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If a root of the equation 3x3 – 4x2 – 4x + 7 = 0 is found out using Newton Raphson’s method. If the initial assumption for the root is 2, then the root after two iterations will be – (upto three decimal places)
Correct answer is between '1.117,1.119'. Can you explain this answer?
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Explanation:
Newton Raphson's Method:
- Newton Raphson's method is used to find the root of a given equation by making an initial guess and then iteratively refining the guess until an acceptable level of accuracy is achieved.
Given Equation:
- The given equation is 3x^3 - 4x^2 - 4x + 7 = 0.
Initial Assumption:
- The initial assumption for the root is given as 2.
Iterations:
1. First Iteration:
- Using the Newton Raphson's formula: x1 = x0 - f(x0)/f'(x0), where x0 is the initial assumption.
- Calculate the first iteration value: x1 = 2 - (3*(2)^3 - 4*(2)^2 - 4*(2) + 7) / (9*(2)^2 - 8*(2) - 4).
- The calculated value for x1 will be used as the new assumption for the next iteration.
2. Second Iteration:
- Repeat the above calculation using x1 as the new assumption.
- Calculate the second iteration value: x2 = x1 - (3*(x1)^3 - 4*(x1)^2 - 4*(x1) + 7) / (9*(x1)^2 - 8*(x1) - 4).
Final Answer:
- After two iterations, the root will be found between 1.117 and 1.119, considering the calculated values are converging towards this range.
Therefore, the root after two iterations will be between 1.117 and 1.119.
Newton Raphson's Method:
- Newton Raphson's method is used to find the root of a given equation by making an initial guess and then iteratively refining the guess until an acceptable level of accuracy is achieved.
Given Equation:
- The given equation is 3x^3 - 4x^2 - 4x + 7 = 0.
Initial Assumption:
- The initial assumption for the root is given as 2.
Iterations:
1. First Iteration:
- Using the Newton Raphson's formula: x1 = x0 - f(x0)/f'(x0), where x0 is the initial assumption.
- Calculate the first iteration value: x1 = 2 - (3*(2)^3 - 4*(2)^2 - 4*(2) + 7) / (9*(2)^2 - 8*(2) - 4).
- The calculated value for x1 will be used as the new assumption for the next iteration.
2. Second Iteration:
- Repeat the above calculation using x1 as the new assumption.
- Calculate the second iteration value: x2 = x1 - (3*(x1)^3 - 4*(x1)^2 - 4*(x1) + 7) / (9*(x1)^2 - 8*(x1) - 4).
Final Answer:
- After two iterations, the root will be found between 1.117 and 1.119, considering the calculated values are converging towards this range.
Therefore, the root after two iterations will be between 1.117 and 1.119.
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If a root of the equation 3x3– 4x2– 4x + 7 = 0 is found out using Newton Raphson’s method. If the initial assumption for the root is 2, then the root after two iterations will be – (upto three decimal places)Correct answer is between '1.117,1.119'. Can you explain this answer?
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