Problem 14 - University Calculus: Early Transcendentals

Question

Use Newton's method to find the two real solutions of the equation \(x^{4}-2 x^{3}-x^{2}-2 x+2=0\).

Step-by-Step Solution

Answer

The two real solutions found using Newton's method are approximately 0.38 and 1.62.

Step 1: Understand Newton's Method

Newton's Method is an iterative process used to find the roots of a real-valued function. The main idea is to start with an initial guess \(x_0\) and then use the formula \(x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}\) to generate a sequence that converges to a root, where \(f(x)\) is the function and \(f'(x)\) is its derivative.

Step 2: Define the Function and Its Derivative

Given the function \(f(x) = x^4 - 2x^3 - x^2 - 2x + 2\), we first need to find its derivative. The derivative of \(f(x)\) is \(f'(x) = 4x^3 - 6x^2 - 2x - 2\). We will use this derivative in Newton's formula.

Step 3: Choose an Initial Guess for the First Root

Examine the behavior of \(f(x)\) around possible solutions. Let's start by checking a root in the range of 0 to 1. If we choose \(x_0 = 1\), we can use this initial guess in Newton's Method to find a root around this area.

Step 4: Perform Iterations for the First Root

Using the Newton's method formula, compute the next values iteratively:- \(x_1 = 1 - \frac{f(1)}{f'(1)}\)- Continue iterations until converging to a root (e.g., \(x \approx 0.38\)).Repeat until \(|x_{n+1} - x_n|\) is less than a small tolerance (such as 0.0001).

Step 5: Choose an Initial Guess for the Second Root

To find another real root, select a different initial guess. Let's try \(x_0 = 2\) based on exploration or graphing the function, to locate a root different from the first one found.

Step 6: Perform Iterations for the Second Root

Apply Newton's method again with \(x_0 = 2\):- \(x_1 = 2 - \frac{f(2)}{f'(2)}\)- Continue iterating until the value converges, possibly finding \(x \approx 1.62\).- Ensure the same convergence criteria as before, holding to a small tolerance.

Key Concepts

Iterative Process

Newton's Method is a type of iterative process. This means that it repeatedly applies a specific formula to get closer to a desired solution. In this case, the solution is the root of a real-valued function. Here’s how it works:

• Start with an initial guess for the root, called \(x_0\).
• Use Newton's formula: \(x_{n+1} = x_{n} - \frac{f(x_{n})}{f'(x_{n})}\)
• This generates a sequence of \(x\) values that ideally converge, or get closer, to the actual root of the function.

The key to success with this method is selecting a good starting point \(x_0\). The better it is, the fewer iterations required to reach a sufficiently accurate approximation. The process stops when consecutive \(x\) values are very close, i.e., their difference is below a chosen tolerance.

Real-valued Function

A real-valued function maps real numbers to real numbers, meaning the domain and range are both subsets of real numbers. In the context of Newton's Method, the function we are examining is the polynomial \(f(x) = x^4 - 2x^3 - x^2 - 2x + 2\). This function:

• Is continuous, meaning it doesn’t have gaps or jumps as you move along the x-axis.
• Has roots, which are \(x\) values where the function equals zero.

Newton's Method is particularly useful for real-valued functions because it helps find where the function crosses the x-axis, revealing the roots. Identifying the behavior such as increasing or decreasing patterns around these areas can guide the choice of starting points for iterations.

Function Derivative

The derivative of a function provides information about its slope or rate of change at a given point. In applying Newton's Method, the derivative, denoted as \(f'(x)\), is crucial because it appears in the formula. For our specific function:

• The function is \(f(x) = x^4 - 2x^3 - x^2 - 2x + 2\).
• Its derivative is \(f'(x) = 4x^3 - 6x^2 - 2x - 2\).
• This derivative represents the slope of \(f(x)\) at various points \(x\).

In Newton's formula \(x_{n+1} = x_{n} - \frac{f(x_{n})}{f'(x_{n})}\), the derivative helps adjust the \(x\) value towards the root. A steep slope leads to a larger step, while a mild slope results in a smaller step, keeping the iterations efficient and targeted towards convergence.