- #1

- 1,444

- 2

**1. Homework Statement**

Consider the followign function [itex] f(x) = x^-5 [/itex]

[itex] a=1 [/itex]

[itex] n=2 [/itex]

[itex]0.8 \leq x \leq 1.2 [/itex]

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 [itex] f(x) ≈ T_{n}(x) [/itex] when x lies in the interval

**2. Homework Equations**

[tex] f(x) = f(a) + f'(a)(x-a) +\frac{f''(a)}{2!}(x-a)^2 [/tex]

[tex] R_k(x) =\frac{f^{(k+1)}(\xi_L)}{(k+1)!} (x-a)^{k+1} [/tex]

**3. The Attempt at a Solution**

Part a is simple:

[tex] T_2(x) = 1 - 5(x-1) + 15(x-1)^2 [/tex]

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

[tex] R_2(x) \leq | \frac{M}{3!}(x-1)^3 | [/tex]

Since [tex] f^{(3)} (x) = -210x^{-8} [/tex]

and this is decreasing, we use x = 0.8 and we use [itex] M = -210(0.8)^{-8} [/itex]

[tex] R_2(x) \leq |\frac{-210(0.8)^{-8}}{3!} (0.8-1)^3 [/tex]

and the result of the above is 1.6689

Is the above correct? Thanks for your help!