Problem 61 - A Graphical Approach to Precalculus with Limits

Question

Use a graphing calculator to solve each equation. Give solutions to the nearest hundredth. $$e^{x}=x^{2}$$

Step-by-Step Solution

Answer

The solutions are approximately \(x \approx 0.15\) and \(x \approx 1.91\).

Step 1: Understand the Equation

The equation to solve is \(e^x = x^2\). We need to find values of \(x\) where the expressions \(e^x\) and \(x^2\) are equal. Solutions will be rounded to the nearest hundredth.

Step 2: Determine Graphing Calculator Setup

Set up the graphing calculator to plot the functions \(y_1 = e^x\) and \(y_2 = x^2\). These functions will be plotted on the same graph to identify their points of intersection.

Step 3: Graph the Functions

In the graphing calculator, graph the functions \(y_1 = e^x\) and \(y_2 = x^2\). Adjust the viewing window to ensure the potential intersections are visible. A starting window range for the x-axis could be from -2 to 2.

Step 4: Identify Points of Intersection

Examine the graph to locate where the graphs of \(y_1 = e^x\) and \(y_2 = x^2\) intersect. Note down the approximate x-values where the functions overlap.

Step 5: Use Calculator's Intersection Feature

Use the graphing calculator’s intersection feature to accurately find the x-values of intersections to the nearest hundredth. Follow the calculator's instructions to use the 'intersect' function accurately.

Step 6: Write Down Solutions

The values of \(x\) found using the intersection feature, rounded to the nearest hundredth, are the solutions to the equation \(e^x = x^2\). These values represent where \(e^x\) equals \(x^2\).

Key Concepts

Exponential Functions

Exponential functions are a type of mathematical function that grows rapidly. They are often used to model exponential growth or decay, such as population growth or radioactive decay. The general form of an exponential function is \(f(x) = a \cdot e^{bx}\), where \(e\) is Euler's number, approximately 2.7182818. This constant plays a crucial role in calculus, especially in differential equations.
In our given problem, we have the function \(e^x\), which is a standard exponential function with a base \(e\) and an exponent \(x\). This indicates that as \(x\) increases, \(e^x\) increases swiftly. Conversely, as \(x\) decreases, \(e^x\) approaches zero but never becomes negative. Understanding how exponential functions behave helps in analyzing their intersection points with other functions, like quadratic functions.

Quadratic Functions

Quadratic functions are polynomial functions of degree two, typically written in the standard form \(f(x) = ax^2 + bx + c\). They produce parabolic graphs that can either open upwards or downwards depending on the sign of \(a\). These functions represent various real-life scenarios such as projectile motion and area comparisons.
In our exercise, we look at the quadratic function \(x^2\), which is a basic quadratic representing a parabola opening upwards with its vertex at the origin \((0,0)\). This function grows slowly at first, especially when compared to exponential functions, but starts increasing more noticeably at larger \(x\) values. Recognizing the growth patterns of quadratic functions enriches understanding of potential intersections with exponential functions.

Intersections

Finding the intersection between two functions involves locating the points where they share the same \(y\)-values for the same \(x\)-values. Graphically, this is where the lines or curves of the graphs cross each other.
When looking at our original problem, \(e^x = x^2\), we aimed to find points of intersection between the exponential graph \(e^x\) and the quadratic graph \(x^2\). By plotting both functions on a graph using a graphing calculator, it's possible to visually identify these intersection points. These points are solutions to the equation because they satisfy both \(e^x\) and \(x^2\) at the same \(x\).
Being able to identify these points is critical as it relates directly to solving equations where functions equal each other.

Equation Solving

Solving equations involving different types of functions, like exponential and quadratic, requires a good understanding of their properties. When solving \(e^x = x^2\), one effective method involves graphical representation, where a graphing calculator is used to find points of intersection.
Steps to solve such equations effectively include:

• Graphing both functions on the same coordinate plane. This visual assistance can help avoid algebraic complexity.
• Adjusting the viewing window to ensure all potential intersections are visible, commonly starting from a range that logically fits the problem context.
• Utilizing the calculator's intersection feature to determine precise solutions, ensuring solutions are accurate to the stated precision.

This graphical approach simplifies the task of solving complex equations by focusing on visualization rather than complex algebraic manipulation.