{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. Step 1: Understand the problem. We want to approximate the reciprocal of a number 'a' using Newton's method. The function we will use is f(x) = 1/x - a, where 'a' is the number whose reciprocal we want to find.
2. Step 2: Set up Newton's method. Newton's method is an iterative process given by the formula: x_{n+1} = x_n - f(x_n)/f'(x_n). We need to find the derivative of f(x) = 1/x - a.
3. Step 3: Calculate the derivative f'(x). The derivative of f(x) = 1/x - a is f'(x) = -1/x^2. This will be used in the Newton's method formula.
4. Step 4: Choose a starting value x_0. Since we are approximating the reciprocal of 7, a reasonable starting value might be x_0 = 0.1, as it is close to the reciprocal of 7.
5. Step 5: Apply Newton's method iteratively. Use the formula x_{n+1} = x_n - (1/x_n - 7)/(-1/x_n^2) to update x_n. Continue iterating until the approximation reaches eight digits of accuracy.

Key Concepts

Newton's Method

Newton's Method is an iterative numerical technique used to find approximate solutions to equations. It starts with an initial guess and refines it using the formula x_{n+1} = x_n - f(x_n)/f'(x_n), where f is the function and f' is its derivative. This method is particularly useful for finding roots of functions, such as approximating reciprocals.

Function and Derivative

In calculus, a function represents a relationship between inputs and outputs, while its derivative measures the rate of change of the function at any given point. For the function f(x) = 1/x - a, the derivative f'(x) = -1/x^2 is essential for applying Newton's Method, as it helps determine the slope of the tangent line at the current approximation.

Convergence and Accuracy

Convergence refers to the process by which an iterative method approaches a final value or solution. In the context of Newton's Method, achieving eight digits of accuracy means that the approximation is refined until the result is stable to that level of precision. Understanding how quickly and accurately the method converges is crucial for effectively using it to approximate values.