Problem 49 - Calculus with Concepts in Calculus

Question

Plot the graph of \(f\) and note all asymptotes. Then use the zoom features of the calculator to approximate the relative extreme values.. $$ f(x)=\frac{1}{x}+\frac{1}{x-1}+\frac{1}{x-2} $$

Step-by-Step Solution

Answer

The vertical asymptotes are at \(x = 0, 1, 2\); horizontal asymptote is at \(y = 0\). Use graphing tools to approximate relative extremes.

Step 1: Identify the Function

The given function is \( f(x) = \frac{1}{x} + \frac{1}{x-1} + \frac{1}{x-2} \). We need to consider the behaviour of each term in the function separately to identify any asymptotes.

Step 2: Recognize Vertical Asymptotes

The vertical asymptotes occur where the function is undefined, which is at the values of \(x\) that make the denominator zero. For \( \frac{1}{x} \), the asymptote is at \( x = 0 \); for \( \frac{1}{x-1} \), the asymptote is at \( x = 1 \); and for \( \frac{1}{x-2} \), the asymptote is at \( x = 2 \).

Step 3: Plot the Graph

Use a graphing calculator or software to plot \( f(x) = \frac{1}{x} + \frac{1}{x-1} + \frac{1}{x-2} \). Observe the behavior of the graph near the vertical asymptotes at \( x = 0 \), \( x = 1 \), and \( x = 2 \).

Step 4: Identify Horizontal Asymptotes

Consider the limits as \( x \to \pm\infty \). Since each term tends to zero as \( x \to \infty \), the horizontal asymptote of \( f(x) \) is \( y = 0 \).

Step 5: Use Zoom to Find Relative Extremes

Use the zoom feature of the graphing calculator to closely inspect the graph for any relative minima or maxima between the vertical asymptotes. Specifically, check intervals like \((-\infty, 0)\), \((0, 1)\), \((1, 2)\), and \((2, \infty)\).

Step 6: Approximate Relative Extrema

Through zooming in on different segments of the graph and using the calculator's trace or extreme finding functions, approximate the coordinates of any relative minima or maxima. Make note of these points as the relative extreme values of the function.

Key Concepts

Vertical Asymptotes

Vertical asymptotes are lines where a function becomes undefined. These often occur when the denominator of a fraction is zero. For the function \( f(x) = \frac{1}{x} + \frac{1}{x-1} + \frac{1}{x-2} \), each term contributes a vertical asymptote.
The term \( \frac{1}{x} \) makes the function undefined at \( x = 0 \), resulting in a vertical asymptote there.
Similarly, the term \( \frac{1}{x-1} \) leads to an asymptote at \( x = 1 \), and \( \frac{1}{x-2} \) causes an asymptote at \( x = 2 \).
The graph rapidly rises or drops near these points, never touching the line, forming a vertical asymptote. This behavior is visible upon graphing.

• Vertical asymptote at \( x = 0 \)
• Vertical asymptote at \( x = 1 \)
• Vertical asymptote at \( x = 2 \)

Recognizing these asymptotes is essential for understanding the behavior of the function graph.

Horizontal Asymptotes

Horizontal asymptotes indicate the value that a function approaches as \( x \) moves towards infinity or negative infinity. In the case of the function \( f(x) = \frac{1}{x} + \frac{1}{x-1} + \frac{1}{x-2} \), each component term \( \frac{1}{x}, \frac{1}{x-1}, \frac{1}{x-2} \) approaches zero as \( x \) tends to either positive or negative infinity.
This implies that the overall function \( f(x) \) trends closer to \( y = 0 \) as \( x \) becomes very large in magnitude.
Therefore, the graph of \( f(x) \) flattens out, settling near the horizontal line \( y = 0 \) at the extremities.
Understanding this trend helps identify the overall behavior of the function for very large values of \( x \).

• Horizontal asymptote at \( y = 0 \) as \( x \to \pm\infty \)

Relative Extrema

Relative extrema are points where a function reaches local maximum or minimum values. These extrema are not necessarily the absolute highest or lowest points on the graph, but rather are peaks or troughs relative to surrounding values.
To find these in the function \( f(x) = \frac{1}{x} + \frac{1}{x-1} + \frac{1}{x-2} \), one could use the graph's zoom function to inspect smaller intervals between and around vertical asymptotes.
Zooming in helps detect dips (minima) or bumps (maxima) that aren't visible on a broader scale.

• Check between each pair of vertical asymptotes, e.g., intervals like \((-fty, 0)\), \((0, 1)\), \((1, 2)\), and \((2, fty)\).
• Use the calculator's trace or extrema functions to find precise relative minimum or maximum values.

These relative extrema provide critical points indicating how the function behaves locally across its domain.

Graphing Functions

Graphing functions is a valuable tool for visualizing mathematical equations. It helps illustrate key features of a function like asymptotes and extrema. For the function \( f(x) = \frac{1}{x} + \frac{1}{x-1} + \frac{1}{x-2} \), graphing is crucial to understanding its behavior near asymptotes.
This function will have sharp increases or decreases near its vertical asymptotes at \( x = 0, 1, \) and \( 2 \), with a general flattening towards \( y = 0 \) for larger values of \( x \).
By graphing, you can distinguish how these asymptotes influence the direction and steepness of the graph.
To observe subtle features, use the zoom feature on calculators or graphing software. This helps find and focus on relative decisions and anomalies not obvious at first glance.

• Plot the function using a graphing tool to accurately observe behavior at and between asymptotes.
• Adjust the viewing window and zoom for clearer insight into relative extrema.

Graphing facilitates a better comprehension of how a function behaves in different parts of its domain.