Problem 84 - Essential Calculus

Question

In Exercises \(75-86\), use a computer algebra system to analyze the graph of the function. Label any extrema and/or asymptotes that exist. $$ f(x)=\frac{2 \sin 2 x}{x} $$

Step-by-Step Solution

Answer

The function \(f(x)=\frac{2 \sin 2x}{x}\) doesn't have any vertical asymptotes. The x-axis, y = 0, is the horizontal asymptote. The extrema can be found by determining the critical points from the derivative of the function.

Step 1: Analyze the function and find vertical asymptotes

For rational functions like this one, a vertical asymptote can be found wherever the denominator is zero and the numerator is not simultaneously zero. The denominator, x, equals zero at the point x=0. However, at x = 0, sin(2x) will also be zero, so there is no vertical asymptote.

Step 2: Find the limit for horizontal asymptotes

We need to analyze the limit of the function when x approaches infinity or minus infinity. After the calculation, the limit results in zero when x approaches both infinity and minus infinity. Therefore, the x-axis (y = 0) is the horizontal asymptote.

Step 3: Find the extrema

The extrema of the function can be found where its derivative equals to zero or does not exist. Find the derivative of the function and solve for x. Then substitute these values into the original function to get the respective y values. This helps to locate the extrema on the graph of the function.

Key Concepts

Computer Algebra System

A Computer Algebra System (CAS) is a software tool designed to facilitate algebraic calculations. It helps perform complex manipulations on algebraic expressions and can handle tasks such as symbolic integration, differentiation, and solving equations. Using a CAS:

• Speeds up calculations.
• Ensures accuracy.
• Allows for easy manipulation of functions.

In our exercise, a CAS would aid in computing derivatives and limits, both integral to identifying extrema and asymptotes. By inputting the function, many of the labor-intensive and error-prone tasks of hand calculations are reduced, leading to quicker and more reliable results.

Asymptotes

Asymptotes are lines that a graph approaches but never actually reaches. They help illustrate the behavior of a function as it tends towards infinity. There are mainly two types of asymptotes:

• Vertical Asymptotes: These occur when a function tends towards infinity at certain x-values where the denominator of a rational function is zero. However, in our function \(f(x)=\frac{2 \sin 2 x}{x}\), the potential vertical asymptote at x = 0 is nullified as the numerator is also zero at that point.
• Horizontal Asymptotes: These occur when a function tends to a constant value as x goes to infinity or negative infinity. For this function, the horizontal asymptote is y = 0, as the function approaches 0 at both ends of the x-axis.

Understanding asymptotes provides a clearer idea of how a function behaves at its extremities.

Graph Analysis

Graph analysis involves examining the structural features of a function's graph. This includes studying:

• Shape: The symmetry and curvature.
• Intercepts: Points where the graph crosses the axes.
• Asymptotic behavior: As previously discussed.
• Continuity: Whether breaks or gaps exist in the graph.

For the function \(f(x)=\frac{2 \sin 2 x}{x}\), we have already highlighted the lack of vertical asymptotes and the presence of a horizontal asymptote. Intercepts and regions where the graph flattens or rises steeply are crucial for drawing the graph accurately. All these factors together provide a comprehensive picture of a function's graph.

Extrema

Extrema refer to the maximum or minimum points on a function's graph. These points indicate where the function has peaks (maximum) or troughs (minimum). To find extrema:

• Compute the derivative of the function.
• Set the derivative equal to zero to find critical points.
• Substitute these points back into the original function to find the y-values of the extrema.

In our function \(f(x)=\frac{2 \sin 2 x}{x}\), a CAS can simplify solving for extrema by handling derivative calculations swiftly. Knowing where a function peaks or troughs helps illustrate the dynamic nature of its graph.

Rational Functions

Rational functions are functions represented as the ratio of two polynomials. They are characterized by:

• Having potential vertical asymptotes where the denominator is zero.
• Exhibiting polynomial-like behavior but with more complex dynamics due to division.
• Involving limits to define behavior at infinity, often leading to horizontal asymptotes.

Our function \(f(x)=\frac{2 \sin 2 x}{x}\) highlights these complexities, having been simplified to not exhibit typical vertical asymptotes, while still behaving like a rational function with a horizontal asymptote. Understanding rational functions is key to mastering graph behavior and transformations.