APPLIC ATIONS OF TAYLOR POLYNOMIALS In this section we explore two types of applications of Taylor polynomials. First we look

11.11 APPLIC ATIONS OF TAYLOR POLYNOMIALS In this section we explore two types of applications of Taylor polynomials. First we look

at how they are used to approximate functions–– computer scientists like them because polynomials are the simplest of functions. Then we investigate how physicists and engi- neers use them in such fields as relativity, optics, blackbody radiation, electric dipoles, the velocity of water waves, and building highways across a desert.

APPROXIMATING FUNCTIONS BY POLYNOMIALS Suppose that f 共x兲 is equal to the sum of its Taylor series at a:

f 共n兲 共a兲

f 共x兲 苷 n 共x ⫺ a兲

n苷 兺 0 n !

In Section 11.10 we introduced the notation T n 共x兲 for the th partial sum of this series n

and called it the th-degree Taylor polynomial of at . Thus n f a

f 共i兲 共a兲

共x兲 苷 i

i苷 兺

共x ⫺ a兲 ⫹ 共x ⫺ a兲 2 ⫹⭈⭈⭈⫹

n ! Since is the sum of its Taylor series, we know that f T n 共x兲 l f 共x兲 as nl⬁ and so T n can

be used as an approximation to : f f 共x兲 ⬇ T n 共x兲 .

Notice that the first-degree Taylor polynomial

y=´ y=T£(x) y=T™(x)

T 1 共x兲 苷 f 共a兲 ⫹ f ⬘共a兲共x ⫺ a兲

y=T¡(x)

is the same as the linearization of f at a that we discussed in Section 3.10. Notice also that T 1 and its derivative have the same values at a that f and f⬘ have. In general, it can be

shown that the derivatives of T n at agree with those of up to and including derivatives a f

of order (see Exercise 38). n

To illustrate these ideas let’s take another look at the graphs of y苷e x and its first few Taylor polynomials, as shown in Figure 1. The graph of T 1 is the tangent line to y苷e x

at

; this tangent line is the best linear approximation to e 共0, 1兲 x near 共0, 1兲 . The graph

of T 2 is the parabola y苷 1⫹x⫹x 2 兾2 , and the graph of T 3 is the cubic curve y苷 1⫹x⫹x 2 兾2 ⫹ x 3 兾6 , which is a closer fit to the exponential curve y苷e x than . T 2 The next Taylor polynomial T 4 would be an even better approximation, and so on. The values in the table give a numerical demonstration of the convergence of the Taylor

FIGURE 1

共x兲 x to the function y苷e . We see that when x苷 0.2 the convergence is T 2 共x兲

x苷 0.2 x苷 3.0

polynomials T n

very rapid, but when x苷 3 it is somewhat slower. In fact, the farther is from 0, the more x T 4 共x兲

slowly converges T n to x 共x兲 e .

T 6 共x兲 1.221403 19.412500 When using a Taylor polynomial T n to approximate a function , we have to ask the f T 8 共x兲

1.221403 20.009152 questions: How good an approximation is it? How large should we take to be in order to n T 10 共x兲

1.221403 20.079665 achieve a desired accuracy? To answer these questions we need to look at the absolute e x

value of the remainder:

ⱍ R n 共x兲 苷 ⱍ ⱍ f n 共x兲 ⱍ

CHAPTER 11 INFINITE SEQUENCES AND SERIES

There are three possible methods for estimating the size of the error: