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Chapter 4: Problem 8

Write the formula for Newton's method and use the given initial approximation to compute the approximations \(x_{1}\) and \(x_{2}\). $$f(x)=x^{3}-2 ; x_{0}=2$$

Short Answer

Expert verified
The first two approximations using Newton's method are \(x_{1} = \frac{3}{2}\) and \(x_{2} = \frac{25}{18}\).

Step by step solution

01

Find the derivative of the function f(x)

In this exercise, the given function is \(f(x) = x^3 - 2\). To find its derivative, use the power rule for differentiation: \(f'(x) = 3x^2\).
02

Apply Newton's method formula for the first approximation (\(x_{1}\))

Using the Newton's method formula, calculate the first approximation \(x_{1}\) with the initial approximation \(x_{0}=2\): $$x_{1} = x_0 - \frac{f(x_0)}{f'(x_0)}$$ Substitute \(f(x) = x^3 - 2\) and \(f'(x) = 3x^2\) into the formula along with \(x_0 = 2\): $$x_{1} = 2 - \frac{(2^3 -2)}{3(2^2)} = 2 - \frac{6}{12} = \frac{3}{2}$$ Thus, the first approximation \(x_{1} = \frac{3}{2}\).
03

Apply Newton's method formula for the second approximation (\(x_{2}\))

Now, compute the second approximation \(x_{2}\) using the previously found value of \(x_{1} = \frac{3}{2}\): $$x_{2} = x_1 - \frac{f(x_1)}{f'(x_1)}$$ Substitute \(f(x) = x^3 - 2\) and \(f'(x) = 3x^2\) into the formula along with \(x_1 = \frac{3}{2}\): $$x_{2} = \frac{3}{2} - \frac{\left(\frac{3}{2}\right)^3 - 2}{3\left(\frac{3}{2}\right)^2} = \frac{3}{2} - \frac{\frac{27}{8}-2}{\frac{27}{4}} = \frac{3}{2} - \frac{\frac{11}{8}}{\frac{27}{4}} = \frac{3}{2} - \frac{2}{9} = \frac{25}{18}$$ Thus, the second approximation \(x_{2} = \frac{25}{18}\). In conclusion, using Newton's method, the first two approximations of the given function \(f(x) = x^3 - 2\) are \(x_{1} = \frac{3}{2}\) and \(x_{2} = \frac{25}{18}\).

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