Question

Using the bisection method, find the root of the following function:

f(x)=cos(2x) - x

Use the following initial values: xl=0 and xu=2
NOTE: Perform only 3 iterations.

Answer 1

Find a root of an equation f(x)=cos(2x)-x between 0 and 2, using Bisection method

Solution:
Here cos(2x)-x=0

Let f(x)=cos(2x)-x

Here

x	0	1	2
f(x)	1	-1.4161	-2.6536

Here f(0)=1>0 and f(1)=-1.4161<0

∴ Root lies between 0 and 1


1st iteration :

Here f(0)=1>0 and f(1)=-1.4161<0

∴ Now, Root lies between 0 and 1

x0=0+12=0.5

f(x0)=f(0.5)=cos(1)-0.5=0.0403>0


2nd iteration :

Here f(0.5)=0.0403>0 and f(1)=-1.4161<0

∴ Now, Root lies between 0.5 and 1

x1=0.5+12=0.75

f(x1)=f(0.75)=cos(1.5)-0.75=-0.6793<0


3rd iteration :

Here f(0.5)=0.0403>0 and f(0.75)=-0.6793<0

∴ Now, Root lies between 0.5 and 0.75

x2=0.5+0.752=0.625

f(x2)=f(0.625)=cos(1.25)-0.625=-0.3097<0