00:01
For this problem, we want to find the third order tailor polynomial for function f of x equals 1, given the information for the values of f, f, f, double, prime, and f triple prime, evaluated at 1.
00:17
Now we recall the nth order, tailor polynomial for f centered at x equals a, is given by f of x that's equal to f of a plus f prime of a times x minus a plus f double prime of a times x minus a squared over two factorial and so on which is the same as summation from n equals zero to infinity of the nth derivative of f evaluated at a over n factorial times x minus a raised a power of n.
01:01
So if we're looking for the third order tailor polynomial, we need until the third derivative.
01:08
So now our f of x should be f of 1 plus f prime of 1 times x minus 1 plus f minus 1 plus f double prime of 1 over 2 factorial times x minus 1, raise the power of 2, plus f triple prime of 1 over 3 factorial times x minus 1 cubed.
01:39
Using the given information, this is just 2 plus 3 times x minus 1 plus 7 over 2 factorial is 2 times x minus 1, plus 7 over 2 minus 1 squared plus f triple prime of 1 which is 5 over 3 factorial which is 6 times x minus 1 raised a power of 3 and now we'll have to use this to approximate the value of f of 1 .3 so f of 1 .3 this is approximately 2 plus 3 times 1 .3 minus 1 plus 7 over 2 times 1 .3 minus 1, raise of power of 2, plus 5 over 6 times 1 .3 minus 1, raise the power of 3.
02:41
That's equal to 3 .2375.
02:47
And then for the next problem, we want to find the 4th degree mclaurian polynomial for f of x that's equal to 1 over 1 minus x.
02:58
Now we recall nth degree macloryn polynomial for a function is given by f of x that's equal to f of 0 plus f prime of 0 times x plus f double prime of 0 over 2 factorial times x squared plus f triple prime of 0 over 3 factorial times x raised to the 3 power and so on, which is the same as summation from n equal 0 to infinity of the nth derivative evaluated at 0 over n factorial times x raised a power of n.
03:41
So if it's the fourth degree, then we need until the fourth derivative of this function.
03:48
So first you want to let f of x equal to 1 over 1 minus x or that's 1 minus x raise to a power of negative 1.
03:58
So f of 0 is equal to 1.
04:03
F prime of x will be the negative of 1 minus x raised a power of negative 2 times negative 1.
04:14
That's 1 minus x raised to a negative 2.
04:19
So f prime of 0 is equal to 1 as well.
04:26
F double prime of x is equal to negative 2 times 1 minus x raise the power of negative 3 times negative 1.
04:38
That's 2 times 1 minus x raised a power of negative 3.
04:42
So f double prime of 0 is equal to 2.
04:46
F triple prime of x is equal to negative 6 times 1 minus x raised a power of negative 4 times negative 1.
04:57
That's equal to 6 times 1 minus x raised a power of negative 4.
05:02
So f triple prime at 0 is equal to 6.
05:07
And then f fourth derivative is equal to negative 24 times 1 minus x raised a power of negative 5 times negative 1.
05:19
That's 24 times 1 minus x raised a power of negative 5.
05:26
So f fourth derivative at 0 is equal to 24.
05:31
So our f of x, the fourth degree, mclauri polynomial, is equal to f of 0, which is 1, plus f prime of 0, which is also 1, times x, that's x, plus f double prime of 0, which is 2, over 2 factorial, times x squared, plus f triple prime, which is 6 over 3 factorial, times x squared, plus f triple prime, which is 6 over 3 factorial, times x cubed, plus, we have 24 over 4 factorial times x to the 4th, that's equal to 1 plus x squared plus x cubed plus x to the 4th...