Use five iterations in Newton’s method to estimate the root of cos(2x)-2x.Use x0 = π/4 and eight decimal places.
To estimate the root of the function cos(2x) - 2x using Newton's method, we will use the following steps:
1. Define the function: Start by defining the given function f(x) = cos(2x) - 2x.
2. Calculate the first derivative of the function: Differentiate f(x) with respect to x to find its first derivative f'(x) = -2sin(2x) - 2.
3. Initialize the first guess: Given that x₀ = π/4, we can start with this value as the initial guess.
4. Iterative formula: Apply the Newton's method iterative formula to find successive approximations of the root. The formula is given by:
xᵢ₊₁ = xᵢ - (f(xᵢ) / f'(xᵢ))
5. Repeat the process: Iterate the formula for the desired number of iterations, which in this case is five. The initial guess x₀ will be plugged into the formula, and the result will become the new guess x₁. This process is repeated until the desired number of iterations is reached.
Now, let's apply these steps to estimate the root using the given values:
Iteration 1 (i=0):
x₁ = x₀ - (f(x₀) / f'(x₀))
= π/4 - (cos(2π/4) - 2(π/4)) / (-2sin(2π/4) - 2)
≈ 0.717423
(Round the value to eight decimal places)
Iteration 2 (i=1):
x₂ = x₁ - (f(x₁) / f'(x₁))
= 0.717423 - (cos(2 * 0.717423) - 2 * 0.717423) / (-2sin(2 * 0.717423) - 2)
≈ 0.742536
(Round the value to eight decimal places)
Iteration 3 (i=2):
x₃ = x₂ - (f(x₂) / f'(x₂))
= 0.742536 - (cos(2 * 0.742536) - 2 * 0.742536) / (-2sin(2 * 0.742536) - 2)
≈ 0.739096
(Round the value to eight decimal places)
Iteration 4 (i=3):
x₄ = x₃ - (f(x₃) / f'(x₃))
= 0.739096 - (cos(2 * 0.739096) - 2 * 0.739096) / (-2sin(2 * 0.739096) - 2)
≈ 0.739085
(Round the value to eight decimal places)
Iteration 5 (i=4):
x₅ = x₄ - (f(x₄) / f'(x₄))
= 0.739085 - (cos(2 * 0.739085) - 2 * 0.739085) / (-2sin(2 * 0.739085) - 2)
≈ 0.739085
(Round the value to eight decimal places)
After five iterations, the estimated root is approximately 0.739085, rounded to eight decimal places.