Problem 29 - Thomas Calculus

Question

Curves that are nearly flat at the root Some curves are so flat that, in practice, Newton's method stops too far from the root to give a useful estimate. Try Newton's method on \(f(x)=(x-1)^{40}\) with a starting value of \(x_{0}=2\) to see how close your machine comes to the root \(x=1 .\) See the accompanying graph.

Step-by-Step Solution

Answer

Newton's method progresses slowly and stops far from 1 due to the function's flatness.

Step 1: Understand Newton's Method

Newton's method is an iterative process to find the root of a function. Given a function \( f(x) \), the iterative formula is \( x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \). You start with an initial approximation \( x_0 \) and apply the formula repeatedly to get closer estimates of the root.

Step 2: Define the Function and Its Derivative

The function given is \( f(x) = (x - 1)^{40} \). To apply Newton's method, we also need the derivative, \( f'(x) \). The derivative is \( 40(x - 1)^{39} \).

Step 3: Apply Newton's Method

Start with \( x_0 = 2 \) and apply Newton's iteration formula:\[x_{n+1} = x_n - \frac{(x_n - 1)^{40}}{40(x_n - 1)^{39}}\]Simplify the formula:\[x_{n+1} = x_n - \frac{x_n - 1}{40}\] Calculate \( x_1 \):\[x_1 = 2 - \frac{2 - 1}{40} = 2 - 0.025 = 1.975\]

Step 4: Iterative Calculations

Continue with the iterative process using the simplified formula:- \( x_2 = 1.975 - \frac{1.975 - 1}{40} = 1.950625 \)- \( x_3 = 1.950625 - \frac{1.950625 - 1}{40} = 1.926859375 \)Repeat the process several times and observe how \( x_n \) moves towards 1, though the progress becomes slower due to the equation's flatness near the root.

Step 5: Assess Closeness to Actual Root

After several iterations, you'll notice that \( x_n \) is indeed approaching 1, but the progress is slow and becomes negligible due to how flat the function \( (x-1)^{40} \) is near \( x=1 \).

Key Concepts

Iterative Process

The iterative process is a fundamental concept in Newton's method. It refers to the repetition of a specific procedure to gradually approach a solution, in this case, the root of a function. In mathematical terms, this involves using a formula repeatedly to improve the accuracy of the estimate of the root. Newton's method employs an iterative formula given by:\[x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}\]This iterative technique starts with an initial guess, denoted as \(x_0\), and uses the formula to find successive approximations \(x_1, x_2, x_3,\) and so on. Each iteration brings us closer to the actual root. In our example with \(f(x) = (x - 1)^{40}\) and a starting point of \(x_0 = 2\), we observe how each successive value of \(x\) gets closer to 1. By continuously refining our approximation, we hone in on the root, highlighting the power of iterative processes in numerical methods.

Function Root

In mathematics, the root of a function refers to the value(s) of \(x\) where the function equals zero. Essentially, it is where the graph of the function intersects the x-axis. For the function \(f(x) = (x-1)^{40}\), the root is at \(x = 1\). This is because substituting \(x = 1\) into the function results in:\[f(1) = (1-1)^{40} = 0\]It's important to understand that finding the root of a function can be difficult when the function curves very flatly, as in this case. Here, Newton's method helps us locate this root through iterative calculations. The goal is to find the exact point along the x-axis where the function becomes zero, using smart guesses and calculus-based adjustments.

Derivative Calculation

The derivative of a function is crucial in Newton's method as it provides the rate of change or slope of the function at any given point. For the function \(f(x) = (x-1)^{40}\), the derivative is calculated as:\[f'(x) = 40(x - 1)^{39}\]This derivative tells us how steep or flat the curve of the function is near our current approximation of the root. It is used in the iterative formula:\[x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}\]A correctly calculated derivative is fundamental, as a mistake can significantly affect the convergence of the method. In our case, it helps determine how large the adjustment to \(x_n\) should be, in order to improve the accuracy of the root approximation with each iteration.

Convergence Analysis

Convergence analysis in the context of Newton's method examines how and if the iterative process actually approaches the function's root. The speed of convergence can vary widely depending on the nature of the function and the chosen initial guess. For the function \((x-1)^{40}\), we observed that convergence is relatively slow due to the function's near-flat sections around the root \(x=1\). In each iteration, the values of \(x_n\) move closer to the actual root, but progress diminishes noticeably due to the function's shape.Factors such as the flatness of the curve significantly influence convergence. A flatter curve at the root results in smaller step sizes per iteration, making it harder to quickly reach the true root. Thus, convergence analysis not only helps in understanding the efficiency of Newton’s method in specific applications but also guides adjustments to the method for improved performance.