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

18.4 Taylor polynomials of the nth order 517
6 p 2 (t) =1+2t −4.5t 2
( )
7 (a)A1 +x+ x2
2
(b) Cx
(c) A+(A+C)x+ Ax2
2
TechnicalComputingExercises18.3
Many technicalcomputing languageshave the
capability ofproducingaTaylor polynomial ofa
function.Insome languagesthisis available asa
defaultwhereasin others, such asMATLAB ® ,thisis
offered via a toolboxfunctionwhichmayneedto be
loaded orpurchased separately. Ifthe software you
are usingdoes not have the capability to produce the
Taylor seriesautomaticallyyou maychoose to dothis
partmanuallyandthenplot the resultsusingthe
technicalcomputing language.
1 (a) Calculate the second-order Taylor polynomial,
p 2 (x),generatedbyy(x) =x 3 aboutx = 0.
(b) Drawy(x)andp 2 (x)for −2 x2.
2 (a) Calculate the second-order Taylor polynomial,
p 2 (x),generated byy(x) = sinx aboutx = 0.
(b) Drawy(x)andp 2 (x)for −2 x 2.
3 (a) Calculate the second-order Taylor polynomial,
( )
1
p 2 (x),generated byy(x) = sin aboutx = 3.
x
(b) Drawy(x)andp 2 (x)for1 x 5.
4 (a) Calculate the second-order Taylor polynomial,
p 2 (x),generated byy(x) = e cosx aboutx = 0.
(b) Drawy(x)andp 2 (x)for −2 x 2.
18.4 TAYLORPOLYNOMIALSOFTHEnTHORDER
If we know y and its first n derivatives evaluated at x = a, that is y(a), y ′ (a),
y ′′ (a), ...,y (n) (a), thenthenth-orderTaylorpolynomial, p n
(x), maybewrittenas
p n
(x)=y(a) +y ′ (a)(x −a) +y ′′ (x −a)2
(a) +y (3) (x −a)3
(a)
2! 3!
+···+y (n) (x −a)n
(a)
n!
This provides an approximation toy(x). The polynomial and its firstnderivatives evaluated
atx = a match the values ofy(x) and its firstnderivatives evaluated atx = a,
thatis
p n
(a) =y(a)
p ′ n (a) =y′ (a)
p ′′ n (a) =y′′ (a)
.
p (n)
n (a) =y(n) (a)