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Consider the following equation: f(x) = x^2 - 2. Using Newton's method, what is the value of x0 that will converge to one of the roots of the equation if started from x0 = 2? (Note: df denotes the derivative of f)
- a)0.585786437626905
- b)1.414213562373095
- c)-1.414213562373095
- d)-0.585786437626905
Correct answer is option 'C'. Can you explain this answer?
Verified Answer
Consider the following equation: f(x) = x^2 - 2. Using Newtons method,...
To find the value of x0 that will converge to one of the roots of the equation f(x) = x^2 - 2 using Newton's method, we need to find the root. By initializing x0 = 2 and performing the iterations of Newton's method, the approximation will converge to the root x = -1.414213562373095.
Most Upvoted Answer
Consider the following equation: f(x) = x^2 - 2. Using Newtons method,...
Understanding the Equation and Its Roots
To solve the equation \( f(x) = x^2 - 2 \), we need to find its roots. The roots occur where \( f(x) = 0 \):
- This leads to \( x^2 = 2 \), yielding \( x = \sqrt{2} \) and \( x = -\sqrt{2} \).
- The approximate values of these roots are \( 1.414213562373095 \) and \( -1.414213562373095 \).
Newton's Method Overview
Newton's method is an iterative numerical technique used to find roots of a function. It utilizes the formula:
\[ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \]
where \( f' \) is the derivative of \( f \).
Calculating the Derivative
For our function:
- \( f'(x) = 2x \)
Applying Newton's Method
Starting from \( x_0 = 2 \):
1. Calculate \( f(x_0) \):
\[ f(2) = 2^2 - 2 = 2 \]
2. Calculate \( f'(x_0) \):
\[ f'(2) = 2 \cdot 2 = 4 \]
3. Update using Newton's formula:
\[ x_1 = 2 - \frac{2}{4} = 2 - 0.5 = 1.5 \]
4. Repeat the process:
- Calculate \( f(1.5) \) and \( f'(1.5) \).
- Continue iterating until convergence to one of the roots.
Convergence to the Correct Root
Through iteration, starting from \( x_0 = 2 \), the method will converge to the root \( x = -\sqrt{2} \approx -1.414213562373095 \). The negative root is reached due to the nature of the function and the initial guess which leads the iteration towards the left side of the x-axis.
Thus, the correct answer is option 'C': \( -1.414213562373095 \).
To solve the equation \( f(x) = x^2 - 2 \), we need to find its roots. The roots occur where \( f(x) = 0 \):
- This leads to \( x^2 = 2 \), yielding \( x = \sqrt{2} \) and \( x = -\sqrt{2} \).
- The approximate values of these roots are \( 1.414213562373095 \) and \( -1.414213562373095 \).
Newton's Method Overview
Newton's method is an iterative numerical technique used to find roots of a function. It utilizes the formula:
\[ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \]
where \( f' \) is the derivative of \( f \).
Calculating the Derivative
For our function:
- \( f'(x) = 2x \)
Applying Newton's Method
Starting from \( x_0 = 2 \):
1. Calculate \( f(x_0) \):
\[ f(2) = 2^2 - 2 = 2 \]
2. Calculate \( f'(x_0) \):
\[ f'(2) = 2 \cdot 2 = 4 \]
3. Update using Newton's formula:
\[ x_1 = 2 - \frac{2}{4} = 2 - 0.5 = 1.5 \]
4. Repeat the process:
- Calculate \( f(1.5) \) and \( f'(1.5) \).
- Continue iterating until convergence to one of the roots.
Convergence to the Correct Root
Through iteration, starting from \( x_0 = 2 \), the method will converge to the root \( x = -\sqrt{2} \approx -1.414213562373095 \). The negative root is reached due to the nature of the function and the initial guess which leads the iteration towards the left side of the x-axis.
Thus, the correct answer is option 'C': \( -1.414213562373095 \).
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Consider the following equation: f(x) = x^2 - 2. Using Newtons method, what is the value of x0 that will converge to one of the roots of the equation if started from x0 = 2? (Note: df denotes the derivative of f)a)0.585786437626905b)1.414213562373095c)-1.414213562373095d)-0.585786437626905Correct answer is option 'C'. Can you explain this answer?
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