sin(1/2)𝑋+1/2ln𝑥=1 in [5, 20] using the regula falsi method

To solve the equation \( \sin\left(\frac{1}{2}x\right) + \frac{1}{2} \ln x = 1 \) in the interval \([5, 20]\) using the Regula Falsi method (also known as the False Position method), follow these steps:

### 1. **Define the function:**
The equation can be rewritten as a function \( f(x) \) where:
\[
f(x) = \sin\left(\frac{1}{2}x\right) + \frac{1}{2} \ln x - 1
\]

### 2. **Initial Bracketing:**
You need to identify two points \( a \) and \( b \) within the interval \([5, 20]\) such that \( f(a) \) and \( f(b) \) have opposite signs, i.e., \( f(a) \times f(b) < 0 \). This indicates that there is a root between \( a \) and \( b \).

### 3. **Regula Falsi Formula:**
Given two points \( a \) and \( b \), the root \( c \) is approximated by the formula:
\[
c = b - \frac{f(b) \times (b - a)}{f(b) - f(a)}
\]
Then evaluate \( f(c) \).

### 4. **Iterative Process:**
- If \( f(a) \times f(c) < 0 \), then the root lies between \( a \) and \( c \). Set \( b = c \).
- If \( f(b) \times f(c) < 0 \), then the root lies between \( c \) and \( b \). Set \( a = c \).
- Repeat the process until the value of \( c \) converges within a desired tolerance level or the maximum number of iterations is reached.

### 5. **Implementation:**
Let's calculate the root using Python. I'll walk through the calculation.

The root of the equation \( \sin\left(\frac{1}{2}x\right) + \frac{1}{2} \ln x = 1 \) in the interval \([5, 20]\) using the Regula Falsi method is approximately \( x = 19.8852 \). This result was obtained after 5 iterations.

Would you like more details on the process or have any questions?

Here are 5 related questions for further exploration:
1. How does the Regula Falsi method compare to the Bisection method?
2. What is the significance of choosing the initial interval \([5, 20]\)?
3. How does the choice of tolerance \( \text{tol} \) affect the solution's accuracy?
4. Can the Regula Falsi method be applied to non-continuous functions?
5. What happens if \( f(a) \times f(b) \geq 0 \) in the Regula Falsi method?

**Tip:** Always check if the function changes sign over your interval before applying methods like Regula Falsi, as this is crucial for finding a root.

Learn how to solve the equation sin(1/2)x + (1/2)ln(x) = 1 in the interval [5, 20] using the Regula Falsi method. This problem involves trigonometry, logarithms, and numerical methods, providing a detailed step-by-step solution. Suitable for advanced college-level students and professionals.

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