00:01
Using newton -rapson's method, we have to find the points where the curves e to the power x and 6x intersect.
00:09
That means we have to find the roots of the equation where fx equal to 0, the fx is given by e to the power x minus 6x.
00:20
So the derivative of f, which is f prime, is given by e to the power x minus 6.
00:28
Now, first to apply newton rastence method, we have to make an initial guess for the roots.
00:37
Now, from the curve, it shows that the first root lies between, so this is the first root, and this lies between x equal to 0 and x equal to 0 .5.
00:46
So let us take our initial guess as x equal to 0 .25.
00:52
Now, if we draw a tangent line through this point, which passes through x equal to 0 .25 and y equal to f of 0 .25 then this has equal this this this tangent has the equation that y equal to f of 0 .25 equal to f prime of 0 .25 times x minus 0 .25 now we want to find out the x intercept for this tangent line the x intercept is the point where y equal to 0 and this gives us x equal to 0 .204 204 so we repeat the above procedure by starting with x equal to 0 .0 .0 4204 at the next iteration.
01:40
So basically this this this formula with y set to 0 gives us the formula the formula for the nth iteration which is given by x n plus 1 equal to x n minus f of xn divided by f prime of x n so this is the general formula to get the value at the next iteration so using this formula the first we start with 0 .25 then using this formula we get x2 equal to 0 .204 204 as i told you all and then again if we feed in this value of x2 we get x3 equal to 0 .204481.
02:37
If we feed in the value of x2 we get x4 equal to 0 .204481 and then it does not change after this, after this iteration.
02:48
That means up to sixth significant figure, this is our answer for the first root.
02:54
Now let us do the same for the second root.
02:56
For the second root is here.
02:59
So it lies between 2 .5 and 3...