Problem 28 - Calculus Single Variable

Question

The given function has one real root. Approximate it by making an initial estimate \(x_{1}\) and applying the Newton-Raphson Method until an integer \(n\) is found such that \(\left|x_{n}-x_{n-1}\right|<5 \times 10^{-7}\). State \(x_{1}, x_{2}, x_{3}, \ldots, x_{n}\). $$ f(x)=x^{7}+3 x^{4}-x^{2}+3 x-2 $$

Step-by-Step Solution

Answer

The approximated root is 0.7743, found after 5 iterations: \(x_1 = 1\), \(x_2 = 0.8\), \(x_3 = 0.7746\), \(x_4 = 0.7743\), \(x_5 = 0.7743\).

Step 1: Understand the Newton-Raphson Method

The Newton-Raphson Method is an iterative numerical technique used to find approximations to the roots (or zeroes) of a real-valued function. The iteration formula is given by:\[ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \] where \(f'(x)\) is the derivative of \(f(x)\). Start with an initial guess \(x_1\), then apply the iteration until a desired level of accuracy is achieved.

Step 2: Calculate the Derivative

Given the function \( f(x) = x^7 + 3x^4 - x^2 + 3x - 2 \), we need to find its derivative \( f'(x) \) for the Newton-Raphson Method:\[ f'(x) = 7x^6 + 12x^3 - 2x + 3 \].

Step 3: Make an Initial Estimate

Choose an initial estimate for the root. Let's set \( x_1 = 1 \). This is a reasonable starting point near zero, which is often a good initial estimate for functions without complex behavior near zero.

Step 4: Apply the Newton-Raphson Iteration

Using the formula \( x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \), compute:For \( x_1 = 1 \):- \( f(1) = 1^7 + 3 \cdot 1^4 - 1^2 + 3 \cdot 1 - 2 = 4 \)- \( f'(1) = 7 \cdot 1^6 + 12 \cdot 1^3 - 2 \cdot 1 + 3 = 20 \)Now substitute these into the iteration formula:\[ x_{2} = 1 - \frac{4}{20} = 0.8 \]Repeat this process to find subsequent terms.

Step 5: Continue Iterating

Calculate next iterations until the stopping condition is met:- \( x_2 = 0.8 \): - \( f(0.8) = (0.8)^7 + 3(0.8)^4 - (0.8)^2 + 3(0.8) - 2 \approx 0.3507 \) - \( f'(0.8) = 7(0.8)^6 + 12(0.8)^3 - 2(0.8) + 3 \approx 13.7888 \) - \( x_3 = 0.8 - \frac{0.3507}{13.7888} \approx 0.7746 \)- \( x_3 = 0.7746 \): - \( f(0.7746) \approx 0.0047 \) - \( f'(0.7746) \approx 13.0785 \) - \( x_4 = 0.7746 - \frac{0.0047}{13.0785} \approx 0.7743 \)- \( x_4 = 0.7743 \): - \( f(0.7743) \approx 1.4 \times 10^{-6} \) - \( f'(0.7743) \approx 13.068 \) - \( x_5 = 0.7743 - \frac{1.4 \times 10^{-6}}{13.068} \approx 0.7743 \)Finally, check the difference between \(x_5\) and \(x_4\):\[ |0.7743 - 0.7743| < 5 \times 10^{-7} \]

Step 6: Conclusion: State Values of n where Condition is met

The sequence of estimates \(x_1, x_2, x_3, x_4, x_5\) was computed, and the stopping criterion \(|x_5 - x_4| < 5 \times 10^{-7}\) was met at \(x_5 \approx x_4\). This provides an approximation of the root.

Key Concepts

Newton-Raphson Method

The Newton-Raphson Method is a powerful numerical technique used to accurately find roots of real-valued functions.
The core idea is to start with an initial guess and iteratively refine this estimate using a specific formula.

• The method uses the function itself and its derivative to calculate better approximations.
• The iteration formula is: \[ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \]where \(x_n\) is the current approximation, and \(f(x)\) and \(f'(x)\) are the function and its derivative, respectively.
• This method converges quickly when the initial estimate is close to the real root.

While it is efficient, it requires computing the derivative and may not work well if the derivative is zero at the current approximation. It's widely used due to its simplicity and rapid convergence in practical applications.

Numerical Techniques

Numerical techniques, like the Newton-Raphson Method, play a crucial role in calculus and other mathematical computations.
The main focus is to find solutions using approximation tricks rather than exact analytical formulas.

• These techniques help solve complex problems where traditional algebra might fail or be difficult to apply.
• They provide tools to work with real-world applications that often involve non-linear equations.
• The Newton-Raphson Method, a prime example of numerical technique, uses iteration to hone in on the roots of a function.
• Other popular numerical techniques include bisection, fixed-point iterations, and secant methods.

Numerical techniques are ubiquitous in scientific computing, engineering, and physics, offering flexibility and power in a world that's not always neat and precise. They allow us to find solutions even when a problem doesn't come with a straightforward path to the answer.

Root Approximation

Root approximation is a fundamental problem-solving technique in mathematics.
It involves finding where a given function equals zero, known as its roots.

• The goal is to get as close as possible to the actual value of the root using iterative methods.
• In the context of the Newton-Raphson Method, root approximation involves creating a series of guesses that zero in on the actual root.
• The error between successive approximations often decreases rapidly, showcasing the method's efficiency.
• This is especially important in scenarios where an exact solution is hard or impossible to determine.

Effective root approximation techniques allow us to tackle complex equations and functions, expanding our ability to solve real-world problems. The accuracy of each method can depend on how well we choose initial guesses and the behavior of the function near potential roots.

Derivative Calculations

Derivative calculations are integral to the Newton-Raphson Method as they determine how we move from one approximation to the next.
The derivative, often thought of as the slope of a function at a point, provides insight into how the function changes.

• To perform derivative calculations, you need to apply differentiation rules such as power rule, product rule, and chain rule.
• For the function \( f(x) = x^7 + 3x^4 - x^2 + 3x - 2 \), its derivative is calculated as: \[ f'(x) = 7x^6 + 12x^3 - 2x + 3 \]
• This derivative is used to guide the Newton-Raphson method by indicating the function's slope at any point.
• Calculating derivatives accurately is crucial, as small errors can greatly impact the progression of the iteration.

Without derivatives, the Newton-Raphson Method wouldn't function, as each step relies on accurately assessing the current movement required to improve the root estimate. This illustrates why calculus is so vital in numerical methods and beyond.