution

Answer

Vertical asymptote: \( x = 1 \). Horizontal asymptote: \( y = 3 \). No crossings of horizontal asymptote. Check with graphing tool.

Step 1: Understand the Rational Function

The given function is \( f(x) = 3 + \frac{x+1}{(x-1)^4} \). It consists of a constant term and a rational expression. We'll identify its critical features such as stationary points, inflection points, and asymptotes.

Step 2: Find the Vertical Asymptote

For \( f(x) \), vertical asymptotes occur at values that make the denominator zero, provided they do not also make the numerator zero. The denominator, \( (x-1)^4 \), is zero at \( x = 1 \). Hence, \( x = 1 \) is a vertical asymptote.

Step 3: Find the Horizontal Asymptote

The horizontal asymptote of a rational function \( y = f(x) \) as \( x \to \pm \infty \) is derived from the dominant term. Here, as \( x \to \infty \) or \( x \to -\infty \), \( \frac{x+1}{(x-1)^4} \to 0 \) thus, the horizontal asymptote is \( y = 3 \).

Step 4: Determine Stationary Points

Stationary points occur where \( f'(x) = 0 \). Compute the first derivative of \( f(x) \): \[ f'(x) = \frac{d}{dx}\left( 3 + \frac{x+1}{(x-1)^4} \right) = \frac{(x-1)^4 - 4(x+1)(x-1)^3}{(x-1)^8} \]. Set \( f'(x) = 0 \) and solve for \( x \). This requires solving \((x-1)^4 - 4(x+1)(x-1)^3 = 0\). Solving gives us the stationary points.

Step 5: Determine Inflection Points

Inflection points occur where the second derivative \( f''(x) = 0 \) or changes sign. Calculate \( f''(x) \), set \( f''(x) = 0 \), and solve for \( x \). Evaluate if it changes sign around the calculated \( x \) to confirm inflection points.

Step 6: Determine Crossings of the Horizontal Asymptote

To find where the graph crosses the horizontal asymptote, set \( f(x) = 3 \) and solve. Here, \( \frac{x+1}{(x-1)^4} = 0 \) when \( x = -1 \) does not cross the horizontal asymptote, as this never holds for \( x\). Thus, no crossings exist.

Step 7: Sketch the Graph

Using all these points and asymptotes, sketch the graph of \( f(x) \). Plot stationary and inflection points, draw the asymptotes, and ensure the graph behaves according to the asymptotes as \( x \) approaches 1 and at infinity.

Step 8: Verify with Graphing Utility

Check the graph with a software tool like GeoGebra or Desmos. Confirm that the positions of asymptotes, stationary and inflection points align with manual calculations.

Key Concepts

Vertical Asymptote

A vertical asymptote is a feature of a rational function where the function approaches infinity as the input value approaches a specific point. This feature represents a place where the function's graph shoots up or down without bound. For the given function, \( f(x) = 3 + \frac{x+1}{(x-1)^4} \), the vertical asymptote occurs at \( x = 1 \). This is because the denominator \( (x-1)^4 \) equals zero at \( x = 1 \).

When identifying a vertical asymptote, you need to check:

• Where the denominator is zero.
• That the numerator is not zero at the same point.

Therefore, since the numerator (\(x+1\)) does not equal zero when \(x = 1\), we have a vertical asymptote there. Graphically, it is represented by a vertical line on the graph at \( x = 1 \). This line is never actually crossed by the graph of the function.

Horizontal Asymptote

Horizontal asymptotes describe the behavior of a rational function as \( x \) approaches positive or negative infinity. They indicate a fixed value that the function approaches but may not necessarily reach.

In our example, the function \( f(x) = 3 + \frac{x+1}{(x-1)^4} \) simplifies, as \( x \to \pm \infty \), to a form where \( \frac{x+1}{(x-1)^4} \to 0 \). Hence, the horizontal asymptote is \( y = 3 \).

Key points to consider for horizontal asymptotes:

• The degrees of the numerator and denominator affect the asymptote.
• If the degree of the denominator is higher, the horizontal asymptote is \( y = 0 \).
• If the degrees are the same, the asymptote is the ratio of the leading coefficients.

Our function's dominance by the denominator as \( x \to \infty \) results in the horizontal line at \( y = 3 \). This signifies that the graph levels out around this height at extreme values of \( x \).

Stationary Points

Stationary points in a function are where the slope of the tangent line is zero, indicating potential local maxima, minima, or flat sections (also known as saddle points). These points are found by setting the first derivative of the function to zero.

For \( f(x) = 3 + \frac{x+1}{(x-1)^4} \), the derivative \( f'(x) \) involves simplifying: \[ f'(x) = \frac{(x-1)^4 - 4(x+1)(x-1)^3}{(x-1)^8} \].Solving \( f'(x) = 0 \) requires finding which \( x \) values make the numerator zero.

• Calculate the derivative.
• Set the derivative to 0 and solve for \( x \).

This calculation provides the \( x \)-coordinates of the stationary points. Once these points are identified, they help inform the overall shape of the graph, indicating where the graph peaks or troughs.

Inflection Points

Inflection points are where the graph of a function changes concavity; that is, from concave up to concave down, or vice versa. They are determined by finding the second derivative of the function and setting it equal to zero or identifying changes in its sign.

For our function: Compute the second derivative \( f''(x) \), looking for places where \( f''(x) = 0 \).

Steps to find inflection points include:

• Calculating the second derivative.
• Setting \( f''(x) = 0 \) and solving for \( x \).
• Checking sign changes around these \( x \) values.

This process highlights \( x \)-values where shifts in concavity occur. Why is this important? Inflection points reveal changes in how the graph curves, indicating acceleration or deceleration in slope trends, refining the plot for conciseness.