1549901369-Elements_of_Real_Analysis__Denlinger_

330 Chapter 6 • Differentiable Functions

That is ,

n J(k)(a) k

Tn(x) = 2:-k,-(x - a) ,

k=O

(4)

where j<^0 J = f and J(k) denote the kth derivative of f.

It is the purpose of this section to investigate the relationship between
a function f and its Taylor polynomials. We shall see that when f is "well
behaved," f(x) is closely approximated by Tn(x) for values of x close to a. We
shall understand better what this means as we progress through this section.

Example 6.5.2 Find the 4th Taylor polynomial of the function f(x) = sinx
about 0.

Solution. Let f(x) = sinx. Then

Thus,

j<^0 l(x) = sinx ::::} j<^0 l(O) = sinO = O;

J'(x) = cosx ::::} f'(O) = cosO = l;

f"(x) = -sinx ::::} f"(O) = -sinO = O;

f"'(x) = -cosx::::} fm(o) = - cosO = -1;

j<^4 l(x) = sinx ::::} f(^4 )(0) = sinO = 0.

!" (0) fl// (0) 1<^4 ) (0)
T 4 (x) = f(O)+ f'(O)(x-0)+
21

(x-0)^2 +-

3

,-(x-0)^3 +· · ·+-

4

,-(x-0)^4

0 -1 0

= 0 + 1 · x + -x^2 + - x^3 + - x^4

2! 3! 4!

= X - tX^3. 0

Example 6.5.3 Find the 4th Taylor polynomial of the function
f(x) = 3 + 5x^2 - 4x^3 + x^4 about 0.

Solution. Let f(x) = 3 + 5x^2 - 4x^3 + x^4. Then

j<^0 l(x) = 3 + 5x^2 - 4x^3 + x^4 ::::} j<^0 l(O) = 3;

f'(x) = lOx - 12 x^2 + 4x^3 ::::} f'(O) = O;

f"(x) = 10 - 24x + 12x^2 ::::} f"(O) = 10;

f'"(x) = -24 + 24x ::::} f'"(O) = -24;

j<^4 l(x) = 24 ::::} j<^4 l(O) = 24.