Question

Two consecutive estimates of the root of a function \( f(x) \) obtained using the Newton-Raphson method are \( x_i = 8.5 \) and \( x_{i+1} = 13.5 \), and the value of the function at \( x_i \) is 15. The numerical value of the first derivative of the function evaluated at \( x_i \) is _________ (in integer).

Hint:

The Newton-Raphson method is used to iteratively find the roots of a function. The first derivative at the current estimate is calculated using the formula \( f'(x_i) = \frac{f(x_i)}{x_i - x_{i+1}} \). Ensure that the correct values are substituted when solving for the derivative.

Answer 1

The Newton-Raphson method for root finding is given by: \[ x_{i+1} = x_i - \frac{f(x_i)}{f'(x_i)}. \] We are given: - \( x_i = 8.5 \), - \( x_{i+1} = 13.5 \), - \( f(x_i) = 15 \). We can rearrange the formula to solve for the first derivative \( f'(x_i) \): \[ f'(x_i) = \frac{f(x_i)}{x_i - x_{i+1}}. \] Substituting the known values: \[ f'(x_i) = \frac{15}{8.5 - 13.5} = \frac{15}{-5} = -3. \] Thus, the numerical value of the first derivative of the function evaluated at \( x_i \) is -3.
Answer: -3.