# Taylor polynomial remainder term

1. Feb 27, 2013

### stunner5000pt

1. The problem statement, all variables and given/known data
Consider the followign function $f(x) = x^-5$
$a=1$
$n=2$
$0.8 \leq x \leq 1.2$

a) Approximate f with a tayloy polynomial of nth degree at the number a = 1
b) use taylor's inequality to estimate the accuracy of approximation $f(x) ≈ T_{n}(x)$ when x lies in the interval

2. Relevant equations
$$f(x) = f(a) + f'(a)(x-a) +\frac{f''(a)}{2!}(x-a)^2$$
$$R_k(x) =\frac{f^{(k+1)}(\xi_L)}{(k+1)!} (x-a)^{k+1}$$

3. The attempt at a solution

Part a is simple:
$$T_2(x) = 1 - 5(x-1) + 15(x-1)^2$$

Since we have found the taylor polynomial at n = 2 the remainder:

$$R_2(x) \leq | \frac{M}{3!}(x-1)^3 |$$

Since $$f^{(3)} (x) = -210x^{-8}$$
and this is decreasing, we use x = 0.8 and we use $M = -210(0.8)^{-8}$

$$R_2(x) \leq |\frac{-210(0.8)^{-8}}{3!} (0.8-1)^3$$

and the result of the above is 1.6689

Is the above correct? Thanks for your help!

2. Feb 28, 2013

### eumyang

Technially,
$f^{(3)} (x) = -210x^{-8}$
is increasing on the interval. But since we need
$M = max \left| f^{(k+1)}(\xi_L) \right|$
x = 0.8 is the correct value of xi.

I also got 1.6689 as the estimated error.

3. Feb 28, 2013

### stunner5000pt

Ah yes, didn't consider the negative sign - thanks for your help!