Problem 80 - Calculus Single Variable

Question

Plot the given region. \(\left\\{(x, y): 1 \leq x^{2}-2 x-\sqrt{3} y^{2}+2 \sqrt{3} y\right.\) and \(\left.y \leq \sqrt{5}+\sqrt{3} x-x^{2}\right\\}\) Plot the parabola with Cartesian equation \(y=2+x+x^{2}\)

Step-by-Step Solution

Answer

Identify and plot the region formed by intersecting the described inequalities and plot the parabola separately.

Step 1: Understanding the Inequality

The given region is described by two inequalities: 1. \(1 \leq x^{2} - 2x - \sqrt{3}y^{2} + 2\sqrt{3}y\)2. \(y \leq \sqrt{5} + \sqrt{3}x - x^{2}\)Our goal is to find the set of points \((x, y)\) that satisfy both these conditions. This will involve interpreting and manipulating these inequalities.

Step 2: Simplifying the First Inequality

Rearrange the first inequality: \(x^{2} - 2x + 2\sqrt{3}y - \sqrt{3}y^{2} \geq 1\). We need to complete the square to better understand the region described by this inequality.

Step 3: Completing the Square for First Inequality

Complete the square for the \(x\) terms and \(y\) terms: - For \(x\): \((x - 1)^{2} - 1\) - For \(y\): \(-\sqrt{3}(y^{2} - \frac{2y}{\sqrt{3}}) = -\sqrt{3}((y - \frac{1}{\sqrt{3}})^{2} - \frac{1}{3})\).So, the inequality becomes \((x-1)^{2} - \sqrt{3}(y - \frac{1}{\sqrt{3}})^{2} \geq \frac{4}{3}.\)

Step 4: Simplifying the Second Inequality

The second inequality is straightforward: \(y \leq \sqrt{5} + \sqrt{3}x - x^{2}\).This is the equation of a downward-facing parabola.”

Step 5: Full Region Description

Now combine the results: 1. The first inequality represents a region outside of an ellipse-like shape after completing the square.2. The second condition, a parabola, limits \(y\) on both sides. Thus, we are looking for the intersection of these regions.

Step 6: Plotting the Region and Parabola

1. Plot the parabola \(y = 2 + x + x^{2}\) for reference. It is a parabolic curve opening upwards, shifted from the origin.2. Plot \(y \leq \sqrt{5} + \sqrt{3}x - x^{2}\), a downward-facing parabola.3. The region satisfying \((x-1)^{2} - \sqrt{3}(y - \frac{1}{\sqrt{3}})^{2} \geq \frac{4}{3}\) depicts a separated space on the plane.The overlap of the regions under these constraints provides the solution.

Key Concepts

Inequality Regions

In multivariable calculus, inequality regions represent a set of points that satisfy given inequality conditions. It’s like the special territory on a map that follows certain rules. For our exercise, the challenge is to identify points

• where the first inequality is met: \(1 \leq x^{2} - 2x - \sqrt{3}y^{2} + 2\sqrt{3}y\)
• And the second constraint : \(y \leq \sqrt{5} + \sqrt{3}x - x^{2}\).

To better grasp these regions, picture them as a combination of plots or areas on a graph. You need to check every point (x, y) to confirm it meets both inequalities. The overlapping area within the defined boundaries is the solution region, and reflects status quo in both conditions. Simplifying these inequalities helps to visualize what the region looks like,
making it easier to sketch the overlapping portions on graph paper.

Parabolic Curves

Parabolas are curves that look like a U or an upside-down U, depending on their orientation. In this exercise, we interpret both an upward and a downward parabola:

• The parabola represented by the equation \(y = 2 + x + x^{2}\) opens upwards.
• Meanwhile, the inequality \(y \leq \sqrt{5} + \sqrt{3}x - x^{2}\) describes a downward opening parabola.

By assessing these forms, you observe how they interplay and intersect. The first parabola starts off any point by being shifted from origin due to its linear components.
While analyzing these curves, notice shifts and changes not just in shape but also position on the coordinate plane, dictated by the constants and coefficients in each equation.

Completing the Square

Completing the square is a method used in algebra to simplify quadratic equations, revealing more about the graph of the function. In our task, we apply this to a mixed quadratic inequality:

• For x: turns "\(x^{2} - 2x\)" into "\((x - 1)^{2} - 1\)"
• For y: "\(-\sqrt{3}(y^{2} - \frac{2y}{\sqrt{3}} )\)"to "\(-\sqrt{3}((y - \frac{1}{\sqrt{3}})^{2} - \frac{1}{3})\)".

Completing the square, thus, reshapes quadratic expressions into a form that's easier to compare, facilitating easier identification of shifts and transformations. In essence, this trick plays a critical role in identifying the basic "shape" of the inequality within the coordinate plane, particularly in determining boundary outlines before cross-referencing with the parabola.