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

508 Chapter 18 Taylor polynomials, Taylor series and Maclaurin series
18.2 LINEARIZATIONUSINGFIRST-ORDERTAYLOR
POLYNOMIALS
Suppose we know that y is a function of x and we know the values of y and y ′ when
x =a,thatisy(a)andy ′ (a)areknown.Wecanusey(a)andy ′ (a)todeterminealinear
polynomial which approximates toy(x). Let this polynomial be
p 1
(x) =c 0
+c 1
x
We choose the constantsc 0
andc 1
so that
p 1
(a) =y(a)
p ′ 1 (a) =y′ (a)
that is, the values of p 1
and its first derivative evaluated atx = a match the values ofy
and itsfirst derivative evaluated atx =a. Then,
p 1
(a) =y(a) =c 0
+c 1
a
p ′ 1 (a) =y′ (a) =c 1
Solving forc 0
andc 1
yields
Thus,
c 0
=y(a) −ay ′ (a)
c 1
=y ′ (a)
p 1
(x) =y(a) −ay ′ (a) +y ′ (a)x
p 1
(x) =y(a) +y ′ (a)(x −a)
p 1
(x) isthefirst-order Taylor polynomialgenerated byyatx =a.
y
O
Q
Figure18.1
Graphical representation
ofafirst-orderTaylor
polynomial.
a
x
The function,y(x), isoften referred toas thegenerating function. Note that p 1
(x) and
its first derivative evaluated atx =a agree withy(x) and its first derivative evaluated at
x=a.
First-order Taylor polynomials can also be viewed from a graphical perspective.
Figure 18.1 shows the function, y(x), and a tangent at Q where x = a. Let the equation
ofthe tangent atx =abe
p(x)=mx+c
The gradient of the tangent is, by definition, the derivative ofyatx = a, that isy ′ (a).
So,
p(x) =y ′ (a)x +c
The tangent passes through the point (a,y(a)), and so
y(a) =y ′ (a)a +c
that isc =y(a) −y ′ (a)a. The equation of the tangent isthus
p(x) =y ′ (a)x +y(a) −y ′ (a)a
p(x) =y(a) +y ′ (a)(x −a)