Question

Use Newton's Method to approximate all roots of the given functions accurate to 3 places after the decimal. If an interval is given, find only the roots that lie in that interval. Use technology to obtain good initial approximations. $$ f(x)=x^{17}-2 x^{13}-10 x^{8}+10 \text { on }(-2,2) $$

Short answer

The roots of the function in the interval \((-2,2)\) are approximately \(-1.833\), \(-0.669\), and \(0.818\).

Step-by-step solution

Understanding Newton's Method

Newton's Method is an iterative technique used to approximate the roots of a real-valued function. Given a function \(f(x)\) and its derivative \(f'(x)\), we use an initial guess \(x_0\) and iterate using the formula: \[ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \]_until a sufficiently accurate value is reached.

Calculate the Derivative

For the function \( f(x) = x^{17} - 2x^{13} - 10x^{8} + 10 \), we need to find its derivative \( f'(x) \). The derivative is calculated as: \[ f'(x) = 17x^{16} - 26x^{12} - 80x^{7} \]This derivative will be used to apply Newton's iterative formula.

Use Technology to Find Good Initial Approximations

Using graphing technology or software, explore the function within the interval \((-2, 2)\). Identify potential approximate roots based on where the graph crosses the x-axis. Suppose potential approximations found are: \(-1.5, -0.5, 1\).

Apply Newton's Method Iteratively

Starting with \(x_0 = -1.5\), calculate:\[ x_1 = x_0 - \frac{f(x_0)}{f'(x_0)} \]Repeat this process iteratively until the result stabilizes and achieves an accuracy of three decimal places. Repeat similar calculations for \(x_0 = -0.5\) and \(x_0 = 1\).

Achieve Convergence for Desired Accuracy

Continue the iterative process for each starting value until the difference between successive approximations is less than 0.001, indicating that three decimal place accuracy is achieved. Confirm the results using technology.

Key concepts

Iterative Technique

Newton's Method is a fascinating iterative technique used to approximate the roots of non-linear equations. The method revolves around repeating a specific calculation, known as iteration, to gradually get closer to the solution. Starting with an initial guess, the technique uses a process to improve the approximation in each step.

Each iteration relies on the formula:

• \[x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}\]

This formula represents advancing from one approximation to the next by leveraging both the function value \(f(x_n)\) and its derivative \(f'(x_n)\). \(x_n\) represents our current approximation, and \(x_{n+1}\) is the next. As this process is repeated, the approximation becomes increasingly accurate and moves closer to the true root.

The beauty of an iterative technique like this lies in its dynamism—continually honing in on the root with each repetition.

Approximation of Roots

The main goal of Newton's Method is the approximation of roots of a function. A root of a function \(f(x)\) is a value \(x\) such that \(f(x) = 0\). Approximating roots becomes particularly useful when we deal with complex functions that are difficult to solve analytically.

Newton's Method gives us a practical way to find these roots by using an initial guess and refining it using the iterative process mentioned previously.

• Start with an initial guess, \(x_0\).
• Through successive iterations, improve on this guess \(x_1, x_2, \ldots\) until the value stabilizes.
• The stabilization indicates convergence to an approximate root.

This method allows us to approximate roots with desirable accuracy. In practice, continuing the iteration until changes between successive approximations are minimal ensures precision—such as finding a value accurate to three decimal places, for example.

Derivative Calculation

The derivative calculation is a crucial aspect of Newton's Method. It provides the necessary slope information at each approximation point, which influences the adjustment made to move closer to the root.

For the function \( f(x) = x^{17} - 2x^{13} - 10x^{8} + 10 \), the derivative \( f'(x) \) is calculated as follows:

• First term: derivative of \( x^{17} \) is \( 17x^{16} \).
• Second term: derivative of \(-2x^{13} \) is \(-26x^{12} \).
• Third term: derivative of \(-10x^{8} \) is \(-80x^{7} \).

Altogether, the derivative is

• \[ f'(x) = 17x^{16} - 26x^{12} - 80x^{7} \]

This derivative must be recalculated at each iteration since it guides the iterative adjustment to approximate the root accurately.

Graphing Technology

Incorporating graphing technology offers a significant advantage when using Newton's Method. It aids in selecting good initial approximations, which are critical for the success of the method. By visualizing the function, you can pinpoint where it crosses the x-axis—these points signify potential roots.

Using graphing tools or software assists with:

• Identifying intervals where the function changes sign.
• Estimating initial guesses by spotting where the curve meets or nears the x-axis.

For the function given in our problem \(f(x)=x^{17}-2x^{13}-10x^{8}+10\), graphing technology helped identify initial approximations like \(-1.5, -0.5, \) and \(1\), which we used as starting points for Newton's iterations.

This technology saves time and improves the precision of the initial guess, effectively guiding the iterative process to converge on accurate roots.