Answer:
After the fifth iteration of the Bisection Method we have that the root is x = 0.7344 and the absolute error is [tex]\Delta x=0.0047[/tex]
Step-by-step explanation:
This is the algorithm for the Bisection method:
Suppose we need a root for f(x) = 0 and we have an error tolerance of ε
We know
[tex]f(x) = x-cos \:x[/tex]
How close the value of c gets to the real root depends on the value of the tolerance we set for the algorithm. For that reason, we choose an error tolerance of ε = 0.01.
Applying the steps of the Bisection Method you get:
1. There is a root between [0,1] because:
[tex]f(0) = 0-cos \:0=-1\\f(1) = 1-cos \:1=0.4597[/tex]
2. Define [tex]c=\frac{0+1}{2} =0.5[/tex]
3. [tex]2-0.5\geq 0.01[/tex]
4. Because [tex]f(0)\cdot f(0.5)=0.3776[/tex] we set 0.5 as the new a and we return to step 1.
The bisection algorithm is detailed in the following table.
To find the absolute error we use the value of the Bisection Method and the value we find using WolframAlpha.
The root that the Bisection Method give us is x = 0.7344 and the root that WolframAlpha give us is x = 0.7391. Therefore
[tex]\Delta x= |0.7344-0.7391|= 0.0047[/tex]
