I need to calculate the taylor polynomial T2(x) at 1 for the function
f(x) = x/1+2x
I have the following, with some blanks can anyone help?
For this function
f(x) = x/1+2x, f(1) = 1/3
F'(x) = ? f'(1) = ?
F"(x) = ? f"(1) = ?
Hence the Taylor polynomial of degree 2 at 1 for f is
T2(x) = 1/3 ? (x-1) ? (x-1)^2
Then I need to show that T2(x) approximates f(x) with an error less than 1/30 on the interval [0.5,1.5].
I have with I=[0.5,1.5], a=1,r=1,n=2.
1.First, f^(3) (c) = ?
2. Thus |f^(3) (c)|= ? for c E [0.5,1.5]
so we take M =
3. Using the remainder estimate (M/(n+1)!r^n+1,
|f(x)-T2(x)| = |R2(x)|
equalor more than M/(2+1)! r^2+1
=1/3! * ? * 1^3
= ? = ? for x E [0.5,1.5]
Thus T2(x) approximates f(x) with an error less than 1/30 on [0.5,1.5].