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).
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).
Show Hint
Solution and Explanation
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.
Top Questions on Metallurgical
- The hydrostatic stress for the stress tensor provided below is __________ MPa (in integer).
\[ \begin{bmatrix} 150 & 0 & 0 \\ 0 & -100 & 100 \\ 0 & 100 & 250 \end{bmatrix} \ \text{MPa} \]- GATE MT - 2025
- Metallurgical Engineering
- Metallurgical
- 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).
- GATE MT - 2025
- Metallurgical Engineering
- Metallurgical
- Temperatures at two sides of a 0.4 m thick copper plate are 1000°C and 500°C. Assuming steady state, one-dimensional conductive heat transfer through the wall and ignoring end-effects, the magnitude of the heat flux through the wall is _________ \( \times 10^5 \, {W m}^{-2} \) (in integer). Given: Thermal conductivity of copper \( k = 400 \, {W m}^{-1} {K}^{-1} \).
- GATE MT - 2025
- Metallurgical Engineering
- Metallurgical
- In polycrystalline Ni, Nabarro-Herring diffusion creep was found to be the rate controlling creep mechanism at a certain temperature. At that temperature, if the steady state strain rate is \( 10^{-8} \, {s}^{-1} \) at a stress of 10 MPa, the steady state strain rate of \( 10^{-9} \, {s}^{-1} \) will be obtained at a stress value of _________ MPa (in integer). Assume that the same creep mechanism is rate controlling during the creep deformation.
- GATE MT - 2025
- Metallurgical Engineering
- Metallurgical
- The value of \[ \lim_{x \to 0} \frac{6(x - \sin x)}{x^3} \] is ........ (in integer).
- GATE MT - 2025
- Metallurgical Engineering
- Metallurgical
Questions Asked in GATE MT exam
- Let \( p_1 \) and \( p_2 \) denote two arbitrary prime numbers. Which one of the following statements is correct for all values of \( p_1 \) and \( p_2 \)?
Despite his initial hesitation, Rehman’s ____________ to contribute to the success of the project never wavered.
Select the most appropriate option to complete the above sentence.Which one of the following options is correct for the given data in the table?
- If \( P e^x = Q e^{-x} \) for all real values of \( x \), which one of the following statements is true?
Based only on the conversation below, identify the logically correct inference:
“Even if I had known that you were in the hospital, I would not have gone there to see you”, Ramya told Josephine.- GATE CS - 2025
- GATE IN - 2025
- GATE MT - 2025
- GATE NM - 2025
- GATE TF - 2025
- Logical and Analytical Reasoning Skills