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The next iterative value of the root of x2 − 4 = 0 using the Newton-Raphson method, if the initial guess is 3, is _________
- a)10.22
- b)2.016
- c)2.0236
- d)2.167
Correct answer is option 'D'. Can you explain this answer?
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The next iterative value of the root of x2 4 = 0 using the Newton-Rap...
Newton-Raphson Method:
The Newton-Raphson method is an iterative method used to find the root of a function. It uses the tangent line approximation to iteratively approach the root of the function. The formula for the Newton-Raphson method is given by:
x_(n+1) = x_n - (f(x_n)/f'(x_n))
where x_(n+1) is the next iterative value of the root, x_n is the current value of the root, f(x_n) is the value of the function at x_n, and f'(x_n) is the derivative of the function at x_n.
Given Equation:
The equation given is x^2 - 4 = 0. To find the root of this equation, we need to rewrite it as a function:
f(x) = x^2 - 4
Applying the Newton-Raphson Method:
Let's find the next iterative value of the root using the Newton-Raphson method with an initial guess of x_0 = 3.
1. Calculate the value of the function at x_0:
f(x_0) = (3)^2 - 4 = 5
2. Calculate the derivative of the function:
f'(x) = 2x
3. Calculate the value of the derivative at x_0:
f'(x_0) = 2(3) = 6
4. Substitute the values into the Newton-Raphson formula:
x_(n+1) = x_n - (f(x_n)/f'(x_n))
x_(1) = 3 - (5/6) = 3 - 0.8333 = 2.1667
5. Round the value to the nearest decimal place:
x_(1) ≈ 2.167
Therefore, the next iterative value of the root using the Newton-Raphson method, with an initial guess of 3, is approximately 2.167.
Conclusion:
The correct answer is option D) 2.167.
The Newton-Raphson method is an iterative method used to find the root of a function. It uses the tangent line approximation to iteratively approach the root of the function. The formula for the Newton-Raphson method is given by:
x_(n+1) = x_n - (f(x_n)/f'(x_n))
where x_(n+1) is the next iterative value of the root, x_n is the current value of the root, f(x_n) is the value of the function at x_n, and f'(x_n) is the derivative of the function at x_n.
Given Equation:
The equation given is x^2 - 4 = 0. To find the root of this equation, we need to rewrite it as a function:
f(x) = x^2 - 4
Applying the Newton-Raphson Method:
Let's find the next iterative value of the root using the Newton-Raphson method with an initial guess of x_0 = 3.
1. Calculate the value of the function at x_0:
f(x_0) = (3)^2 - 4 = 5
2. Calculate the derivative of the function:
f'(x) = 2x
3. Calculate the value of the derivative at x_0:
f'(x_0) = 2(3) = 6
4. Substitute the values into the Newton-Raphson formula:
x_(n+1) = x_n - (f(x_n)/f'(x_n))
x_(1) = 3 - (5/6) = 3 - 0.8333 = 2.1667
5. Round the value to the nearest decimal place:
x_(1) ≈ 2.167
Therefore, the next iterative value of the root using the Newton-Raphson method, with an initial guess of 3, is approximately 2.167.
Conclusion:
The correct answer is option D) 2.167.
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The next iterative value of the root of x2 4 = 0 using the Newton-Raphson method, if the initial guess is 3, is _________a)10.22b)2.016c)2.0236d)2.167Correct answer is option 'D'. Can you explain this answer?
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