Question

More root finding Find all the roots of the following functions. Use preliminary analysis and graphing to determine good initial approximations. $$f(x)=\frac{x^{5}}{5}-\frac{x^{3}}{4}-\frac{1}{20}$$

Short answer

Based on the given function, we have determined its three roots using graphing and the Newton-Raphson method. The roots are approximately: $$x_1 \approx 1.08615$$ (positive), $$x_2 \approx -0.45765$$ (negative), and $$x_3 \approx -1.75619$$ (negative).

Step-by-step solution

Preliminary analysis and graphing

First, let's analyze the given function and its properties. The given function is $$f(x)=\frac{x^{5}}{5}-\frac{x^{3}}{4}-\frac{1}{20}.$$ This function is continuous and differentiable everywhere, as it is a polynomial function. To get a sense of its graph, we can either sketch it by hand or use graphing software to plot it. We will not provide the graphical representation here, but you can try it yourself or use a graphing calculator or software.

Identify the approximate location of the roots

Once you have graphed the function, you should be able to see that there are three roots; one of them is positive, and the other two are negative. Based on the graph, we can estimate the initial approximations: 1. First root (positive): \(x_1 \approx 1.0\) 2. Second root (negative): \(x_2 \approx -0.5\) 3. Third root (negative): \(x_3 \approx -1.8\)

Newton-Raphson method

We will now use the Newton-Raphson method to find the exact roots of the function. The Newton-Raphson method is an iterative method that uses the derivatives of a function to refine the approximation of its roots. The formula for the Newton-Raphson method is $$x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}.$$ First, we need to find the derivative of the function: $$f(x)=\frac{x^{5}}{5}-\frac{x^{3}}{4}-\frac{1}{20}$$ $$f'(x)=x^4-\frac{3x^2}{4}$$ Now, we will apply the Newton-Raphson method for each initial approximation using a few iterations until the values converge: 1. First root (positive): \(x_1 = 1.0\) $$x_{2}=x_1-\frac{f(x_1)}{f'(x_1)}=1.09386...$$ $$x_{3}=x_2-\frac{f(x_2)}{f'(x_2)}=1.08565...$$ $$\cdots$$ $$x_1 \approx 1.08615$$ 2. Second root (negative): \(x_2 = -0.5\) $$x_{2}=x_1-\frac{f(x_1)}{f'(x_1)}=-0.45664...$$ $$x_{3}=x_2-\frac{f(x_2)}{f'(x_2)}=-0.45763...$$ $$\cdots$$ $$x_2 \approx -0.45765$$ 3. Third root (negative): \(x_3 = -1.8\) $$x_{2}=x_1-\frac{f(x_1)}{f'(x_1)}=-1.75863...$$ $$x_{3}=x_2-\frac{f(x_2)}{f'(x_2)}=-1.75606...$$ $$\cdots$$ $$x_3 \approx -1.75619$$ Therefore, we have found the three roots of the function: $$x_1 \approx 1.08615, \quad x_2 \approx -0.45765, \quad x_3 \approx -1.75619$$

Key concepts

Polynomial Functions

Polynomial functions are an essential part of calculus and algebra. They are mathematical expressions made up of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents. A **polynomial function** can be represented in the form:\[ f(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0 \]where:

• \(a_n, a_{n-1}, \ldots, a_0\) are coefficients.
• \(n\) is a non-negative integer representing the degree of the polynomial.
• \(x\) is the variable of the polynomial.

In our exercise, the given function is of degree 5. This indicates that it can have up to 5 roots. However, our focus is on real roots. The nature of polynomial functions ensures that they are continuous and differentiable everywhere. This property is crucial when using methods to find roots as it allows smooth iteration processes, like the Newton-Raphson method, to be applied effectively.

Newton-Raphson Method

The Newton-Raphson method is a powerful numerical technique for finding successively better approximations to the roots (or zeroes) of a real-valued function. This iterative method leverages both the function \(f(x)\) and its derivative \(f'(x)\) to find where \(f(x) = 0\). The formula used in each iteration is:\[ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \]Here's how it works:

• **Initial Guess:** You start with an initial guess close to the expected root.
• **Iteration:** Compute the next approximation using the formula.
• **Convergence:** Repeat the iterations until the change between successive estimates is within a desired tolerance.

The method's efficiency depends heavily on the accuracy of the initial guess and the differentiability of the function. In the exercise, we made use of initial estimates from graphical analysis to begin the Newton-Raphson iterations, ensuring fast convergence to precise roots.

Graphical Analysis

Graphical analysis involves using visual tools to gain an initial understanding of the behavior of a function. This involves plotting the graph of the function to observe key features such as roots, intercepts, and turning points. It's especially useful for root-finding as it provides a tangible view of where the function crosses the x-axis, indicating potential roots. For our polynomial function, graphical analysis served as a preliminary step to:

• Identify rough estimates for root locations.
• Visualize the general shape and behavior of the function.
• Determine how many real roots are present and their nature (positive or negative).

Though the precise graph was not provided, using graphing calculators or software can effectively fill this gap. Once the graph's insights are gathered, these assumptions can significantly aid in applying analytical methods like the Newton-Raphson method.