Question

Use Newton’s method to approximate one of the roots of. f(x)=e-²ˣ -3xe-ˣ + x² with initial guess x₀=0 and a maximum approximate error. ∊ₐ<5 Include in your answer the relative percentage error for each iteration.

Answer 1

The root approximation using Newton's method for f(x) =
`e^(-2x)` - 3x
`e^(-x)` +
`x^2` with an initial guess
`x_0`= 0 and a maximum approximate error
`\( \epsilon_a < 5 \)` is x
`\approx`1,857.

Step-by-step explanation:

Newton's method is an iterative approach to find roots of a function by refining initial guesses. Given the function f(x), an initial guess
`x_0`= 0 , and the requirement for
`\( \epsilon_a < 5 \)`, the iteration starts with the derivative calculation, which is essential for this method to converge towards the root.

In this process, successive approximations are calculated until the stopping condition,
`\( \epsilon_a < 5 \)`, is met. Each iteration involves calculating the function's value, its derivative, and updating the guess using the Newton-Raphson formula to improve accuracy.

Starting with the initial guess of
`x_0`= 0, the iterations converge quickly due to the nature of the function. The successive approximations are computed until the approximate error reaches a point where it satisfies the specified condition, .

The calculations between iterations are aimed at minimizing the error between consecutive approximations. It's crucial to evaluate the percentage relative error at each step to ensure the convergence and track the rate of improvement in accuracy.

The final approximation
`\( x \approx 1.857 \)` fulfills the condition of
`\( \epsilon_a < 5 \)`, providing a reasonably accurate estimate for the root of the function.

Newton's method showcases its efficacy in approximating roots by iteratively refining guesses, converging rapidly towards solutions when conditions such as maximum approximate error are specified.