00:01
In this problem, we are given this equation, f of x equal to, and i am defining this f of x here, f of x equal to x cubed minus 6x minus 4 equal to 0.
00:15
We are going to use the bi -section method, and we are going to find a real root between 2 and 3.
00:24
And we will achieve that by using only 5 iterations.
00:29
So we are going to, we should prepare some table for that.
00:35
And i prepared this table that will mimic a spreadsheet application.
00:42
Because i mean if we are given such a problem, we should always be able to solve it using some exile or, you know, google sheets like that.
00:53
So that will mimic that behavior.
00:57
And on this table, on this studio table, i will show you the algorithm of what we are doing here.
01:07
So i have my columns a to g and i have my rows 1 to 6 and let me write down my parameters here.
01:21
So i will use this k parameters to just keep track of iterations from 1 to 5 like that.
01:34
And in the b column i will have my...
01:39
My lower limit of the bisection method and in the c column i will have the upper limit.
01:46
So the whole idea of this bisection method is just taking some interval, dividing it into two and depending on some condition you will focus on either the upper portion or the lower portion so the up and lower limits of the interval you have will change at each step.
02:09
So that's why we have to clearly, we should clearly keep track of our intervals, these endpoints, upper and lower endpoints.
02:21
And the most convenient or traditional notation is called them a and b.
02:30
So a is the lower limit and b is the upper one.
02:33
And here i am going to define the c parameter which will just equal to a, plus b divided by 2 so this is the midpoint of the interval for an iteration then in the e column i will evaluate my function f of x at the points of a and here i will evaluate my function at the point of c and the condition is given by this product f of a times f of c being larger, being greater or smaller than zero.
03:21
So we are going to check the sign of this product to determine our next interval in the next iteration.
03:34
And we should also see that this product getting smaller and smaller at each each other.
03:40
Step if it's not getting smaller than we are doing something wrong and hopefully it won't be decay here and this is the whole essence of this method basically so in the first step first iteration we are we will just use the given interval so our a and b values will be simply two and three respectively so okay we define the function here, the c function.
04:20
So it will be a plus b over 2.
04:25
So in terms of cells, we can write this b2 plus c2 over 2.
04:39
And for this e column for this cell e2, we just evaluate this a value here using the actual function i mean we have i don't want right but i mean it's too obvious so we have b two cubed minus six times b two minus four so very similarly we have the same function for this b value over here for for us c value over here so in the last step we take the product so e 2 times f2 just like that then now comes a critical point we are going to write a condition here in the cell b3 it will be an if condition so if this product here given in the cell g2 if this is a positive number then we take this value take the c value from the previous iteration as our new lower limit.
06:22
So we take this value as our lower limit.
06:29
And if it's not the case, then we just keep our original value from the previous line.
06:39
So that is how this method works.
06:43
Very similarly or maybe in a very opposite way to get this new upper limit, we again look at this product.
06:58
F of a times f of c we say that if this expression is smaller than is as negative then we update this value by the c value from the previous duration so d2 or if this quantity is if this product is a positive number we keep the original the same upper limit from the previous iteration...