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Use Newton's method with initial approximation $ x_1 = 1 $ to find $ x_2 $, the second approximation to the root of the equation $ x^4 - x - 1 = 0 $. Explain how the method works by first graphing the function and its tangent line at $ (1, -1) $.

$$

x_{2}=\frac{4}{3} \approx 1.333

$$

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Campbell University

Oregon State University

Harvey Mudd College

Boston College

The first thing we can do is differentiate the equation being given toe end up with four X cubed minus one. Now use an equation except to his ex of one minus F of X Juan over f prime of X ones. This is why we different cheated above, which is 4/3 which is 1.33 Therefore, we can graft us to look like this.