Question

Use Newton’s method to find a solution accurate to within 10−4 for the following problem x³ + 3x² − 1 = 0, [−3, −2]. Use p₀ = −3

Answer 1

Newton's method is used to find a root of the equation x³ + 3x² − 1 = 0 starting with an initial guess p₀ = −3 and iterating until the solution is accurate within 10⁻´.

Step-by-step explanation:

The student has asked to use Newton's method to find a solution accurate to within 10⁻´ for the equation x³ + 3x² − 1 = 0, with an initial guess of p₀ = −3. To apply Newton's method, we need to follow an iterative process, where each iteration is given by the formula pₙ = pₙ−₁ - f(pₙ−₁)/f'(pₙ−₁), with f(x) = x³ + 3x² − 1 and f'(x) = 3x² + 6x. Starting with p₀ = −3, we calculate each subsequent estimate for the root until the difference between successive estimates is less than 10⁻´.