Question

A tangent question Verify by graphing that the graphs of \(y=\sin x\) and \(y=x / 2\) have one point of intersection, for \(x>0\) whereas the graphs of \(y=\sin x\) and \(y=x / 9\) have three points of intersection, for \(x>0 .\) Approximate the value of \(a\) such that the graphs of \(y=\sin x\) and \(y=x / a\) have exactly two points of intersection, for \(x>0\).

Short answer

Based on the given functions and graphs, we can conclude that an approximate value of \(a\) for the functions \(y=\sin x\) and \(y=\frac{x}{a}\) to have exactly two intersection points for \(x>0\) is \(a \approx 8\).

Step-by-step solution

Graph the Functions and Find the Intersection Points

For this step, we use a graphing software, such as Desmos or Grapher. After entering the functions \(y=\sin x\), \(y= \frac{x}{2}\) and \(y= \frac{x}{9}\), we'll notice the following: - There is one intersection point for the functions \(y=\sin x\) and \(y= \frac{x}{2}\). This confirms the first part of the problem statement. - There are three intersection points for the functions \(y=\sin x\) and \(y= \frac{x}{9}\). This confirms the second part of the problem statement.

Approximation for \(a\) for \(y=\sin x\) and \(y=\frac{x}{a}\) functions to have two intersection points

Visually analyzing the graphs, we can notice that if we "stretch" the graph of \(y= \frac{x}{9}\) a little bit, we might be able to find the conditions for which the functions have exactly two points of intersection. Thus, we will choose an \(a\) that is smaller than 9. For example, let's try \(a=8\) for \(y= \frac{x}{8}\). By graphing the functions \(y=\sin x\) and \(y= \frac{x}{8}\), we see that they have exactly two intersection points for \(x>0\). Experimenting with other values nearby (like \(a=7\), \(a=6\)), we can conclude that having \(a=8\) achieves the desired two intersection points for the functions \(y=\sin x\) and \(y=\frac{x}{8}\). So, an approximate value of \(a\) that produces the desired two intersection points is \(a \approx 8\).

Key concepts

Intersection Points of Graphs

Understanding the intersection points of graphs is a fundamental aspect of algebra and calculus. It is where two or more functions share common points on the coordinate plane. When graphing trigonometric functions alongside linear functions, such as graphing \(y=\sin x\) and \(y=\frac{x}{2}\), their intersection points can be visually identified or calculated algebraically when the equations are set equal to each other.

In the exercise, the use of graphing software simplifies the process by allowing a prompt visualization of where the graphs intersect. For \(x>0\), one intersection point for \(y=\sin x\) and \(y=\frac{x}{2}\) confirms that the linear function crosses the sine wave only once in the given interval. Similarly, three intersection points for \(y=\sin x\) and \(y=\frac{x}{9}\) suggest the sine wave oscillates through the linear function three times. The concept of intersection points is crucial for applications across physics, engineering, and economics, where finding common solutions between variable relationships is often required.

Approximation Methods in Calculus

Approximation methods are pivotal in calculus for estimating values and solving problems that might be otherwise impossible to tackle analytically. In the context of the textbook exercise, we approximate the value of \(a\) in the equation \(y=\frac{x}{a}\) to find out how it should be adjusted to achieve exactly two intersection points with the graph of \(y=\sin x\).

One common approach is trial and error, graphically testing various values for \(a\) and observing the outcomes. Another approach is using methods like Newton's method or the bisection method to iterate closer to an accurate value. Looking at the graph, one can determine a tighter interval where the accurate value of \(a\) might lie, and through successive refinement, pinpoint an approximation to solve our specific problem. This process reflects the power of approximation in calculus, allowing us to handle real-world problems with complex models where exact solutions are not readily available or too complex to find.

Sine Function Properties

The sine function, denoted by \(\sin\), exhibits specific properties that are key to understanding its behavior on a graph. It is periodic with a period of \(2\pi\) radians, meaning it repeats its values every \(2\pi\) units along the x-axis. The amplitude, or the peak value of the wave, is 1, and the sine wave oscillates between -1 and 1.

The function has roots (or zeros) at integer multiples of \(\pi\), and it reaches its maximum and minimum at odd multiples of \(\frac{\pi}{2}\). In the exercise, observing these sine wave properties assists in analyzing the number of intersection points with linear functions such as \(y=\frac{x}{a}\). Adjusting the slope of the linear function changes the rate at which it intersects the sine wave—shaping our understanding of how the two functions interact on the graph and the intervals where they intersect.

Function Transformation

Function transformation involves altering the appearance of a graph without changing its fundamental properties or shape. Standard transformations include shifting, stretching, compressing, and reflecting a graph along the axes. For trigonometric functions like the sine function, these transformations can affect the period, amplitude, and phase shift, altering where and how the function intersects with others on the graph.

In our exercise, the transformation of interest is stretching the linear graph of \(y=\frac{x}{a}\) by changing the value of \(a\). As \(a\) decreases, the graph stretches vertically, which affects the number of intersection points with the graph of \(y=\sin x\). Intuitive understanding of function transformation assists in predicting and explaining the interaction between \(y=\sin x\) and \(y=\frac{x}{a}\), leading to the correct approximation of \(a\) to yield the desired number of intersection points. This is especially useful in the fields of engineering and physics, where understanding the effects of transformations can aid in the design of systems and the interpretation of signals.