Problem 60 - University Calculus: Early Transcendentals

Question

Use your graphing utility. Graph the rational function \(y=\left(2-x^{2}\right) / x^{2} .\) Then graph \(y=\) \(\cos \left(2 \sec ^{-1} x\right)\) in the same graphing window. What do you see? Explain.

Step-by-Step Solution

Answer

The graphs differ significantly, showing different behaviors and domains, particularly note asymptotes and undefined sections.

Step 1: Understanding the Rational Function

The function \( y = \frac{2 - x^2}{x^2} \) is a rational function where the numerator is \( 2 - x^2 \) and the denominator is \( x^2 \). It may have vertical asymptotes at points where the denominator is zero, particularly at \( x = 0 \).

Step 2: Graph the Rational Function

Use a graphing utility to plot the function \( y = \frac{2 - x^2}{x^2} \). You should observe the behavior of the graph near \( x = 0 \) and note any asymptotes or intercepts. The graph will extend towards infinity as it approaches the vertical asymptote at \( x = 0 \), and it will cross the y-axis at the point where \( y = 2 \).

Step 3: Understanding the Trigonometric Function

The function \( y = \cos(2 \sec^{-1} x) \) involves inverse trigonometric functions. The inverse secant function, \( \sec^{-1}(x) \), means that its domain is \( x \leq -1 \) or \( x \geq 1 \).

Step 4: Graph the Trigonometric Function

On the same graphing utility, plot the function \( y = \cos(2 \sec^{-1} x) \). Observe its behavior, especially how it aligns with or diverges from the rational function's graph. Note the range of \( x \) values where the graph exists.

Step 5: Analyzing the Graphs Together

In the graphing window, both functions should appear. You may observe points where the two graphs intersect or run parallel. Note if their behavior is significantly different or if they possess any symmetry, particularly around vertical lines or asymptotes.

Key Concepts

Rational Functions

Rational functions are fractions where both the numerator and the denominator are polynomials. In our problem, the rational function is given by \(y = \frac{2 - x^2}{x^2}\). This means:

• The numerator is \(2 - x^2\), which is a polynomial of degree 2.
• The denominator is \(x^2\), also a polynomial of degree 2.

The degree of the numerator and the denominator can influence the shape of the graph. A key feature to look for in rational functions is the presence of vertical asymptotes, which occur at values of \(x\) that make the denominator zero—in this case, \(x = 0\). As \(x \) approaches these values, the graph tends to rise or fall sharply, hence creating an asymptote.

Understanding these basic components helps to predict the overall behavior of the function graphically.

Trigonometric Functions

Trigonometric functions are essential in various fields of mathematics and sciences. In this scenario, we're dealing with the cosine function combined with an inverse function, specifically \(y = \cos(2 \sec^{-1} x)\). The distinct aspect of this function is the alteration of a trigonometric identity by an inverse function, meaning:

• The secant function's range impacts the applicability of the graph; \(\sec^{-1}(x)\) is only defined for \(|x| \ge 1\).
• Once the angle is determined by \(\sec^{-1}(x)\), it is doubled as indicated by \(2 \sec^{-1}(x)\) before the cosine is taken.

This construction leads to unique wave patterns when plotted against the rational function, which may show specific intersection points due to their respective ranges or amplitudes.

Vertical Asymptotes

Vertical asymptotes are vital in understanding rational functions. They occur where the function tends to infinity, specifically when the denominator is zero. For the function \(y = \frac{2 - x^2}{x^2}\), a vertical asymptote appears at \(x = 0\). This means:

• As \(x\) approaches zero from either the left or the right, the function's value will increase or decrease infinitely.
• This results in dividing the graph into different regions where the function's behavior changes dramatically at \(x=0\).

Graphically, it's represented by a line that the graph approaches but never actually touches. Understanding vertical asymptotes gives insight into the graph's overall behavior and also helps in comparing it to other function types.

Inverse Trigonometric Functions

Inverse trigonometric functions allow us to determine angles from known trigonometric ratios. In our case, the function \(y = \cos(2 \sec^{-1} x)\) includes the inverse secant \(\sec^{-1}(x)\). It's crucial to note:

• The principal values for \(\sec^{-1}(x)\) are defined for \(|x| \ge 1\), translating into angles that contribute to \(\cos(2 \theta)\) once computed.
• In this graphing exercise, the double angle formula makes things complex, as \(2 \sec^{-1}(x)\) effectively stretches and compresses the resulting cosine wave.

These nuances often result in complex intersections or parallel trajectories when comparing against other plotted graphs like rational functions, challenging students to analyze and understand how trigonometric inversions interact with algebraic graphs.