Calc1231W9T3 - Lagrange formula for the remainder
Tuesday, 18 September 2018
10:22 PM
Created with Microsoft OneNote 2016.
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The Taylor polynomial is a nice simple way to approximate a function. However simplicity comes at the cost of accuracy Lagrange's form for the remainder gives crude but easy-to-calculate
bounds for this accuracy
By Taylor's theorem, given a point = a and a function f with n + 1 continuous derivatives we can write
f@) = pn(æ) + Rn+l(c),
where pn is the nth degree Taylor polynomial about x = a, and the remainder term is given by the integral
f(n+l) — t)ndt.
1
Rn+l (x)
n!
This is exactly the error between f and pn, but it is difficult to look at, and difficult to compute. Another expression for the remainder term is Lagrange's form for the remainder
Rn+l (x) = f(n+l) ( C)
for some c between a and If we can find a bound for f (n +1) (c) then this can be used to bound the remainder term Rn+l (x).
ex then the 3rd Talyor polynomial for f about x = 0 is
For example, consider x in the range [—0.01, 0]. If f@)
Lagrange's form for the reminder term is
4
4!
where
O (a function of c).
Where c is a real number where —0.01 < < c < 0. The maximum value of f (4) (c) is 17 which is achieved when
Using this we can bound the error term](Calc1231W9T3%20-%20Lagrange%20formula%20for%20the%20remainder_files/image001.png)
