Question

Use Newton's method with the first approximation c1 = 0.6 to find c3, the third approximation to the real zero of the function.

a) c3 = c1 - f(c1)/f'(c1)
b) c3 = c1 - 2f(c1)/f''(c1)
c) c3 = c1 - f(c1)/2f'(c1)
d) c3 = c1 - f(c1)/f''(c1)

Answer 1

To find c3, the third approximation of the real zero of a function using Newton's method and starting with c1 = 0.6, the correct formula is c3 = c1 - f(c1)/f'(c1), per option (a). The correct option is (a).

Step-by-step explanation:

The provided question asks how to use Newton's method to find the third approximation c3 of the real zero of a function, given the first approximation c1 = 0.6. The correct formula to use here is c3 = c1 - f(c1)/f'(c1), making option (a) the valid choice.

To apply Newton's method, we require both the function f(x) and its derivative f'(x). However, the function is not explicitly given in the question, thus this answer cannot proceed without the function to which Newton's method would be applied. Without this information, we can conclude that to apply Newton's method, one calculates subsequent approximations using the formula cn = cn-1 - f(cn-1)/f'(cn-1). After the first iteration with c1, we would continue with c2 = c1 - f(c1)/f'(c1) and then c3 = c2 - f(c2)/f'(c2).