Question

Calculate two iterations of Newton's Method to approximate a zero of the function using the given initial guess. (Round your answers to four decimal places.)

f(x) = x2 − 2, x1 = 1.4
n
xn
f(xn)
f ′(xn)
f(xn)
f ′(xn) xn − f(xn)
f ′(xn)
1 2

Answer 1

To approximate a zero of the function using Newton's Method, we can calculate two iterations with the given initial guess using the formula xn+1 = xn - f(xn)/f'(xn).

Step-by-step explanation:

To calculate two iterations of Newton's Method to approximate a zero of the function, we need to use the formula:

xn+1 = xn - f(xn)/f'(xn)

Given that f(x) = x2 - 2 and x1 = 1.4, we can calculate the following:

1. f(1.4) = (1.4)2 - 2 = 0.96
2. f'(1.4) = 2(1.4) = 2.8
3. x2 = 1.4 - 0.96/2.8 = 1.116
4. f(1.116) = (1.116)2 - 2 = -0.2281
5. f'(1.116) = 2(1.116) = 2.232
6. x3 = 1.116 - (-0.2281)/2.232 = 1.2277