Use the Newton method to approximate the indicated zero of the function. Continue with the iteration
until two successive approximations differ by less than 0,0001
the zero of: f(x)=x^3+2x^2+x-7 between x=1 and x=2, x0=1
To use the Newton method to approximate the zero of the function f(x) = x^3 + 2x^2 + x - 7, we need to follow these steps:
Step 1: Start with an initial approximation, x0 = 1, within the interval [1, 2].
Step 2: Calculate f'(x), the derivative of f(x):
f'(x) = 3x^2 + 4x + 1.
Step 3: Use the Newton method iteration formula:
x1 = x0 - (f(x0) / f'(x0)).
Step 4: Calculate f(x0):
f(x0) = (1)^3 + 2(1)^2 + (1) - 7 = -3.
Step 5: Calculate f'(x0):
f'(x0) = 3(1)^2 + 4(1) + 1 = 8.
Step 6: Use the iteration formula to find x1:
x1 = 1 - (-3 / 8) = 1.375.
Step 7: Calculate f(x1):
f(x1) = (1.375)^3 + 2(1.375)^2 + (1.375) - 7 = -0.852.
Step 8: Repeat steps 5-7 until two successive approximations differ by less than 0.0001.
Keep repeating steps 5 to 7 until we achieve the desired accuracy.