Problem 54 - Calculus with Concepts in Calculus

Question

Sketch the region in the plane satisfying the given conditions. \(y \leq 0\)

Step-by-Step Solution

Answer

Shade the entire region below the x-axis in the coordinate plane.

Step 1: Understanding the Condition

The inequality given is \( y \leq 0 \). This means we need to consider points where the \( y \)-coordinate of any point in the plane is less than or equal to zero. Essentially, we are looking for the entire region that includes the x-axis and everything below it in the coordinate plane.

Step 2: Identifying Boundaries

The boundary for this inequality is the line \( y = 0 \), which is the x-axis itself. Since the inequality is \( y \leq 0 \), the x-axis is included in the solution as part of the region we are interested in.

Step 3: Describing the Region

The region that satisfies \( y \leq 0 \) includes all the points on and below the x-axis. This would cover the entirety of the lower half of the plane, where y is any negative number or zero. Graphically, this spans from \( -\infty \) to \( \infty \) horizontally along the x-axis, and from the x-axis downwards along the y-direction to \( -\infty \).

Step 4: Sketch the Region

To sketch this region, draw a solid line on the x-axis (which is \( y = 0 \)) to indicate it is included in the solution. Then shade all the area below this line to show that any point where the \( y \)-coordinate is negative (or zero) is included. This represents the required region.

Key Concepts

Coordinate Plane Regions

When dealing with coordinate plane regions, we are examining areas within a two-dimensional Cartesian coordinate system. In mathematics, these regions typically stem from a set of conditions or constraints like inequalities. These constraints delineate specific parts of the plane that can be shaded or highlighted, showcasing where certain conditions hold true.
For instance:

• An inequality like \( y \leq 0 \) implies we focus on areas where the \( y \)-value of any given point does not exceed zero.
• Regions could span limitless areas, constrained sections, or single lines based on the designated inequality.
• Understanding these regions is crucial in visualizing solutions and interpreting graphical data accurately.

Within coordinate plane regions, different inequalities will sculpt different areas. They guide us visually, enabling us to "see" solutions beyond numerical computations.

X-Axis Boundary

The x-axis serves as a crucial boundary line in coordinate geometry. It is defined by the equation \( y = 0 \). Whenever inequalities involve comparing \( y \) values to zero, the x-axis becomes a significant reference point.
The importance of the x-axis boundary includes:

• Providing a clear demarcation line between positive and negative \( y \)-values.
• Acting as a limit for inequalities that include zero or exclude zero from the solution region.
• Helping define solutions visually and offering a straightforward boundary in sketches.

In the given inequality \( y \leq 0 \), the x-axis (\( y = 0 \)) is incorporated in the solution set, serving as the uppermost limit of the region. Each point on this line satisfies the condition, making it a viable part of the solution.

Graphical Representation of Inequalities

Graphical representation is a powerful tool for illustrating inequalities' solutions on the coordinate plane. It translates mathematical conditions into visual depictions, boosting comprehension and analysis skills.
Here's how graphical representation functions:

• Lines and Shading: Sketched lines symbolize the boundaries where inequalities change. Shading depicts the entire region fulfilling the inequality condition.
• Inclusion and Exclusion: A solid line indicates the boundary is part of the feasible solution (\( \leq \) or \( \geq \)), while dashed lines mean it's not included (\(<\) or \(>\)).
• Showing Solutions: By shading areas below the x-axis for \( y \leq 0 \), we identify all points that meet this condition. The shading technique helps differentiate between fulfilled and unfulfilled areas.

This visual approach makes it easier to identify if a specific point meets the conditions set by the inequality, offering a clear and immediate understanding of mathematical constraints.