Final answer:
The Secant Method is an iterative numerical method used to find the roots of a function. It estimates the location of the root by linear interpolation. Four iterations of the Secant Method are performed using the given initial values.
Step-by-step explanation:
The Secant Method is an iterative numerical method used to find the roots of a function. It is based on linear interpolation between two initial points to estimate the location of the root. The formula for each iteration is:
x(i+1) = x(i) - (f(x(i))*(x(i)-x(i-1))/(f(x(i))-f(x(i-1))))
To perform four iterations of the Secant Method, we start with two initial values x(0) and x(1) and use the formula to calculate x(2), x(3), and x(4).
Iteration 1:
x(0) = 0.0000
x(1) = 8.0000
f(x(0)) = 0.0148
f(x(1)) = 65.67
x(2) = x(1) - (f(x(1))*(x(1)-x(0))/(f(x(1))-f(x(0))))
x(2) = 8.0000 - (65.67*(8.0000-0.0000))/(65.67-0.0148)
x(2) = 0.0148
Iteration 2:
x(1) = 8.0000
x(2) = 0.0148
f(x(1)) = 65.67
f(x(2)) = 1.3209
x(3) = x(2) - (f(x(2))*(x(2)-x(1))/(f(x(2))-f(x(1))))
x(3) = 0.0148 - (1.3209*(0.0148-8.0000))/(1.3209-65.67)
x(3) = -50.13
Iteration 3:
x(2) = 0.0148
x(3) = -50.13
f(x(2)) = 1.3209
f(x(3)) = 101.364
x(4) = x(3) - (f(x(3))*(x(3)-x(2))/(f(x(3))-f(x(2))))
x(4) = -50.13 - (101.364*(-50.13-0.0148))/(101.364-1.3209)
x(4) = 2.1883
Iteration 4:
x(3) = -50.13
x(4) = 2.1883
f(x(3)) = 101.364
f(x(4)) = 1.3209
The table is now complete with four iterations of the Secant Method.