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To solve the equation 2 sin x = x by Newton-Raphson method, the initial guess was chosen to be x = 2.0. Consider x in radian only. The value of x (in radian) obtained after one iteration will be closest to
- a)-8.101
- b)1.901
- c)2.099
- d)12.101
Correct answer is option 'B'. Can you explain this answer?
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Most Upvoted Answer
To solve the equation 2 sin x = x by Newton-Raphson method, the initia...
Concept:
The iterative formula for Newton Raphson method is given as,
[NOTE: Take the trignometric terms in Radian while using scitific calculator for this type of numericals]
Calculation:
Given:
f(x) = 2 sin x - x
∴ f'(x) = 2 cos x - 1
Initial guess is x0 = 2.0
The first iteration by Newton Raphson method is given by,
⇒ x1 = 1.901
The iterative formula for Newton Raphson method is given as,
[NOTE: Take the trignometric terms in Radian while using scitific calculator for this type of numericals]
Calculation:
Given:
f(x) = 2 sin x - x
∴ f'(x) = 2 cos x - 1
Initial guess is x0 = 2.0
The first iteration by Newton Raphson method is given by,
⇒ x1 = 1.901
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Community Answer
To solve the equation 2 sin x = x by Newton-Raphson method, the initia...
Explanation:
Newton-Raphson Method:
- The Newton-Raphson method is an iterative technique for finding the roots of a real-valued function.
- It uses the first few terms of the Taylor series of the function at a starting point to approximate the root.
Given Equation:
- The equation to be solved is 2 sin(x) = x.
Initial Guess:
- The initial guess for the root is x = 2.0.
Iteration:
- To find the next approximation, we use the formula: x_new = x_old - f(x_old) / f'(x_old), where f(x) is the given function and f'(x) is its derivative.
- For the given equation, f(x) = 2 sin(x) - x and f'(x) = 2 cos(x) - 1.
Calculations:
- Substituting x_old = 2.0 into the formula, we get:
x_new = 2.0 - (2 sin(2.0) - 2.0) / (2 cos(2.0) - 1)
x_new ≈ 1.901
Conclusion:
- The value of x obtained after one iteration using the Newton-Raphson method is closest to 1.901, which corresponds to option (b).
Newton-Raphson Method:
- The Newton-Raphson method is an iterative technique for finding the roots of a real-valued function.
- It uses the first few terms of the Taylor series of the function at a starting point to approximate the root.
Given Equation:
- The equation to be solved is 2 sin(x) = x.
Initial Guess:
- The initial guess for the root is x = 2.0.
Iteration:
- To find the next approximation, we use the formula: x_new = x_old - f(x_old) / f'(x_old), where f(x) is the given function and f'(x) is its derivative.
- For the given equation, f(x) = 2 sin(x) - x and f'(x) = 2 cos(x) - 1.
Calculations:
- Substituting x_old = 2.0 into the formula, we get:
x_new = 2.0 - (2 sin(2.0) - 2.0) / (2 cos(2.0) - 1)
x_new ≈ 1.901
Conclusion:
- The value of x obtained after one iteration using the Newton-Raphson method is closest to 1.901, which corresponds to option (b).
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To solve the equation 2 sin x = x by Newton-Raphson method, the initial guess was chosen to be x = 2.0. Consider x in radian only. The value of x (in radian) obtained after one iteration will be closest toa)-8.101b)1.901c)2.099d)12.101Correct answer is option 'B'. Can you explain this answer?
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