Question

Find the root of the function f(x) = e^(-2x)x + xcos(x) - 4sin(2x) on the interval [1, 2] with the error tolerance |f(x)| ≤ 10⁻⁵ by: (a) Bisection method (b) Regula Falsi method.

Answer 1

To find the root of the function f(x) = e^(-2x)x + xcos(x) - 4sin(2x) on the interval [1, 2] with an error tolerance of |f(x)| ≤ 10⁻⁵, you can use the (a) Bisection method or (b) Regula Falsi method.

Step-by-step explanation:

To find the root of the function f(x) = e^(-2x)x + xcos(x) - 4sin(2x) on the interval [1, 2] with an error tolerance of |f(x)| ≤ 10⁻⁵:

(a) Bisection method:

Calculate f(1) and f(2) to determine if there is a root within the interval [1, 2].

Calculate f(c), where c is the midpoint of the interval [1, 2].

Repeat steps 1 and 2 until the error tolerance is met or the interval is small enough.

(b) Regula Falsi method:

Calculate f(1) and f(2) to determine if there is a root within the interval [1, 2].

Calculate c using the formula c = (a*f(b) - b*f(a)) / (f(b) - f(a)), where a = 1, b = 2.

Calculate f(c) and update the interval [a, c] or [c, b] based on the sign of f(c).

Repeat steps 1-3 until the error tolerance is met or the interval is small enough.