Problem 58 - Calculus with Concepts in Calculus

Question

Sketch the region in the plane satisfying the given conditions. \(x>0\) and \(y<0\)

Step-by-Step Solution

Answer

The region is in the fourth quadrant, to the right of the y-axis and below the x-axis, excluding the axes.

Step 1: Understand the Conditions

We have two conditions to sketch: \(x > 0\) and \(y < 0\). The condition \(x > 0\) specifies the region to the right of the \(y\)-axis, not including the axis itself. The condition \(y < 0\) specifies the region below the \(x\)-axis, not including the axis itself.

Step 2: Sketch the Coordinate Axes

Draw the coordinate system with the \(x\)-axis horizontally and the \(y\)-axis vertically. Label the positive direction of the \(x\)-axis to the right and the negative \(y\)-axis downward.

Step 3: Identify the Region for \(x > 0\)

Shade or highlight the region to the right of the \(y\)-axis. This represents \(x > 0\). Ensure that this shaded region does not include the \(y\)-axis itself.

Step 4: Identify the Region for \(y < 0\)

Similarly, shade or highlight the region below the \(x\)-axis. This represents \(y < 0\). Ensure that this shaded region does not include the \(x\)-axis itself.

Step 5: Combine the Regions

The final region that satisfies both conditions \(x > 0\) and \(y < 0\) is where the two shaded regions overlap. Identify this overlap, which will be in the fourth quadrant of the coordinate plane, excluding the axes.

Step 6: Label the Region

Clearly label the region in the fourth quadrant where both conditions are met. This will be a region entirely within the fourth quadrant, excluding the axes.

Key Concepts

Quadrants

The coordinate plane is divided into four regions called quadrants. These divisions help us identify the location of points based on the signs of their coordinates.

• The first quadrant is where both coordinates are positive, meaning points in this area have values of \(x > 0\) and \(y > 0\).
• The second quadrant lies where points have negative \(x\) coordinates and positive \(y\) coordinates, i.e., \(x < 0\) and \(y > 0\).
• In the third quadrant, both values are negative: \(x < 0\) and \(y < 0\).
• The fourth quadrant, which applies to our exercise, is where the \(x\) values are positive and the \(y\) values are negative (\(x > 0\), \(y < 0\)).

Understanding these quadrants is essential when graphing inequalities on a coordinate plane.

Inequalities

Inequalities describe a range of values rather than an exact number, and they guide us in shading the correct regions in the coordinate plane.

• For \(x > 0\), we consider all points to the right of the \(y\)-axis but excluding the axis itself, as the inequality is strict (does not include equality).
• For \(y < 0\), the points located below the \(x\)-axis are included in the solution, except for the axis itself.

Hence, when both inequalities are considered together, they lead to a specific segment of the plane, namely in the fourth quadrant.

Graphical Representation

A graphical representation helps visualize algebraic conditions like inequalities and understand their location on the coordinate system.
To represent \(x > 0\) and \(y < 0\) graphically:

• First, the line \(x = 0\) which constitutes the \(y\)-axis, serves as the boundary for \(x > 0\). Points to the right of this line satisfy the inequality.
• Next, the line \(y = 0\) that forms the \(x\)-axis, acts as the boundary for \(y < 0\). Points below this line align with the condition.
• Overlay these two conditions on the graph, and the resulting region is an intersection bounded by these axes, specifically in the fourth quadrant.

Through this graphical approach, it becomes clear how the conditions intersect and form the solution's visual representation.

Coordinate System

The coordinate system, also known as the Cartesian coordinate plane, is a two-dimensional plane defined by a horizontal line (the \(x\)-axis) and a vertical line (the \(y\)-axis).
This system allows for the precise plotting of points and visualization of relationships between equations or inequalities.
In the context of this exercise:

• Each point is denoted as an ordered pair \((x, y)\), where \(x\) represents the horizontal position, and \(y\) the vertical.
• The origin \((0, 0)\) is where the two axes intersect, serving as the reference point for the four quadrants.
• The axes themselves are not included in solutions involving strict inequalities like \(x > 0\) or \(y < 0\), as these express conditions strictly greater or lesser without equality.

By mastering the coordinate system, recognizing and interpreting areas on the plane becomes intuitive, a vital skill for solving and understanding mathematical problems.