Question

Use Newton's method to approximate a solution of the equation e⁻ˣ=-4 x+5, starting with the initial guess indicated.

[ x₁=4 .; x₂=; x₃= ]

Answer 1

To approximate the solution to the equation e^(-x) = -4x + 5 using Newton's method, start with an initial guess and repeat the steps until reaching an acceptable approximation.

Step-by-step explanation:

To use Newton's method and approximate a solution to the equation e^(-x) = -4x + 5, we start with an initial guess. Let's start with x₁ = 4. We repeat the following steps until we reach an acceptable approximation:

1. Calculate the function value and the derivative at the current guess, x₁.
2. Use the formula x₂ = x₁ - f(x₁)/f'(x₁) to find the next guess.
3. Repeat step 2 until the change in the guess is small enough.

Using Newton's method, we have x₂ = 4 - (e^(-4) + 4(4) - 5)/(e^(-4) + 4) which simplifies to x₂ ≈ 3.967. Now, we can repeat step 2 to find the next guess, x₃.

Using Newton's method again, x₃ = 3.967 - (e^(-3.967) + 4(3.967) - 5)/(e^(-3.967) + 4) which simplifies to x₃ ≈ 3.964.