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

18.2 Linearization using first-order Taylor polynomials 509
This is the first-order Taylor polynomial. We see that the first-order Taylor polynomial
issimplythe equation of the tangent toy(x) wherex =a.
Clearly,forvaluesofxneartox =athevalueofp 1
(x)willbeneartoy(x);p 1
(x)isa
linearapproximationtoy(x).Intheneighbourhoodofx =a,p 1
(x)closelyapproximates
y(x), but being linear isamuch easier function todeal with.
Example18.1 Afunction,y, and its first derivative areevaluated atx = 2.
y(2)=1 y ′ (2)=3
(a) State the first-order Taylor polynomial generated byyatx = 2.
(b) Estimatey(2.5).
Solution (a) p 1
(x) =y(2) +y ′ (2)(x −2) =1+3(x −2) = −5 +3x
(b) We use the first-order Taylor polynomial toestimatey(2.5):
p 1
(2.5) = −5 +3(2.5) = 2.5
Hence,y(2.5) ≈ 2.5.
Example18.2 Find a linear approximation toy(t) =t 2 neart = 3.
Solution We require the equation of the tangent toy = t 2 att = 3, that is the first-order Taylor
polynomial aboutt = 3.Note thaty(3) = 9 andy ′ (3) = 6.
p 1
(t) =y(a) +y ′ (a)(t −a) =y(3) +y ′ (3)(t −3)
=9+6(t−3)
=6t−9
Att = 3, p 1
(t) and y(t) have an identical value. Near tot = 3, p 1
(t) and y(t) have
similarvalues, forexample p 1
(2.8) = 7.8,y(2.8) = 7.84.
18.2.1 Linearization
It is a frequent requirement in engineering to obtain a linear mathematical model of a
systemwhichisbasicallynon-linear.Mathematicallyandcomputationallylinearmodels
arefareasiertodealwiththannon-linearmodels.Themainreasonforthisisthatlinear
modelsobeytheprincipleofsuperposition.Itfollowsthatiftheapplication,separately,
of inputs u 1
(t) and u 2
(t) to the system produces outputs y 1
(t) and y 2
(t), respectively,
thentheapplicationofaninputu 1
(t) +u 2
(t)willproduceanoutputy 1
(t) +y 2
(t).This
isonly trueforlinear systems.
The value of this principle is that the effect of several inputs to a system can be calculated
merely by adding together the effects of the individual inputs. This allows the
effect of simple individual inputs to the system to be analysed and then combined to
evaluate the effect of more complicated combinations of inputs. A few examples will
help clarifythese points.