Problem 36 - Calculus Single Variable

Question

Sketch the given region. \(\left\\{(x, y): x^{2}

Step-by-Step Solution

Answer

Shade the region above the parabola \( y = x^2 \).

Step 1: Understand the Inequality

The given inequality is \( x^2 < y \). This means we are interested in the region where the value of \( y \) is greater than the square of \( x \). This helps us determine which side of the parabola defined by \( y = x^2 \) we are interested in.

Step 2: Identify the Boundary

The boundary of this region is the parabola \( y = x^2 \). This parabola opens upwards and has its vertex at the origin \( (0,0) \). Our task is to find out which part of the plane is above this parabolic curve.

Step 3: Determine the Region Relative to the Parabola

For any point \( (x, y) \) to satisfy \( x^2 < y \), it needs to be located strictly above the parabola \( y = x^2 \). This means the region is the area above and away from the curve, including all points where \( y > x^2 \).

Step 4: Sketch the Graph

Begin by drawing the parabola \( y = x^2 \) on the coordinate plane. This curve will not be part of the solution since the inequality does not include equality. Next, shade the entire region that lies above this parabola to indicate all points satisfying \( x^2 < y \). Ensure not to include the curve itself in the shaded area.

Key Concepts

Coordinate Plane

The coordinate plane is like a big piece of graph paper where you plot points, lines, and shapes. It has two main lines, also known as the axes. The horizontal line is called the x-axis, and the vertical one is the y-axis. Together, they help us find specific points by using pairs of numbers, called coordinates.

• The origin is the center point where the x-axis and y-axis cross, at coordinate (0, 0).
• Each point on the plane is described by a pair of numbers, \(x, y\), where \(x\) tells how far left or right the point is, and \(y\) tells how far up or down it is.

When sketching graphs like parabolas, understanding the coordinate plane is essential. This graph ensures that every parabola and inequality is plotted accurately in the correct location, making it easier to analyze determined regions.

Graphing Inequalities

Graphing inequalities involves shading a part of the graph to show where an inequality is true. In our exercise, we work with the inequality \(x^2 < y\). Instead of a line or curve like equalities, inequalities cover more space, known as a region. When dealing with inequalities, start by drawing the boundary line or curve, as if it was an equality. In this case, it's the parabola \(y = x^2\). Since it is not equal, only look above this curve.

• If the inequality is \< (or >), the line or curve isn't part of the solution itself, so graph it with a dashed line to show this.
• If the inequality is \leq (or \geq), then the line or curve is part of the solution, so you would use a solid line.

Finally, choose a point not on the boundary to test if it satisfies the inequality. If it does, then the entire region that includes this point is shaded. This shading clearly shows the range of solutions, marking where \(y\) values are greater than \(x^2\).

Parabola

A parabola is a U-shaped curve that can open upwards or downwards. The basic form of a parabola's equation is \(y = x^2\), with its vertex at the origin (0,0). In this context, the parabola acts as a boundary for the inequality we are graphing.Some quick facts about parabolas:

• A parabola will always be symmetric around its vertex.
• The orientation (up or down) is determined by the sign of the quadratic term; positive \(x^2\) opens upwards.

For the exercise, the parabola \(y = x^2\) isn't included in the solution because the inequality \(x^2 < y\) excludes it, focusing only on the space above. By plotting this parabola correctly, students can easily visualize how it guides the area we shade for this exercise. Identifying and understanding this boundary ensures precise graphing and better comprehension of inequalities involving parabolic relations.