00:01
Function f of x equal to x power 1 by 5 and we are asked to find the tailor polynomial of degree 4 for this one.
00:11
Now i am writing the general tailor polynomial for any f of x function.
00:17
Taylor series for f of x at x equal to some point a, any point a.
00:28
We can general represent f of x as a function of series, a series as a, of a plus x minus a f dash of a means first derivative of f of x at point a plus x minus a whole square divided by two factorial f double dash of a plus x minus a whole cube by three factorial f triple dash of a plus x minus a whole power 4 by 4 factorial 4 derivative of a now here since we are asked good to find the polynomial of degree 4 x minus a whole power 4 gives rise to the polynomial of degree 4 the terms which are greater than x minus a whole power 4 will give the polynomial of degree 5 6 7 and more so we will neglect all those and we will take up to here only so in this first we have to find f of a next f dash of a double dash of a, f triple dash of a, f 4 dash of a.
01:40
Now let us try to find these values here.
01:43
And after finding these values, we can directly substitute them to get our final answer.
01:48
Now first, we have given the function f of x equal to x power 1 by 5.
01:56
Okay.
01:58
And you all know, i am writing one more formula here.
02:01
Derivative of x power n is nothing but n into x power n minus 1.
02:07
So see it is in the form x power n where n equal to 1 by 5.
02:10
So for while finding the derivative is we will use this formula.
02:15
So this is f of x.
02:17
So i am now finding f dash of x first derivative.
02:23
So by using this formula it becomes 1 by 5 x power 1 by 5 minus 1.
02:35
That equal to 1 by 5 x power minus 4 by 5.
02:38
So this is the f dash of x.
02:41
Now let us find f double dash of x.
02:44
So derivative of f dash of x.
02:47
So already we have 1 by 5 here.
02:50
And derivative of x power minus 4 by 5 is n minus 4 by 5 into x power minus 4 by 5 minus 1.
02:58
So it becomes minus 4 by 25 x power minus 9 by 5.
03:06
Okay.
03:06
Now, f triple dash of x.
03:11
Okay, f triple dash of x is derivative of f double dash of x.
03:16
If we derivative, we have minus 4 by 25 constant into derivative of x power minus 9 by 5...