Use Newton’s Method to approximate 3^(√7) to four decimal places.
Use x1 = 2 as your seed. Round off intermediate iterates to five decimal places
To use Newton's Method to approximate 3^(√7) to four decimal places with a seed of x1 = 2, follow these steps:
Step 1: Find the derivative of the function f(x) = 3^(√x).
The derivative of f(x) with respect to x can be found using the chain rule:
f'(x) = (d/dx)(3^(√x)) = (3^(√x)) * (ln(3)/2√x).
Step 2: Set up the recursive formula for Newton's Method.
The recursive formula for Newton's Method is:
x_(n+1) = x_n - f(x_n)/f'(x_n).
Step 3: Start with the initial seed x1 = 2 and apply the recursive formula.
Plug in x1 = 2 into the recursive formula:
x2 = x1 - f(x1)/f'(x1).
Step 4: Calculate f(x1) and f'(x1).
f(x1) = 3^(√2),
f'(x1) = (3^(√2)) * (ln(3)/2√2).
Step 5: Plug in the values of f(x1) and f'(x1) into the recursive formula.
x2 ≈ x1 - (3^(√2))/( (3^(√2)) * (ln(3)/(2√2)) ).
Step 6: Calculate x2 using the formula in Step 5.
x2 ≈ 2 - (3^(√2))/( (3^(√2)) * (ln(3)/(2√2)) ).
Step 7: Round off x2 to five decimal places.
x2 ≈ 2 - (3^(√2))/( (3^(√2)) * (ln(3)/(2√2)) ).
Repeat steps 4-7 using x2 as the new seed (x1) until the desired accuracy is achieved.