Problem 139 - Thomas Calculus in SI Units

Question

The equation \(x^{2}=2^{x}\) has three solutions: \(x=2, x=4,\) and one other. Estimate the third solution as accurately as you can by graphing.

Step-by-Step Solution

Answer

The third solution is approximately \(x = 0.5\).

Step 1: Understanding the Problem

We need to find the third solution for the equation \(x^2 = 2^x\). The known solutions are \(x = 2\) and \(x = 4\). To find the third, we will use graphing to identify where the curves \(f(x) = x^2\) and \(g(x) = 2^x\) intersect.

Step 2: Graph the Functions

Plot the functions \(f(x) = x^2\) and \(g(x) = 2^x\) on the same set of axes. Observe where these two curves intersect. Known intersection points include \(x = 2\) and \(x = 4\). Look for any other point of intersection.

Step 3: Estimate the Third Solution by Inspection

By closely examining the graph, observe that the curves intersect at a third point that is not at the known solutions. Estimate this point as occurring at a value of \(x\) somewhat near \(0.5\).

Step 4: Confirm with Additional Calculation

To be more precise, check a few values around the approximate point \(x = 0.5\). Calculate and compare \(x^2\) and \(2^x\) at \(x = 0.5\). Check if they get closer as \(x\) is slightly less than 0.5 to ensure the estimation is accurate.

Step 5: Finalize the Estimation

Through the calculations near \(x = 0.5\), confirm \(x \approx 0.5\) as the third intersection point, because both \(f(x)\) and \(g(x)\) values are nearly identical at \(x = 0.5\). This implies the estimated third solution is approximately at this point.

Key Concepts

Graphing Methods

Graphing methods are a powerful way to visualize and solve equations, especially when finding solutions involves understanding how two functions interact. To use graphing methods:

• Start by plotting the graphs of both functions involved in the equation on the same set of axes. In our exercise, this means plotting the functions \(f(x) = x^2\) and \(g(x) = 2^x\).
• Pay close attention to where the two graphs intersect, as the intersection points represent the solutions to the equation. Here, intersection points would be where \(x^2\) equals \(2^x\).
• Utilize graphing technology or graphing paper to accurately plot these curves, aiding in noticing where they cross each other.

By visually assessing these plots, we can estimate solutions that might be difficult to calculate purely algebraically. This approach is especially useful when the exact numerical solutions are non-integer or need approximation.

Intersection Points

Intersection points are crucial as they represent the solutions to equations like \(x^2 = 2^x\). To comprehend how they work in our context:

• Analyze the intersection of the graphs \(f(x) = x^2\) and \(g(x) = 2^x\) to find the points where these expressions are equal.
• Examine known solutions like \(x = 2\) and \(x = 4\), where the curves clearly meet.
• Identify any additional intersection points by inspecting the graph for any other points where the two lines meet, which helps in estimating unknown solutions.

In this problem, upon careful examination, a point of intersection approximates around \(x = 0.5\). This implies that the functions are close to equal at this point, making it a new solution to the equation. Understanding intersection points can essentially reveal where two different mathematical relationships equalize.

Function Analysis

Function analysis involves understanding the behavior and characteristics of functions, like \(x^2\) and \(2^x\), to solve equations. Here's how to approach it:

• Evaluate the growth patterns of \(x^2\) and \(2^x\). For instance, while \(x^2\) grows quadratically, \(2^x\) exhibits exponential growth, which is steeper, especially at larger values of \(x\).
• Analyze their rates of change and how they behave as \(x\) increases or decreases. This helps understand where and why they might intersect.
• When these functions are graphed, notice how \(x^2\) tends to flatten out while \(2^x\) rises sharply. This informs us about potential point intersections, contributing to estimating unknown solutions.

Through function analysis, we get insights into how these expressions' behaviors might lead to a crossover — pinpointing where they equal each other and thus, their solution. Even subtle insights like the disparity in growth rates can help predict where and at which values these might intersect, aiding in finding the desired solutions.