Problem 82 - Calculus of a Single Variable

Question

Analyzing a Graph Using Technology In Exercises \(75-82,\) 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 \(f(x) = \frac{2 \sin 2x}{x}\) has extrema which appear as peak points on the plotted graph and also has a vertical asymptote at \(x=0\) which is evident as the function approaches positive or negative infinity when \(x\) approaches 0.

Step 1 - Identification of Function Components

Recognize the function \(f(x)=\frac{2 \sin 2x}{x}\) to be analyzed, which is a combination of the sine function (a periodic function) and a rational function.

Step 2 - Plot the Function

Use a computer algebra system to plot the function \(f(x)=\frac{2 \sin 2x}{x}\). Make sure to carefully adjust the domain and range to clearly observe the behavior of the function.

Step 3 - Identify and Label Extrema

Search for the maximum and minimum points on the graph. These are extrema. Label them on the plot.

Step 4 - Identify and Label Asymptotes

Notice that when \(x\) approaches 0, the function goes to infinity or minus infinity. This is due to the rational function part. So the vertical asymptote is at \(x=0\). Label this on the plot.

Key Concepts

Computer Algebra System

Computer Algebra Systems (CAS) are incredibly powerful tools that allow us to perform complex mathematical operations using technology. When it comes to analyzing graphs of functions like \( f(x)=\frac{2 \sin 2x}{x} \), CAS can be indispensable. Using CAS, students can easily plot and adjust the graph of a function. This helps in visualizing how different components of the function interact with one another, such as the oscillatory behavior of the sine function and the rational aspect represented here. Key functions of CAS include:

• Graph plotting and manipulation: Adjust the domain and range to highlight important features.
• Automatic calculation: Compute derivatives, integrals, or other features to understand function behavior.
• Navigational controls: Use zooming and panning to focus on regions of interest for detailed analysis.

CAS not only simplifies calculation but also enhances comprehension of complex functions, making it a staple in graph analysis exercises.

Extrema Identification

Identifying extrema involves determining the high and low points on a graph, known as maximums and minimums, respectively. For the function \( f(x)=\frac{2 \sin 2x}{x} \), the task of finding extrema is crucial for understanding its overall behavior. When the graph is created using a Computer Algebra System, you can look for these extrema points:

• Local Maximum: Peaks where the function goes up then down in a small region.
• Local Minimum: Valleys where the function goes down then up in a small region.

Finding extrema typically involves taking the derivative of the function and setting it equal to zero to find critical points. This process is greatly simplified by CAS, which can not only plot these points but also compute them quickly, offering a clear visual representation.

Asymptotes

Asymptotes are lines that the graph of a function approaches but never actually touches. For the function \( f(x)=\frac{2 \sin 2x}{x} \), there is a vertical asymptote at \( x=0 \). Why does this asymptote exist?- As \( x \) approaches zero, the term \( \frac{2 \sin 2x}{x} \) causes the function to potentially go to infinity due to division by a small number (or theoretically zero).- Therefore, the graph shoots up or down indefinitely, which is indicated by the asymptote.In summary, identifying asymptotes helps clarify regions where the function behaves atypically, showcasing limits to the graph even as it stretches indefinitely toward or away from these lines. This insight is gleaned visually and numerically when using CAS.

Sine Function Behavior

Understanding the sine function within the context of graph analysis is crucial. Here, the sine function \( \sin 2x \) dictates the oscillatory nature of \( f(x)=\frac{2 \sin 2x}{x} \).Key points about the sine function behavior in this context:

• Periodic Oscillation: The sine component repeats its values in regular intervals, influencing the repeating peaks and troughs on the graph.
• Amplitude Effects: The coefficient 2 before \( \sin \) alters how far the peaks and valleys of these oscillations occur.

While the sine function adds complexity to \( f(x) \), its regular nature means that using a CAS highlights this periodic pattern over the domain. This helps predict the overall shape and behavior, particularly when combined with the rational division by \( x \), shaping how these oscillations resolve near asymptotes.