Question

In Exercises \(1-4,\) complete two iterations of Newton's Method for the function using the given initial guess. $$ f(x)=x^{2}-3, \quad x_{1}=1.7 $$

Short answer

After two iterations of Newton's method, the values obtained for \(x_2\) and \(x_3\) are \(x_2 = 1.720588235294118\) and \(x_3 = 1.7320508100147274\).

Step-by-step solution

Calculate the derivative of f

The derivative of \(f(x)=x^2-3\) with respect to x, written as \(f'(x)\), is calculated as \(2x\).

Apply Newton's Method to find \(x2\)

Substitute \(x_1 = 1.7\) into the formula for Newton's method, \(x_{2} = x_{1} - \frac{f(x_{1})}{f'(x_{1})}\), to find \(x_{2}\). So, \(x_2 = 1.7 - \frac{{(1.7)^2 - 3}}{2*(1.7)} = 1.720588235294118\).

Apply Newton's Method again to find \(x3\)

Substitute \(x_2 = 1.720588235294118\) into the Newton's method formula to find \(x_3\). Hence, \(x_3 = x_2 - \frac{{f(x_{2})}}{{f'(x_{2})}} = 1.720588235294118 - \frac{{(1.720588235294118)^2 - 3}}{2*(1.720588235294118)} = 1.7320508100147274\).

Key concepts

Iterative Methods

Iterative methods are techniques used to solve mathematical problems by repeatedly applying a formula or algorithm to approach a desired result. In the context of calculus, they varyingly provide solutions to equations that might be difficult or impossible to solve analytically.

Newton's Method, featured in the textbook exercise, is a classic example of an iterative process. The method starts with an initial guess, which is then refined incrementally to get closer to the actual root of an equation. The beauty of this approach lies in its simplicity and efficiency; with each iteration, the estimate typically becomes more accurate.

When learning iterative methods, it's crucial to understand that the success of the process can depend on the initial guess. A poor initial guess might lead to slower convergence or even divergence. In the exercise given, an initial guess of 1.7 is used to find the roots of the function by employing Newton's Method. This guess is subsequently improved through iterations to approach the actual solution.

Numerical Analysis

Numerical analysis deals with the development and justification of algorithms for obtaining numerical solutions to mathematical problems. It is a fundamental area of mathematics and computer science that provides tools for solving real-world problems where exact answers are either impossible or impractical to calculate.

Newton's Method is rooted in numerical analysis. It falls under the umbrella of numerical root finding techniques, where we cannot derive roots explicitly. The convergence of this method might not be guaranteed; thus, a good understanding of convergence criteria and error estimation is necessary.

Numerical analysis encourages testing the sensitivity of solutions to initial values and algorithm parameters. For learners, playing with different initial guesses and examining their impact on the solution path can provide deep insights into the stability and efficiency of the method.

Root Finding Algorithms

Root finding algorithms are designed to calculate numerical solutions to the equation f(x) = 0. These roots represent the values of x for which the function crosses the x-axis. Newton's Method, as described in the exercise, is one of these algorithms, and it is particularly known for its fast convergence rates when close to the actual root.

The algorithm uses a function and its derivative to iteratively move towards the root. The initial guess is crucial, but so is the choice of the function. Some functions can cause the method to fail due to local minima or inflection points.

Furthermore, students should be made aware that root finding algorithms, while powerful, have limitations and can fail under certain conditions. Discussions about convergence criteria, such as whether the sequence of approximations is bounding and whether successive approximations are getting sufficiently closer to each other, can illustrate these conditions. By understanding these aspects, students can better evaluate the appropriateness and potential success of a root finding algorithm like Newton's Method.