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The Newton-Raphson method is to be used to determine the reciprocal of the number x = 4. If we start with the initial guess 0.20 then after the first iteration the reciprocal is
- a)0.23
- b)0.24
- c)0.25
- d)0.26
Correct answer is option 'B'. Can you explain this answer?
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Most Upvoted Answer
The Newton-Raphson method is to be used to determine the reciprocal of...
Newton-Raphson Method:
The Newton-Raphson method is an iterative numerical method used to find the roots of a given equation. It is based on the principle of linear approximation and uses the tangent line to iteratively approach the root.
Reciprocal of a Number:
The reciprocal of a number x is defined as 1/x. In this case, we are tasked with finding the reciprocal of x = 4.
Initial Guess:
In order to apply the Newton-Raphson method, we need to start with an initial guess. In this case, the initial guess is given as 0.20.
Iteration Process:
The Newton-Raphson method involves iteratively improving the initial guess until an acceptable level of accuracy is achieved. Let's go through the iteration process step by step:
Step 1:
We start with the initial guess: x0 = 0.20
Step 2:
The equation that we want to find the root of is: f(x) = 1/x - 4 = 0
Step 3:
To apply the Newton-Raphson method, we need to find the derivative of f(x) with respect to x. The derivative of 1/x is -1/x^2.
Step 4:
We can now apply the Newton-Raphson formula to update the guess:
x1 = x0 - (f(x0) / f'(x0))
= 0.20 - ((1/0.20 - 4) / (-1/0.20^2))
= 0.20 - ((1/0.20 - 4) / (-1/0.04))
= 0.20 - ((5 - 4) / (-25))
= 0.20 + (1 / 25)
= 0.20 + 0.04
= 0.24
Step 5:
We have now completed the first iteration and obtained a new guess x1 = 0.24.
Conclusion:
After the first iteration of the Newton-Raphson method, the reciprocal of x = 4 is approximately 0.24. Therefore, the correct answer is option 'B'.
The Newton-Raphson method is an iterative numerical method used to find the roots of a given equation. It is based on the principle of linear approximation and uses the tangent line to iteratively approach the root.
Reciprocal of a Number:
The reciprocal of a number x is defined as 1/x. In this case, we are tasked with finding the reciprocal of x = 4.
Initial Guess:
In order to apply the Newton-Raphson method, we need to start with an initial guess. In this case, the initial guess is given as 0.20.
Iteration Process:
The Newton-Raphson method involves iteratively improving the initial guess until an acceptable level of accuracy is achieved. Let's go through the iteration process step by step:
Step 1:
We start with the initial guess: x0 = 0.20
Step 2:
The equation that we want to find the root of is: f(x) = 1/x - 4 = 0
Step 3:
To apply the Newton-Raphson method, we need to find the derivative of f(x) with respect to x. The derivative of 1/x is -1/x^2.
Step 4:
We can now apply the Newton-Raphson formula to update the guess:
x1 = x0 - (f(x0) / f'(x0))
= 0.20 - ((1/0.20 - 4) / (-1/0.20^2))
= 0.20 - ((1/0.20 - 4) / (-1/0.04))
= 0.20 - ((5 - 4) / (-25))
= 0.20 + (1 / 25)
= 0.20 + 0.04
= 0.24
Step 5:
We have now completed the first iteration and obtained a new guess x1 = 0.24.
Conclusion:
After the first iteration of the Newton-Raphson method, the reciprocal of x = 4 is approximately 0.24. Therefore, the correct answer is option 'B'.
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Community Answer
The Newton-Raphson method is to be used to determine the reciprocal of...
According to the Newton-Rapson method we have;
CALCULATION:
As we know;
Now,
Now putting all the values we have;
xn+1 = 2 × 0.20 − 0.202 × 4
⇒ xn+1 = 0.24
Hence option (2) is the correct answer.
CALCULATION:
As we know;
Now,
Now putting all the values we have;
xn+1 = 2 × 0.20 − 0.202 × 4
⇒ xn+1 = 0.24
Hence option (2) is the correct answer.
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The Newton-Raphson method is to be used to determine the reciprocal of the number x = 4. If we start with the initial guess 0.20 then after the first iteration the reciprocal isa)0.23b)0.24c)0.25d)0.26Correct answer is option 'B'. Can you explain this answer?
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The Newton-Raphson method is to be used to determine the reciprocal of the number x = 4. If we start with the initial guess 0.20 then after the first iteration the reciprocal isa)0.23b)0.24c)0.25d)0.26Correct answer is option 'B'. Can you explain this answer? for Civil Engineering (CE) 2024 is part of Civil Engineering (CE) preparation. The Question and answers have been prepared according to the Civil Engineering (CE) exam syllabus. Information about The Newton-Raphson method is to be used to determine the reciprocal of the number x = 4. If we start with the initial guess 0.20 then after the first iteration the reciprocal isa)0.23b)0.24c)0.25d)0.26Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for Civil Engineering (CE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The Newton-Raphson method is to be used to determine the reciprocal of the number x = 4. If we start with the initial guess 0.20 then after the first iteration the reciprocal isa)0.23b)0.24c)0.25d)0.26Correct answer is option 'B'. Can you explain this answer?.
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