Question

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

Short answer

After two iterations of Newton's method, the approximate root is \(x \approx 1.225\).

Step-by-step solution

Calculate \(f(x)\) and \(f'(x)\) for \(x=1\)

Substitute \(x=1\) into \(f(x)=2x^2-3\) and its derivative \(f'(x)=4x\).So, \(f(1)=2*(1)^2-3 =-1\) and \(f'(1)=4*(1)=4\).

Apply Newton's formula for \(x_{2}\)

Using the Newton's formula, we get \(x_{2}=x_1 - \frac{f(x_1)}{f'(x_1)}= 1 - \frac{-1}{4}=1.25\).

Calculate \(f(x)\) and \(f'(x)\) for the new \(x=1.25\)

Substitute \(x=1.25\) into \(f(x)=2x^2-3\) and its derivative \(f'(x)=4x\).So, \(f(1.25)=2*(1.25)^2-3 =0.125\) and \(f'(1.25)=4*(1.25)=5\).

Apply Newton's formula for \(x_{3}\)

Applying Newton's Method again, we get \(x_{3}=x_2 - \frac{f(x_2)}{f'(x_2)}= 1.25 - \frac{0.125}{5}=1.225\).

Key concepts

Iterative Methods

Iterative methods are a cornerstone of numerical analysis, concerned with finding approximations to complex mathematical problems that are otherwise challenging to solve exactly. The essence of an iterative method is that it starts with an initial guess, and then improves this guess step-by-step.

One of the most popular iterative methods is Newton's Method, also known as Newton-Raphson Method, used for finding successively better approximations to the roots (or zeros) of a real-valued function. The method's beauty lies in its simplicity and efficiency, often converging to an accurate solution in just a few iterations. Students typically encounter this technique in calculus or numerical methods courses, where they learn the foundational skills of approximation and analysis.

Key Principles

• Start with an initial guess.
• Use a formula to find a better approximation.
• Repeat the process until the solution is sufficiently accurate.

Numerical Root Finding

Numerical root finding is the process of determining the roots of a function numerically, rather than analytically. Analytical solutions might not exist or are too complicated to find for many functions, making numerical techniques essential.

Newton's Method is a powerful tool in this domain, helping students solve calculus problems that may seem intractable at first glance. By iteratively applying Newton's formula, one can home in on a function's root with an astonishing degree of precision. The method requires calculating the function's value and its derivative at each iteration, which, in turn, improves the current estimate of the root.

Benefits of Numerical Root Finding

• Enables solving equations without analytical solutions.
• Improves understanding of the behavior of functions.
• Essential for real-world applications where exactness is impractical.

Calculus Problem Solving

Calculus problem solving often involves understanding the behavior of functions and finding their characteristic points, such as maxima, minima, and roots. In a typical calculus class, students learn a variety of techniques to analyze functions, one of which is Newton's Method for root finding.

When solving calculus problems, it is crucial to conceptualize the problem, visualize the function, and carefully execute computational steps. Newton's Method provides a systematic approach to narrow down the exact value of a function's root, by iteratively approximating where the function crosses the x-axis. Its application demonstrates the practical use of derivatives and reinforces the fundamental theorem of calculus in linking the concept of derivatives to actual problem-solving.

Steps to Success in Calculus

• Understand the function and its properties.
• Visualize the problem.
• Apply appropriate calculus techniques.
• Iterate toward the solution methodically.

Derivative Calculation

Derivative calculation is an essential operation not just in calculus but also in various numerical methods, including Newton's Method. The derivative represents the slope of the tangent line to a function's graph at a particular point, essentially giving the rate at which the function's value is changing at that point. In Newton's Method, the derivative informs us of how to adjust our current estimate to get closer to the root.

During each iteration, to move to the next approximation, students must calculate the function's derivative at the current guess. This derivative evaluation typically reveals whether the next approximation will be above or below the current one, guiding the iterative process in a systematic manner.

Importance of Derivative Calculation

• Indicates the direction to adjust our approximation.
• Helps in understanding the function's behavior around the current estimate.
• Integral to the convergence of Newton's Method.