Question

With the initial guess of \( x_0 = 1, x_1 = 2 \), the first iteration value using secant method for \( f(x) = x^2 + 5x + 4 \) will be _______.

• A. 0.5
• B. -0.25
• C. -0.75
• D. -0.5

Hint:

The Secant Method can be thought of as a linear interpolation between two points. It is often faster than bisection but can fail to converge if the initial guesses are too far from the actual root.

Answer 1

The Correct Option is B

Concept:
The Secant Method is an iterative root-finding algorithm that uses a succession of roots of secant lines to better approximate a root of a function \( f(x) \). Unlike the Newton-Raphson method, it does not require the calculation of derivatives. The formula for the next iteration \( x_{n+1} \) is: \[ x_{n+1} = x_n - f(x_n) \frac{x_n - x_{n-1}}{f(x_n) - f(x_{n-1})} \]

Step 1: Evaluate the function at the initial points.
Given \( f(x) = x^2 + 5x + 4 \), we calculate \( f(x_0) \) and \( f(x_1) \):
• At \( x_0 = 1 \): \( f(1) = (1)^2 + 5(1) + 4 = 1 + 5 + 4 = 10 \)
• At \( x_1 = 2 \): \( f(2) = (2)^2 + 5(2) + 4 = 4 + 10 + 4 = 18 \)

Step 2: Apply the Secant Method formula.
To find the first iteration value \( x_2 \): \[ x_2 = x_1 - f(x_1) \frac{x_1 - x_0}{f(x_1) - f(x_0)} \] Substitute the values into the formula: \[ x_2 = 2 - 18 \cdot \frac{2 - 1}{18 - 10} \] \[ x_2 = 2 - 18 \cdot \frac{1}{8} \]

Step 3: Perform the final calculation.
\[ x_2 = 2 - \frac{18}{8} = 2 - 2.25 \] \[ x_2 = -0.25 \]