Problem 43 - Precalculus: Mathematical for Calculus

Question

Use a graphing calculator to graph the solution of the system of inequalities. Find the coordinates of all vertices, correct to one decimal place. $$\left\\{\begin{array}{l}y \leq 6 x-x^{2} \\\x+y \geq 4\end{array}\right.$$

Step-by-Step Solution

Answer

Vertices are approximately (1.0, 3.0) and (3.5, 0.5).

Step 1: Graph the First Inequality

Let's start by graphing the first inequality, \( y \leq 6x - x^2 \), on the graphing calculator. This is a downward opening parabola that intersects the y-axis at the origin. The region below this curve is the solution set where \( y \) is less than or equal to this expression.

Step 2: Graph the Second Inequality

Now, graph the second inequality, \( x + y \geq 4 \). This is a straight line with a slope of -1, intersecting the y-axis at 4. The region above and to the right of this line is the solution set where \( x + y \) is greater than or equal to 4.

Step 3: Find Intersection Points

Use the graphing calculator's 'intersect' function to find the points where the two graphs intersect. This will give us the boundaries of the feasible region.

Step 4: Determine Vertices of the Feasible Region

The vertices of the solution region are where the boundary lines (either curves or straight lines) intersect with each other and possibly the axes. Check and calculate the intersections more precisely based on the equations.

Step 5: Identify and list the Coordinates

Based on the intersections identified, list the coordinates of the vertices: the exact points where the solution region changes direction. Ensure all coordinates are precise up to one decimal place.

Key Concepts

System of Inequalities

A system of inequalities comprises two or more inequalities that are evaluated simultaneously. The goal when working with inequalities is to find the common solution set that satisfies all inequalities in the system.
For instance, consider the inequalities given:

• \( y \leq 6x - x^2 \)
• \( x + y \geq 4 \)

In this scenario, the task is to determine where both conditions hold true. Graphing each inequality provides a visual representation of the regions they cover, allowing us to find the overlapping area that represents the solution to the system.
Understanding the nature of the individual lines or curves is essential. Each inequality can define a line, half-plane, or curve, and their intersections allow us to pinpoint accurate solutions.

Feasible Region

The feasible region is an essential aspect when dealing with a system of inequalities. It represents the set of all possible solutions that satisfy all inequalities involved. This region is typically visualized as the overlapping area or section on a graph where each inequality's conditions are met.
For the inequalities we have:

• \( y \leq 6x - x^2 \): This forms a parabola opening downwards.
• \( x + y \geq 4 \): This forms a straight line sloping downward.

The feasible region is where these graphically represented inequalities intersect. To find it, we shade the portion of the graph that satisfies each inequality and then identify the shared area. This overlapping area symbolizes all the potential solutions, making it crucial for decision-making in problems involving constraints.

Intersection Points

Intersection points are vital in determining the feasible region for a system of inequalities. These points are where the graphs of individual inequalities meet, marking the boundaries of the solution set.
For our set of inequalities, we use a graphing tool to find these intersections precisely:

• The parabola \( y \leq 6x - x^2 \) will intersect with the line \( x + y \geq 4 \) at specific points.

These intersections can be found analytically or using digital tools like graphing calculators, which provide more speed and accuracy. Knowing how to locate these intersection points assists in defining the edges of the feasible region, allowing for the accurate determination of potential solutions.

Vertices

Vertices of the feasible region are the corner points where the boundary of the feasible region changes direction. They are found at the intersection of the lines or curves that form the boundaries of the solution region.
In our exercise, once the intersections are identified, these become candidates for vertices:

• Accurate graphing and calculation are essential to finding these vertices correctly.
• Each vertex must be recorded to a precision of one decimal place as required.

Their coordinates precisely define the shape and extent of the feasible region. Thus, accurately determining these vertices is fundamental in obtaining the correct solution set, particularly in optimization problems where one needs to maximize or minimize a function over this region.