00:01
In this question, we need to use the bisection method to solve for one real root of the equation sin of x minus x cube equals to zero on the closed interval, sorry this is open interval 0 .5, 1 until the percentage error is less or equals to 15 percent.
00:23
So, let us see how we are going to do this part here.
00:27
So, using the bisection method, we are going to solve this.
00:32
So, first of all, what i am going to do, i will try to find what is f of 0 .5 and this is my function f of x which is given here.
00:43
So, f of 0 .5 if i calculate it is sin of 0 .5 minus 0 .5 cube.
00:51
So, this comes out to be equals to nearly 0 .8544, it is greater than 0.
00:58
Then i try to find f of 1 which is sin of 1 minus 1 cube, it comes out to be minus 0 .16 which is less than 0.
01:07
So, they have the opposite sign.
01:10
So, i would say that hence one root it lie in the interval 0 .5, 1.
01:21
Then i define a variable x1 which is the mean of these two values of x which is 0 .5 plus 1 over 2 which is equals to 0 .75.
01:33
Then we evaluate what is f of x1.
01:36
So, evaluating we get this is 0 .26 which is greater than 0 and we already have f of 1 to be less than 0.
01:45
Again, they are having the opposite sign.
01:47
So, i would say that hence the root must lie in this interval 0 .75, 1.
01:59
Then again we define another variable x2 which is the mean of these two x 0 .75 plus 1 over 2 which is equals to 0 .875.
02:10
Then i try to find the value at x2 which is equals to 0 .0976 greater than 0 and we already have f of 1 less than 0.
02:21
Again, they have the opposite sign.
02:23
So, we say that root must lie in 0 .875, 1.
02:39
So, i try to now find what is x3 0 .875 plus 1 over 2 which is coming out to be 0 .9375.
02:50
I calculate the value of x3 here.
02:55
So, f of x3 it comes out to be minus 0 .01789.
03:01
Now, this is negative correct and already we have f of x2 as greater than 0.
03:08
So, they have the opposite sign.
03:10
So, i define another variable which is x4 equals to the mean of these two 0 .875, 0 .9375 over 2 which is equals to it is equals to 0 .90625...