Problem 64 - Calculus Single Variable

Question

Sketch the set. \(\\{(x, y): 0

Step-by-Step Solution

Answer

The set is a triangular region on the xy-plane bounded by the lines: \( y = x \), \( y = 2x \), and \( x = \frac{1}{2} \).

Step 1: Understand the Conditions

The set is defined as \( \{(x, y): 0 < x \leq y < 2x \leq 1 \} \). This means we must find the region on the xy-plane satisfying these conditions: \( 0 < x \), \( x \leq y \), \( y < 2x \), and \( 2x \leq 1 \).

Step 2: Simplify and Analyze Each Condition

The first condition \( 0 < x \) simply means that \( x \) is greater than zero and does not include zero. The condition \( x \leq y \) implies that points lie on or above the line \( y = x \). The condition \( y < 2x \) implies that points lie below the line \( y = 2x \). Lastly, \( 2x \leq 1 \) is the same as \( x \leq \frac{1}{2} \), which restricts \( x \) to being less than or equal to 0.5.

Step 3: Identify the Feasible Region

Now let's consider how these inequalities interact: the values of \( x \) are in the interval \( (0, \frac{1}{2}] \). For a given \( x \), \( y \) must satisfy \( x \leq y < 2x \). These constraints plot a triangular region bounded between the lines \( y = x \) (inclusive), \( y = 2x \) (exclusive), and the vertical line \( x = \frac{1}{2} \).

Step 4: Sketch the Region

On a graph, start by drawing the lines \( y = x \) and \( y = 2x \). Shade the region above or on the line \( y = x \) but below \( y = 2x \). The restriction \( 2x \leq 1 \) limits this region to extend only up to \( x = \frac{1}{2} \). The point \( (\frac{1}{2}, 1) \) is where the line \( y = 2x \) intersects with \( x = \frac{1}{2} \), completing the right boundary where \( y < 1 \). This results in a triangular region with vertices at \( (0,0), (\frac{1}{2}, \frac{1}{2}), \text{and} (\frac{1}{2}, 1) \).

Key Concepts

Inequalities in Parametric Equations

Inequalities play a crucial role when dealing with parametric equations, particularly in interpreting the feasible regions of a given set.
When you have a set of inequalities such as \(0 < x\), \(x \leq y\), \(y < 2x\), and \(2x \leq 1\), you're essentially determining a specific area on the coordinate plane where all these conditions hold true.
Each inequality not only imposes a condition on possible values of \(x\) and \(y\), but also contributes in aligning these values within a confined range.

• For example, \(0 < x\) suggests that \(x\) is positive, excluding zero.
• The condition \(x \leq y\) indicates that for any \(x\), \(y\) must be equal or larger, marking a range that begins at \(y=x\).
• Meanwhile, \(y < 2x\) confines \(y\) to be strictly less than twice the value of \(x\), keeping \(y\) from surpassing the threshold created by \(y=2x\).
• The final condition, \(2x \leq 1\), limits \(x\) to the maximum of \(0.5\), completing the boundary of this set.

These inequalities, when combined, form a triangular region on the graph, illustrating which coordinate pairs \((x, y)\) are viable.

Graphing Parametric Inequalities

To properly visualize the region defined by a set of parametric inequalities, graphing becomes essential.
Graphing helps identify overlaps and intersections among the conditions placed by each inequality, making it easier to comprehend their collective effect.
Let's explore this using the predefined inequalities from the set:

• First, plot the base lines \(y = x\) and \(y = 2x\) on the coordinate plane. These lines serve as boundaries for the feasible region, where one side \(y \geq x\) is inclusive, and the other \(y < 2x\) is exclusive.
• Next, mark the limit \(x = \frac{1}{2}\) set by the inequality \(2x \leq 1\). This forms a vertical stopping point further restricting the x-values.
• Finally, locate the intersection points like \((0,0)\), \(\left( \frac{1}{2}, \frac{1}{2} \right)\), and \(\left( \frac{1}{2}, 1 \right)\). These points mark vertices of the triangular region where all inequalities shine through simultaneously.

Shading the area that lies above the \(y=x\) line and below the \(y=2x\) line while stopping at \(x=\frac{1}{2}\) will perfectly outline the specified region.

Set Notation and Interpretation

Set notation is a powerful mathematical tool used to define collections of elements based on given conditions.
This notation helps clearly communicate and establish rules on what is included within a set, especially when managing inequalities.
In the expression \(\{(x, y): 0 < x \leq y < 2x \leq 1\}\), each part tells us something about the permissible coordinates.

• The curly braces \(\{\}\) indicate that we are listing all possible pairs \((x, y)\) satisfying the described inequalities.
• The colon \(:\) separates the element list from the conditions they must satisfy, acting as the dividing line between who is in the set and the terms they must satisfy.
• The combination of inequalities inside indicates that both \(x\) and \(y\) must simultaneously fulfill each condition to be included.

This concise representation informs us about which specific region of the coordinate plane we are considering, illustrating how the power of set notation empowers easy comprehension and communication of mathematical ideas.