Answer:
Down
Step-by-step explanation:
The initial guess is given as x1 = 0. We start by evaluating the function at this point to find y1: y1 = (0)^3 − 2(0) − 2 = -2.
Find the derivative of the function, which is dy/dx = 3x^2 - 2. We can then use this derivative to update our initial guess and find x2:
x2 = x1 - y1 / (dy/dx) at x1
x2 = 0 - (-2) / (3(0)^2 - 2) = -1.0
We repeat the process by evaluating the function at x2 to find y2: y2 = (-1.0)^3 − 2(-1.0) − 2 = -1.0. Then, we update our guess to find x3:
x3 = x2 - y2 / (dy/dx) at x2
x3 = -1.0 - (-1.0) / (3(-1.0)^2 - 2) = -1.4
We once again evaluate the function at x3 to find y3: y3 = (-1.4)^3 − 2(-1.4) − 2 ≈ 0.6176. We update our guess to find x4:
x4 = x3 - y3 / (dy/dx) at x3
x4 = -1.4 - 0.6176 / (3(-1.4)^2 - 2) ≈ -1.36559139785
The iterations continue until the desired level of accuracy is achieved. In this case, x4 is an approximation of the root of the function y = x^3 − 2x − 2.