Question:
Given \( x_0 \neq 0 \), the iteration
\[
x_{n+1} = \frac{1}{2} \left( -\frac{9}{x_n} + x_n \right), n \geq 0 { is a}
\]
Given \( x_0 \neq 0 \), the iteration
\[
x_{n+1} = \frac{1}{2} \left( -\frac{9}{x_n} + x_n \right), n \geq 0 { is a}
\]
Show Hint
When solving for roots using Newton's method, the iteration formula is typically in the form \( x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \). Recognizing this pattern helps identify the method used.
Updated On: Jun 19, 2025
- Newton's method for \( f(x) = -9 + x^2 \)
- Fixed point iteration for \( f(x) = \frac{9 + x^2}{2x} \)
- Newton's method for \( f(x) = 9 + x^2 \)
- Fixed point iteration for \( f(x) = 9 + x^2 \)
Show Solution
Verified By Collegedunia
The Correct Option is C
Solution and Explanation
The given iteration is: \[ x_{n+1} = \frac{1}{2} \left( -\frac{9}{x_n} + x_n \right) \] This is a form of Newton's method. In general, Newton's method is used to find successively better approximations to the roots of a real-valued function. The standard form of Newton's method is: \[ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \] In our case, the iteration formula matches the form of Newton's method, with \( f(x) = 9 + x^2 \).
Therefore, the correct answer is option (3), which corresponds to Newton's method for \( f(x) = 9 + x^2 \).
Was this answer helpful?
0
0
Top AP PGECET Engineering Mathematics Questions
- The Boolean function \( \sim (\sim P \land Q) \land (\sim (\sim P \land \sim Q)) \lor (P \lor R) \) is equal to the Boolean function:
- There are three cards in a box. Both sides of one card are black, both sides of one card are red, and the third card has one black side and one red side. We pick a card at random and observe only one side. What is the probability that the opposite side is the same colour as the one side we observed?
- Evaluation of the integral \(\int_{2}^{4} x^2 \, dx\) using the trapezoidal rule (with two equal segments) gives a value of .......
- Let \( f(x) = x^3 - x - 3 \). In finding a positive real root of \( f(x) = 0 \) using Newton-Raphson method, if the starting guess \( x_0 = 2 \), then the numerical value of the root \( x_1 \) after the first iteration is .........
- If \( \vec{F} = x(x^2 + y^2 + z^2) \, \hat{i} + 2y(x^2 + y^2 + z^2) \, \hat{j} + 3z(x^2 + y^2 + z^2) \, \hat{k} \), then \( \text{div} \, \vec{F} \) at \( (1, 1, 1) \) is equal to ...........
View More Questions
Top AP PGECET Numerical Methods Questions
- Let \( f(x) = x^3 - x - 3 \). In finding a positive real root of \( f(x) = 0 \) using Newton-Raphson method, if the starting guess \( x_0 = 2 \), then the numerical value of the root \( x_1 \) after the first iteration is .........
- Let $f(x) = x^3 + 2x^2 + 3x - 1$. Newton-Raphson method is used to find the root of the equation $f(x) = 0$. If the initial guess is $x_0 = 1$, then the value of $x$ after first iteration is ...........
- The values of a function \( f(x) \) at discrete values of \( x \) are given in the following table:
\[ \begin{array}{|c|c|c|c|c|c|} \hline x & 0 & 1 & 2 & 3 & 4 \\ \hline f(x) & 1 & 4 & 8 & 10 & 15 \\ \hline \end{array} \] Using Trapezoidal rule, the value of \( \int_0^4 f(x)\,dx \) is ...........
- The Newton-Raphson method is used to find the root of the equation
\[ f(x) = e^{-x} - x \] If the initial guess for the root is 0, then the estimate of the root after the first iteration is ............ - Using Newton-Raphson method to solve $f(x) = e^{-x} - x$ with initial guess $x_0 = 0$, find the next iteration.
Top AP PGECET Questions
- If 125V is applied across a 250V, 100W incandescent bulb, the power consumed will be _____ .
- The power consumed by a coil is 300 watts when connected to a 30V DC source and 108 watts when connected to a 30V AC source. The reactance of the coil is ____ .
The supply voltage magnitude \( |V| \) of the circuit shown below is ____ .
A two-port network is defined by the relation
\(\text{I}_1 = 5V_1 + 3V_2 \)
\(\text{I}_2 = 2V_1 - 7V_2 \)
The value of \( Z_{12} \) is:- Which of the following theorems is the dual of Norton’s theorem?
View More Questions