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518 Chapter 18 Taylor polynomials, Taylor series and Maclaurin series
Example18.5 Giveny(0) = 1,y ′ (0) = 1,y ′′ (0) = 1,y (3) (0) = −1,y (4) (0) = 2,obtainafourth-order
Taylor polynomial generated byyaboutx = 0.Estimatey(0.2).
Solution Inthis examplea = 0 and hence
p 4
(x) =y(0) +y ′ (0)x +y ′′ (0) x2
2! +y(3) (0) x3
3! +y(4) (0) x4
4!
=1+x+ x2
2 − x3
6 + x4
12
The Taylor polynomial can be used toestimatey(0.2):
p 4
(0.2) = 1 +0.2 + (0.2)2
2
y(0.2) ≈ 1.2188
− (0.2)3
6
+ (0.2)4
12
= 1.2188
Example18.6 (a) Calculatethe first-,second-,third-and fourth-orderTaylor polynomialsgenerated
byy(x) = e x aboutx = 0.
(b) Ploty = e x and the Taylor polynomials for −2 x2.
Solution (a) We have
and
y(x) =y ′ (x) =y ′′ (x) =y (3) (x) =y (4) (x) =e x
y(0) =y ′ (0) =y ′′ (0) =y (3) (0) =y (4) (0) =1
Thus the Taylor polynomials, p 1
(x),p 2
(x),p 3
(x) and p 4
(x), are given by
p 1
(x) =y(0)+y ′ (0)x =1+x
p 2
(x) =y(0) +y ′ (0)x +y ′′ (0) x2
2 =1+x+x2 2
p 3
(x) =y(0) +y ′ (0)x +y ′′ (0) x2
2 +y(3) (0) x3
3! =1+x+x2 2 + x3
6
p 4
(x) =y(0) +y ′ (0)x +y ′′ (0) x2
2 +y(3) (0) x3
3! +y(4) (0) x4
4!
=1+x+ x2
2 + x3
6 + x4
24
(b) The graphs ofy = e x and the Taylor polynomials are shown in Figure 18.5. Note
that the Taylor polynomials become better and better approximations to e x as the
order of the polynomial increases.
Example18.7 Given thatysatisfiesthe equation
y ′′ −(y ′ ) 2 +2y=x 2 (18.4)
and alsothe conditions
y(0)=1 y ′ (0)=2