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Solve the equation x = 10 cos (x) using the Newton-Raphson method. The initial guess is x = ℼ/4 The value of the predicted root after the first iteration, up to second decimal, is _______.(Answer up to two decimal places)
Correct answer is '1.56'. Can you explain this answer?
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Solve the equation x = 10 cos (x) using the Newton-Raphson method. Th...
By Newton-Raphson method; the iterative formula for finding approximate root at (n+1)th iteration is
Putting n = 0; then
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Solve the equation x = 10 cos (x) using the Newton-Raphson method. Th...
Introduction:
The Newton-Raphson method is an iterative numerical method used to find the root of a given equation. In this case, we are given the equation x = 10 cos(x) and we need to find the root using the Newton-Raphson method with an initial guess of x = π/4.
Newton-Raphson Method:
The Newton-Raphson method is based on the idea of approximating the root of an equation by using the tangent line to the curve at a given point. The formula for the Newton-Raphson iteration is:
x(n+1) = x(n) - f(x(n))/f'(x(n))
where x(n) is the current estimate 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).
Applying Newton-Raphson Method:
1. Given equation: x = 10 cos(x)
2. Let's calculate the derivative of the equation:
- Differentiating both sides with respect to x, we get:
1 = -10 sin(x)
- Rearranging the equation, we get:
sin(x) = -1/10
- Taking the inverse sine on both sides, we get:
x = arcsin(-1/10)
- Evaluating the inverse sine, we get:
x ≈ -0.1002 radians (approximately)
3. Let's calculate the value of f(x) and f'(x) at the initial guess x = π/4:
- f(x) = x - 10 cos(x)
f(π/4) = π/4 - 10 cos(π/4) ≈ 0.7854 - 7.0711 ≈ -6.2857
- f'(x) = 1 + 10 sin(x)
f'(π/4) = 1 + 10 sin(π/4) = 1 + 10/√2 ≈ 8.5355
4. Applying the Newton-Raphson iteration formula, we get:
x(1) = π/4 - (-6.2857)/8.5355 ≈ 1.5610
5. Rounding the value to two decimal places, we get the predicted root after the first iteration as 1.56.
Conclusion:
Using the Newton-Raphson method with an initial guess of x = π/4, we found the predicted root after the first iteration to be approximately 1.56. The method involves iteratively improving the estimate of the root by using the tangent line to the curve at each iteration. By plugging the initial guess into the iteration formula and performing the necessary calculations, we obtained the predicted root value.
The Newton-Raphson method is an iterative numerical method used to find the root of a given equation. In this case, we are given the equation x = 10 cos(x) and we need to find the root using the Newton-Raphson method with an initial guess of x = π/4.
Newton-Raphson Method:
The Newton-Raphson method is based on the idea of approximating the root of an equation by using the tangent line to the curve at a given point. The formula for the Newton-Raphson iteration is:
x(n+1) = x(n) - f(x(n))/f'(x(n))
where x(n) is the current estimate 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).
Applying Newton-Raphson Method:
1. Given equation: x = 10 cos(x)
2. Let's calculate the derivative of the equation:
- Differentiating both sides with respect to x, we get:
1 = -10 sin(x)
- Rearranging the equation, we get:
sin(x) = -1/10
- Taking the inverse sine on both sides, we get:
x = arcsin(-1/10)
- Evaluating the inverse sine, we get:
x ≈ -0.1002 radians (approximately)
3. Let's calculate the value of f(x) and f'(x) at the initial guess x = π/4:
- f(x) = x - 10 cos(x)
f(π/4) = π/4 - 10 cos(π/4) ≈ 0.7854 - 7.0711 ≈ -6.2857
- f'(x) = 1 + 10 sin(x)
f'(π/4) = 1 + 10 sin(π/4) = 1 + 10/√2 ≈ 8.5355
4. Applying the Newton-Raphson iteration formula, we get:
x(1) = π/4 - (-6.2857)/8.5355 ≈ 1.5610
5. Rounding the value to two decimal places, we get the predicted root after the first iteration as 1.56.
Conclusion:
Using the Newton-Raphson method with an initial guess of x = π/4, we found the predicted root after the first iteration to be approximately 1.56. The method involves iteratively improving the estimate of the root by using the tangent line to the curve at each iteration. By plugging the initial guess into the iteration formula and performing the necessary calculations, we obtained the predicted root value.
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Solve the equation x = 10 cos (x) using the Newton-Raphson method. The initial guess is x = ℼ/4 The value of the predicted root after the first iteration, up to second decimal, is _______.(Answer up to two decimal places)Correct answer is '1.56'. Can you explain this answer?
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Solve the equation x = 10 cos (x) using the Newton-Raphson method. The initial guess is x = ℼ/4 The value of the predicted root after the first iteration, up to second decimal, is _______.(Answer up to two decimal places)Correct answer is '1.56'. Can you explain this answer? for Electrical Engineering (EE) 2024 is part of Electrical Engineering (EE) preparation. The Question and answers have been prepared according to the Electrical Engineering (EE) exam syllabus. Information about Solve the equation x = 10 cos (x) using the Newton-Raphson method. The initial guess is x = ℼ/4 The value of the predicted root after the first iteration, up to second decimal, is _______.(Answer up to two decimal places)Correct answer is '1.56'. Can you explain this answer? covers all topics & solutions for Electrical Engineering (EE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Solve the equation x = 10 cos (x) using the Newton-Raphson method. The initial guess is x = ℼ/4 The value of the predicted root after the first iteration, up to second decimal, is _______.(Answer up to two decimal places)Correct answer is '1.56'. Can you explain this answer?.
Solve the equation x = 10 cos (x) using the Newton-Raphson method. The initial guess is x = ℼ/4 The value of the predicted root after the first iteration, up to second decimal, is _______.(Answer up to two decimal places)Correct answer is '1.56'. Can you explain this answer? for Electrical Engineering (EE) 2024 is part of Electrical Engineering (EE) preparation. The Question and answers have been prepared according to the Electrical Engineering (EE) exam syllabus. Information about Solve the equation x = 10 cos (x) using the Newton-Raphson method. The initial guess is x = ℼ/4 The value of the predicted root after the first iteration, up to second decimal, is _______.(Answer up to two decimal places)Correct answer is '1.56'. Can you explain this answer? covers all topics & solutions for Electrical Engineering (EE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Solve the equation x = 10 cos (x) using the Newton-Raphson method. The initial guess is x = ℼ/4 The value of the predicted root after the first iteration, up to second decimal, is _______.(Answer up to two decimal places)Correct answer is '1.56'. Can you explain this answer?.
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