find the least positive root of the equation tanx =x to an accuracy of 0.0001 by Newtown raphson method.
The correct answer and explanation is :
To find the least positive root of the equation tan(x) = x to an accuracy of 0.0001 using the Newton-Raphson method, we proceed as follows:
1. Understanding the Newton-Raphson Method:
The Newton-Raphson method is an iterative technique for solving equations of the form f(x) = 0. Starting with an initial guess x₀, the method updates the guess using the formula:
xₙ₊₁ = xₙ – f(xₙ) / f'(xₙ)
This process is repeated until the difference between successive approximations is less than the desired accuracy.
2. Defining the Function and Its Derivative:
We are tasked with solving tan(x) = x, which can be rewritten as:
f(x) = tan(x) – x
The derivative of f(x) is:
f'(x) = sec²(x) – 1
3. Selecting an Initial Guess:
Graphical analysis or numerical methods suggest that the least positive root lies near x = 4.5. Therefore, we choose x₀ = 4.5 as our initial guess.
4. Applying the Newton-Raphson Iteration:
Using the iterative formula, we compute successive approximations:
- Iteration 1:
- f(4.5) = tan(4.5) – 4.5 ≈ 0.4877
- f'(4.5) = sec²(4.5) – 1 ≈ 5.7080
- x₁ = 4.5 – 0.4877 / 5.7080 ≈ 4.1457
- Iteration 2:
- f(4.1457) = tan(4.1457) – 4.1457 ≈ 0.0420
- f'(4.1457) = sec²(4.1457) – 1 ≈ 4.7148
- x₂ = 4.1457 – 0.0420 / 4.7148 ≈ 4.1290
- Iteration 3:
- f(4.1290) = tan(4.1290) – 4.1290 ≈ 0.0015
- f'(4.1290) = sec²(4.1290) – 1 ≈ 4.6798
- x₃ = 4.1290 – 0.0015 / 4.6798 ≈ 4.1285
- Iteration 4:
- f(4.1285) = tan(4.1285) – 4.1285 ≈ 0.0000
- f'(4.1285) = sec²(4.1285) – 1 ≈ 4.6790
- x₄ = 4.1285 – 0.0000 / 4.6790 ≈ 4.1285
5. Conclusion:
After four iterations, the value of x converges to approximately 4.1285. The difference between successive approximations is less than 0.0001, indicating that the desired accuracy has been achieved. Therefore, the least positive root of the equation tan(x) = x, accurate to four decimal places, is approximately 4.1285.
6. Explanation of Convergence:
The rapid convergence of the Newton-Raphson method in this case is due to the choice of a suitable initial guess (x₀ = 4.5) and the nature of the function f(x) = tan(x) – x. The method’s quadratic convergence ensures that the number of correct digits approximately doubles with each iteration, leading to a swift approach to the true root.
7. Practical Considerations:
While the Newton-Raphson method is efficient, it requires the computation of both the function and its derivative at each iteration. Additionally, the method’s convergence can be sensitive to the initial guess, especially if the derivative is small or the function behaves non-linearly near the root. In such cases, alternative methods or a more careful selection of the initial guess may be necessary.