{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) = e⁻ˣ - x; x₀ = ln 2

Verified step by step guidance

1. Newton's method is an iterative technique for finding successively better approximations to the roots (or zeroes) of a real-valued function. The formula for Newton's method is: <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>x</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><msub><mi>x</mi><mi>n</mi></msub><mo>-</mo><mfrac><mrow><mi>f</mi><mo>(</mo><msub><mi>x</mi><mi>n</mi></msub><mo>)</mo></mrow><mrow><mi>f</mi><mo>'</mo><mo>(</mo><msub><mi>x</mi><mi>n</mi></msub><mo>)</mo></mrow></mfrac></mrow></math>.
2. First, compute the derivative of the function f(x) = e^-x - x. The derivative, f'(x), is: <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>-</mo><msup><mi>e</mi><mo>-</mo><mi>x</mi></msup><mo>-</mo><mn>1</mn></mrow></math>.
3. Using the initial approximation x₀ = ln(2), substitute x₀ into the Newton's method formula to find x₁: <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>x</mi><mn>1</mn></msub><mo>=</mo><mi>ln</mi><mo>(</mo><mn>2</mn><mo>)</mo><mo>-</mo><mfrac><mrow><mi>f</mi><mo>(</mo><mi>ln</mi><mo>(</mo><mn>2</mn><mo>)</mo><mo>)</mo></mrow><mrow><mi>f</mi><mo>'</mo><mo>(</mo><mi>ln</mi><mo>(</mo><mn>2</mn><mo>)</mo><mo>)</mo></mrow></mfrac></mrow></math>.
4. Calculate f(ln(2)) and f'(ln(2)) using the expressions for f(x) and f'(x). Substitute these values into the formula to compute x₁.
5. Repeat the process using x₁ to find x₂: <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>x</mi><mn>2</mn></msub><mo>=</mo><msub><mi>x</mi><mn>1</mn></msub><mo>-</mo><mfrac><mrow><mi>f</mi><mo>(</mo><msub><mi>x</mi><mn>1</mn></msub><mo>)</mo></mrow><mrow><mi>f</mi><mo>'</mo><mo>(</mo><msub><mi>x</mi><mn>1</mn></msub><mo>)</mo></mrow></mfrac></mrow></math>. Calculate f(x₁) and f'(x₁), then substitute these values to find x₂.

Key Concepts

Newton's Method

Newton's Method is an iterative numerical technique used to find approximate solutions to equations of the form f(x) = 0. The method uses the derivative of the function to refine guesses, starting from an initial approximation. The formula for the method is x₁ = x₀ - f(x₀)/f'(x₀), where x₀ is the current approximation, and f'(x₀) is the derivative evaluated at x₀.

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 adjustment to the current approximation.

Initial Approximation

The initial approximation is the starting value used in iterative methods like Newton's Method. A good initial approximation can significantly affect the convergence speed and accuracy of the method. In this case, x₀ = ln(2) serves as the starting point for calculating subsequent approximations x₁ and x₂, which will help in finding the root of the function f(x).