Question

Use analytical methods and/or a graphing utility en identify the vertical asymptotes (if any) of the following functions. $$q(s)=\frac{\pi}{s-\sin s}$$

Short answer

Question: Find the approximate values of s where vertical asymptotes occur for the function $$\frac{\pi}{s-\sin s}$$. Answer: After analyzing the given function, we found that there are vertical asymptotes at approximately $$s \approx -8.15, 1.93, 10.90, \dots$$, with an infinite number of solutions for s.

Step-by-step solution

Identify when the denominator equals zero

To find the vertical asymptotes, we need to find the values of \(s\) when the denominator of the function equals zero: $$s - \sin s = 0$$

Solve for s

This equation is difficult to solve analytically, but we can approximate the solutions using a graphing utility. Graph the function \(s - \sin s\) and find the points where it intersects the x-axis. Upon plotting the function, we notice that there are infinitely many solutions to the equation, however, we can list some of the first few approximate solutions: $$s \approx -8.15, 1.93, 10.90, \dots$$

Verify the presence of vertical asymptotes

Now that we have the values of \(s\) where the denominator is zero, we need to confirm that the function approaches infinity at these points. To check this, let's examine the limit of the function as \(s\) approaches each of the identified values: $$\lim_{s\to -8.15}\frac{\pi}{s-\sin s}$$ The function approaches infinity, indicating a vertical asymptote at \(s \approx -8.15\). Similarly, we can check for other values: $$\lim_{s\to 1.93}\frac{\pi}{s-\sin s}$$ The function approaches infinity, indicating a vertical asymptote at \(s \approx 1.93\). $$\lim_{s\to 10.90}\frac{\pi}{s-\sin s}$$ The function approaches infinity, indicating a vertical asymptote at \(s \approx 10.90\).

Summary

Using a graphing utility, we have found several points where the function has vertical asymptotes at approximately \(s \approx -8.15, 1.93, 10.90, \dots\). However, since the equation \(s - \sin s = 0\) has an infinite number of solutions, it is important to note that there are likely more vertical asymptotes not listed here.

Key concepts

Analytical Methods in Calculus

Understanding vertical asymptotes involves grasping the concept of limits, which is a cornerstone of analytical methods in calculus. An asymptote represents a line that a graph approaches but never actually touches or crosses. Analyzing functions to determine asymptotes is a vital skill, as it provides insight into the behavior of functions as they tend toward infinity or approach undefined points. When we use analytical methods to find vertical asymptotes, we are essentially determining the values at which the function becomes unbounded, typically due to division by zero. For example, the function \(q(s)=\frac{\pi}{s-\sin s}\) has potential vertical asymptotes wherever the denominator equals zero, which would be where \(s-\sin s=0\). To analyze this function for vertical asymptotes, students must seek to solve this equation for \(s\), which is often achieved through advanced calculus techniques like Newton's method, or graphically, which brings us to the utility of graphing tools in understanding calculus problems.

By mastering analytical methods, students not only become proficient in finding vertical asymptotes but also deepen their understanding of calculus and its application to real-world problems.

Graphing Utility

Graphing utilities, such as calculators or software programs, are indispensable tools when tackling complex functions, especially in calculus and trigonometry. These tools offer a visual representation of the behavior of functions, which can greatly enhance understanding. For instance, when you plot the function \(s-\sin s\) using a graphing utility, you can clearly see where the graph crosses the x-axis. These intersections point to the values of \(s\) where the denominator of \(q(s)\) becomes zero—the potential locations of vertical asymptotes.

By complementing analytical methods with graphing utilities, students gain a more tangible connection with the abstract nature of calculus. It helps them to visualize the limits and behavior of functions that are otherwise difficult to comprehend through equations alone. This improves their intuition for future problem-solving.

Limits and Continuity

The concepts of limits and continuity are tightly knit, leading to a deeper understanding of calculus. Limits describe the behavior of a function as the input values approach a certain point. Continuity, on the other hand, means that the function is unbroken - it can be drawn without lifting a pencil from the paper. Vertical asymptotes are directly linked to limits, as they represent the points at which the limit of the function does not exist or is infinite. For example, investigating \(\lim_{s\to -8.15}\frac{\pi}{s-\sin s}\) shows the function's behavior as \(s\) approaches -8.15. Since, in this case, the limit yields infinity, we assert that there is a vertical asymptote at \(s\approx -8.15\).

Through practicing the calculation of limits, students develop a systematic approach to identifying discontinuities and asymptotes, thereby fostering a solid foundation for further calculus concepts.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, often appear in calculus problems and possess properties that influence the behavior of other functions. In the exercise at hand, \(\sin s\) is a significant component of the denominator. The sine function oscillates between -1 and 1, and its periodic nature can cause difficulties in solving equations analytically since it can intersect with the identity function \(s\) infinitely many times, at various intervals. Understanding the properties of trigonometric functions, such as their periodicity and range, can offer insights into why certain functions may have multiple or infinite asymptotes. By analyzing and grasping the relationship between these trigonometric functions and other elements within a larger function, students can better predict and understand the overall function's behavior, such as the presence and location of vertical asymptotes.

A deeper dive into trigonometry not only assists in dealing with functions involving sine, cosine, and other trigonometric ratios but significantly enhances problem-solving skills in calculus.