Problem 46 - Precalculus: Mathematical for Calculus

Question

Use the graphical method to find all solutions of the system of equations, correct to two decimal places. $$\left\\{\begin{array}{l}y=e^{x}+e^{-x} \\\y=5-x^{2}\end{array}\right.$$

Step-by-Step Solution

Answer

The intersections are at approximately (0.69, 4.52) and (-1.79, 1.42).

Step 1: Understand the Problem

We need to find all solutions for the system of equations: 1. \( y = e^x + e^{-x} \)2. \( y = 5 - x^2 \)By using the graphical method, we'll plot these two functions on the same set of axes and identify their intersection points, which will give us the solutions to the system.

Step 2: Plot the First Function

Plot the function \( y = e^x + e^{-x} \). This is a classic hyperbolic cosine curve that is symmetric around the y-axis. It increases exponentially for both positive and negative values of \( x \) and has a minimum value of 2 at \( x = 0 \).

Step 3: Plot the Second Function

Plot the function \( y = 5 - x^2 \). This is a downward opening parabola with its vertex at \( (0, 5) \). It intersects the y-axis at 5 and opens symmetrically around the y-axis.

Step 4: Identify Intersection Points

On the graph, look for the points where the curve \( y = e^x + e^{-x} \) intersects with the parabola \( y = 5 - x^2 \). These intersection points represent the solutions to the system of equations.

Step 5: Read Intersection Points from the Graph

Based on the graph, the intersection points between \( y = e^x + e^{-x} \) and \( y = 5 - x^2 \) are approximately \( (0.69, 4.52) \) and \( (-1.79, 1.42) \), which are rounded to two decimal places.

Key Concepts

Systems of Equations

A system of equations involves two or more equations that have common solutions. In this context, they are equations with the variables, usually representing certain relationships or conditions. For example, in our given problem, we have:

• First equation: \(y = e^x + e^{-x}\)
• Second equation: \(y = 5 - x^2\)

The goal is to find the set of values for \(x\) and \(y\) that satisfy both equations simultaneously. The very essence of solving systems of equations graphically is to plot all equations on the same axes. The common solutions are found at the intersection points of the graphs.
Using the graphical method means that you can visually observe where the equations meet, thus providing an intuitive way of understanding the solutions. This method is beneficial when precision is not as critical, and it helps in grasping the overall behavior of the functions involved.

Hyperbolic Functions

Hyperbolic functions appear in the form of combinations of exponential functions and have properties similar to the ordinary trigonometric functions. In our system, the function \(y = e^x + e^{-x}\) is an example of a hyperbolic function. Specifically, it represents the hyperbolic cosine function, written as \(\cosh(x)\) in mathematical notation.

• \(\cosh(x)\) gives symmetrical curves about the y-axis.
• The graph has a characteristic U-shape, increasing as \(x\) moves away from zero in both the positive and negative directions.
• The minimum value of \(\cosh(x)\) is at \(x = 0\) and equals 2.

These properties make \(y = e^x + e^{-x}\) important for understanding the intersection points when compared against different functions on the same scale. It's good to note how hyperbolic functions in equations result in sophisticated curves, offering numerous potential points for intersection based upon other functions plotted.

Intersection Points

Intersection points are essential in the graphical solution of a system of equations. They represent the values of \(x\) and \(y\) that satisfy both equations simultaneously. In terms of graphing, these are the points where the graphs of the functions cross one another. In our problem, we are looking for these specific points on the graphs of \(y = e^x + e^{-x}\) and \(y = 5 - x^2\) as they are plotted together.

• The first intersection point appears at approximately \((0.69, 4.52)\).
• The second intersection point is around \((-1.79, 1.42)\).

These solutions are accurate to two decimal places, allowing us to know the exact points where both equations agree. Identifying these points through visual inspection is the cornerstone of solving systems via the graphical method, providing both a unique understanding and a simple practical solution.