Question:
The Newton-Raphson method is applied to determine the solution of \( f(x) = 0 \) where \( f(x) = x - \cos(x) \). If the initial guess of the solution is \( x_0 = 0 \), the value of the next approximation \( x_1 \) is _________ (round off to two decimal places).
The Newton-Raphson method is applied to determine the solution of \( f(x) = 0 \) where \( f(x) = x - \cos(x) \). If the initial guess of the solution is \( x_0 = 0 \), the value of the next approximation \( x_1 \) is _________ (round off to two decimal places).
Updated On: Nov 25, 2025
Hide Solution
Verified By Collegedunia
Correct Answer: 0.99
Solution and Explanation
The Newton-Raphson method is given by the formula:
\[
x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}
\]
We are given \( f(x) = x - \cos(x) \), so:
\[
f'(x) = 1 + \sin(x)
\]
At \( x_0 = 0 \):
\[
f(0) = 0 - \cos(0) = -1, \quad f'(0) = 1 + \sin(0) = 1
\]
Now, using the Newton-Raphson formula to find \( x_1 \):
\[
x_1 = 0 - \frac{-1}{1} = 1
\]
Thus, the next approximation is \( x_1 = 1.0 \).
Was this answer helpful?
0
0
Top Questions on Solutions of non-linear algebraic equations
Questions Asked in GATE IN exam
- Consider the function \( f(z) = \frac{1}{(z+1)(z+2)(z+3)} \). The residue of \( f(z) \) at \( z = -1 \) is _________.
- GATE IN - 2022
- Residue Theorem
- In the circuit shown, the load is driven by a sinusoidal ac voltage source \( V_1 = 100 \angle 0^\circ \, \text{V} \) at 50 Hz. Given \( R_1 = 20 \, \Omega \), \( C_1 = \left(\frac{1000}{\pi}\right) \, \mu\text{F} \), \( L_1 = \left(\frac{20}{\pi}\right) \, \text{mH} \), and \( R_2 = 4 \, \Omega \), the power factor is _________ (round off to one decimal place)
- GATE IN - 2022
- Transient analysis of RLC circuits with ac excitation
- In the circuit shown, the capacitance \( C_0 = 10 \, \mu\text{F} \) and inductance \( L_0 = 1 \, \text{mH} \) and the diode is ideal. The capacitor is initially charged to 10 V and the current in the inductor is initially zero. If the switch is closed at \( t = 0 \, \text{s} \), the voltage \( V_C(t) \) (in volts) across the capacitor at \( t = 0.5 \, \text{s} \) is _________ (round off to one decimal place)
- GATE IN - 2022
- Diodes: Characteristics and Applications
- A 1 kHz sine-wave generator having an internal resistance of 50Ω generates an open-circuit voltage of 10 Vp-p. When a capacitor is connected across the output terminals, the voltage drops to 8 Vp-p. The capacitance of the capacitor (in microfarads) is _________.
- GATE IN - 2022
- Characteristics of ideal and practical operational amplifiers
- In the Wheatstone bridge circuit shown, \( R_1 = 1.5 \, k\Omega \) and \( R_2 = R_3 = R_4 = 1 \, k\Omega \). The switch is initially open and the voltage between the points C and D is \( V_{CD} \). Upon closing the switch at \( t = 0 \), the resistance in the arm AD changes by an amount \( \delta R_1 \), and the voltage between C and D changes by \( \delta V_{CD} \). The sensitivity of the bridge in volt/kilohm, defined as \( \left| \frac{\delta V_{CD}}{\delta R_1} \right| \), is _________.
- GATE IN - 2022
- Bridges and Potentiometers
View More Questions