Problem 31 - Calculus Early Transcendentals: Pearson New International Edition

Question

\(\\{(x, y): x>0, y<\sin (1 / x)\\}\)

Step-by-Step Solution

Answer

The set consists of all points \((x, y)\) with \(x > 0\) and \(y < \sin(1/x)\).

Step 1: Understand the Domain

Interpret the domain condition \(x > 0\). This indicates that the solutions are only valid when \(x\) is positive.

Step 2: Understand the Range Condition

Analyze \(y < \sin (1/x)\). The condition instructs that \(y\) must be less than the value of \(\sin(1/x)\). As \(x\) approaches zero from the positive side, \(1/x\) becomes very large, causing \(\sin(1/x)\) to oscillate between -1 and 1.

Step 3: Combine Conditions

The set consists of pairs \((x,y)\) where \(x\) is positive and \(y\) is strictly less than \(\sin(1/x)\). This means for any given positive \(x\), \(y\) can take any value up to but not including \(\sin(1/x)\).

Step 4: Visualize the Solution

Visualize the graph: for each positive \(x\), \(\sin(1/x)\) oscillates quickly as \(x\) approaches zero, and the solutions for \(y\) are those beneath the corresponding \(\sin(1/x)\) value. This creates a series of wavy regions in the positive half of the XY-plane.

Key Concepts

Sine Function

The sine function, a fundamental element of trigonometry, is pivotal in understanding periodic phenomena. It takes an angle as input and gives an output value that varies within a certain range. For any angle - The output of the sine function is a number between -1 and 1. This implies that you can never have a sine value of 1.2 or -1.5. - This output range is particularly useful in modeling waves, such as sound or light waves.The expression \[ \sin(\frac{1}{x}) \] involves the sine function applied to \( \frac{1}{x} \), a reciprocal function. As \( x \) approaches 0 from the positive side, \( \frac{1}{x} \) grows extremely large, causing \( \sin(\frac{1}{x}) \) to oscillate rapidly between -1 and 1. This oscillation reflects the periodic nature of the sine function, which repeats every \( 2\pi \) radians (or 360 degrees). The closer \( x \) is to zero, the faster the oscillation happens.

Domain and Range in Calculus

In calculus, knowing the domain and range of a function is essential. Let's dive a bit deeper into these terms:- **Domain**: This is all possible input values of a function. For the given function, the domain is \( x > 0 \). This simply means that no matter how close it gets to zero, \( x \) must remain positive for the function to stay defined. Think of it as the rule book: only positive values of \( x \) can come into play here.- **Range**: For a given \( x \) within the domain, the range is all potential output values a function can produce. In this case, the condition \( y < \sin(\frac{1}{x}) \) dictates the range. It shows that the value of \( y \) can be any number less than \( \sin(\frac{1}{x}) \). Since \( \sin(\frac{1}{x}) \) oscillates between -1 and 1, this restriction creates complex and interesting behavior as \( x \) gets smaller but positive. Understanding domain and range helps in graphing and analyzing the behavior of functions, making it a crucial part of calculus and mathematical analysis.

Graphical Analysis of Functions

Graphical analysis is like putting together a puzzle to see a complete picture of the function's behavior. Visualizing equations graphically makes it simpler to comprehend complex relationships:For \((x, y)\) given in the problem, the graphical analysis provides wealth insights:- The graph has \( x > 0 \), meaning that it only exists on the right side of the y-axis where \( x \) values are positive. - The function \( y < \sin(1/x) \) means that the graph of \( y \) lies below the oscillating curve of \( \sin(1/x) \).As \( x \) edges closer to zero, \( \sin(1/x) \) oscillates wildly between -1 and 1. - For every specific positive \( x \), the permissible values of \( y \) are those below the sine function, showcasing a signature wave-like pattern.- These oscillations result in wavy bands on the graph, progressively narrowing as \( x \) narrows toward zero.This analysis not only helps in visual representation but also aids in comprehending how such functions behave, predict changes, and enables the solving of more complex mathematical models.