Question

Taking the endpoints of the last interval obtained in part a) above as the initial
approximations, perform two iterations of the secant method to approximate the
root.

Answer 1

As follows from question 110503, we were given an equation $f(x)=x^3-5x^2+1$ and we established that its smallest positive root belongs to the interval

$$[x_0,x_1]=[x_{i-1},x_i]=[0.469,0.5].$$

Use the secant method to approximate the root:

$$x_{i+1}=x_i-\frac{f(x_i)\cdot(x_i-x_{i-1})}{f(x_i)-f(x_{i-1})}=\\ \space\\ =0.5-\frac{-0.125\cdot(0.5-0.469)}{(-0.125)-0.0034}=0.4698.$$

Use this root for the second iteration, where $x_i=0.4698,x_{i-1}=0.469$:

$$x_{i+1}=0.4698-\frac{1.303\cdot10^{-4}\cdot(0.4698-0.469)}{1.303\cdot10^{-4}-0.0034}=0.46983.$$

Check the value of the function:

$$f(0.46983)=9.23\cdot10^{-6}.$$