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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, \(x_1\) and \(x_2\), for the function \(f(x) = x^2 - 6\) with an initial approximation \(x_0 = 3\). Answer: The first two approximations using Newton's method for the given function and initial approximation are \(x_1 = \frac{3}{2}\) and \(x_2 = \frac{7}{4}\).

Step by step solution

01

Write Newton's method formula

Newton's method is an iterative method used to find the approximations of the roots of a function. The formula for Newton's method is: $$x_{n+1} = x_{n} - \frac{f(x_{n})}{f'(x_{n})}$$ where \(x_{n}\) is the current approximation, \(f(x_{n})\) is the value of the function at \(x_{n}\), and \(f'(x_{n})\) is the derivative of the function evaluated at \(x_{n}\).
02

Determine the derivative of the given function

We are given the function \(f(x)=x^{2}-6\). To apply Newton's method, we need to find its derivative. The derivative of \(f(x)\) is: $$f'(x) = 2x$$
03

Apply Newton's formula for the initial approximation

Now we have the function, its derivative, and the initial approximation. We can apply Newton's method formula to find the first two approximations, \(x_{1}\) and \(x_{2}\). For \(x_{1}\), we use the initial approximation \(x_{0}=3\): $$x_{1} = x_{0} - \frac{f(x_{0})}{f'(x_{0})} = 3 - \frac{(3)^{2}-6}{2(3)}$$ Calculating the values, we get: $$x_{1} = 3 - \frac{3}{2} = \frac{3}{2}$$
04

Apply Newton's formula for the second approximation

For \(x_{2}\), we use the previous approximation \(x_{1}=\frac{3}{2}\): $$x_{2} = x_{1} - \frac{f(x_{1})}{f'(x_{1})} = \frac{3}{2} - \frac{(\frac{3}{2})^{2}-6}{2(\frac{3}{2})}$$ Calculating the values, we get: $$x_{2} = \frac{3}{2} - \frac{-\frac{3}{4}}{3} = \frac{3}{2} + \frac{1}{4} = \frac{7}{4}$$ The first two approximations using Newton's method for the given function and initial approximation are \(x_{1} = \frac{3}{2}\) and \(x_{2} = \frac{7}{4}\).

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Most popular questions from this chapter

Verify the following indefinite integrals by differentiation. These integrals are derived in later chapters. $$\int x^{2} \cos x^{3} d x=\frac{1}{3} \sin x^{3}+C$$

Interpreting the derivative The graph of \(f^{\prime}\) on the interval [-3,2] is shown in the figure. a. On what interval(s) is \(f\) increasing? Decreasing? b. Find the critical points of \(f .\) Which critical points correspond to local maxima? Local minima? Neither? c. At what point(s) does \(f\) have an inflection point? d. On what interval(s) is \(f\) concave up? Concave down? e. Sketch the graph of \(f^{\prime \prime}\) f. Sketch one possible graph of \(f\)

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