{Use of Tech} Write the formula for Newton’s method and use the given initial approximation to compute the approximations x₁ and x₂.


f(x) = x² - 6; x₀ = 3

Verified step by step guidance

1. Newton's method is an iterative process used to approximate the roots of a real-valued function. The formula for Newton's method is: <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>_</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>=</mo><mi>x</mi><mo>_</mo><mi>n</mi><mo>-</mo><mfrac><mrow><mi>f</mi><mo>(</mo><mi>x</mi><mo>_</mo><mi>n</mi><mo>)</mo></mrow><mrow><mi>f</mi><mo>'</mo><mo>(</mo><mi>x</mi><mo>_</mo><mi>n</mi><mo>)</mo></mrow></mfrac></math>.
2. First, we need to find the derivative of the function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>6</mn></math>. The derivative is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mn>2</mn><mi>x</mi></math>.
3. Using the initial approximation <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>_</mo><mn>0</mn><mo>=</mo><mn>3</mn></math>, calculate <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>_</mo><mn>0</mn><mo>)</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mo>(</mo><mi>x</mi><mo>_</mo><mn>0</mn><mo>)</mo></math>.
4. Substitute these values into the Newton's method formula to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>_</mo><mn>1</mn></math>: <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>_</mo><mn>1</mn><mo>=</mo><mi>x</mi><mo>_</mo><mn>0</mn><mo>-</mo><mfrac><mrow><mi>f</mi><mo>(</mo><mi>x</mi><mo>_</mo><mn>0</mn><mo>)</mo></mrow><mrow><mi>f</mi><mo>'</mo><mo>(</mo><mi>x</mi><mo>_</mo><mn>0</mn><mo>)</mo></mrow></mfrac></math>.
5. Repeat the process using <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>_</mo><mn>1</mn></math> to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>_</mo><mn>2</mn></math>: <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>_</mo><mn>2</mn><mo>=</mo><mi>x</mi><mo>_</mo><mn>1</mn><mo>-</mo><mfrac><mrow><mi>f</mi><mo>(</mo><mi>x</mi><mo>_</mo><mn>1</mn><mo>)</mo></mrow><mrow><mi>f</mi><mo>'</mo><mo>(</mo><mi>x</mi><mo>_</mo><mn>1</mn><mo>)</mo></mrow></mfrac></math>.

Key Concepts

Newton's Method

Newton's Method is an iterative numerical technique used to find approximate solutions to equations. It starts with an initial guess and refines it using the formula x₁ = x₀ - f(x₀)/f'(x₀), where f'(x) is the derivative of the function. This method is particularly effective for functions that are continuous and differentiable near the root.

Derivative

The derivative of a function measures how the function's output changes as its input changes. It is a fundamental concept in calculus that provides the slope of the tangent line to the function at any given point. In the context of Newton's Method, the derivative is crucial for determining the direction and magnitude of the adjustments made to the initial approximation.

Initial Approximation

An initial approximation is the starting value used in iterative methods like Newton's Method. The choice of this value can significantly affect the convergence of the method to the actual root. A good initial approximation should be close to the actual root to ensure that the iterations lead to a successful and rapid convergence.