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

512 Chapter 18 Taylor polynomials, Taylor series and Maclaurin series
The first-order polynomial, valid around an operating pointI = 0.5,is
p 1
(I) =P(0.5) +P ′ (0.5)(I −0.5)
NowP(0.5) = (0.5) 2 10 = 2.5,P ′ (0.5) = 2(0.5)(10) = 10,andso
p 1
(I) =2.5 +10(I −0.5) =10I −2.5
It is interesting to compare this linear approximation with the true curve for values
ofI around the operating point. Table 18.1 shows some typical values. Figure 18.4
shows a graph of the power dissipated in the resistor,P, against the current flowing
intheresistor,I.Noticethatthelinearapproximationisquitegoodwhenclosetothe
operating pointbut deteriorates further away.
Table18.1
A comparison oflinear approximations with truevalues.
I (A) TruevalueofP (W) ApproximatevalueofP (W)
0.5 2.5 2.5
0.499 2.49001 2.49
0.501 2.51001 2.51
0.49 2.401 2.4
0.51 2.601 2.6
0.4 1.6 1.5
0.6 3.6 3.5
1.0 10 7.5
P
10
8
6
4
Operating point
P = 10 I 2
p 1 (I) = 10 I – 2.5
2
0
0.2 0.4 0.6 0.8 1
I
Figure18.4
We see that the tangential approximation is
good when close to the operating point.
EXERCISES18.2
1 Calculate the first-orderTaylor polynomial generated
byy(x) = e x about
(a) x=0 (b) x=2 (c) x=−3
2 Calculate the first-orderTaylor polynomial generated
byy(x) = sinxabout
(a) x=0 (b) x=1 (c) x=−0.5
3 Calculate the first-orderTaylor polynomial generated
byy(x) = cosxabout
(a)x=0
(b)x=1
(c)x = −0.5