Starting with an initial guess of x=2, use Newton’s method to approximate (Third root of 7). Stop the iterations when your approximations converge to four decimal places of accuracy. Compare with the approximation provided by your calculator
I'm so stuck
grab you calculator. start with 2^3, that should be 8, so 2 is too hight. So now punchin
1.9^3, 6.86 that is too low. So now try
1.95^3 7.41, so it is too high, so punchin
1.93^3, 7.19, still too hight, so punchin
1.92^3 7.08 still too hight
punchin
1.913^3 7.0008 still very slightly too high, but getting close to four decimal places...
punchin
1.9128^3 6.998 too low, so punchin
1.9129^3 6.9996, so up again
punchin
1.91293^3 6.999987 that is 7 to four decimal places. Then compare it to your calculator 7^(1/3) results.
To approximate the third root of 7 using Newton's method, we'll need to find the zero of the function f(x) = x^3 - 7. The third root of 7 is also equivalent to 7^(1/3).
Newton's method is an iterative process that involves repeatedly refining an initial guess to approximate the zero of a function. Here are the steps to apply Newton's method:
1. Define the function f(x) = x^3 - 7.
2. Take the derivative of f(x) with respect to x to find f'(x), which is equal to 3x^2.
3. Choose an initial guess for the root, let's say x₀ = 2.
4. Use the formula for Newton's iteration: xᵢ₊₁ = xᵢ - f(xᵢ)/f'(xᵢ).
5. Perform the iteration until the approximations converge to four decimal places of accuracy.
Let's go through the iterations step by step:
Iteration 1:
x₀ = 2
x₁ = x₀ - f(x₀) / f'(x₀)
= 2 - (2^3 - 7) / (3 * (2^2))
= 2 - (8 - 7) / (3 * 4)
= 2 - 1 / 12
≈ 1.917
Iteration 2:
x₁ ≈ 1.917
x₂ = x₁ - f(x₁) / f'(x₁)
= 1.917 - (1.917^3 - 7) / (3 * (1.917^2))
= 1.917 - (7.238 - 7) / (3 * 3.672)
= 1.917 - 0.004 / 11.016
≈ 1.912
Iteration 3:
x₂ ≈ 1.912
x₃ = x₂ - f(x₂) / f'(x₂)
= 1.912 - (1.912^3 - 7) / (3 * (1.912^2))
= 1.912 - (6.749 - 7) / (3 * 3.654)
= 1.912 - (-0.251) / 10.962
≈ 1.913
Continue the iterations until the approximations converge to four decimal places of accuracy. Compare the final approximation to the value provided by your calculator.
Repeat the above-mentioned steps until you achieve the desired accuracy.