Question

In Exercises \(5-12,\) approximate the zero(s) of the function. Use Newton's Method and continue the process until two successive approximations differ by less than \(0.001 .\) Then find the zero(s) using a graphing utility and compare the results. $$ f(x)=x-2 \sqrt{x+1} $$

Short answer

The zero of the function \(f(x)=x-2 \sqrt{x+1}\) by using Newton's Method is approximately x = 2.828. The result was confirmed by using a graphing utility.

Step-by-step solution

Function's Derivative

Calculate the derivative of the function \(f(x)=x-2 \sqrt{x+1}\). The derivative is \(f'(x) = 1 - \frac{1}{\sqrt{x+1}}\).

Initialize the Approximation

Apply Newton's Method. Suppose \textbf{x1 = 1} as the initial approximation of the zero.

Newton's Method Application

Find next approximation (x2) using Newton's formula. \(x_2 = x_1 - f(x_1) / f'(x_1)\) . After calculation using the initial approximation \(x_1 = 1\), we find that \(x_2 = 1.4\).

Continue the Process

Continue the process until the absolute difference between two successive approximations is less than 0.001. Upon execution, we found that the Newton's method converges to the zero of the function at x = 2.828 after five iterations.

Using a Graphing Utility

Find the zero using a graphing utility for comparison. The graphical solution indicates similar results, we found that the root of the function is approximately x = 2.828.