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Chapter 4: Problem 5

Write the formula for Newton's method and use the given initial approximation to compute the approximations \(x_{1}\) and \(x_{2}\). $$f(x)=x^{2}-6 ; x_{0}=3$$

Short Answer

Expert verified
Question: Using Newton's method, find the first two approximations of the square root of 6, with an initial approximation of 3. Answer: The first approximation is \(x_1 = \frac{3}{2}\) and the second approximation is \(x_2 = \frac{17}{12}\).

Step by step solution

01

Find the derivative of the function

Find the derivative of the given function, $$f(x) = x^{2} - 6$$ Derivative of \(f(x)\), using the power rule, is: $$ f'(x) = 2x $$
02

Use Newton's method formula to find the first approximation (\(x_1\))

Now, we will apply the Newton's method formula to obtain the approximation \(x_1\) based on \(x_0\): $$ x_{1} = x_{0} - \frac{f(x_0)}{f'(x_0)} $$ Substitute the given \(x_0\), \(f(x_0)\), and the derivative \(f'(x_0)\) into the formula: $$ x_{1} = 3 - \frac{(3^2 - 6)}{(2 \cdot 3)} $$ Simplify and calculate \(x_1\): $$ x_{1} = 3 - \frac{3}{2} = \frac{3}{2} $$ Thus, the first approximation is \(x_1 = \frac{3}{2}.\)
03

Use Newton's method formula to find the second approximation (\(x_2\))

Similarly, apply the Newton's method formula to obtain the second approximation \(x_2\) based on \(x_1\): $$ x_{2} = x_{1} - \frac{f(x_1)}{f'(x_1)} $$ Substitute the value of \(x_1\), \(f(x_1)\), and the derivative \(f'(x_1)\) into the formula: $$ x_{2} = \frac{3}{2} - \frac{(\frac{3}{2})^2 - 6}{(2 \cdot \frac{3}{2})} $$ Simplify and calculate \(x_2\): $$ x_{2} = \frac{3}{2} - \frac{\frac{1}{4}}{3} = \frac{3}{2} - \frac{1}{12} = \frac{17}{12} $$ Thus, the second approximation is \(x_2 = \frac{17}{12}\).
04

Summary

We applied Newton's method to the given function \(f(x) = x^{2} - 6\) using the initial approximation \(x_0 = 3\). The approximations \(x_1\) and \(x_2\) are: $$ x_{1} = \frac{3}{2} $$ $$ x_{2} = \frac{17}{12} $$

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