Problem 32 - Calculus Single Variable

Question

The polynomial \(f(x)=x^{11}+x^{7}-1\) has one root \(\gamma\) in the interval [0,1] . Find this root to five decimal places using the Newton-Raphson Method.

Step-by-Step Solution

Answer

The root \( \gamma \) is approximately 0.90708, found using Newton-Raphson iterations.

Step 1: Understanding the Problem

We need to find a root of the polynomial \( f(x) = x^{11} + x^7 - 1 \) within the interval [0, 1] to five decimal places using the Newton-Raphson method.

Step 2: Initialize Newton-Raphson Method

The Newton-Raphson method requires an initial guess. Since the root is in [0,1], choose an initial guess such as \( x_0 = 0.5 \). Also, recall the iterative formula: \( x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \).

Step 3: Derive the Derivative

Compute the derivative of \( f(x) = x^{11} + x^7 - 1 \). The derivative is \( f'(x) = 11x^{10} + 7x^6 \).

Step 4: Iteration 1

Using \( x_0 = 0.5 \), calculate \( f(x_0) = 0.5^{11} + 0.5^7 - 1 \) and \( f'(x_0) = 11(0.5^{10}) + 7(0.5^6) \). Update \( x_1 = x_0 - \frac{f(x_0)}{f'(x_0)} \).

Step 5: Continue Iterations

Repeat the iteration using the formula \( x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \) until the difference between consecutive values of \( x \) is less than \( 10^{-5} \), indicating sufficient precision.

Step 6: Checking Convergence

Check if the difference \( |x_{n+1} - x_n| < 0.00001 \). If so, stop iterating, if not, continue with the next iteration.

Step 7: Result and Verification

Once convergence is achieved, the latest value returned as \( x_n \) is the approximate root to five decimal places. Verify by substituting back into \( f(x) \) to ensure \( f(x_n) \) is close to zero, indicating the accuracy of the root.

Key Concepts

Polynomial Root Finding

Finding roots of polynomials is like pinpointing where the graph of a polynomial function crosses the x-axis. These roots are solutions to the equation \( f(x) = 0 \). For the polynomial \( f(x) = x^{11} + x^7 - 1 \), we want to find a value \( \gamma \) such that when \( x = \gamma \), the function equals zero. This process is crucial because it helps in understanding the behavior of polynomial functions across different ranges.
In this specific problem, we focus on the interval \([0, 1]\) to locate one of these crossings. The challenge is to determine the root with great precision - up to five decimal places. Such precision is often necessary in fields ranging from engineering to computer science where small inaccuracies can lead to significant errors in outcomes.

Numerical Methods

Numerical methods are practical tools designed to find solutions to mathematical problems that are difficult or impossible to solve algebraically. The Newton-Raphson method is a popular numerical technique for finding roots of real-valued functions quickly and with high precision.
This method involves an iterative process starting from an initial guess, which is refined to approach a true root. The basic idea is to use the tangent line at a given point to approximate the root, then repeat the process to get closer and closer to the actual value. Specifically, this is done by using the formula:

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

Each step of the iteration uses the derivative of the function, \( f'(x) \), which indicates the slope of the tangent line at any point \( x_n \). The efficiency and speed of the Newton-Raphson method make it exceptionally useful for root-finding tasks, provided the function behaves nicely and an appropriate initial guess is chosen.

Calculus Concepts

Calculus forms the backbone of many numerical methods used in root finding, particularly through the use of derivatives. The derivative, as seen in the Newton-Raphson method, plays a pivotal role. It represents the slope or rate of change of a function and is critical when approximating roots by providing the direction and magnitude in which to adjust guesses.
For the polynomial \( f(x) = x^{11} + x^7 - 1 \), the derivative \( f'(x) = 11x^{10} + 7x^6 \) is derived using basic differentiation rules - highlighting the power of calculus in breaking down complex functions into manageable components. Understanding how derivatives work helps us navigate more complicated equations and is essential for ensuring the Newton-Raphson iterations are valid and will converge to a meaningful solution.