Problem 52 - Precalculus with Limits

Question

In Exercises 47-52, use a graphing utility to graph the solution set of the system of inequalities. $$\left\\{\begin{array}{c}{y \leq e^{-x^{2} / 2}} \\ {y \geq 0} \\ {-2 \leq x \leq 2}\end{array}\right.$$

Step-by-Step Solution

Answer

The solution set of the system is the common region represented by the overlap of the graphs of all three inequalities. It is the region where \(y \leq e^{-x^{2} / 2}\), \(y \geq 0\), and \(-2 \leq x \leq 2\).

Step 1: Graph the First Inequality

First, let's graph the inequality \(y \leq e^{-x^{2} / 2}\). The graph would contain all points (x, y) such that y is less than or equal to \(e^{-x^{2} / 2}\).

Step 2: Graph the Second Inequality

Next, we graph the inequality \(y \geq 0\). All values of y greater than or equal to 0 are included in this graph.

Step 3: Graph the Third Inequality

Finally, let's plot the inequality \(-2 \leq x \leq 2\). This includes all values of x between -2 and 2, inclusive.

Step 4: Find the Common Region

The solution set of the system is the region where all three inequalities overlap. This is the region that satisfies all the given constraints.

Key Concepts

Graphing Utilities

Graphing utilities are powerful tools that help us visually represent mathematical expressions and make sense of complex systems by converting them into clear, easy-to-understand graphs. These tools can be software like Desmos, GeoGebra, or graphing calculators that allow input of mathematical equations and inequalities. By supporting dynamic interaction, graphing utilities help explore relationships between variables and directly visualize effects of changes.

For the provided exercise, we use a graphing utility to input each inequality and observe their combined effects on the coordinate plane. This involves entering equations like \( y \leq e^{-x^2 / 2} \), \( y \geq 0 \), and the constraints on \( x \) as part of creating a comprehensive graphical solution. The software highlights areas where these conditions are met, showing exactly where the solution set resides on the graph.

Inequalities

Inequalities describe a relationship where two expressions are not equal, represented by symbols like \( \leq \), \( \geq \), \( < \), and \( > \). In algebra and calculus, inequalities allow us to define regions of interest and boundaries.

In the example exercise, we have inequalities like \( y \leq e^{-x^2/2} \). This indicates the y-values that are less than or equal to the calculated function \( e^{-x^2/2} \). Understanding inequalities involves recognizing how these relationships limit values on the plane, either below, above, or in-between specific lines or curves.

It's crucial to correctly identify shading areas for each inequality when sketching them on a graph. This informs the solution area, showing where every condition is simultaneously satisfied.

Solution Sets

A solution set in the context of graphing inequalities is a region on the graph where all inequalities in the system hold true. It's where every inequality constraint is met satisfactorily by a group of solutions.

For the given system of inequalities, after graphing each inequality individually, we find a common area on the graph that satisfies all parts of the system. For any point within this region, plugging the \( x \) and \( y \) values into the inequalities will confirm they satisfy each condition.

Visualizing solution sets helps solve complex problem sets since they provide a direct view of all possible solutions. This region often reveals itself through overlapping shaded areas when graphing each inequality.

Systems of Inequalities

Systems of inequalities consist of multiple inequality expressions linked together. The goal when graphing these systems is to find the solution region where all inequalities intersect and overlap.

In exercises like the one provided, we're working with systems of inequalities such as \( y \leq e^{-x^2/2} \), \( y \geq 0 \), and \(-2 \leq x \leq 2 \). Each inequality contributes a unique constraint to the graph, and effectively working with these systems involves graphing each inequality to observe the regions of overlap.

• \( y \leq e^{-x^2/2} \) defines an upper boundary dictated by the exponential function.
• \( y \geq 0 \) adds a clear lower boundary along the x-axis.
• \(-2 \leq x \leq 2 \) limits the horizontal spread of the solution set from \( x = -2 \) to \( x = 2 \).

Graphing the solution involves finding the shared area of all these constraints, representing all possible pairs \((x, y)\) that satisfy the system.