{Use of Tech} Approximating reciprocals To approximate the reciprocal of a number a without using division, we can apply Newton’s method to the function f(x) = 1/x - a.​​ 
b. Apply Newton’s method with a = 7 using a starting value of your choice. Compute an approximation with eight digits of accuracy. What number does Newton’s method approximate in this case?  

Verified step by step guidance

1. Define the function f(x) = 1/x - a, where a = 7. This function will help us find the root, which corresponds to the reciprocal of 7.
2. Choose an initial guess for x, such as x_0 = 0.1. This value should be reasonably close to the expected reciprocal (1/7 ≈ 0.142857).
3. Apply Newton's method using the formula x_{n+1} = x_n - f(x_n)/f'(x_n). First, compute f'(x) = -1/x^2.
4. Calculate f(x_0) and f'(x_0) using your initial guess, then substitute these values into the Newton's method formula to find x_1.
5. Repeat the process with the new value x_1 to find x_2, and continue iterating until the approximation stabilizes to eight digits of accuracy.