082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

18.4. Taylor polynomials of the nth order 519
y(x), p i (x)
8
7
6
5
4
3
2
p
1 2 (x)
0 y(x)
p 1 (x)
–1
–2.0 –1.5 –1.0 –0.5 0
x
(a)
y(x)
p 2 (x)
p 1 (x)
0.5 1.0 1.5 2.0
y(x), p i (x)
Figure18.5
Agraph ofy = e x andfour Taylor polynomials.
8
7
6
5
4
3
2
1
p 4 (x)
y(x)
0 p 3 (x)
–1
–2.0 –1.5 –1.0 –0.5 0
(b)
x
p 4 (x)
y(x)
p 3 (x)
0.5 1.0 1.5 2.0
use a third-order Taylor polynomial toestimatey(0.5).
Solution To write down the third-order Taylor polynomial about x = 0 we require y(0), y ′ (0),
y ′′ (0) andy (3) (0). FromEquation (18.4),
So,
y ′′ (x) =x 2 + {y ′ (x)} 2 −2y(x) (18.5)
y ′′ (0) =0+ {y ′ (0)} 2 −2y(0) =2
To findy (3) (0), Equation (18.5) is differentiated w.r.t.x:
y (3) (x) = 2x +2y ′ (x)y ′′ (x) −2y ′ (x)
y (3) (0) = 2y ′ (0)y ′′ (0) −2y ′ (0) =4
The Taylor polynomial may now be written as
p 3
(x) =y(0) +y ′ (0)x +y ′′ (0) x2
2 +y(3) (0) x3
6
=1+2x+x 2 + 2x3
3
We use p 3
(x) as anapproximation toy(x):
thatis,
p 3
(0.5) = 1 +1+0.25 +0.0833 = 2.333
y(0.5) ≈ 2.333
EXERCISES18.4
1 Afunction,y(x),hasy(0) = 3,y ′ (0) = 1,
y ′′ (0) = −1andy (3) (0) =2.
(a) Obtainathird-orderTaylor polynomial, p 3 (x),
generated byy(x) aboutx = 0.
(b) Estimatey(0.2).
2 Afunction,h(t),hash(2) = 1,h ′ (2) = 4,
h ′′ (2) = −2,h (3) (2) =1andh (4) (2) =3.
(a) Obtainafourth-orderTaylor polynomial, p 4 (t),
generated byh(t)aboutt = 2.
(b) Estimateh(1.8).