{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

검증된 단계별 안내

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:

x_{n + 1} = x_{n} - \frac{f (x_{n})}{f' (x_{n})}

First, compute the derivative of the function f(x) = e^-x - x. The derivative, f'(x), is:

- e^{- x} - 1

Using the initial approximation x₀ = ln(2), substitute x₀ into the Newton's method formula to find x₁:

x_{1} = \ln (2) - \frac{f (\ln (2))}{f' (\ln (2))}

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.

Repeat the process using x₁ to find x₂:

x_{2} = x_{1} - \frac{f (x_{1})}{f' (x_{1})}

. Calculate f(x₁) and f'(x₁), then substitute these values to find x₂.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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).

{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