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

18.5 Taylor’s formula and the remainder term 521
(b) p 3 (x) =1−2x 2 ,
p 4 (x) =1−2x 2 + 2x4
3 ,
p 5 (x) =1−2x 2 + 2x4
3
4x 3
8 (a)
3 −2x2 +2x− 1 3
(b) 1.816
5x 4
(c)
12 − x3
3 + x2
2 + x 3 + 1 12
(d) 1.8194
9 (a) 0.7240 (b) 0.7240
10 (a) 4.2187 (b) 6.18
TechnicalComputingExercises18.4
1 Drawy(x) = 1 for1 x8. On the same axesdraw
x
the fourth-orderTaylor polynomial generated byy(x)
aboutx = 3.
3 Drawy(x) = lnx for0.5 x10.On the same axes,
draw the third-, fourth- andfifth-orderTaylor
polynomials generatedbyy(x)aboutx = 1.
2 Drawy = tanx for −1 x1. Using the same axes,
drawthe fifth-orderTaylor polynomial generatedby
y(x)aboutx = 0.
18.5 TAYLOR’SFORMULAANDTHEREMAINDERTERM
So far we have found Taylor polynomials of orders 1, 2, 3, 4 and so on. Example 18.6
suggeststhatthegeneratingfunction,y(x),andtheTaylorpolynomialsareincloseagreementforvalues
ofxneartothe pointwherex =a. Itisreasonable toask:
‘HowaccuratelydoTaylorpolynomialsgeneratedbyy(x)atx =aapproximateto
y atvaluesofxotherthana?’
‘IfmoreandmoretermsareusedintheTaylorpolynomialwillthisproduceabetter
and better approximation toy?’
To answerthesequestionsweintroduceTaylor’s formulaand the remainder term.
Suppose p n
(x) is an nth-order Taylor polynomial generated by y(x) about x = a.
ThenTaylor’s formulastates:
y(x) = p n
(x) +R n
(x)
whereR n
(x) iscalled theremainderofordernand isgiven by
remainder ofordern =R n
(x) = y(n+1) (c)(x −a) n+1
for somenumbercbetweenaandx.
(n +1)!
Theremainderofordernisalsocalledtheerrorterm.Theerrortermineffectgives
thedifferencebetweenthefunction,y(x),andtheTaylorpolynomialgeneratedbyy(x).
ForaTaylorpolynomialtobeacloseapproximationtothegeneratingfunctionrequires
the error termtobesmall.