False Position Method
The following equations has a root in the indicated interval. Using False Position Method,
generate the first four elements in the sequence that eventually converges to the root
2e^−x/2 + ln x = 1, (0.1,1)

The **False Position Method** (also called the Regula Falsi method) is a numerical method used to solve equations of the form \( f(x) = 0 \), where \( f(x) \) is a continuous function. It iteratively refines the estimate of the root based on the values of \( f(x) \) at two points that bracket the root.

### Given equation:
\[
2e^{-x/2} + \ln(x) = 1
\]
We need to rewrite this equation in the form:
\[
f(x) = 2e^{-x/2} + \ln(x) - 1 = 0
\]

We are also given the interval \((0.1, 1)\), which means the root lies between these two values. Let's now apply the False Position Method to generate the first four approximations of the root.

### Step 1: Calculate \( f(x) \) at the endpoints of the interval
We first evaluate the function \( f(x) \) at the endpoints \( x_0 = 0.1 \) and \( x_1 = 1 \).

#### 1. Evaluate \( f(0.1) \):
\[
f(0.1) = 2e^{-0.1/2} + \ln(0.1) - 1 = 2e^{-0.05} + \ln(0.1) - 1
\]
#### 2. Evaluate \( f(1) \):
\[
f(1) = 2e^{-1/2} + \ln(1) - 1 = 2e^{-0.5} + 0 - 1
\]

### Step 2: Calculate the first approximation \( x_2 \) using the False Position formula:
The False Position formula for the next approximation is:
\[
x_2 =

The following equation has a root in the indicated interval. Using False Position Method, generate the first four elements in the sequence that eventually converges to the root: 2e^−x/2 + ln(x) = 1, (0.1, 1).

Apply the False Position Method to solve the equation 2e^(-x/2) + ln(x) = 1 within the interval (0.1, 1). This numerical method uses the Intermediate Value Theorem to iteratively approximate the root. Learn how to compute the first four elements of the sequence.

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