Lagrange remainder

[Melissa00]

Hi

How do you pick the right values for ε and x in the lagrange remainder?

Here is an example:
f(x)=x/(x+1) with x₀=2 and the interval [1,3]

The second degree taylor polynomial is T₂(x,f,2)= 2/3+(x-2)/9-(x-2)²/27

Okay so I know that f(x)=T_n(x,f,x₀) + R_n+1(x,f,x₀)

with (of course this should be an absolute value... sorry!)


Hence with all the info above:

Adding the 3rd derivative of f(x) into this:

(Of course, they're all absolute values... Sorry for that.)

Now about the part where I have to pick/find x and ϵ...
Am I right by saying that x_max=3 because of the interval? And pick x so that the numerator's value is maximal as well? So x=1 or x=3

What to do for ϵ?

 

[HallsofIvy]

The remainder formula gives the difference between f(x) and its "nth order Taylor polynomial" expansion: |f(x)- P_n(x-a)|. The "x" in the formula is the value of x at which you are calculating the value. As for, \(\displaystyle \epsilon\), you can't find its actual value, normally. If you could, you wouldn't need the whole polynomial. Just find the first degree value and add the "error" to get an exact value. The best you can do is replace \(\displaystyle \epsilon\) with the maximum possible value of f on the interval [a, x] (or [x, a]) to get an upper bound on the error. If "M" is that maximum then we know that the error cannot be larger than the value with that M replacing \(\displaystyle \epsilon\).