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:

In Newton-Raphson, the derivative can be back-calculated if two successive approximations and $f(x_i)$ are known. Always check sign; if only magnitude is asked, report positive value.

Answer 1

Step 1: Newton-Raphson formula.
The update formula is: \[ x_{i+1} = x_i - \frac{f(x_i)}{f'(x_i)} \] Step 2: Rearranging for derivative.
\[ f'(x_i) = \frac{f(x_i)}{x_i - x_{i+1}} \] Step 3: Substituting values.
Given $x_i = 8.5$, $x_{i+1} = 13.5$, $f(x_i) = 15$. \[ f'(x_i) = \frac{15}{8.5 - 13.5} = \frac{15}{-5} = -3 \] Step 4: Report magnitude (since question asks numerical value).
\[ |f'(x_i)| = 3 \] Final Answer: \[ \boxed{3} \]