Problem 90 - Algebra and Trigonometry

Question

\(87-90\) . Sketch the region given by the set. $$ \left\\{(x, y) | 2 x

Step-by-Step Solution

Answer

Sketch the region inside the circle \(x^2 + y^2 \leq 4\) and outside \((x-1)^2 + y^2 = 1\).

Step 1: Understand the Set Description

The set is defined by two conditions: \(2x < x^2 + y^2\) and \(x^2 + y^2 \leq 4\). The first condition describes the region outside a parabola that opens to the right, and the second describes the area inside a circle centered at the origin with radius 2.

Step 2: Sketch the Parabola

Sketch the inequality \(x^2 + y^2 > 2x\). By rearranging, you can write it as \((x-1)^2 + y^2 > 1\), which is a circle centered at (1,0) with radius 1. However, we are interested in the area outside this smaller circle.

Step 3: Sketch the Circle

Sketch the inequality \(x^2 + y^2 \leq 4\). This is a circle centered at the origin (0,0) with radius 2. We are interested in the region inside this circle.

Step 4: Identify the Region of Interest

The region of interest is the intersection of the two conditions: inside the larger circle \(x^2 + y^2 = 4\) and outside the smaller circle \((x-1)^2 + y^2 = 1\). This forms a ring-like area which is not centered around the larger circle.

Step 5: Sketch the Region

Combine the sketches to mark the actual region. Start by drawing both the larger circle with radius 2 and the smaller circle centered at (1,0) with radius 1 on the same graph. The region you want is inside the larger circle but outside the smaller one.

Key Concepts

Graphing Inequalities

When graphing inequalities, it's crucial to understand how they differ from standard equations. An equation gives a precise line or curve on a graph, while an inequality represents a region. This region could be above, below, or on either side of a line or curve.

For example, consider the inequality \(2x < x^2 + y^2\). Instead of a single curve, it represents a region in the plane where all points satisfy this condition. The boundary is the curve obtained by changing the "<" to "=", giving \(2x = x^2 + y^2\).

This inequality describes a region outside a specific parabola-shaped curve, meaning any point outside lies in this region. To sketch these regions, it's helpful to first draw the boundary curve, then decide where the region of interest lies relative to this boundary.

• Draw the boundary curve by solving the equation equivalent.
• Determine which side of the curve satisfies the inequality using test points.
• Shade or highlight the region satisfying the inequality on the graph.

This method helps delineate areas such as those inside a circle or outside a parabola.

Parabola and Circle Intersection

Investigating the intersection between a parabola and a circle involves identifying where the regions defined by their inequalities overlap. These intersections are crucial to solving problems involving set boundaries, like in the provided exercise.

The parabola defined by \(2x < x^2 + y^2\) can be rewritten to emphasize its structure by rearranging to \((x-1)^2 + y^2 > 1\). This represents not strictly a parabola but part of a round region, specifically outside a smaller circle centered at (1,0) with radius 1.

The circle, defined by \(x^2 + y^2 \leq 4\), represents a standard circular region centered at the origin with a 2-unit radius. Visualizing these on a graph, you will shade both outside the smaller circle and inside the larger one resulting in a ring-shaped area.

Key steps for finding such intersections include:

• Graphing each inequality separately to understand their shapes and locations.
• Locating and highlighting their intersection region where both conditions are true.
• Using these visual tools enhances understanding, allowing you to determine the area fulfilling both conditions.

Such an analysis ensures you accurately depict the area required by multiple conditions.

Set Theory in Mathematics

Set theory provides the language and framework for working with collections of objects, essential in describing regions defined by inequalities. Sets are simply collections that might contain numbers, objects, or even regions on a plane.

In the exercise, the region of interest is represented by a set notation: \(\left\{(x, y) | 2x < x^2 + y^2 \leq 4\right\}\). This notation shows a collection of all \(x, y\) points satisfying both given conditions.

Set operations like unions, intersections, and complements can describe and manipulate these collections:

• Intersection: Involves finding the common elements between two sets, such as the overlapping region between shapes like in our example.
• Union: Combines all elements from multiple sets, though not used explicitly here, important in set representation.
• Complement: Includes elements not in a subset, relevant for defining areas outside a boundary.

Each operation has specific symbolism and meaning that enrich the understanding and representation of mathematical ideas. Using set theory, you can capture complex conditions within concise mathematical language, offering a precise way to discuss relationships between different types of regions or objects.