MATH 152 Chapter 10.9

« previous | Monday, November 15, 2010 | next »

Error Analysis in Taylor Polynomials

We can find the Taylor series of any differentiable function $f\left(x\right)=\sum _{n=0}^{\mathrm{\infty }}{c}_n{(x-a)}^n$, where $c_n=\frac{f^{\left(n\right)}\left(a\right)}{n!}$.

We can approximate the function with a finite polynomial by taking the Nth degree Taylor Polynomial of $f$ at $x=a$:

$T_N\left(x\right)=\sum _{n=0}^N\frac{f^{\left(n\right)}\left(a\right)}{n!}{(x-a)}^n$

Remainder

Everything else past the finite polynomial that we chopped off:

$\begin{array}{rl}{R}_N\left(x\right)& =\sum _{n=N+1}^{\mathrm{\infty }}\frac{f^{\left(n\right)}\left(a\right)}{n!}{(x-a)}^n\\ & =f\left(x\right)-T_N\left(x\right)\end{array}$

We can estimate $R_N$; this shows how far off we are at any point on the given interval (of $x$) when we stop at the Nth degree Taylor Polynomial:

Alternating Series

$\begin{array}{rl}\left|R_N\left(x\right)\right|& \le a_{N+1}\\ & \le \frac{f^{(N+1)}\left(a\right)}{(N+1)!}{|x-a|}^{N+1}\end{array}$

Taylor's Inequality

If $\left|f^{(N+1)}\left(x\right)\right|\le M$ on a given interval (in other words, if we can get an upper bound on the derivative), then

$\left|R_N\left(x\right)\right|\le \frac{M}{(N+1)!}{|x-a|}^{N+1}$

Note: given; do not memorize

Example

Use a 3rd degree Taylor polynomial at $a=1$ to approximate $f\left(x\right)=\sqrt{x}$ on the interval [0.9, 1.1] and determine the accuracy of your results using the remainder theorem

$n$	$f^{\left(n\right)}\left(x\right)$	$f^{\left(n\right)}\left(1\right)$	$\frac{f^{\left(n\right)}\left(1\right)}{n!}{(x-1)}^n$
0	$\sqrt{x}$	1	1
1	$\frac{1}{2}{x}^{-1/2}$	$\frac{1}{2}$	$\frac{1/2}{1!}{(x-1)}^1$
2	$-\frac{1}{4}{x}^{-3/2}$	$-\frac{1}{4}$	$-\frac{1/4}{2!}{(x-1)}^2$
3	$\frac{3}{8}{x}^{-5/2}$	$\frac{3}{8}$	$\frac{3/8}{3!}{(x-1)}^3$

$T_3\left(x\right)=1+\frac{1}{2}(x-1)-\frac{1}{8}{(x-1)}^2+\frac{1}{16}{(x-1)}^3$

• $N=3$
• $a=1$
• $x\in [0.9,1.1]$ . . . Which value of $x$ maximizes the error bound?
  • $M=\left|f^{\left(4\right)}\left(x\right)\right|$ (max. of next derivative on the interval of $x$)
  • M_max occurs at $x=0.9$: $\frac{15/16{(0.9)}^{7/2}}{4!}{|0.9-1|}^4$

$\left|R_3\left(x\right)\right|\le \frac{15/16{(0.9)}^{7/2}}{4!}{|0.9-1|}^4$

Example

Determine the degree of the Taylor Polynomial needed to approximate $\sin (0.1)$ to within 0.00001 accuracy.

$\sin x=\sum _{n=0}^{\mathrm{\infty }}{(-1)}^n\frac{x^{2n+1}}{(2n+1)!}$

$\begin{array}{rl}|s-s_N|& \le \left|a_{N+!}\right|<.00001\\ \frac{{(0.1)}^{2N+3}}{(2N+3)!}& <.00001\\ \text{use tech. to solve}N=1\end{array}$