Question

Using the two-curve graphical procedure over the interval 0 to 5) show if the function f(x)= √x+sinx−x in the simple fixed-point iteration method converges or diverges using an initial value of x=0.5.

Answer 1

To determine if the function f(x)= √x+sinx−x converges with an initial value of x=0.5 using fixed-point iteration, graph the function, apply iterations starting from x=0.5, and check for convergence by ensuring the derivative of the iterating function is less than 1 in magnitude over the interval.

Step-by-step explanation:

The question asks about using the fixed-point iteration method to determine if the function f(x)= √x+sinx−x converges or diverges with an initial guess of x=0.5 over the interval [0, 5]. In fixed-point iteration, the function is rewritten in the form x = g(x), and we iterate using x_{n+1} = g(x_n). To check for convergence, we can graphically represent the function and perform iterations.

The convergence can be determined by repeatedly applying the function g(x) starting from x=0.5 and checking whether the values approach a fixed point without diverging.

To assess whether the iteration converges, one should check if the function meets the convergence criteria, which typically involve the derivative of g(x) being less than 1 in absolute value in the interval of interest, suggesting that the method will likely converge near a fixed point.