Problem 30 - Calculus Early Transcendentals

Question

An important question about many functions concerns the existence and location of fixed points. A fixed point of \(f\) is a value of \(x\) that satisfies the equation \(f(x)=x ;\) it corresponds to a point at which the graph off intersects the line \(y=x\). Find all the fixed points of the following functions. Use preliminary analysis and graphing to determine good initial approximations. $$f(x)=\tan \frac{x}{2} \text { on }(-\pi, \pi)$$

Step-by-Step Solution

Answer

Question: Find the fixed point of the function \(f(x) = \tan\frac{x}{2}\) on the interval \((-\pi, \pi)\). Answer: The fixed point of the function on the given interval is \(x = 0\).

Step 1: Identify the function and equation

We are given the function \(f(x) = \tan\frac{x}{2}\) and we need to find the fixed points, i.e., the values of \(x\) such that \(f(x) = x\). Our objective is to solve the equation: $$\tan\frac{x}{2} = x$$

Step 2: Graph the function

Graph the function \(f(x) = \tan\frac{x}{2}\) and the line \(y = x\) on the interval \((-\pi, \pi)\). The points of intersection of these two graphs represent fixed points because they represent the points where \(f(x) = x\).

Step 3: Observe the graph

Look for any values that approximate where the graphs intersect. We can see that the function \(f(x) = \tan\frac{x}{2}\) intersects the line \(y = x\) at approximately \(x = 0\).

Step 4: Find the fixed points

Using our observation from the graph, we know that the fixed point is approximately at \(x = 0\). To find the exact value, plug the approximation into the equation and solve for \(x\): $$\tan\frac{x}{2} = x$$ $$\tan\frac{0}{2} = 0$$ Since the tangent of 0 is 0, we can conclude that \(x=0\) is a fixed point of the function. Thus, the fixed point of the function \(f(x)=\tan\frac{x}{2}\) on the interval \((-\pi,\pi)\) is \(x=0\).

Key Concepts

Graphical Analysis

Graphical analysis is a powerful tool to visually understand functions and their behaviors. It can be particularly helpful in identifying fixed points, which are the solutions where the function equals the variable itself, leading to an intersection on the graph with the line \( y = x \). By plotting \( f(x) = \tan\frac{x}{2} \) on the interval \( (-\pi, \pi) \) alongside the line \( y = x \), we can easily observe where these two graphs meet.

• The graph of \( f(x) = \tan\frac{x}{2} \) will appear as a curve with vertical asymptotes wherever the tangent function becomes undefined.
• The line \( y = x \) is a simple straight line cutting diagonally through the origin.
• The intersection point, which is also the graphical representation of the fixed point of the function within the specific interval, can be directly seen in the plot.

Using graphical analysis allows for preliminary visual approximation of solutions before mathematical validation.

Intersection of Graphs

The intersection of graphs represents the solution to the equation set by aligning your function with a benchmark line, commonly \( y = x \). This intersection is where the function's output equals its input, yielding a fixed point. For \( f(x) = \tan\frac{x}{2} \), this task involves setting \( \tan\frac{x}{2} = x \) and determining where these outputs coincide with the line.

• Start by drawing both graphs—\( \tan\frac{x}{2} \) and \( y = x \)— and look for overlapping areas.
• These intersection points are where the function literally meets itself, indicating a balance between the function value and the input value \( x \).

In the specific problem on the interval \((-\pi, \pi)\), visual inspection identifies the potential fixed points, which analytical methods can subsequently verify more rigorously.

Trigonometric Functions

Trigonometric functions like the tangent function, commonly explored in mathematical problems, have distinctive properties and behaviors. They map angles or arc lengths onto ratios of right triangle sides or points on the unit circle. For \( f(x) = \tan\frac{x}{2} \):

• This function computes the tangent of half of \( x \), modifying the typical periodic behavior and vertical asymptotes of the tangent function.
• The tangent function has vertical asymptotes at every odd multiple of \( \frac{\pi}{2} \). Consequently, \( \tan\frac{x}{2} \) will have these asymptotes stretched to different scales within the interval \( (-\pi, \pi) \).

Understanding these characteristics helps ascertain where the graphs intersect with the line \( y = x \), yielding approximate fixed points. For example, as derived from both the graph and calculations, \( x = 0 \) is where \( \tan\frac{x}{2} = x \), identifying a fixed point.